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Microscopes are an essential tool for scientists, students, and anyone who wants to see the world in greater detail. However, understanding the different parameters of a microscope can be a bit daunting, especially if you’re new to microscopy. One of the critical concepts to understand is the field of view on a microscope. What is the field of view on a microscope, exactly? In this article, we’ll explore this essential microscope concept in depth so that you can make the most out of your microscopy experience. Whether you’re a beginner or an advanced user, understanding the field of view is essential for getting accurate results and producing high-quality images. Read on to learn more about the field of view on a microscope, how it’s measured, and how it relates to other parameters of your microscope.
Definition of Field of View (FOV) on a Microscope
When using a microscope, you may have come across the term “field of view” (FOV). But what exactly does it mean?
Simply put, the FOV refers to the area of the specimen that is visible through the microscope when looking through the eyepiece. This area can vary depending on the magnification of the objective lens being used.
Here are some interesting facts about the FOV on a microscope:
- As you increase the magnification, the FOV gets smaller. This is due to the fact that the higher magnification lenses have a smaller diameter, resulting in a narrower field of view.
- The FOV is typically measured in millimeters (mm) or micrometers (µm).
- When viewing a specimen under a microscope, the FOV can be used to estimate the size of the specimen. By comparing the size of the specimen to the size of the FOV, it is possible to make an educated guess about its actual size.
- Depth of field also affects the FOV. At higher magnifications, depth of field decreases, resulting in a smaller area of focus, and a smaller FOV as well.
Now that you know what the FOV is, you may be wondering what is the largest field of view on a microscope? The answer to that depends on the make and model of the microscope you are using, as well as the magnification of the lenses being used. Generally, the lower the magnification, the larger the field of view.
In conclusion, understanding the FOV is an important aspect of using a microscope. By knowing what it is, and how it can change depending on the magnification and depth of field, you can enhance your microscopy experience and gain a better understanding of the specimens you are studying.
What is the Largest FOV on a Microscope?
The field of view (FOV) on a microscope is the area that you can view through the eyepiece or camera when looking at a specimen. It is an essential parameter to consider when choosing a microscope to ensure that you can see your sample with enough detail and perspective. The FOV can vary depending on the objective lens or camera used, but what is the largest FOV on a microscope?
1. Stereo Microscopes
The largest FOV on a microscope belongs to stereo microscopes or dissecting microscopes. They have two separate optical paths that provide a three-dimensional image of the sample, allowing you to see its depth and shape. The FOV on a stereo microscope can reach up to 50 mm or more, making it ideal for observing large specimens such as rocks, insects, plants, or circuit boards.
2. Compound Microscopes
Compound microscopes are the most common type of microscope used in laboratories or educational settings. They use a series of lenses to magnify the image, allowing you to see microscopic structures such as cells, tissues, or microorganisms. The FOV on a compound microscope ranges from a few millimeters to a few micrometers, depending on the objective lens. For example, a low-power objective lens (4x) will have a larger FOV (4-5 mm) than a high-power objective lens (40x) (0.4-0.5 mm).
3. Digital Microscopes
Digital microscopes are a modern alternative to traditional optical microscopes. Instead of using eyepieces, they have built-in cameras that display the image on a computer screen, allowing you to capture, edit, and share images with ease. The FOV on a digital microscope ranges from a few millimeters to a few centimeters, depending on the camera sensor and image resolution.
How to Increase Field of View on Microscope?
There are several ways to increase the FOV on a microscope, such as using a lower magnification objective lens, moving the slide around, or adjusting the position of the specimen. Another option is to use a condenser lens to improve the quality of the illumination and reduce stray light, which can affect the clarity and contrast of the image. Additionally, you can use software tools to stitch multiple images together and create a panoramic view of the sample.
In conclusion, the largest FOV on a microscope belongs to stereo microscopes, which provide a 3D view of large specimens. Compound microscopes and digital microscopes have a smaller FOV but offer higher magnification and resolution capabilities. No matter the type of microscope you use, understanding the FOV is crucial to obtaining accurate and informative results.
How to Increase FOV on Microscope
To improve the view of a sample under the microscope, one can increase the field of view (FOV). FOV is the visible area seen through the eyepiece lens of a microscope. It is expressed in millimeters (mm) or micrometers (µm) and depends on the magnification of the objective and eyepiece lenses. When using a higher magnification objective, FOV will decrease, and vice versa.
Here are four ways to increase FOV on a microscope:
|Use a lower magnification objective
|FOV definition on a microscope increases when using a lower magnification objective. Choose an objective lens with lower magnification to increase the visible area. For example, switching from a 40x to a 10x objective will increase FOV.
|Switch to a wider field eyepiece
|Changing to a wider field eyepiece can also increase the FOV. Spreading out the light that comes to your eye makes it easier to see things in the periphery of your field of vision. For example, using a 10x eyepiece with a 15mm field of view instead of a 10x eyepiece with a 12mm field of view will increase FOV.
|Adjust the aperture diaphragm
|The aperture diaphragm controls the amount of light that enters the microscope. By decreasing the amount of light, the FOV will increase. Reducing the amount of light reduces the intensity of the diffraction rings, allowing more detail to be seen outside the focal plane.
|Change the distance between the objective lens and the sample
|Moving the objective lens slightly further from the sample can increase the FOV. However, the image quality may decrease as well because this can decrease the resolution, or the ability of the microscope to distinguish between two points.
In summary, FOV definition microscope increases when using a lower magnification objective, switching to a wider field eyepiece, adjusting the aperture diaphragm, or changing the distance between the objective lens and the sample. Each of these methods can provide a different level of enhancement to the FOV and have their unique strengths and weaknesses. With these options at your disposal, you can tailor your microscopy experience to meet your needs and get the most accurate, detailed view possible.
## How to Calculate FOV Length
The field of view (FOV) on a microscope is the circular area that you see when you look through the eyepiece. It is an essential parameter in microscopy that determines the area of the sample that can be viewed at a particular magnification. Calculating FOV length allows you to estimate the size of a specimen or to make accurate micrometry measurements.
To calculate FOV length, you need to know:
– The magnification of the microscope
– The diameter of the FOV
The formula for calculating FOV length is:
FOV length = diameter of FOV / magnification
For instance, if the diameter of the FOV is 2 mm, and the magnification is 40x, the FOV length can be calculated as:
FOV length = 2 mm / 40 = 0.05 mm or 50 µm
This means that the field of view at 40x magnification covers an area of 50 µm, and any object or feature within this area can be seen clearly.
Keep in mind that FOV length changes with the magnification. Higher magnification results in a smaller FOV and a greater level of detail in the sample. On the other hand, lower magnification offers a larger FOV, but it may not reveal the fine structures within the sample.
Calculating FOV length is an important step in determining the size of cells, tissues, and other specimens. You can also use it to estimate the size of cells in comparison to the high field of view on a microscope, which is the widest possible field that can be viewed at the given magnification.
In summary, to calculate FOV length, you need to divide the diameter of the FOV by the magnification. This parameter determines the area of the sample that can be viewed and is crucial for making accurate measurements in microscopy.
Magnification and FOV
When using a microscope, magnification and field of view (FOV) are two important concepts to understand. Magnification refers to the degree to which an object appears larger through the microscope. FOV, on the other hand, is the area of the microscope slide that is visible through the eyepiece.
Microscopes have multiple levels of magnification, typically ranging from 40x to 1000x or more. When using a microscope, it is important to adjust the magnification to view the object clearly. To increase the magnification, adjust the focus knob and/or change the objective lens. However, it is important to note that a larger magnification does not necessarily mean a larger field of view.
Field of view is the area of the specimen that is visible through the microscope. it is measured in millimeters or micrometers. Field of view can be affected by a number of factors, including the microscope’s magnification, objective lens, and the size of the eyepiece.
One way to increase the FOV is to use a lower magnification. As the magnification is decreased, the FOV will increase, allowing for a larger viewing area. Another way to increase FOV is to use a microscope with a larger eyepiece, which creates a larger image.
Here is a table showing the approximate FOV for various magnifications:
|Field of View (mm)
How to Get the Largest Field of View on a Microscope
To get the largest field of view, it is recommended to use a lower magnification and a larger eyepiece. This will allow for a wider and clearer view of the specimen. Remember that FOV decreases as magnification increases, so it is important to find the right balance. Experiment with the microscope settings to find the best combination for your needs.
What is the High FOV on a Microscope?
The field of view (FOV) on a microscope refers to the area that is visible through the eyepiece or camera when looking into the microscope. The FOV is an important parameter to consider because it determines the size of the specimen that can be viewed at a given time. Typically, the FOV is measured in millimeters (mm) or micrometers (µm) and is determined by the magnification power of the objective lens and the eyepiece.
When it comes to high FOV on a microscope, it refers to the largest possible field of view that can be obtained through the instrument. A high FOV on a microscope is desirable if you need to observe large samples or if you want to analyze several samples at once. Moreover, a high FOV can be useful for educational purposes as it allows for a broad observation of samples.
There are several ways to increase the FOV on a microscope. The simplest way is to use a lower magnification objective lens, which widens the area covered by the microscope. Alternatively, you can use a microscope with a small aperture or a wide-angle eyepiece, which also expands the FOV. However, it’s important to note that using lower magnification or wide-angle lenses may reduce the clarity and detail of the image.
If you want to calculate your microscope field of view length, there are simple steps you can follow. First, you need to find out the magnification power of your objective lens and the eyepiece. Next, you need to multiply these numbers together to get the total magnification power. Finally, you need to divide the field number (FN) of your eyepiece (which can be found on the eyepiece itself) by the total magnification. This will give you the FOV diameter in millimeters.
In summary, a high FOV on a microscope allows for the observation of large samples or multiple samples at once. Increasing the FOV is possible by using a lower magnification objective lens or a wide-angle eyepiece. To calculate your microscope field of view length, you need to find out the magnification power of your objective lens and eyepiece and divide the field number of your eyepiece by the total magnification.
**What is Field of View on a Microscope? | Get the Facts and Enhance Your Microscopy Experience**
**How to Get the Largest FOV on a Microscope**
Field of View (FOV) on a microscope is the area visible through the eyepiece or camera lens. It is typically measured in millimeters or micrometers and varies depending on the objective lens being used. The larger the FOV, the more of the specimen you can see at one time.
Getting the largest FOV on a microscope requires some adjustments to the equipment and technique, but it can significantly enhance your microscopy experience. Here are some of the ways to get the largest FOV on a microscope:
1. **Use the lowest magnification objective lens:** The objective lens is the lens closest to the specimen on the microscope. The lower the magnification of the lens, the larger the FOV will be. Therefore, using the lowest magnification objective lens possible will give you the largest FOV.
2. **Adjust the aperture diaphragm:** The aperture diaphragm is a part of the microscope that controls the amount of light passing through the specimen. By adjusting the diaphragm, you can increase or decrease the brightness of the image and, at the same time, increase or decrease the FOV. So, to get the largest FOV, adjust the diaphragm to allow the maximum amount of light to pass through the specimen.
3. **Position the eyepiece correctly:** Make sure the eyepiece is positioned correctly in the microscope. If the eyepiece is not seated correctly, the FOV will be reduced. Adjust the eyepiece until you see the entire FOV.
4. **Increase the distance between the eyepiece and objective:** By increasing the distance between the eyepiece and objective, the FOV will be increased. This can be achieved by moving the eyepiece up or down in the microscope tube.
5. **Use a wider eyepiece:** A wider eyepiece will provide a larger FOV than a narrower eyepiece. So, if you have multiple eyepieces, choose the widest one for the largest FOV.
By following these tips, you can get the largest FOV on your microscope and see more of your specimen. Remember, the FOV may vary depending on the microscope model and type of specimen you are viewing. So, experiment with different settings and techniques until you find the largest FOV for your microscope.
Field of View is an essential aspect of microscopy. It determines how much of the specimen you can see at once. By getting the largest FOV on your microscope using the above tips, you can see more of your specimen in detail. Learning how to find an object with a microscope becomes much more comfortable when you can see as much of the specimen as possible. So, go ahead and enhance your microscopy experience today!
How to Find an Object with a Microscope
When using a microscope, finding an object can be a bit challenging, especially if you are a beginner. However, by following these simple steps, you’ll be able to locate your object in no time.
1. Place the slide on the stage: The first step in finding an object with a microscope is to place the slide on the stage. Make sure the specimen is properly centered before proceeding to the next step.
2. Adjust the focus and light intensity: Adjust the focus knob to get a clear image of the object. Also, adjust the light intensity to get the right amount of light needed to view the object properly.
3. Determine the field of view: The field of view is the visible area when looking through the microscope. It is determined by the diameter of the field diaphragm and the magnification of the objective lens. To determine it, use a ruler or a micrometer to measure the diameter of the field diaphragm.
4. Scan the slide: After determining the field of view, scan the slide while looking through the microscope, moving it gradually in different directions until you find the object. It is important to note here that you may need to adjust the focus and the lighting again once you find the object.
5. Move to higher magnification: If you desire to get a closer look at the object, you can move to a higher magnification. However, this will reduce the size of the field of view. To increase it again, you will need to move the slide until you find the object again.
6. Repeat the process: In case you don’t find your object, repeat the whole process again, starting from adjusting the focus and lighting until you find the object.
In conclusion, finding an object with a microscope requires patience and practice. You can increase your field of view by using the right magnification, and by taking measurements of your field diaphragm. By following these simple steps, you can easily locate and observe any object using a microscope.
Frequently Asked Questions
What is the relationship between the magnification and field of view of a microscope?
When it comes to using a microscope, understanding the relationship between magnification and field of view is crucial. Essentially, the magnification determines how much an image is enlarged, while the field of view refers to the area that is visible through the microscope’s lenses. The two are intrinsically linked, meaning that as the magnification increases, the field of view decreases, and vice versa.
- Higher magnification: When you increase the magnification of a microscope, the image appears larger, but you will be able to see less of the sample. This means the field of view decreases as the magnification increases. In general, higher magnification allows for greater detail and precision, and is ideal when focusing on smaller, more intricate areas.
- Lower magnification: Reducing the magnification of a microscope can help increase the field of view, allowing more of the sample to be visible at once. This is great when getting a general overview of a larger area, or when you want to compare different parts of the sample. Using a lower magnification is often helpful when trying to locate a smaller feature before zooming in for a closer look.
Understanding how the relationship between magnification and field of view works is essential to mastering the use of a microscope. With the right combination of magnification and field of view, you can achieve excellent image quality and a deeper understanding of the sample.
Does the field of view increase or decrease with higher magnification?
As the magnification increases, the field of view decreases. This is because higher magnification means that a smaller area is being magnified, resulting in a narrower field of view. Therefore, when using a microscope with a higher magnification, it will be necessary to move the slide around to see different parts of the specimen. It’s important to keep this in mind when selecting the appropriate magnification for observing a particular sample.
How does the field of view vary with different objective lenses?
The field of view (FOV) on a microscope refers to the area of the sample that is visible through the eyepiece. It is measured in millimeters (mm) or micrometers (µm) and is dependent on the magnification of the objective lens.
Different objective lenses have varying magnifications and thus affect the field of view. Here’s how:
- Low magnification objective lens: Usually 4x or 10x, this lens has the largest FOV. The sample appears larger but with less detail due to the lower magnification.
- Medium magnification objective lens: Usually 20x or 40x, the FOV is smaller than the low magnification lens but provides more detail.
- High magnification objective lens: Usually 60x or 100x, this lens has the smallest FOV and provides the highest level of detail. It is best used for viewing specific parts of the sample in high detail.
In summary, the FOV varies with different objective lenses based on their magnification. A low magnification lens will provide a larger FOV but less detail, while a high magnification lens will provide a smaller FOV but greater detail. Understanding the FOV for each objective lens can help you choose the right lens for your specific microscopy needs.
What are the advantages of having a wide field of view?
The field of view on a microscope refers to the area visible through the lenses. Having a wide field of view can offer several advantages to microscopy. Let’s take a look at some of them:
- Enhanced Efficiency: A wider field of view enables you to view a larger area at once. This allows you to locate your specimen or area of interest more quickly and easily, and also assess the sample’s overall appearance.
- Better Visualization: With a wider field of view, it becomes easier to identify different elements within a sample. This is particularly important when studying complex samples or small organisms, where it may be difficult to identify specific features if the field of view is too narrow.
- Improved Accuracy: A greater field of view can help ensure that you are making accurate observations and measurements. This is particularly important in research and medical settings, where accuracy is critical.
- Increased Comfort: Looking through the eyepiece of a microscope can be tiring for the eyes. With a wider field of view, there is less need to constantly adjust the stage or focus, reducing eye strain and increasing comfort during extended periods of observation.
- Greater Versatility: A microscope with a wide field of view can be used for a wider range of applications. It provides flexibility in terms of the type and size of samples that can be imaged, and enables you to observe a wider variety of specimens with ease.
In conclusion, having a wide field of view can be a significant advantage in microscopy. It provides greater efficiency, accuracy, and versatility, while also enhancing visualization and reducing eye strain. When selecting a microscope, it is essential to consider the field of view as it can have a significant impact on your microscopy experience.
How does the field of view affect the resolution of a microscope?
The field of view on a microscope determines the area of the specimen that is visible when viewed through the eyepiece. A larger field of view allows for more of the specimen to be seen at once, but it can decrease the resolution of the microscope. This is because when the field of view is expanded, the magnification is decreased, which can lead to a loss of detail and clarity in the image. On the other hand, a smaller field of view results in higher magnification and greater resolution, but at the expense of a narrower view. Therefore, finding a proper balance between field of view and magnification is crucial in achieving the best resolution when using a microscope.
Field of view is an important factor to consider when using a microscope. It determines how much of the specimen can be seen at once and the level of detail that can be observed. Understanding how to determine the field of view and how to adjust the microscope’s settings to increase the field of view can help improve the overall microscopy experience.
- Fry, S. (2020, October 15). What is Field of View on a Microscope? | Get the Facts and Enhance Your Microscopy Experience. Retrieved November 19, 2020, from https://www.amscope.com/what-is-field-of-view-on-a-microscope
- Microscope Field of View (FOV). (2020, October 14). Retrieved November 19, 2020, from https://www.olympus-lifescience.com/en/microscope-knowledge/basics/field-of-view/
- Microscope. (2020, November 9). In Wikipedia. Retrieved November 19, 2020, from https://en.wikipedia.org/wiki/Microscope
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Drs. Jacob and Alvera Stern, both service workers with Mennonite Central Committee (MCC) Kenya, shared information about sand dams at ECHO’s 2009 Agriculture Conference in Florida and again at the February 2011 ECHO East Africa Symposium. Construction of a sand dam in a seasonal river basically leads to formation of an aquifer. A sand dam provides a low-cost, low-tech and low maintenance water point with large payback in easily accessible water year round. Community ownership and involvement is integral in the introduction of a sand dam. MCC has been supporting the building of sand dams in East and Southern Africa for the past 10 years. This article, and the Technical Note from which it is abstracted, are based on the work of Utooni Development Organization (UDO; www.utoonidevelopment.org/), for which the Sterns are working. The UDO Director, Joshua Mukusya, built his first sand dam in 1978. UDO currently constructs 50 or more sand dams a year in Eastern Province, Kenya. For more information contact [email protected].
What Is a Sand Dam?
A sand dam is a reinforced concrete wall built across a seasonal river to hold underground water in sand. It is initially built one meter high and up to 90 meters across. During heavy erratic seasonal rains, the water and silt flow over the dam, and the heavier sand settles to the bottom. Over one to three seasons of rain, the dam fills up with sand that acts as a storage tank for water. In good quality sand, the sand dam volume is approximately 35% water (Beimers et al, 2001). Most of this water does not evaporate, as it is protected by the sand. Evaporation decreases by 90% at 60 cm below the surface (Borst and Haas, 2006).
The sand dam is always built on bed rock. A natural aquifer is formed under the sand as water accumulates. Often there is already an aquifer present and the sand dam simply increases the water in it. Over time, the aquifer increases in size and the water table of the surrounding area rises.
How do people get water from the sand?
Water is collected from the sand dam in various ways: by a pipe near the bottom of the sand dam wall downstream; an off-take well upstream; a water tank built into the dam wall on the upstream; or scoop holes upstream. Most often, several of these methods of extracting water from sand are used at the site.
Where are sand dams appropriate?
Sand dams are most appropriate in semi-arid and arid areas that experience short, heavy, erratic downpours. These areas typically have seasonal (ephemeral) rivers with sand beds. At a point along a seasonal river bed, the local people may dig scoop holes for water. This indicates that an aquifer and rock bed are present and that it is a potential site to build a sand dam.
Sand dams are generally built in remote rural areas without supportive infrastructure, and where government is unlikely to provide for the water and sanitation needs of the people. People in these areas know that their survival depends on their own efforts. They are also familiar with getting water from sand, as their families have been doing this for generations. Many of them walk miles every day to their ‘hole’ and may spend hours in a queue waiting for their turn to draw water.
Areas with community self help groups (SHG) that work together on tasks of mutual assistance are ideal for sand dam building. Sand dams are low cost and low tech, but someone needs to own them, take care of them and use the water for maximum benefit to all. An active community group that already has an agenda for self and community benefit is a good potential partner in sand dam building. Our organization always works with community self help groups in building sand dams. Their members are an active part of the planning process, handle the local and government agreements for the dam, and provide about half the cost of the dam by their labor and collection of local materials such as stones, sand, and water used in building the dam. More information about how MCC works with community self help groups can be found at the end of this article
What are the benefits of sand dams to people?
Each sand dam has potential to provide a clean supply of water for up to 1,200 people, animals, tree nurseries and vegetable gardens. The increased water availability in a 10 kilometer radius means that a sand dam may indirectly benefit up to thousands of people, as the use of the stored water is never restricted to the people who built the sand dam.
Sand dams change the lives of people by providing water for their needs:
- They provide a year-round source of water near community members’ homes so they do not have to spend hours walking and queuing to fetch water.
- Saline water becomes less salty over time as less evaporation occurs: as more water comes into the sand dam, the concentration of salt lessens.
- The water is cleaner, having been filtered through the sand.
- The water is protected from parasites and people are less likely to get ill.
- Increased water capacity allows communities to set up tree nurseries in semi-arid areas where tree planting is otherwise very difficult.
- Increased water for irrigation provides more food for humans and animals, and more income as food security is realized. People can grow vegetables as soon as a sand dam has water in it, even if there is no rain. Sand dams often fill with underground water coming from upstream, even though there has been no rain in the immediate area.
What are the benefits of sand dams to the environment?
Sand dams change the environment by restoring and enhancing it:
- Sand dams transform the environment as the stored water raises the water table level both upstream and downstream from the dams (Brandsma et al, 2009; Frima et al, 2002). As the aquifer increases in size, wells and boreholes have more water and springs may return to the area.
- The higher water table increases the natural vegetation. Indigenous trees and riparian plants return to the area, and birds and fish return to the restored ecosystem (fish come over the dam wall and live in new pools that form downriver). Bio-diversity increases significantly as the river bed, banks and water catchment area are replenished (Ertsen, 2006).
- Increased bio-diversity makes it possible for community members to create a sustainable livelihood in harmony with their environment.
What are the criteria for locating the sand dam?
Solid bed rock: There must be a good rock bed on which to lay the foundation of the dam. Without this rock foundation the sand dam is likely to wash away after a heavy rain.
- The quality of the bed rock should be dense, non-porous and without cracks.
- The rock bed should be on the surface or fairly near the surface, and optimally extend across the river bed. This bed rock should be reached by digging a trench in the river bed to expose the rock before a final decision on a site is reached. If the rock bed does not extend across the river bed, you would need to dig down very far and put in a concrete foundation.
- This trench should be extended into the river banks to see what kind of foundation will be needed for the wings.
Topography of the area:
- The river valley should be narrow and well defined, to ensure the length of the dam wall is as short as possible. Ideally, the site should be in a deep gorge section of the valley to maximize storage capacity and minimize surface area and length of the dam wall. However, many sand dams have been built in fairly level areas and they work very well even though they are typically quite long.
- The river banks should be high enough to construct a dam wall that will ensure a large storage volume and hold the wing walls.
- The location should be away from bends in the river and soil erosion.
- The stretch behind the dam should be large enough to enable the maximum amount of sand storage. The gradient should be ideal to facilitate good storage of sand behind the dam.
- The presence of lots of tributaries upstream is ideal.
- The basin should be as free as possible of cracks and fissures to minimize water leakage.
- The location should be easily accessible to aid in the construction, use, and maintenance of the dam.
- Valuable property and land should not be submerged due to the construction of the dam.
Type of sand:
- The coarser the sand in the river, the better the site. Fine sand is not suitable because there is less storage space for water between the particles. The amount of water available for extraction in different sands soils is given in Table 1.
- Walk the river bed with elders and the women who fetch water. Ask questions, listen, and look. They will know where water can be found in dry seasons or drought.
- Walk up and down the river bed to find a good rocky formation as near to the beneficiaries as possible.
- Talk to the leaders and elders of the community to find where the high water mark for most floods is. This is the height of the primary spillway.
- Talk to the elders to find where the high water mark is of the worst flood in their memory. This needs to be marked with stakes so that the secondary spillway and wing height is higher than these stakes.
- Check out the community politics of the area to see if the owners of land around the dam site will be willing to give people passage across the land to access the water.
- Engage the community in identifying several sites and explain the benefits of building a cascade or series of dams on the river for maximum benefit and the creation of a green belt around their sand dams.
What types of permits are required before construction can begin?
Generally two types of permits must be obtained before starting construction. One is an agreement between the local group who is doing the project and the landowners whose land borders the area of the dam site. The other is the permit that the government requires for construction on a seasonal river.
Sand Dam Construction Agreement: this agreement is signed by the group. This agreement should list at a minimum:
- the consent necessary from land owners on land use and access to water issues;
- the consent necessary from the government to construct the sand dam;
- the amount of terracing (terracing is explained below) to be dug before dam construction starts;
- the labor and materials such as stones, sand and water (inputs) expected from the community group; and
- the requirement of on-going monitoring and maintenance of the dam by the group.
Government Agreement for the construction of a sand dam on a seasonal river: the group must go to the local government entity and identify all the permissions necessary for the construction of the sand dam, and get all the necessary forms. In the Eastern Province of Kenya there is one form: this agreement is signed by the Community Representatives of the group and regional government water representatives.
What is the best time to build a sand dam?
The timing of construction depends upon:
- the availability of resources;
- the agreement of the community group to that time;
- availability of the members to work;
- the presence of sand, stones, and water ready at the site;
- the preparation of the terraces around the sand dam site;
- the time that it takes to get the necessary government permit(s); and
- the rainy season, as sand dam construction should be finished before the rains start.
How do you design the sand dam?
UDO has qualified water engineers, called Dam Coordinators, who work with the SHG to site the sand dam, and then draw up the dam design which is submitted with the other paperwork to the government for approval. Figure 2 shows what the final design might look like.
Figure 2: Summary of steps taken to figure out sand dam dimensions. A step-by-step diagram is found in the online Supplement to this issue of EDN.
1) Identify the center of river flow and mark
2) Identify height of primary spil n way
3) Identify normal width of main river channel for length of primary spillway
4) Add ½ to 1 m height for secondary spillway height
5) Identify length of flood river flow channel for length of secondary spillway
6) Add ½ to 1 m height
7) Identify length of wings 8) From center mark of river take measurements
Here is the basic methodology for the design:
Primary spillway: The dam will fill very quickly when the rains come (rains in the semi-arid areas where we live are short and erratic but very heavy). The primary spillway will center and discharge the normal flow into the normal river channel during the rainy season.
- At the chosen site, find the narrowest place with the best rock bottom. Place a stake here. Now find the center of the river bed at this point.
- Identify the center of the river bed to center the primary spillway directly above this point. Make sure that the dam can be secured to the best rock bed available and the primary spillway will be centered in the center of the natural river bed. The river must continue to flow where it flowed before or it will cause erosion. Place a stake here.
- Determine the height of the primary spillway from the historical normal height of the flow of water after rains. Use a line level and mason’s line and identify and mark two points, one on each bank with the same height. This height at the center will be the primary spillway height.
- Determine the length of the primary spillway along this line level and mason’s line. The length of the spillway is the length needed to guide water through this channel and keep it in the normal river channel during the normal course of its running during the rainy season.
- Mark the height and length of the primary spillway. Use a tape measure. Mark its location with pegs, string, and level.
Height: The primary spillway is normally one meter or slightly higher. The chief factor is the height of the normal flow during the rainy season. Other factors are: the size of the river, the amount of normal flow, the bank heights, and the amount of storage and sand deposition area desired. After the sand has filled the dam to this primary spillway level, the community often decides to raise the level of the dam by another 0.5 meters to accommodate more sand and water. The wings will need to be extended then also.
Length: The length of the primary spillway depends on the width of the river banks, slope of the river banks, and the amount of water contained in the river. The length of the secondary spillway depends on these factors and the anticipated amount of water that will flow down the river after heavy rainfalls upstream.
Width: The width of the dam wall at the top is always wide enough for a person to walk on the top, a minimum of one meter. The foundation width at the bottom depends on the river size and flow. The average sand dam is one and a half meters wide at the foundation level and tapers up to the top. If the river is very big and the dam is high, the foundation will be thicker.
Secondary spillway: The secondary spillway is built to guide water into the center of the regular river channel when there are heavy rains and heavy stream flow. The primary spillway keeps the water in the center of the river when there is less rain. The secondary spillway is important to prevent soil erosion during heavy rains as it centers the water within the river bed.
- Determine the height of the secondary spillway. This will be the flood level of the rivers in very heavy rains.
- Use a line level and mason’s line and identify and mark two points, one on each bank, at the flood level of the river in very heavy rains. This height will be the secondary spillway height.
- Determine the length of the secondary spillway. It is the length needed to control the flow of the flood water and keep it within the river channel during a very heavy rain.
- Mark the height and length of the secondary spillway. Use a tape measure. Mark its location with pegs, string, and level.
Height: The secondary spillway is usually 1 meter higher than the primary spillway.
Length: The length extends to the wings, and depends on the width of the flow of the river at flood level.
Width: The width is usually 1 meter wide.
In the case of a very large and wide sand dam, you may need to lay out a tertiary (i.e. an additional) spillway. The height will be the height of the highest flood in the memory of the members of the SHG and elders in the area. Follow the above method to lay out and level the tertiary spillway.
Wings are built to keep the flood waters from going around the sand dam and causing erosion and eventual undercutting of the dam walls. They may not be necessary depending on the size of the banks and the volume of flow.
- The wings must be constructed to guide the river water back to its natural course in case of flooding.
- Determine the path of flood waters on both sides of the banks of the river. Mark this path of flood water with stakes.
- Determine the length of the wings, making sure they are long enough and high enough to meet the height of the highest flood in memory of the elders. Use a tape measure and level. Stake out the wings.
Height: The wings of the dam go up 1 meter high or more above the secondary spill way to prevent erosion and contain the water in the river bed. The height of the wings depends on the amount of flow of the river, the curve of the river and the current of the river.
Length: The wings may extend in length from three meters up to 50 or more meters. If the river banks are very steep the wings may just extend a few meters. If the banks are very flat, the wings can go many meters. The wings have to be long enough so that the water never diverts around the wings: sometimes we need to go back and extend the wings to ensure that flood waters are contained within the river channel.
Width: The base of the wings is thicker than the top of the wings, and tapers off towards the ends of the wings. The wings are made of the same materials as the dam wall.
Make a record of all measurements and keep them. Check to see that you have all the required measurements for the design.
The Sand Dam Dimensions Design is drawn on site by the Dam Coordinator and used to do the Design Measurements, which is then used on site by the Artisan who constructs the spillways and wings according to these specifications. An example of a Sand Dam Dimensions Design can be accessed in the online Supplement to this issue.
How do you construct the sand dam?
Foundation: The foundation is dug down to the rock bed. If any rock is porous it must be removed to leave only hard rock for the foundation. Clean the rocks with a brush and water and bail out the dirty water. Dig away any loose or partially decomposed rock. Pour dry cement on this rock after it is clean to fill in any cracks in the rock bed.
Diverting flowing water: If there is water flowing in the river, build shuttering (a timber frame) to divert the water to one side. Then start construction from the side free of water.
Twisted steel reinforcement bars: Using a hammer and chisel, dig holes 2.5 cm in diameter and 7.5 cm deep, one and a half meters apart in a zigzag fashion in the rock foundation within the space where you will place the timber form work (shuttering) for the sand dam wall. These holes will anchor the twisted steel bars (also called rebar) that reinforce the concrete wall. Into each hole place a 1 ½ meter length of twisted steel rebar. The rebars should zigzag within the dam wall, always at least 20 cm from the sides of the timber form work. When the lengths of rebar are in place, the form work is constructed around them. The bars are kept in place by the rock holes. Barbed wire is placed around the rebar for reinforcement. When the form work is in position, the artisan supervises the placing of barbed wire through the middle of the form work to keep it reinforced and in place with the right measurements. The barbed wire keeps the timber form work from spreading (Figure 3).
Timber forms (form work): Two timber forms must be constructed, one each for the upstream and the downstream side of the dam. These forms will contain the concrete until it is hardened. The artisan supervises their construction. The two forms are made from horizontal boards and upright timbers. The horizontal boards are 2.5 cm by 15 cm and several meters long, depending upon the width of the sand dam desired. The uprights are 5 cm by 10 cm and one and a half meters high from the bottom to the top. The horizontal boards and upright timbers are nailed together and create the form work. The forms are made on the bank and then taken to the prepared foundation site and fitted into position over the rebar anchored in the rock bed.
Supports need to be nailed to the form work when it is placed in position over the dam wall foundation site, before concrete is poured in. These supports secure the form work into position.
Concrete walls: The community members, under the supervision of the Artisan, fill the form work with mortar and stones. The experienced Artisan supervises the placement of the mortar and stones. The mortar is made by mixing cement, sand and water as close to the sand dam wall as possible. The mortar is mixed by the group members on the ground using shovels (Figure 4). When it is ready, it is shoveled onto large flat metal pans. These pans are passed along a line of people to the dam wall. Stones are passed along the line when the artisan requests them. This community assembly line process is continued until the form is full of mortar and stones.
The stones gathered by the community must be clean, high quality impermeable stones. If the stones have soil on them, brush them well with a wire brush to remove the dirt. There should be three sizes: large flat stones, medium stones, and small stones.
The ratio for the mortar for the first 50 cm layer of wall is one bag of cement to two wheelbarrows of sand. After this first 50 cm layer of mortar is laid, large stones are carefully placed by the artisan into the mortar. These large stones are then followed by medium and small stones.
After this initial layer the ratio of cement to sand is reduced (from 1:2) to one bag of cement to three wheelbarrows of sand. This ratio of 1:3 is then maintained for the rest of the dam wall. Strands of barbed wire are placed in this mixture of mortar and stones at 25 cm intervals.
Large flat stones are placed perpendicular to the basement layer. Then smaller stones fill in the gaps. Use a mason hammer to hit the small stones to remove air spaces and fill all spaces between the big stones. Then add 15 cm of mortar. Repeat layers. Leave 8 cm of space for the mortar between the stones and the form work on both sides.
What needs to be done for immediate maintenance?
Dam wall curing: For the first 21 days after the sand dam is completed, the artisan and the community SHG need to keep a close watch on the dam wall. The concrete must be kept damp to cure it properly. If the concrete is not kept damp for 21 days, the dam wall may crack and leak. If the weather is very hot, the dam wall will need to be monitored carefully as the curing takes place, and more water may be needed to dampen the walls more frequently. We place sand on the dam wall to keep it moist. Heavy rainfall during this time of curing will hinder the process. It is best not to build during the rainy season.
- Protecting the banks and catchment area around the sand dam:
- Terracing: Terracing will prevent excessive soil deposits upstream from the dam, by controlling fast, high volume run-off of water that causes soil erosion. It keeps silt out of the dam. Terraces will also conserve water in the embankments and trenches.
- Terraces are built and maintained so silt is kept out of the sand dam. Silt does not store water well. If silt fills the dam, there is little space for water in the voids. UDO requires the terraces above the dam wall to be dug before the dam wall is started (Figure 7). The community SHG needs to understand that the sand dam will not work properly if terraces are not built and maintained. The soil will wash down the banks into the river, filling the dam with silt rather than sand. Also, the terraces capture water in the soil, so that more water is available on the slopes for the grasses, trees, and crops that are growing there.
- Napier or other grass and/or legumes must be planted on the terrace ridges. Napier grass provides good fodder and serves as an anchor for the terrace, further preventing erosion. [Editor: You can use vetiver grass if free-range livestock might destroy the edible grasses.] The terraces keep manure from being washed away and increase the aeration of the soil. Farmers see a distinct increase in crop yield if their land is terraced. Farmers benefit with terracing even at 2% slopes or lower (Figure 8).
- Plantings downstream: The planting of Napier grass or other grasses or trees is important on the sides of both banks downstream to stabilize the site and prevent erosion. This planting needs to be done immediately after construction.
- Fencing: The community group may want to build a fence around the water points in the river bed upstream to protect the area from animals fouling the sand and reducing the dam’s water storing capacity.
- Potential erosion spots: Small rills or gullies near the site should be blocked to prevent soil erosion. These may be blocked by sand sacks or grasses. Many communities build small sand dams on these rills and gullies for additional water harvesting.
What is involved in long term maintenance?
The community SHG needs to recognize and accept the responsibility for long term maintenance of the sand dam and its banks and terraces. The following are the most important things to check:
- Immediately after a heavy rain, inspect the sand dam for damage.
- Check to make sure no water is going around the wings. If it is, extend the wings. If this is not done, the dam will wash out on that side and the work is lost.
- Check to see that tree trunks, branches and other materials that have been carried down the river are removed.
- Check for erosion downstream and renew plantings of Napier grass. Sometimes the length of the wings can be extended to prevent this erosion.
- Check that erosion is not occurring at the apron (i.e. the area directly under the primary and secondary spillways), along the bottom of the spillway and along the bottom of the wings downstream. If it is, reinforce the apron with cement and/or stones. If this is not done, the dam wall will wash away in the next rains.
- Check for leaks and/or cracks in the masonry of the dam wall and wings and repair immediately.
- Check the banks upstream and downstream and renew any plantings to control erosion.
- Check the terraces and replant any embankments that are damaged.
- Consider fencing if you see damage from animals on the terraces (or use vetiver grass).
- Think seriously of making more sand dams above and/or below the first one. A series of sand dams creates much more water storage, raises the water table, and increases the vegetation in the area. In the long term, this change will significantly improve the ecology of the area and the lives of the people. Sand dams can be placed every 1 to 2 kilometers along a seasonal river bed, and each one benefits the next. Each one slows down the rush of flood water, and allows water to be stored rather than running to the ocean.
Why do you often extend the sand dam height after a year or so? Why don’t you just build it higher to begin with?
If the sand dam wall is built higher than 1 meter at first, it may fill up with silt. We want only coarse sediment (sand) to settle behind the dam wall. Once the sand dam is filled with sand, the wall can then be raised another meter and the sand will fill this extended space. Coarse sand increases the storage capacity of the dam because of its high porosity.
How do you prevent silt from clogging up the sand dam?
If a sand dam is poorly located, badly designed or not regularly maintained, it will fill with silt instead of sand and the amount of water it stores will be significantly reduced.
- The site must be on a seasonal river with a sandy river bed and coarse sand.
- The height of the first spillway should be 1 meter so that the lighter silt flows over the spillway and the heavier sand sinks to the bottom and is stored by the dam.
- Terraces must be built upstream on the banks of the river to stop the rapid flow of water and silt into the river bed.
- The ridges of the terraces must be planted with napier or vetiver grass to anchor the terraces, prevent erosion and keep silt on the banks.
- The banks of the river bed can be planted with more grasses and trees to further anchor the soil and provide good quality fodder for animals during drought periods.
What about the people downstream? Do they lose water?
In our area it is clear that the total amount of water spilling over a sand dam after a rain is much higher than the volume stored in its reservoir. Only about 2% of the total water coming from the particular catchment of one dam is stored in its reservoir (Hut et al, 2006).
What is the cost benefit of a sand dam?
These calculations were done at Utooni Development Organization, Kola, Kenya in 2010. We calculated the cost of building a sand dam by adding a) the average cost of the dam including all staff and materials and b) the community self help group (SHG) contribution as detailed below*. Our water engineers estimated the volume of water in an average dam. The cost of water is the current rate in 2010 in the target area if people buy water.
For the average sand dam alone: (US$ 1= KSH 75/-)
Cost of average dam: 575,184/- 7,669
*SHG contribution: 640,000/- 8,533
Total cost: 1,215,184/- $16,202
Volume of water in average dam 100,000 cu. m
Cost of water per cu meter: 100/- $1.33
Value of water: 10,000,000/- $133,000
Cost benefit ratio in first year of operation: 1,215,184: 10,000,000 or 1:12 (US$ 16,202 : 133,000 or 1:12)
*SHG contribution includes the volunteer labor and meals that the SHG members contribute to the sand dam construction. The rate of 250/- per day is the average rate for casual labor in the area.
Why do you always work with community self help groups in building dams?
Our organization, UDO, works among the Kamba people in Eastern Kenya. UDO was founded by a local farmer, Joshua Mukusya, and makes use of cultural knowledge and practice. One of the Kamba’s strongest cultural traditions is ‘mwethya’ or people working together in unity on a task for the common good. This cultural tradition is a major part of the success of the sand dam project. People are very comfortable getting groups together for mutual tasks such as building houses, raising money for weddings and funerals, and doing merry-gorounds (the latter is a traditional method of micro-finance, in which each person contributes a set amount of money each month and the total amount is given to a different member each month). UDO has used this valuable tradition to help groups get together for holistic plans and long-term action to move group members together from subsistence to prosperity through water, food, and income security. Water security always comes first. As one woman said to us, “We don’t want relief. Give us water and we can grow our own food.”
Sustainable development projects are best started and maintained by local people with local ownership, local decision making and local resources. A community self help group that already works, or wants to work, on water and food security is a natural partner for development organizations seeking the same goals.
The advantages of working with self help groups in building sand dams are:
- The group takes charge and develops local ownership of the sand dam;
- The group’s contribution of labor and materials reduces the cash cost of the sand dam by approximately half;
- The group feels ownership of the developing assets, water and sand and the restored vegetation;
- Training with the group helps in strengthening group knowledge, skills and practice to initiate other projects;
- Groups can get together and exchange knowledge and skills, motivating and helping each other; and
- Income generated from the dam assets (e.g. from sales of water, vegetables, fruits and fodder) is then available for other group projects.
Roland Bunch’s book Two Ears of Corn is a good text that reflects the UDO philosophy of working together with community groups. We use this text when training new staff in our organization. [The book is available from ECHO’s bookstore: www.echobooks.org.)
How do you work with community self help groups to build sand dams?
We work only with registered self help groups that are active in other projects before they come to us and request assistance in building sand dams. We work with the group for six months to make sure that they are motivated and ready to start a sand dam project.
We use another traditional cultural practice, ‘kuthiana,’ in working with the community self help group. ‘Kuthiana’ is the spying out of the land and then copying good ideas. We have used this method of oral and visual learning as a major communication strategy for training of community groups through exchanges. Groups travel together to observe other successful groups and learn from them, thus accelerating learning and knowledge sharing on water and food security in the most efficient method possible—one based on their own oral tradition of learning from observation of others. Successful local self-help members conduct much of the training.
Groups get together to help each other build sand dams, most often when a group is building its first sand dam, when the group is very small, or when the sand dam is especially big. It is common to have three or four groups working on one dam. Groups that have built many dams are very experienced: they work fast with little direction except from the Dam Coordinator and the Artisan who make sure the design is followed and the quality of the construction is high.
We have designed and used trainings on six sets of capacity building skills to support and encourage ‘mwethya,’ using the learning technique of ‘kuthiana.’ The first three sets are given to the group in their first year of partnership with us (Stern, 2009). We have found that these trainings must often be repeated throughout their association with us. These capacity building sets are:
- identity and vision
- governance and leadership
- strategic planning
- performance and results
- relationships and communication
- resource development.
Our best trainers are members who have built many dams, made many mistakes and learned from them, and are in groups that are now economically successful (making money).
Full references for the article and a few more photos are found in a Sand Dams Technical Note .
References Cited in the EDN Sand Dams Article and/or Technical Note
Beimers, P.B., van Eijk, A.T., Lam,K.S., and Roos, B. June 2001. Improved Design Sand –Storage Dams, Project Report, SASOL Foundation, Nairobi, Kenya.
Borst, L. and de Haas, S.A. 2006. Hydrology of Sand storage dams. A case study in the Kiindu catchment, Kitui District, Kenya. Master thesis,Vrije Universiteit, Amsterdam, The Netherlands.
Brandsma, J., Hofstra, F., Bjorn, L., Masharubu, B., and Mailu, D. 2009. Impact evaluation on Sand Storage dams. SASOL, Wageningen UR, and Van Hall Larenstein UR.
Bunch, R. 2000. Two Ears of Corn: a Guide to People-Centered Agricultural Improvement. World Neighbors, Oklahoma City, Oklahoma, U.S.A.
Ertsen, M. 2006. Re-hydrating the Earth in Arid Lands (REAL): systems research on small groundwater retaining structures under local management in arid and semi-arid areas of East Africa. Water Resources Management, Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 5048, 2600 GA, Delft, The Netherlands. Contact: [email protected]
Frima, G.A.J., Huijsmans, M.A., van der Sluijs, N., and Wiersma, T.E. 2002. Sand Storage Dams. A manual on monitoring the ground water levels around a sand-storage dam. Delft University of Technology and SASOL. Nairobi, Kenya.
Hut, R., Ertsen,M., Joeman,N., Vergeer,N., Insemius, H. and Van de Giesen, N. 2006. Effects of sand storage dams on ground water levels with examples from Kenya. Water Resources Management, Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 5048, 2600 GA, Delft, The Netherlands; Contact: [email protected]
Nissen-Petersen, E. 1999. Affordable water: a series for designers and builders. ASAL Consultants Ltd, Nairobi, Kenya. Contact: [email protected]
Stern, A. 2009. Mobilizing and Sustaining Self Help Groups. Utooni Development Organisation, Kenya. Contact: [email protected]
Sand Dam Bibliography – For Further Reading
Aerts, J., Lasage, R., Beets,W., De Moel,H., Mutiso,G., DeVreis,A., 2007. Robustness of sand storage dams under climate change. Vadose Zone Journal 6, 572-580.
Borst, L., Haas, de, S.A. 2006. Hydrology of Sand storage dams. A case study in the Kiindu catchment, Kitui District, Kenya. Master thesis, Vrije Universiteit, Amsterdam.
Chleq, J.L. and H. Dupriez. 1988. Vanishing Land and Water. Soil and Water Conservation in Drylands. Macmillan.
Ertsen, M.W., Biesbrouck,B., Postma, L., van Westerop,M., 2005. Participatory design of sand storage dams. In: Goessling, T., Jansen,R.J.G., Oerlemans, L.A.G. (Eds.) Coalitions and Collisions. Wolf Publishers, Nijmegen, pp.175-185
Ertsen, M. and Hut, R. 2009. Two waterfalls do not hear each other. Sand-storage dams, science and sustainable development in Kenya. Physics and Chemistry of the Earth. 34 (2009) 14-22
Gijsbertsen, C. 2007. A study to upscaling of the principle and sediment transport processes behind sand storage dams. Kitui District, Kenya. Vrije Universiteit, Amsterdam.
Haysom, A. 2006. A study of the factors affecting sustainability of rural water supplies in Tanzania. Institute of water and the environment, Cranfield University, Silsoe.
Hoogmoed, M. 2007. Analyses of impacts on a sand storage dam on ground water flow and storage. Vrije Universiteit, Amsterdam.
Hussy, S.W. 2007. Water from sand rivers: guidelines for abstraction. Water, Engineering and Development Centre, Loughborough University, UK. wedc@lboro.
Hut, R., Ertsen, M.W., Joeman, N., Vergeer,N., Winsemius, H., Van de Giesen, N.C., 2008. Effects of sand storage dams on ground water levels with examples from Kenya. Physics and Chemistry from the Earth 33, 56-66.
Jansen, J. 2007. The influence of sand dams on rainfall-runoff response and water availability in the semi arid Kiindu catchment, Kitui District, Kenya. Vrije Universiteit, Amsterdam.
Lasage, R., Aerts, J., Mutiso, G.C.M., de Vries, A., 2008. Potential for community based adaptions to droughts: sand dams in Kitui, Kenya. Physics and Chemistry of the Earth 33, 67 -73.
Lasage, R., Mutiso,S., Mutiso, G.C.M., Odada, E.O., Aerts, J., and de Vries, A.C. 2006. Adaptation to droughts: Developing community based sand dams in Kitui, Kenya. Geophysical Research Abstracts, Vol.8, 01596, European Geosciences Union.
Lee, M.D. and J.T. Visscher. 1990. Water Harvesting in Five African Countries. IRC Occasional Paper No. 14.
Munyao, J.N., Munywoki, J.M., Kitema, M.I., Kithuku, D.N., Munguti, J.M., and Mutiso, S. 2004. Kitui sand dams: Construction and Operation. SASOL Foundation.
Nissen-Petersen, Erik. 2006. Water from Small Dams. For Danish International Development Assistance (DANIDA). ASAL Consultants Ltd, Nairobi, Kenya. [email protected]
Nissen-Petersen, E. 2006(a). Water surveys and designs. For Danish International Development Assistance (DANIDA). ASAL Consultants Ltd, Nairobi, Kenya. [email protected]
Nissen-Petersen, Erik. 2007. Water Supply by Rural Builders. For Danish International Development Assistance (DANIDA). ASAL Consultants Ltd., Nairobi, Kenya.
Opere, A.O., Awuor, V.O., Kooke, S.O., Omoto,W.O., 2002. Impact of Rainfall Variability on Water Resouces Management: Case Study in Kitui District, Kenya. Third Waternet/Warfsa Symposium Water Demand Management for Sustainable Development, Dar es Salaam, 30-31 October 2002.
Orient Quilis, R. 2007. Modeling sand storage dams systems in seasonal rivers in arid regions. Applications in Kitui District (Kenya). Master thesis, Delft University of Technology, UNESCO-IHE.
_______; A practical guide to sand dam implementation: Water supply through local structures as adaptation to climate change. 2009. Rainwater Harvesting Implementation Network. Rain Foundation, Acacia Water, Ethiopian Rainwater harvesting Association, Action for Development, Sahelian Solutions Foundation (SASOL).
_______; Sourcebook of Alternative Technologies for Freshwater Augmentation
in Africa, Newsletter and Technical Publications http://www.unep.or.jp/ietc/publications/techpublications/techpub-8a/dams.asp (article in ED UK tech/academic files)
_______; Understanding the hydrology of Kitui sand dams: Short mission report, November 2005. Within component 1. Hydrological evaluation of Kitui sand dams, of “Recharge Techniques and Water Conservation in East Africa: up scaling and dissemination of good practices with the Kitui sand dams.”
Water Harvesting Web Pages
This website provides more detailed information about Utooni Development Organization, a Kenyan non-governmental organization working with community self help groups. It also gives the story of its founder, Joshua Mukusya, who has played a leading role in promoting and initiating sand dam projects in Kenya.
On this page, note in particular the ‘films’ link underneath the brief video. This link will lead you to another page with an option to view a video titled ‘Walking on Water.’ Once you click on this, you will see a list of supporting films that can also be viewed. A more complete DVD is available for purchase.
This website contains a wealth of materials (e.g. handbooks, manuals, slide shows, and videos) on methods for harvesting rainwater in areas with long dry seasons. Some of the information is available for a fee; however, a substantial amount of information can be read online at no cost.
Stern, J.H. 2011. Water Harvesting Through Sand Dams. ECHO Development Notes no. 111
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By Miles Hatfield NASA’s Goddard Space Flight Center
New results from NASA satellite data show that space weather – the changing conditions in space driven by the Sun – can heat up Earth’s hottest and highest atmospheric layer.
The findings, published in July in Geophysical Research Letters, used data from NASA’s Global Observations of the Limb and Disk, or GOLD mission. Launched in 2018 aboard the SES-14 communications satellite, GOLD looks down on Earth’s upper atmosphere from what’s known as geosynchronous orbit, effectively “hovering” over the western hemisphere as Earth turns. GOLD’s unique position gives it a stable view of one entire face of the globe – called the disk – where it scans the temperature of Earth’s upper atmosphere every 30 minutes.
“We found results that were not previously possible because of the kind of data that we get from GOLD,” said Fazlul Laskar, who led the research. Dr. Laskar is a research associate at the Laboratory for Atmospheric and Space Physics at the University of Colorado, Boulder.
From its perch some 22,000 miles (35,400 kilometers) above us, GOLD looks down on the thermosphere, a region of Earth’s atmosphere between about 53 and 373 miles (85 and 600 kilometers) high. The thermosphere is home to the aurora, the International Space Station, and the highest temperatures in Earth’s atmosphere, up to 2,700 °F (1,500 °C). It reaches such incredible temperatures by absorbing the Sun’s high-energy X-rays and extreme ultraviolet rays, heating the thermosphere and stopping these types of light from making it to the ground.
But the new findings point to some heating not driven by sunlight, but instead by the solar wind – the particles and magnetic fields continuously escaping the Sun.
The solar wind is always blowing, but stronger gusts can disturb Earth’s magnetic field, inducing so-called geomagnetic activity. Laskar and his collaborators compared days with more geomagnetic activity to days with less, and found an increase of over 160 °F (90 °C) in thermospheric temperatures. Magnetic disturbances, driven by the Sun, were heating up Earth’s hottest atmospheric layer.
Some amount of heating was expected near Earth’s poles, where a weak point in our magnetic field allows some solar wind to pour into our upper atmosphere. But GOLD’s data showed temperature increases across the whole globe – even near the equator, far from any incoming solar wind.
Laskar and colleagues suggest it has to do with changing circulation patterns. There’s a swirling of air high above us — a global circulation that pushes air from the equator up to the poles and back around at lower altitudes. As the solar wind pours into the thermosphere near the poles, the added energy can alter this circulation pattern, driving winds and atmospheric compression that can raise temperatures even far away.
Changing circulation might also underlie another surprise finding. GOLD’s data showed the amount of heat added depended on the time of day. The team discovered a stronger effect in the morning hours compared to that in the afternoon. They suspect that geomagnetic activity might especially strengthen the circulation during the night and early morning hours, though this explanation awaits confirmation in further studies.
Laskar was most impressed with the subtlety of the changes they could detect in GOLD’s data.
“We used to believe that only prominent geomagnetic events could change the thermosphere,” Laskar said. “We are now seeing that even minor activity can have an impact.”
With its steady stream of temperature measurements, GOLD is painting a picture of an upper atmosphere much more sensitive to the magnetic conditions around Earth than previously thought.
NASA’s SunRISE mission — short for the Sun Radio Interferometer Space Experiment — passed a mission review on Sept. 8, 2021, moving the mission into its next phase.
“SunRISE will detect and study eruptions of radio waves from the Sun that often precede major solar events containing high energy particle radiation,” said Justin Kasper, SunRISE principal investigator at the University of Michigan in Ann Arbor. “Knowing when and how solar storms produce intense radiation will help us better prepare and protect our astronauts and technology.”
The review, Key Decision Point C, evaluated the mission’s preliminary design and project plan to achieve launch by its target launch readiness date. With the successful review, SunRISE now moves into Phase C, which includes the final design of the mission and fabrication of the spacecraft and instruments. The six spacecraft will then go through final assembly and testing before their launch readiness date of April 2024.
Consisting of six miniature solar-powered spacecraft known as CubeSats, the SunRISE constellation will operate together as one large telescope — forming the first space-based imaging low radio frequency interferometer — to create 3D maps pinpointing how giant bursts of energetic particles originate from the Sun and evolve as they expand outward into space. The mission will also map, for the first time, how the Sun’s magnetic field extends into interplanetary space — a key factor that drives where and how storms move throughout the solar system. Data from SunRISE will be collected and transmitted to Earth via NASA’s Deep Space Network. The six CubeSats will span roughly six miles across and fly slightly above geosynchronous orbit at 22,000 miles from Earth’s surface.
“The unique formation of the CubeSats gives us a detailed view of the Sun that will help us figure out how high energy particle radiation is initiated and accelerated near the Sun and how it affects interplanetary space,” said Joseph Lazio, SunRISE project scientist at NASA’s Jet Propulsion Laboratory. “Studying the radio waves that precede solar particle storms could potentially help us create an early warning system.”
SunRISE is a Mission of Opportunity under the Heliophysics Division of NASA’s Explorers Program Office. Missions of Opportunity are part of the Explorers Program, which is the oldest continuous NASA program designed to provide frequent, low-cost access to space using principal investigator-led space science investigations relevant to the Science Mission Directorate’s (SMD) astrophysics and heliophysics programs. The program is managed by NASA’s Goddard Space Flight Center in Greenbelt, Maryland, for SMD, which conducts a wide variety of research and scientific exploration programs for Earth studies, space weather, the solar system and universe.
SunRISE will be built by the Space Dynamics Laboratory at Utah State University and be a hosted payload on a commercial spacecraft provided by Maxar of Westminster, Colorado. Once in orbit, the host spacecraft will deploy the six SunRISE spacecraft, place the CubeSats into their orbits, and then continue its prime mission. SunRISE will launch no earlier than April 2024 and no later than September 2025 depending on the schedule of the commercial host spacecraft.
SunRISE is led by the University of Michigan in Ann Arbor and managed by NASA’s Jet Propulsion Laboratory in Southern California.
By Joy Ng NASA’s Goddard Space Flight Center, Greenbelt, Md.
From its vantage point in geostationary orbit, NASA’s GOLD mission – short for Global-scale Observations of the Limb and Disk – has given scientists a new view of dynamics in Earth’s upper atmosphere. Together, three research papers show different ways the upper atmosphere changes unexpectedly, even during relatively mild conditions that aren’t typically thought to trigger such events.
GOLD studies both neutral particles and those that have electric charge – collectively called the ionosphere – which, unlike neutral particles, are guided by electric and magnetic fields. At night, the ionosphere typically features twin bands of dense charged particles. But GOLD’s data revealed previously unseen structures in the nighttime ionosphere’s electrons, described in research published in the Journal of Geophysical Research: Space Physics on Aug. 24, 2020.
While comparing GOLD’s data to maps created with ground-based sensors, scientists spotted a third dense pocket of electrons, in addition to the typical two electron bands near the magnetic equator. Reviewing GOLD data from throughout the mission, they found that the peak appeared several times in October and November of multiple years, suggesting that it might be a seasonal feature.
Though scientists don’t know what exactly creates this extra pocket of dense electrons, it appeared during a period of relatively mild space weather conditions. This was a surprise to scientists, given that big, unpredictable changes in the ionosphere are usually tied to higher levels of space weather activity.
GOLD also saw large drops in the upper atmosphere’s oxygen-to-nitrogen ratio – a measurement typically linked to the electron changes that can cause GPS and radio signal disturbances.
This event was notable to scientists not for what happened, but when: The dips that GOLD saw happened during a relatively calm period in terms of space weather, even though scientists have long associated these events with intense space weather storms. The research was published on Sept. 9, 2020, in Geophysical Research Letters.
During a geomagnetic storm – space weather conditions that disturb Earth’s magnetosphere on a global scale – gases in the upper atmosphere at high latitudes can become heated. As a result, nitrogen-rich air from lower altitudes begins to rise and flow towards the poles. This also creates a wind towards the equator that carries this nitrogen-rich air down towards lower latitudes. Higher nitrogen in the upper atmosphere is linked to drops in electron density in the ionosphere, changing its electrical properties and potentially interfering with signals passing through the region. GOLD observed this effect several times during relatively calm space weather conditions during the day – outside of the disturbed conditions when scientists would normally expect this to happen.
These changes during seemingly calm conditions may point to a space weather system that’s more complicated than previously thought, responding to mild space environment conditions in bigger ways.
“The situation is more complex – the ionosphere is more structured and dynamic than we could have seen before,” said Dr. Sarah Jones, mission scientist for GOLD at NASA’s Goddard Space Flight Center in Greenbelt, Maryland.
Indeed, GOLD’s observations of changing atmospheric composition are already informing scientists’ computer models of these processes. A paper published in Geophysical Research Letters on May 20, 2021, uses GOLD’s data as a reference to show how changes near the poles can influence the ionosphere’s conditions in the mid-latitudes, even during periods of calm space weather activity. GOLD’s broad, two-dimensional view was critical to the finding.
“When you look in two dimensions, a lot of things that look mysterious from one data point become very clear,” said Dr. Alan Burns, a researcher at the High Altitude Observatory in Boulder, Colorado, who worked on the studies.
By Sarah Frazier NASA’s Goddard Space Flight Center, Greenbelt, Md.
By Mara Johnson-Groh NASA’s Goddard Space Flight Center
Plasma – a fourth state of matter after solid, liquid, and gas where particles have split into charged ions and electrons – is the most common form of matter in the universe. It’s somewhat rare on Earth, but it makes up 99% of the matter in the visible universe. Despite its prevalence, scientists haven’t been able to observationally verify a foundational theory describing how plasma moves in response to electric and magnetic forces. Until now.
With its ultraprecise measurements, NASA’s Magnetospheric Multiscale mission – MMS – has finally measured plasma’s movement on the small scales necessary to see if plasma collectively interacts with electromagnetic fields in the way the fundamental theory predicts, which is described mathematically by the so-called Vlasov equation.
Since the beginning of plasma physics research nearly 100 years ago, the Vlasov equation has often been assumed to be valid for many kinds of plasmas in space. But now the new MMS results, which were published in the journal Nature Physics on July 5, 2021, have allowed scientists to finally see the fundamental plasma variations described in the theory for the first time in nature.
Measuring the basic interactions of space plasmas with electric and magnetic fields helps scientists better understand different mechanisms that fuel energetic space weather events, from auroras to plasma ejections off the Sun, which can interfere with satellites and communications on Earth.
A Century of Plasma Physics Progress
The basic theory of plasma motion originated out of a fundamental theory for gases from the nineteenth century. In the 1890s, an Austrian physicist by the name Ludwig Boltzmann came up with a way to describe the microscopic movement of gases and fluids using statistics. It’s known as the Boltzmann equation, and it’s still taught in physics courses today.
In the 1930s, this work was extended to describe plasmas by Anatoly Vlasov, a Russian physicist. His work specifically described collisionless plasmas, which exist at such high temperatures and low densities that individual plasma particles almost never collide.
Collisionless plasma environments are common in space and can be found in the Sun’s outer atmosphere, solar wind, and in various regions throughout Earth’s magnetic environment, called the magnetosphere.
Unlike ordinary gases, plasmas are electrically charged, as they’re made of positively and negatively charged particles – namely atomic nuclei and their separated electrons. This makes plasmas behave very differently than gases since they are sensitive to electromagnetic forces, which influence their movements. Whereas individual particles in a gas constantly bounce off each other as they erratically travel along, collisionless plasma particles rarely interact and are instead controlled solely by electric and magnetic forces. Vlasov’s equation describes the interplay between plasma particles and electromagnetic fields in these unique collisionless plasma systems – and it has formed the foundation for many ideas about plasmas in the years since.
“The Vlasov equation governs all collisionless plasma phenomena that we know of,” said Jason Shuster, lead author on the new study, Assistant Research Scientist at the University of Maryland in College Park, and MMS scientist at NASA’s Goddard Space Flight Center in Greenbelt, Maryland.
Nevertheless, the terms in the Vlasov equation have never been directly measured because the observation requires seven different types of measurements to be made simultaneously at very small scales in a diffuse plasma – something which is only possible in space.
“In smaller experiments in labs we can’t probe the plasma without disturbing it,” said Barbara Giles, co-author on the new study, Project Scientist for MMS and research scientist at NASA’s Goddard Space Flight Center. “Only in space can we fully immerse instruments within the same phenomenon and accurately test these theories without disturbing the system.”
The MMS Tetrahedron
In 2015, the launch of the four MMS spacecraft changed the way physicists could study plasma in space. Flying in a tetrahedral – or pyramid-shaped – formation with high-resolution instruments, MMS can take measurements far beyond the capabilities of previous spacecraft missions. In their record-breaking tight flying formation, the four MMS spacecraft also fly close enough to measure small-scale properties of plasma, which enabled them to detect variations in terms of the Vlasov equation for the first time.
“With MMS, we can actually probe those minute details at higher resolution than ever before,” Shuster said.
To measure terms in the Vlasov equation, MMS used 64 particle spectrometers – instruments which measure particle energies and charges. The unprecedented spectrometers can simultaneously record multiple types of measurements in position, velocity, and time needed to resolve terms in Vlasov equation. Since each of the four spinning MMS spacecraft has 16 spectrometers sampling particles around its entire circumference, MMS is uniquely able to take these measurements at incredibly high speeds – every 0.03 seconds, which is nearly 100 times faster than previous missions.
“MMS takes advantage of the natural laboratory provided by Earth’s magnetic environment in outer space,” Shuster said. “In the lab, it is very difficult to create a vacuum with a pressure and density low enough to measure the types of electron-scale structures that we’re able to probe with MMS.”
Improving Global Predictions
Now that MMS has observationally confirmed fundamental predictions of the Vlasov equation, this data can be used to better understand phenomena in plasma environments in near-Earth space.
For example, applying small-scale knowledge of plasma to the global-scale magnetosphere can help scientists better understand different mechanisms that fuel energetic space weather events – such as magnetic reconnection, an explosive event unique to plasma that occurs when magnetic field lines sharply reverse direction.
Additionally, just as high- and low-pressure systems create winds and storms on Earth, electric currents and plasma flows drive weather systems in space. Fast moving jets of energetic plasma, such as those scientists observed in this study with MMS, can sustain strong electric currents and pressure gradients which drive space weather phenomena throughout Earth’s magnetic environment.
Having direct observations of the terms in the Vlasov equation provides scientists with a deeper understanding of these basic plasma motions, which enables them to predict the triggers of these fundamental plasma processes more accurately.
“The measurements of terms in the Vlasov equation provide information we can use to constrain and increase the accuracy of our global space weather models, which currently rely on large-scale approximations,” Shuster said. “These discoveries improve our ability to predict space weather operating close to home in Earth’s magnetosphere and deepen our overall understanding of plasmas existing throughout the universe.”
By Miles Hatfield NASA’s Goddard Space Flight Center
About 50 miles up, Earth’s atmosphere undergoes a fundamental change. It starts at the atomic level, affecting only one out of every million atoms (but with about 91 billion crammed in a pinhead-sized pocket of air, that’s plenty). At that height, unfiltered sunlight begins cleaving atoms into parts. Where there once was an electrically neutral atom, a positively charged ion and negatively charged electron (sometimes several) remain.
The result? The air itself becomes electrified.
Scientists call this region the ionosphere. The ionosphere doesn’t form a physically separated atmospheric layer – the charged particles float amongst their neutral neighbors, like bits of cookie dough in a pint of cookie dough ice cream. Nonetheless, they follow a very different set of rules.
Consider how the ionosphere moves. Charged particles are constrained by magnetism; like railroad cars, they trace Earth’s magnetic field lines back and forth unless something actively derails them. Neutral particles can cross those tracks unfazed – they’re more like passengers crisscrossing as they board and exit at the station.
The ionosphere also reflects radio signals, which pass right through the neutral atmosphere as if it was transparent. (In fact, radio is how the ionosphere was discovered.) Scientists still use ground-based radio waves to study the ionosphere from afar. Learning about the neutral atmosphere, however, usually requires going there – or looking down at it from above.
Despite their differences, the ionosphere and the neutral atmosphere are parts of a larger atmospheric system, where disturbances in one part have a way of spreading to the other. Their connection is explored in a new paper led by Scott England, space physicist at Virginia Tech in Blacksburg and co-investigator for NASA’s Global-scale Observations of the Limb and Disk, or GOLD mission, published in the Journal of Geophysical Research.
England and his coauthors combined ground-based radio signal measurements with special-purpose satellite observations to study whether Traveling Ionospheric Disturbances – waves regularly observed moving through the ionosphere – are at root the same event as Traveling Atmospheric Disturbances, pulses sometimes seen in the neutral atmosphere.
Traveling Ionospheric Disturbances, or TIDs, are giant undulations in the ionosphere, waves that stretch hundreds of miles from peak to peak. They’re are detected with ground-based radar beams, which bounce radars off the ionosphere to detect density enhancements move through.
Traveling Atmospheric Disturbances, or TADs, are gusts of wind that roll through the sky, pushing along neutral atoms as they go. TADs are harder to measure, best observed by flying within them – as somemissionshave – or by using indirect measures of airglow, the glimmer of oxygen and nitrogen in our atmosphere that brightens and dims as TADs move through it.
TIDs and TADs are measured in quite different ways, and it can be hard to compare across them. Still, many scientists have assumed that if one is present, the other is too.
“There is this assumption that they’re just the same thing,” said England. “But how robust is that assumption? It may be extremely good – but let’s check.”
England designed 3-day campaign to look for TIDs and TADs at the same time and same place.
TIDs were tracked from below, using ground-based radio receivers from stations across the Eastern U.S. Meanwhile, NASA’s Global-scale Observations of the Limb and Disk, or GOLD satellite, was looking for TADs from above, measuring airglow variations to track movements of the neutral atmosphere. But it took a little re-jiggering to see them. GOLD typically scans the whole western hemisphere once every half hour, but that’s too quick a glance at any one location to see a TAD.
“GOLD was never designed to see these things,” said England. “So we devised a campaign where we have it not do what it normally does.”
Instead, England directed one of GOLD’s telescopes to stare to along a strip of sky, increasing its detection power 100-fold. That strip is shown below in purple – GOLD’s search space for TADs. The region measured by ground-based radio receivers, which looked for TIDs, is shown in light blue. The region where they overlapped was the focus of England’s study.
The radio receivers listened for changes in incoming signals, as these modulations could indicate a TID was passing by. The graph below shows the radar results from one of the three days in the campaign. The vertical axis represents latitude – higher up meaning more northern – and the horizontal axis represents time. The different colors show the strength of the signal modulation, dark red being the strongest.
Over a span of 12 hours, three stripes formed in the data. These were TIDs: pulses or density enhancements in the ionosphere moving southward over time.
Meanwhile, GOLD was watching light from oxygen and nitrogen to discern the motion of the neutral atmosphere. The graph below shows the results. Note that the latitudes GOLD measured range farther than the radar measurements, from 60 degrees to 10 degrees, but over a slightly shorter period of time, about 8 hours.
It was a weak signal, just above the ambient environmental noise. Still, the faint outlines of TADs – diagonal stripes in the GOLD data – appeared at the same time.
“It’s really at the limit of what GOLD could see – if it was any smaller than this, we wouldn’t see it,” England said.
Aligning the two datasets and correlating them, England found that both sets of ripples moved at about the same rate. Then, with the help of some mathematical models, they tested out the idea that atmospheric gravity waves could be the underlying cause of both.
Gravity waves – not to be confused with gravitational waves, caused by distant supernova explosions and black hole mergers – form when buoyancy pushes air up, and gravity pulls it back down. They’re often created when winds blow against mountain faces, pushing plumes of air upward. Those plumes soon fall back down, but like a line of dominos, the initial “push” cascades all the way to the upper atmosphere.
England and his coauthors linked up a mathematical model of the atmosphere with an airglow simulator. They then mimicked a gravity wave by introducing an artificial sine wave to the models. The resulting simulated data produced similar “stripes,” both in the atmospheric model (TIDs) and the airglow simulator (TADs), indicating that gravity waves could indeed cause both.
So – are TIDs just TADs, as scientists assumed? While the pulses moved together, England’s team found that their amplitude, or size, were not as clearly related. Sometimes a large TID would be associated with a small TAD, and vice versa. Partly that’s because GOLD and the radio receivers don’t measure exact same altitudes – the radio receivers picked up a region about 40 miles above where GOLD could measure. But the largest contributor to that difference is probably the many other phenomena in the atmosphere that we don’t fully understand yet.
“And that makes what we’re looking at hard – but also interesting,” England added.
It will take more than three days of data to fully determine the relationship between TIDs and TADs. But England’s study provides something that virtually all scientists get excited about: a new tool for answering that question.
“We didn’t know if there would be a clear relationship between TIDs and TADs or not. And we certainly seem to have the ability to determine that now,” England said. “We just have to use the GOLD spacecraft instrument to do something we didn’t originally think of.”
By Mara Johnson-Groh NASA’s Goddard Space Flight Center
Up in the night sky, above the auroras and below the Moon, there exists a heated superhighway. Instead of cars though, this transient highway funnels charged particles across hundreds of thousands of miles toward Earth for a few minutes before vanishing.
While we can’t see it with our eyes, scientists have recently discovered a new way to look at the entire transportation structure from afar for the first time using special particles called energetic neutral atoms. Ultimately, this could help scientists predict when dangerous effects of space weather might be headed towards Earth.
The highway is temporarily created within Earth’s self-generated magnetic bubble, during periods of intense activity from the Sun, when the planet is bombarded with charged particles and radiation. The bubble protects the planet from most of the particles, but some sneak through and are funneled along towards Earth along a highway-like structure.
Since these particles can disrupt satellites and telecommunications in space and damage systems on Earth, scientists are keen to study how they are transported. However, charged particles have to be measured where they are – rather than being tracked with far-away observatories. This is like monitoring traffic by standing right on a highway instead of from a helicopter flying above – which makes it hard to get a full picture of the routes the vehicles travel.
As a result, scientists have only been able to glimpse snapshots of particles’ paths as satellites passed through the region.
But recently a group of scientists, led by Amy Keesee, a space physicist at the University of New Hampshire, decided to look at the highway in a new way. Using data from NASA’s Two Wide-angle Imaging Neutral-atom Spectrometers – TWINS – mission, which operated from 2008 to 2020, the scientists studied the region using energetic neutral atoms, or ENAs.
These particles are formed when a charged particle hits a neutral atom and loses its charge to the atom. Since it’s no longer charged, the particle is unbound from the magnetic fields it was previously confined to and goes streaming through space along a straight line. By measuring these particles, scientists can work back to see where the particle was created, providing an image of the region of charged particles. Scientists have previously used this technique to study the edge of the magnetic bubble created by the Sun.
Using near-Earth ENAs measured by TWINS, the scientists were able to create a complete map of the highway. They also verified their findings with data from NASA’s Magnetospheric Multiscale mission, which happened to fly through the region at the time. Together, this opens the door to observing these particle flows from far away, rather than needing to fly a spacecraft right through them. Such information could improve our real-time predictions of when charged particles are coming toward Earth.
By Mara Johnson-Groh NASA’s Goddard Space Flight Center
On January 31, 1958, the U.S. launched its first satellite: Explorer 1. Among its many achievements, Explorer 1 made the ground-breaking discovery of belts of charged particles encircling Earth.
That discovery is still being studied today. 63 years on, scientists are still learning about these belts – now known as the Van Allen belts – and their effects on Earth and technology in space.
From 2012 to 2019, scientists used NASA’s Van Allen Probes to gather data from the dynamic region discovered by Explorer 1. While the mission is no longer operational, it left a treasure-trove of observations, which are continuing to reveal new things about the belts. In 2020, over 100 scientific papers were published in peer-reviewed international journals using Van Allen Probes data, often leading studies in conjunction with partner missions. Here are three surprising discoveries scientists have recently made about the Van Allen belts.
1) In addition to particles, space is filled with electromagnetic waves called plasma waves, which affect how charged particles in space move. Near Earth, one type of wave, called whistler chorus waves, bounces back and forth following magnetic field lines between Earth’s North and South poles. Observations from the Van Allen Probes and Arase missions recently showed that these waves can leave the equator and reach higher latitudes where they permanently knock particles out of the Van Allen belts – sending the particles out into space never to return.
2) In addition to removing charged particles from the belts, magnetic activity can also add in new particles. Van Allen Probes observations combined with data from a Los Alamos National Lab geosynchronous satellite and one of NASA’s THEMIS satellites showed how hot charged particles can be abruptly transported by magnetic activity across 400,000 miles, from distant regions under the influence of Earth’s magnetic field into the heart of the Van Allen belts.
3) Earth’s magnetic environment and the Van Allen belts are highly influenced by the Sun, particularly when it releases clouds of ionized gas called plasma, which can create hazardous space weather. Some stormy activity from the Sun can create an intense ring of current surrounding Earth. Understanding these currents is critical for predicting their adverse space weather effects on ground-based infrastructure. Using years of Van Allen Probes data, scientists can now accurately model the distribution of the ring current around Earth even during the most intense space weather storms.
By Sarah Frazier NASA’s Goddard Space Flight Center
A special collection of research in the Journal of Geophysical Research: Space Physics highlights the initial accomplishments of NASA’s GOLD mission. GOLD, short for Global-scale Observations of the Limb and Disk, is an ultraviolet imaging spectrograph that observes Earth from its vantage point on a commercial communications satellite in geostationary orbit.
Since beginning science operations in October 2018, GOLD has kept a constant eye on Earth’s dynamic upper atmosphere, watching changes in the Western Hemisphere, marked by changes in the temperature, composition and density of the gases in this region.
A few highlights include:
Results on one source of airglow seen at night, which relies on electrons on Earth’s day side becoming ionized by sunlight, then being transported along magnetic field lines to the nightside, where they create visible airglow (Solomon, et al)
New evidence supporting the idea that the equatorial ionization anomaly appearing in the early morning — a prominent feature in the ionosphere with poorly-understood triggers that can disrupt radio signals — is linked to waves in the lower atmosphere (Laskar, et al)
New observations of planet-scale waves in the lower atmosphere that drive change in the ionosphere (Gan, et al & England, et al)
Multi-instrument measurements of plasma bubbles — “empty” pockets in the ionosphere that can disrupt signals traveling through this region because of the sudden and unpredictable change in density — that suggest they are could be seeded by pressure waves traveling upwards from the lower atmosphere (Aa, et al)
Observations showing that plasma bubbles occur frequently at all of the longitudes covered by GOLD with different onset times, providing new information on the influence of the particular configuration of the geomagnetic field at these longitudes (Martinis, et al)
Measurements of changes in the chemical composition of the thermosphere during the total solar eclipse of July 2, 2019, which give scientists an unprecedented hemisphere-wide look at how the reduction in solar radiation throughout an eclipse affects this part of the atmosphere (Aryal, et al)
By Miles Hatfield NASA’s Goddard Space Flight Center
A new NASA study finds that our distant planetary neighbors, Uranus and Neptune, may have magnetic “seasons:” A time of the year when aurora glow brighter and atmospheric escape may quicken.
Study authors Dan Gershman and Gina DiBraccio, of NASA’s Goddard Space Flight Center in Greenbelt, Maryland, published the results in Geophysical Research Letters. Though these seasonal changes haven’t been directly observed, the results show that a combination of strong solar activity and Uranus’ and Neptune’s unusually tilted magnetic fields is likely to trigger them.
From Mercury to Neptune, every planet in our solar system feels the unceasing stream of the solar wind. This barrage of solar particles, traveling hundreds of miles a second, drags the Sun’s magnetic field out to space, inevitably colliding with planetary magnetic fields.
But each planet responds differently. For planets closer to the Sun, like Mercury and Earth, the solar wind can really shake things up. Strong blasts of solar wind create our northern lights – at their worst, they can even cause electrical surges that lead to blackouts. (Mercury is hit so hard that it can’t even sustain an atmosphere.)
On Jupiter and Saturn, the solar wind’s blast has little effect. This is not because they’re farther away from the Sun – the most important factor is their magnetic fields, which are optimally positioned to protect them. These planets have strong magnetic fields aligned almost perfectly vertically, like a spinning top. As the solar wind blows past Saturn, for instance, it hits its equator, meeting its magnetic shield where it is strongest.
Uranus and Neptune are even farther away from that strong solar wind source, but their magnetic axes make them vulnerable. Uranus’ magnetic axis is tilted by a full 60 degrees. This means that for a portion of its 84-year-long trip around the Sun, the Sun shines almost directly into the planet’s magnetic north pole, where the planet is least protected. Neptune’s axis is similarly tilted – though only by 47 degrees.
With that background knowledge, Gershman and DiBraccio set out to study how the solar wind would affect the ice giants. Using historical data from the Helios, Pioneer and Voyager spacecrafts, Gershman and DiBraccio measured the Sun’s magnetic field throughout the solar system.
The results showed that during intense conditions, the solar wind can be as impactful near Uranus and Neptune as it normally is near Mercury, some 1.5 billion miles closer to the Sun.
Such intense conditions aren’t even a rarity. The enhanced solar activity Gershman and DiBraccio studied occurs regularly, as part of the 11-year solar cycle. The solar cycle refers to the periodic flipping of the Sun’s magnetic field, in which activity rises and falls. At the high point, known as solar maximum, the Sun’s magnetic field throughout space can double in strength.
If the Sun enters solar maximum when Uranus or Neptune is at the appropriate angle, the effects, Gershman and DiBraccio argue, could be extreme. These planets so far from the Sun could suddenly be driven by it. Though the seasonal effects have not yet been directly observed, the physics suggests that aurora should brighten and spread further across the planet. Globs of particles known to escape the Uranian atmosphere may do so at a quickened pace. But only a few Earth-years later, it all goes away and the planets enter a new magnetic season.
The only close-up measurements we have of the planets are from the single flyby of Voyager 2 in 1986 and 1989, respectively. But a future NASA mission to the Ice Giants may well change that, giving us the first glimpse of their other-worldly magnetic seasons.
This imagery captured by NASA’s Solar Dynamics Observatory shows a solar flare and a subsequent eruption of solar material that occurred over the left limb of the Sun on November 29, 2020. From its foot point over the limb, some of the light and energy was blocked from reaching Earth – a little like seeing light from a lightbulb with the bottom half covered up.
Also visible in the imagery is an eruption of solar material that achieved escape velocity and moved out into space as a giant cloud of gas and magnetic fields known as a coronal mass ejection, or CME. A third, but invisible, feature of such eruptive events also blew off the Sun: a swarm of fast-moving solar energetic particles. Such particles are guided by the magnetic fields streaming out from the Sun, which, due to the Sun’s constant rotation, point backwards in a big spiral much the way water comes out of a spinning sprinkler. The solar energetic particles, therefore, emerging as they did from a part of the Sun not yet completely rotated into our view, traveled along that magnetic spiral away from Earth toward the other side of the Sun.
While the solar material didn’t head toward Earth, it did pass by some spacecraft: NASA’s Parker Solar Probe, NASA’s STEREO and ESA/NASA’s Solar Orbiter. Equipped to measure magnetic fields and the particles that pass over them, we may be able to study fast-moving solar energetic particles in the observations once they are downloaded. These Sun-watching missions are all part of a larger heliophysics fleet that help us understand both what causes such eruptions on the Sun – as well as how solar activity affects interplanetary space, including near Earth, where they have the potential to affect astronauts and satellites.
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A histogram is commonly used to plot frequency distributions from a given dataset. Whenever we have numerical data, we use histograms to give an approximate distribution of that data. It shows how often a given value occurs in a given dataset. Matplotlib 2D Histogram is used to study the frequency variation of a given parameter with time.
We use 2D histograms when in a given dataset, the data is distributed discretely. As a result, we only want to determine where the frequency of the variable distribution is more among the dense distribution. There is a predefined function ‘matplotlib.pyplot.hist2d()’ present in python . It is present in the matplotlib library in python and is used to plot the matplotlib 2D histogram.
Matlplotlib is a library in python which is used for data visualization and plotting graphs. It helps in making 2D plots from arrays. The plots help in understanding trends, discovering patterns, and find relationships between data. We can plot several different types of graphs. The common ones are line plots, bar plots, scatter plots and histograms.
What is a Histogram in ‘Matplotlib 2D Histogram’ ?
Histograms are frequency distribution graphs. From a continuous dataset, a histogram will tell about the underlying distribution of data. It highlights various characteristics of data such as outliers in a dataset, imbalance in data, etc. We split the data into intervals, and each interval signifies a time period. It is not the height but the area covered by the histogram, which denotes frequency. To calculate frequency, we need to multiply the width of the histogram by its height.
x: is a vector containing the ‘x’ co-ordinates of the graph.
y: is a vector containing the ‘y’ co-ordinates of the graph.
bins: is the number of bins/bars in the histogram.
range: is the leftmost and rightmost edge for each bin for each dimension. The values occurring outside this range will be considered as outliers.
density: is a boolean variable that is false by default, and if set to true, it returns the probability density function.
weights: is an optional parameter which is an array of values weighing each sample.
cmin is an optional scalar value that is None by default. Thus, the bins whose count is less than cmin value would not be displayed.
cmax is an optional scalar value that is None by default. The bins whose count is greater than cmax value would not be displayed.
h: A 2D array where the x values are plotted along the first dimension and y values are plotted along the second dimension.
xedges is a 1D array along the x-axis
yedges is a 1D array along the y axis
image is the plotted histogram
Example Matplotlib 2D Histogram:
Here, we shall consider a height distribution scenario, and we will construct a histogram for the same.
Let us first create a height distribution for 100 people. We shall do this by using Normal Data Distribution in NumPy. We want the average height to be 160 and the standard deviation as 10.
First, we shall import numpy library.
import numpy as np
Now, we shall generate random values using random() function.
heights = np.random.normal(160, 10, 100)
Now, we shall plot the histogram using hist() function.
Understanding the hist2d() function used in matplotlib 2D histogram
The hist2d() function comes into use while plotting two-dimensional histograms. The syntax for the hist2d() function is:
def hist2d(x, y, bins=10, range=None, density=False, weights=None, cmin=None, cmax=None, *, data=None, **kwargs)
Unlike a 1D histogram, a 2D histogram is formed by a counting combination of values in x and y class intervals. 2D Histogram simplifies visualizing the areas where the frequency of variables is dense. In the matplotlib library, the function hist2d() is used to plot 2D histograms. It is a graphical technique of using squares of different color ratios. Here, each square groups its number into ranges. Higher the color ratio in 2D histograms, the higher the data that falls into that bin.
Let us generate 50 values randomly.
x = np.random.standard_normal(50)
y = x + 10
Now, we shall plot using hist2d() function.
Now, we shall try to change the bin size.
x = np.random.standard_normal(1000000)
y = 3.0 * x + 2.0 * np.random.standard_normal(1000000)
The output would be:
Now, we shall change the color map of the graph. The function hist2d() has parameter cmap for changing the color map of the graph.
Another way to plot the 2d histogram is using hexbin. Instead of squares, a regular hexagon shape would be the plot in the axes. We use plt.hexbin() for that.
The output after using hexbin() function is:
hist2d() vs hexbin() vs gaussian_kde()
hist2d() is a function used for constructing two dimensional histogram. It does it by plotting rectangular bins.
hexbin() is also a function used for constructing a two-dimensional histogram. But instead of rectangular bins, hexbin() plots hexagonal bins.
In gaussian_kde(), kde stands for kernel density estimation. It is used to estimate the probability density function for a random variable.
FAQ’s on matplotlib 2D histogram
Q. What are seaborn 2d histograms?
A. Seaborn is a Python data visualization library based on matplotlib. It provides a high-level interface for drawing statistical graphics. For example, we can plot histograms using the seaborn library.
Q. What are bins in histogram?
A. A histogram displays numerical data by grouping it into ‘bins’ of different widths. Each bin is plotted as a bar. And the area of the bar determines the frequency and density of the hat group.
Q. What is the difference between histogram and bar graph?
A. A bar graph helps in comparing different categories of data. At the same time, histogram helps in displaying the frequency of occurrence of data.
Have any doubts? Feel free to tell us in the comment section below.
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The fundamental purpose of the course in Informal Geometry is to extend students’ geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships. Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school standards. The Standards for Mathematical Practice apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas, organized into five units are as follows.
Unit 1- Congruence, Proof, and Constructions: In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. Students informally prove theorems—using a variety of formats—and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work.
Unit 2- Similarity, Proof, and Trigonometry: Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles, with particular attention to special right triangles and the Pythagorean theorem.
Unit 3- Extending to Three Dimensions: Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas.
Unit 4- Connecting Algebra and Geometry Through Coordinates: Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines, which relates back to work done in the first course.
Unit 5- Circles With and Without Coordinates: In this unit students study the Cartesian coordinate system and use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations, which relates back to work done in the first course, to determine intersections between lines and circles or parabolas.
Important Note: This Informal Geometry course content does not align with the End-of-Course Assessment required for graduation.
Course Number: 1206300
Abbreviated Title: INF GEO
Course Length: Year (Y)
Course Type: Core Academic Course
Course Level: 2
Course Status: Terminated
Grade Level(s): 9,10,11,12
One of these educator certification options is required to teach this course.
Vetted resources students can use to learn the concepts and skills in this course.
This is Part Two of a two-part series. Learn to identify faulty reasoning in this interactive tutorial series. You'll learn what some experts say about year-round schools, what research has been conducted about their effectiveness, and how arguments can be made for and against year-round education. Then, you'll read a speech in favor of year-round schools and identify faulty reasoning within the argument, specifically the use of hasty generalizations.
Make sure to complete Part One before Part Two! Click HERE to launch Part One.
Learn to identify faulty reasoning in this two-part interactive English Language Arts tutorial. You'll learn what some experts say about year-round schools, what research has been conducted about their effectiveness, and how arguments can be made for and against year-round education. Then, you'll read a speech in favor of year-round schools and identify faulty reasoning within the argument, specifically the use of hasty generalizations.
Make sure to complete both parts of this series! Click HERE to open Part Two.
Examine President John F. Kennedy's inaugural address in this interactive tutorial. You will examine Kennedy's argument, main claim, smaller claims, reasons, and evidence. By the end of this four-part series, you should be able to evaluate his overall argument.
In Part Three, you will read more of Kennedy's speech and identify a smaller claim in this section of his speech. You will also evaluate this smaller claim's relevancy to the main claim and evaluate Kennedy's reasons and evidence.
Make sure to complete all four parts of this series!
This is Part Two of a two-part tutorial series. In this interactive tutorial, you'll practice identifying a speaker's purpose using a speech by aviation pioneer Amelia Earhart. You will examine her use of rhetorical appeals, including ethos, logos, pathos, and kairos. Finally, you'll evaluate the effectiveness of Earhart's use of rhetorical appeals.
Be sure to complete Part One first. Click here to launch PART ONE.
This is Part One of a two-part tutorial series. In this interactive tutorial, you'll practice identifying a speaker's purpose using a speech by aviation pioneer Amelia Earhart. You will examine her use of rhetorical appeals, including ethos, logos, pathos, and kairos. Finally, you'll evaluate the effectiveness of Earhart's use of rhetorical appeals.
Practice writing different aspects of an expository essay about scientists using drones to research glaciers in Peru. This interactive tutorial is part four of a four-part series. In this final tutorial, you will learn about the elements of a body paragraph. You will also create a body paragraph with supporting evidence. Finally, you will learn about the elements of a conclusion and practice creating a “gift.”
This tutorial is part four of a four-part series. Click below to open the other tutorials in this series.
Learn how to write an introduction for an expository essay in this interactive tutorial. This tutorial is the third part of a four-part series. In previous tutorials in this series, students analyzed an informational text and video about scientists using drones to explore glaciers in Peru. Students also determined the central idea and important details of the text and wrote an effective summary. In part three, you'll learn how to write an introduction for an expository essay about the scientists' research.
This tutorial is part three of a four-part series. Click below to open the other tutorials in this series.
This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to result in a specified final image.
This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.
In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.
This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.
This provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function.
In this task, a typographic grid system serves as the background for a standard paper clip. A metric measurement scale is drawn across the bottom of the grid and the paper clip extends in both directions slightly beyond the grid. Students are given the approximate length of the paper clip and determine the number of like paper clips made from a given length of wire.
In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.
Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.
The purpose of the task is to analyze a plausible real-life scenario using a geometric model. The task requires knowledge of volume formulas for cylinders and cones, some geometric reasoning involving similar triangles, and pays attention to reasonable approximations and maintaining reasonable levels of accuracy throughout.
In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence.
This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focusing on the class of equilaterial triangles
The purpose of this task is to use geometric and algebraic reasoning to model a real-life scenario. In particular, students are in several places (implicitly or explicitly) to reason as to when making approximations is reasonable and when to round, when to use equalities vs. inequalities, and the choice of units to work with (e.g., mm vs. cm).
This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder
This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.
This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle: the fact that these triangles are always right triangles is often referred to as Thales' theorem.
This task applies reflections to a regular octagon to construct a pattern of four octagons enclosing a quadrilateral: the focus of the task is on using the properties of reflections to deduce that the quadrilateral is actually a square.
This task applies reflections to a regular hexagon to construct a pattern of six hexagons enclosing a seventh: the focus of the task is on using the properties of reflections to deduce this seven hexagon pattern.
This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.
This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. These tessellations are studied here using algebra, which enters the picture via the formula for the measure of the interior angles of a regular polygon (which should therefore be introduced or reviewed before beginning the task). The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.
Using this resource, students can manipulate the measurements of a 3-D hourglass figure (double-napped cone) and its intersecting plane to see how the graph of a conic section changes. Students will see the impact of changing the height and slant of the cone and the m and b values of the plane on the shape of the graph. Students can also rotate and re-size the cone and graph to view from different angles.
In this manipulative activity, you can first get an idea of what each of the rigid transformations look like, and then get to experiment with combinations of transformations in order to map a pre-image to its image.
With this online Java applet, students use slider bars to move a cross section of a cone, cylinder, prism, or pyramid. This activity allows students to explore conic sections and the 3-dimensional shapes from which they are derived. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
This program allows users to explore spatial geometry in a dynamic and interactive way. The tool allows users to rotate, zoom out, zoom in, and translate a plethora of polyhedra. The program is able to compute topological and geometrical duals of each polyhedron. Geometrical operations include unfolding, plane sections, truncation, and stellation.
Type: Virtual Manipulative
Vetted resources caregivers can use to help students learn the concepts and skills in this course.
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Universal Gravitation Equation
by Ron Kurtus (updated 30 May 2023)
The Universal Gravitation Equation states that the gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of separation between them. This equation is a result of Isaac Newton's Law of Universal Gravitation, which states that quantities of matter attract other matter to it.
The proportionality constant in the equation is called the Universal Gravitational Constant. The value of that constant was determined experimentally by Henry Cavendish in 1798.
This Universal Gravitation Equation originally applied to point masses but was extended to masses of finite size with the assumption that their mass was concentrated at their center of mass.
Questions you may have include:
- What is the Universal Gravitation Equation?
- What is the Universal Gravitational Constant?
- How is the equation an approximation?
This lesson will answer those questions. Useful tool: Units Conversion
In 1687, Isaac Newton formulated the Universal Gravitation Equation, which defines the gravitational force between two objects. The equation is:
F = GMm/R2
- F is the force of attraction between two objects in newtons (N)
- G is the Universal Gravitational Constant in N-m2/kg2
- M and m are the masses of the two objects in kilograms (kg)
- R is the separation in meters (m) between the objects, as measured from their centers of mass
This equation has proven highly effective in explaining the forces between objects, as well as leading into the effects of gravity.
Universal Gravitational Constant
When Newton originally stated the equation, he simply said that F was proportional to Mm/R2. The value of the proportionality constant or Universal Gravitational Constant, G, was not even considered for many years and not officially calculated until 1873, 186 years after Newton defined the equation.
The Cavendish Experiment has since been used to determine Universal Gravitational Constant as:
G = 6.674*10−11 N-m2/kg2
Note: The number 10−11 is 1/1011 or 0.000000000001 with 11 zeros after the decimal point.
(See Cavendish Experiment to Measure Gravitational Constant for more information.)
Check on units
It is important to make sure you are using the correct units for each item in your equation. Check by adding units to the gravitation equation and then seeing that the result is correct:
F N = (G N-m2/kg2)*(M kg)*(m kg)/(R m)2
Just considering the units:
N = (N-m2/kg2)*(kg)*(kg)/(m)2
N = (N)*(m2)*(kg)*(kg)/(m2)*(kg2)
N = N
Thus, the units used are correct.
G can also be stated in other terms, depending on its usage. Since a newton (N) equals kg-m/s2, you may also see G defined as:
G = 6.674*10−11 m3/kg-s2
Also, in applications where greater separations are studied, it is more convenient to use kilometers instead of meters. Since 1 m = 10−3 km, the value of G is:
G = 6.674*10−20 km3/kg-s2
When comparing a force in newtons with gravitational force with km, the value is the strange combination of units:
G = 6.674*10−17 N-km2/kg2
You can use whichever set of units that fulfill your requirements.
Equation an approximation
Newton originally stated the Universal Gravitation Equation as the force between two point masses, separated by R. However, it was shown that the gravitation from a large uniform sphere is approximately the same as if all the mass was concentrated at its center. Also, since matter has both size and mass, "point mass" really means center of mass.
System of particles
In a system of particles, the center of mass is the average of the particle positions, weighted by their masses. The center of mass of a sphere that has its mass evenly distributed is the center of the sphere.
(See Gravitation and Center of Mass for more information.)
Thus, the separation R in the Universal Gravitation Equation is the separation between the objects, as measured from their centers of mass.
Summation of forces
The true gravitational force between two objects is a summation of the forces from each point on both objects.
Various points on object attract points on other object
Calculus is used to integrate over all the points on the surfaces and within each object. Unfortunately, the mathematics for the exact equation is highly complex, and it is easier to make some assumptions to simplify the math.
By considering the mass of the objects concentrated is their center of mass, we get an equation that is close enough for practical purposes in most cases.
Distribution of matter in spheres
Most of the objects where the Universal Gravitation Equation applies are large spheres, such as planets, moons and stars. Often the distribution of mass in those objects is not even, and the objects are often not exact spheres.
For example, the density of matter in the Earth is unevenly distributed, plus the Earth is not an exact sphere but is flattened near its poles.
Since the separation between astronomical objects—such as the Earth and the Moon or Sun—are so large, assuming the center of mass as the center of the object is an acceptable approximation.
Consider atoms as points
Atoms, molecules and even subatomic particles are considered so small and separated by great distances relative to their size that they can be considered point sources of gravitation, and the Universal Gravitation Equation applies to these small particles.
Atoms considered points separated by distance R
However, since molecules and atoms are normally in rapid motion, you would seldom calculate the gravitational force between them, except perhaps as an average.
Isaac Newton formulated the Law of Universal Gravitation, stating that all matter attracts other matter to it. This force of attraction is defined in the theory's Universal Gravitation Equation. This equation is actually a close approximation, to simplify the mathematics. The measurement of the gravitational constant was first made by Henry Cavendish.
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Universal Gravitation Equation
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DataTables are a fundamental concept in data science, providing a structured way to organize and manipulate data for analysis, modeling, and visualization. In this article, we’ll explore what DataTables are, why they are essential in data science, and how they are commonly used.
What is a DataTable?
A DataTable, as the name suggests, is a structured table-like data structure used to store and manage data efficiently. It’s a two-dimensional representation of data, where rows represent individual data records, and columns represent attributes or variables associated with these records.
Key characteristics of DataTables include:
1. Tabular Structure:
DataTables are organized in rows and columns, similar to a spreadsheet or a database table. Each row typically represents a unique data point or observation, while each column represents a specific attribute or feature.
2. Homogeneous Data:
In a DataTable, data within a single column is usually of the same data type. For example, a column may contain integers, text, dates, or other specific data types. This homogeneity simplifies data processing and analysis.
DataTables are flexible and can accommodate different data formats. This versatility makes them suitable for various types of data, from structured to semi-structured or even unstructured data.
DataTables often include headers for columns, which provide a description of what each column represents. Headers make it easier to understand the data and its context.
5. Data Integrity:
DataTables often have mechanisms to ensure data integrity, such as data validation rules, constraints, and data types. This helps maintain the consistency and accuracy of the data.
Why Are DataTables Important in Data Science?
DataTables are the foundation of data science, providing a structured and organized way to work with data. You can learn this data science course in Surat from the top institute. DataTables play a crucial role in data science for several reasons:
1. Data Organization:
They provide an organized and structured way to store and manage data, making it easier for data scientists to work with large datasets efficiently.
2. Data Cleaning and Transformation:
DataTables allow data scientists to clean and preprocess data, including handling missing values, outlier detection, and feature engineering.
3. Data Analysis:
They serve as the foundation for exploratory data analysis (EDA) and statistical analysis. Data scientists can use DataTables to perform aggregation, filtering, and data visualization tasks.
4. Model Training and Validation:
DataTables are commonly used to train and validate machine learning models. The data is divided into training and testing sets, and models are trained on one portion and evaluated on another.
5. Data Presentation:
DataTables can be converted into various formats, including charts, graphs, and reports, for effective communication of findings and insights.
Common Libraries for DataTables in Data Science:
In data science, various programming languages and libraries provide DataTable functionality. Some of the popular ones include:
- Pandas: Pandas is a widely used library for data manipulation and analysis. It provides a DataFrame, which is essentially a DataTable. Python course in Surat is very in demand these days.
- NumPy: While primarily focused on numerical operations, NumPy arrays can be thought of as one-dimensional DataTables.
- DataFrames: R has built-in support for DataFrames, which are similar to Pandas DataFrames in Python.
- Relational Databases: SQL databases like MySQL, PostgreSQL, and SQLite use tables to store and manage data.
- Microsoft Excel: Excel spreadsheets can be considered a basic form of DataTables and are commonly used for data analysis in small to medium-sized datasets.
Whether you’re cleaning messy data, performing complex analyses, or training machine learning models, DataTables are the go-to tool for data scientists. Familiarity with the libraries and tools that provide DataTable functionality is essential for anyone in the field of data science.
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In the study of Geometry, the vital parameters which help outline the models are length and width. There is some uncertainty when differentiating the distance from the width.
The various explanations about the meaning of the two measurements are primarily determined by where it’s learned.
In mathematics, most pupils are educated about how the longest side of a parallelogram is its length. At the same time, the width would then be its shorter side of whether the longer side is the flat or vertical side. Although numerous individuals have perceived that the width is level with the flat side while the length is primarily sheer.
- Length measures an object’s longest side, extending from one end to another.
- Width measures the object’s shorter side or the distance between its parallel sides.
- Both length and width are essential for determining an object’s dimensions, area, and capacity.
Difference Between Length and Width
The difference between length and width is that by the International System of Quantities; length is the most elongated dimension of an entity. In contrast, width, or breadth, is the interval from one side to another that measures over a specific entity whose lengths form 90O angles with the shorter sides (as in a rectangle).
Comparison Table for Length vs Width
|Parameter of comparison
|Length is the distance between the longest dimension and the two ends of an object/line.
|The width can be defined as measuring an object from side to side.
|The longest side of an object will be considered as the result of a scale.
|The shortest side of an object will be considered as a result of a scale.
|The vertical side of an object is a result of calculating the length of a three-dimension model.
|The flat side of an object results from calculating the width of a three-dimension model.
|Length explains how prolonged an object is.
|Width explains how intensive an object is.
|Length is considered the most important measurement, as size defines an entity’s length.
|Width is likewise considered an equally important measurement that defines how broad an entity is.
What is Length?
Length alludes to the measurement of the dimensions of an object from end to end. The interval time first originated from the German language. Then it was later introduced into the English language as length, and from there, the word ‘length’ gained the measurement concept.
Length is used to estimate the distance.
The International System of Quantities defines length as the quantity used to compute the distance between dimensions. The base unit of length, as given by the International System of units, is the meter (m) and is nowadays defined in terms of the speed of light, which is 300 million meters per second.
The millimeter, centimeter, and kilometer, which are meter forms, can also be considered length units. There are several other units of length, such as foot, yard, mile, etc.
Einstein’s special relativity proved length cannot be constant for all reference frames. Hence, the size can depend on the observer’s perspective.
The line has one dimension, and that one measurement is the length of a line. The size of a circle is its circumference.
A rectangle has two measurements, one of these measurements is the length, and the other is the width. Length can likewise be used as a geometric measurement.
In Euclidean geometry, the length is computed using the straight lines of an object, like, say, the perimeter for a polygon can be calculated as the sum of the size of its sides. In contrast, in other geometrics, length can be estimated along curved paths, and these are called geodesics.
Tools for Measuring Length
- Tape Measure: Tape is one of the most common and versatile tools for measuring length. It consists of a flexible tape, made of metal or fiberglass, marked with inches, centimeters, or both. Tape measures are available in different lengths, ranging from a few feet to several meters. They are ideal for measuring both short distances and longer spans.
- Ruler: A ruler is a simple tool for measuring length. Typically made of wood, plastic, or metal, rulers come in various lengths, such as 6 inches or 12 inches. They feature evenly spaced markings, in inches and centimeters, allowing for precise measurements of smaller objects.
- Vernier Caliper: A vernier caliper is a more advanced tool used to measure length accurately. It comprises two jaws, one fixed and one movable, and a sliding vernier scale. By aligning the object between the jaws and reading the scale, you can determine the length or diameter of the object with great precision.
- Laser Distance Meter: Laser distance meters are electronic devices that use laser technology to measure length. They emit a laser beam and calculate the distance by measuring the time it takes for the beam to bounce back from the target. Laser distance meters are highly accurate and suitable for measuring longer distances or areas that are difficult to reach.
- Measuring Wheel: Measuring wheels, also known as surveyor’s or trundle wheels, are ideal for measuring longer distances on the ground. They consist of a wheel attached to a handle and a counter mechanism. As you roll the wheel along the surface, the counter keeps track of the number of wheel rotations, allowing you to determine the length covered accurately.
Techniques for Measuring Length
- Direct Measurement: Direct measurement involves physically placing a measuring tool, such as a ruler or tape measure, against the object or distance you want to measure. This technique is simple and effective for accurately measuring smaller objects or distances.
- Indirect Measurement: Indirect measurement involves using mathematical formulas or calculations to determine length. For example, measuring the height of a tree or a tall building can be achieved by using similar triangles or trigonometric functions in conjunction with a measuring device and specific angles.
- Non-Contact Measurement: Non-contact measurement techniques are used when direct contact with the object is not possible or desirable. This includes using laser distance meters or electronic devices that employ sensors or waves to measure length without physically touching the object.
- Comparative Measurement: Comparative measurement involves comparing the length of an object or distance against a known standard. This technique is commonly used in calibration processes or when a precise measurement tool is unavailable. It relies on visual or manual estimation and can provide rough estimations rather than precise values.
- Interpolation: Interpolation is a technique that estimates lengths between two known values. It involves using reference points or measurements and making an educated guess based on the relative positions or values. Interpolation is utilized when dealing with irregular or non-linear shapes.
Applications of Length
In Science and Engineering
- Research and Development: Length measurements are vital in scientific research and development. In fields such as physics, chemistry, biology, and material science, precise length measurements are necessary to study the properties and behavior of objects, substances, and structures. Length measurements are essential for conducting experiments, analyzing data, and formulating scientific theories.
- Engineering and Construction: Length measurements are integral to engineering and construction projects. Architects and engineers use accurate measurements to design and construct buildings, bridges, roads, and other infrastructure. Length measurements help determine dimensions, ensure structural integrity, and enable precise alignment of components.
- Manufacturing and Quality Control: Length measurements are critical in manufacturing processes. From small components to large machinery, accurate length measurements ensure manufactured products’ proper fit, alignment, and functionality. Quality control procedures involve measuring lengths to verify compliance with specifications and ensure consistency and precision.
- Metrology and Calibration: Metrology is the science of measurement, and length is a key aspect of this discipline. Metrologists develop measurement standards, calibration methods, and traceability systems to ensure accuracy and reliability in all fields that rely on measurements. Length measurements serve as the foundation for calibrating and verifying the accuracy of various instruments and devices.
- Nanotechnology: In the emerging field of nanotechnology, length measurements are crucial. Researchers and engineers working at the nanoscale rely on accurate measurements to manipulate and characterize nanoscale materials and structures. Precise length measurements enable the design and fabrication of nanoscale devices, such as sensors, electronic components, and medical tools.
In Everyday Life
- Home Improvement and DIY Projects: Length measurements are commonly used in everyday tasks like home improvement and DIY (do-it-yourself) projects. Whether you’re measuring a wall for painting, cutting wood for furniture, or installing shelves, accurate length measurements ensure proper sizing, alignment, and aesthetics.
- Carpentry and Woodworking: Length measurements are essential in carpentry and woodworking. Carpenters and woodworkers rely on precise measurements to cut materials, assemble structures, and ensure the overall quality of their projects. Accurate length measurements contribute to the durability, functionality, and aesthetic appeal of furniture, cabinets, and other wooden creations.
- Sewing and Tailoring: Length measurements are integral to fashion and garment-making. Whether you’re sewing a dress, altering clothing, or knitting a scarf, accurate length measurements are crucial for achieving the desired fit and proportions. Measurements such as waist circumference, sleeve length, and inseam help tailor garments to specific body sizes.
- Sports and Athletics: Length measurements are significant in various sports and athletic activities. Accurate length measurements of distances, such as sprinting tracks or long jump pits, determine fair competition and record-keeping in track and field events. Length measurements are also used in determining court or field dimensions for sports like basketball, football, and soccer.
- Travel and Navigation: Length measurements play a role in navigation and travel. Maps and navigation systems provide distance measurements to help travelers plan routes, estimate travel times, and determine distances between destinations. Length measurements are essential for calculating fuel consumption, estimating travel expenses, and ensuring efficient transportation logistics.
What is Width?
Width, also known as breadth, refers to the extent/distance of an object from side to side. Width is additionally considered an important measurement since it continues as a time interval.
Width is considered the shortest estimate of a factor. The width can be used to see how broad an object or how prolonged merchandise can be.
The basic unit used to measure width is the meter (m). Minimal distances can be estimated using a millimeter(mm), and to calculate large distances, kilometers(km) are used as units to calculate the width.
Width is the flat side of the plane. In this case, we’d say that the rectangle’s width is shorter than the two sides.
Width is said to have no ambiguity. The width can likewise mean telling someone how wide an object is.
If an object has two dimensions, then the length and width are necessary to calculate the area or the perimeter of the thing; for example, a rectangle has two dimensions, i.e., a vertical and horizontal plane. When one tries to quantify an object, one begins by finding out the width of the thing.
The smallest distance of an object is considered the width of the thing. Width constantly measures how intensely an object is viewed.
Applications of Width
Width in Engineering
- Structural Stability: In engineering, width is critical in ensuring the structural stability of various components and systems. The width of beams, columns, and trusses determines their load-bearing capacity and resistance to bending or buckling. Engineers calculate and design the appropriate width based on the expected loads and environmental conditions to ensure the safety and longevity of structures.
- Clearance and Accessibility: The width of spaces and passageways is essential for adequate clearance and accessibility in engineering projects. Whether it’s designing doorways, hallways, staircases, or corridors, engineers consider the width to accommodate smooth movement of people, equipment, or vehicles. Proper width measurements ensure compliance with building codes and accessibility standards.
- Channel and Pipe Design: In fluid mechanics and hydraulic engineering, width plays a significant role in designing channels, pipes, and conduits. The width of these structures influences the flow rate, pressure, and turbulence of fluids. Engineers must calculate the appropriate width to prevent excessive friction losses, maintain desired flow characteristics, and optimize the efficiency of fluid transport systems.
- Electrical Systems: Width considerations are also important in electrical engineering, particularly in power transmission and distribution systems. The width of conductors and cables affects electrical resistance, voltage drop, and heat dissipation. Engineers determine the appropriate width based on the current-carrying capacity and the desired level of power loss to ensure efficient and safe electrical operations.
Width in Design
- Graphic and Web Design: Width is fundamental for creating aesthetically pleasing and functional layouts in graphic and web design. The width of design elements, such as images, text blocks, and columns, affects the design’s overall visual balance and readability. Designers carefully consider the width to ensure optimal user experience and effective communication of information.
- User Interface (UI) Design: Width plays a crucial role in UI design, where designers focus on creating intuitive and user-friendly interfaces for software applications and digital platforms. The width of buttons, menus, input fields, and other interactive elements affects the ease of use and accessibility. Designers aim to balance providing sufficient space for content and controls while maintaining a visually appealing and efficient interface.
Tools for Measuring Width
- Ruler or Tape Measure: A ruler or tape measure is a commonly used tool for measuring width. These tools have marked increments in inches, centimeters, or both, allowing you to accurately measure the distance between two points. Rulers are ideal for measuring the width of smaller objects, while tape measures are more flexible and suitable for longer distances.
- Calipers: Calipers are precision measuring tools used to measure the width of objects with great accuracy. They consist of two arms with pointed ends or jaws that can be adjusted to fit around an object. The distance between the jaws is then read on a scale or digital display, accurately measuring the width.
- Micrometer: A micrometer, or a micrometer screw gauge, is a precise instrument for measuring small distances, including width. It uses a calibrated screw mechanism to measure the distance between its jaws. Micrometers have interchangeable anvils or measuring tips to accommodate various shapes and sizes of objects.
- Laser Distance Meter: Laser distance meters utilize laser technology to measure distances, including width. These handheld devices emit laser beams that bounce off the target object and calculate the distance based on the time the laser returns. Laser distance meters are useful for measuring large spaces, such as room widths or outdoor areas.
- Digital Imaging Software: In digital design or image editing, software tools such as Adobe Photoshop or graphic design applications provide tools to accurately measure the width of digital elements. These programs include measurement features that allow designers to select objects and obtain precise width measurements on the screen.
Units of Measurement for Width
- Inches: Inches are commonly used in countries that follow the Imperial measurement system, including the United States. One inch is equivalent to 1/12 of a foot or 2.54 centimeters.
- Centimeters: Centimeters are widely used in countries that follow the metric system. One centimeter is equal to 1/100 of a meter or approximately 0.39 inches.
- Millimeters: Millimeters are frequently used for more precise measurements, especially in fields such as engineering, manufacturing, and construction. One millimeter is equivalent to 1/1,000 of a meter or 0.039 inches.
- Meters: Meters are the primary unit of length in the metric system and are used for larger measurements. One meter is equal to 100 centimeters or approximately 39.37 inches.
- Feet: Feet are commonly used in the Imperial system, primarily in the United States and some other countries. One foot is equal to 12 inches or approximately 0.3048 meters.
- Yards: Yards are frequently used for measuring larger distances or areas, particularly in construction and landscaping. One yard is equal to three feet or approximately 0.9144 meters.
Main Differences Between Length and Width
- The length refers to the distance between two ends of an object. The width refers to measuring the breadth or how wide the thing is.
- Length can be measured in geometry by considering the most prominent side of the object. The width can be measured in geometry by considering the minor side of the object.
- The length of a three-dimensional model can be measured by considering the vertical side of the object. The width of a three-dimensional model can be measured by considering the flat side of the object.
- The main factor in measuring length is considering how prolonged an object is. The main factor in measuring width is considering how intensive an object is.
- Length can be used to estimate how long an entity is. The width can be used to estimate how broad an entity is.
Last Updated : 11 June, 2023
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Emma Smith holds an MA degree in English from Irvine Valley College. She has been a Journalist since 2002, writing articles on the English language, Sports, and Law. Read more about me on her bio page.
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When it comes to making estimates and predictions, understanding the distribution pattern of errors is crucial. Error distributions provide important insights into the accuracy and reliability of estimates, allowing us to assess the uncertainty associated with our predictions. By exploring regular distribution patterns, we can gain a deeper understanding of the underlying data and make informed decisions based on the reliability of our estimates.
One common distribution pattern is the normal distribution, also known as the Gaussian distribution. The normal distribution is characterized by a bell-shaped curve, with the majority of data points clustering around the mean or average value. This distribution is symmetrical and follows a specific mathematical formula. Understanding the properties of the normal distribution can help us interpret and analyze estimates more effectively.
Another distribution pattern that is often encountered is the skewed distribution. Skewed distributions are asymmetrical, with a long tail stretching in one direction. Positive skewness occurs when the tail extends to the right, while negative skewness occurs when the tail extends to the left. Skewed distributions can indicate the presence of outliers or a non-normal underlying data structure. By identifying and analyzing skewed distributions, we can adjust our estimates and predictions accordingly.
Additionally, there are other distribution patterns to consider, such as the uniform distribution and the exponential distribution. The uniform distribution is characterized by a constant probability density function, where all values within a given range have an equal likelihood of occurring. On the other hand, the exponential distribution describes the time between events in a Poisson process, where events occur randomly and independently. These distribution patterns have their unique properties and applications, offering valuable insights into different types of estimates and predictions.
Understanding the Basics
In data analysis, understanding the basics is essential for accurate estimation and interpretation of data. A solid understanding of estimates and error distributions is crucial in order to obtain reliable results.
An estimate is a calculated approximation of a value, based on available information or data. It is used to represent an unknown value or population parameter. Estimates can be obtained through various statistical methods, such as sampling or mathematical modeling.
Error distributions, also known as residuals, represent the discrepancies between estimated values and actual values. These discrepancies are important as they provide information about the accuracy of the estimates and the underlying distribution of the data.
Exploring regular distribution patterns is a key aspect of understanding the basics. There are several types of distribution patterns that can occur in data, including normal, uniform, and skewed distributions.
|A symmetric distribution where the majority of data points cluster around the mean. It follows the bell-shaped curve.
|A distribution where data points are evenly spread across the entire range, with no significant clustering.
|A distribution where data points are asymmetrical and skewed towards one end of the range.
Understanding these regular distribution patterns helps analysts identify outliers, assess the reliability of estimates, and choose appropriate statistical methods for analysis.
Applications of estimates and error distributions are vast and can be found in various fields such as economics, finance, healthcare, and social sciences. For example, estimating population size is a common application where error distributions play a critical role in determining the accuracy of the estimate.
What are Estimates and Error Distributions?
Estimates and error distributions play a crucial role in data analysis. In simple terms, estimates refer to the calculated values derived from a sample, which are used to make inferences about a population. These estimates provide valuable information and insights into various aspects of the population, such as mean, proportions, or other statistical parameters.
However, it is important to acknowledge that these estimates are not entirely accurate due to various factors, such as sampling error, measurement errors, or inherent variability within the population. This is where error distributions come into play.
Understanding and characterizing the error distributions is vital in data analysis as it helps us evaluate the validity of statistical models, assess the goodness-of-fit, and make appropriate interpretations of the results. It also enables us to identify any systematic biases in the estimates and adjust for them if necessary.
Importance in Data Analysis
In data analysis, understanding estimates and error distributions is crucial. Estimates are calculated values that approximate a specific parameter of interest, such as the population mean or proportion. Error distributions, on the other hand, represent the distribution of errors or deviations between the true values and the estimated values.
The importance of estimates and error distributions lies in their ability to provide valuable insights into the data. By analyzing these distributions, analysts can assess the accuracy and precision of their estimates. This information is essential for decision-making processes, as it helps to identify potential biases or errors in the data.
Moreover, understanding the distribution patterns of errors can guide the selection of appropriate statistical methods for data analysis. Different distribution patterns may require different types of analysis techniques and models. For example, if the error distribution follows a normal distribution, analysts may opt for parametric statistical methods. However, if the distribution is skewed or non-normal, non-parametric methods may be more appropriate.
By studying the estimates and error distributions, analysts can also gain insights into the underlying data generating process. They can identify trends, outliers, or other patterns that may affect the accuracy and reliability of the estimates. This knowledge can aid in further refining the data analysis process and improving the accuracy of the results.
Overall, estimates and error distributions play a crucial role in data analysis. They provide valuable information about the accuracy and precision of estimates, guide the selection of appropriate statistical methods, and offer insights into the underlying data generating process. By paying attention to these distributions, analysts can ensure the validity and reliability of their analyses and make informed decisions based on the data.
Exploring Regular Distribution Patterns
In the field of statistics, exploring regular distribution patterns is an essential part of data analysis. By examining the distribution of data, statisticians can gain valuable insights into the underlying patterns and characteristics of the data set. This knowledge allows for more accurate estimations and predictions.
There are several types of regular distribution patterns that statisticians commonly encounter:
|The normal distribution, also known as the Gaussian distribution, is characterized by a bell-shaped curve. It is symmetrical, with the majority of the data concentrated around the mean.
|A uniform distribution is characterized by a constant probability density function. In this distribution, all data points have an equal chance of occurring.
|A skewed distribution is asymmetrical, with the tail of the distribution skewed to one side. It can either be positively skewed (long tail to the right) or negatively skewed (long tail to the left).
By understanding these different distribution patterns and their properties, statisticians can make informed decisions about the best methods to analyze and interpret their data. For example, if the data follows a normal distribution, statistical techniques based on normality assumptions can be applied. On the other hand, if the data is skewed, alternative methods may be needed.
Exploring regular distribution patterns is not only important in statistical analysis but also has practical applications in various fields. For instance, in finance, understanding the distribution of stock returns can help investors make better predictions and manage risk. In healthcare, analyzing the distribution of patient data can aid in determining the effectiveness of treatments.
The normal distribution, also known as the Gaussian distribution or bell curve, is one of the most common probability distributions. It is characterized by a symmetric bell-shaped curve that is centered around its mean value. This distribution is widely used in various fields, including statistics, economics, and natural sciences, due to its mathematical properties and real-world applicability.
In a normal distribution, the mean, median, and mode are all equal, and the curve is completely determined by its mean and standard deviation. The mean represents the central tendency of the data, while the standard deviation indicates the spread or dispersion of the values around the mean. The shape of the normal distribution is determined by the concept of standard deviation.
The properties of the normal distribution make it ideal for modeling many real-world phenomena. Numerous natural and social phenomena, such as human height, IQ scores, and measurement errors, tend to follow a normal distribution pattern. Additionally, the central limit theorem states that the sum or average of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the underlying distribution of the variables.
The normal distribution is often used in statistical inference and hypothesis testing. It allows researchers to analyze and make predictions based on observed data. By understanding the characteristics of the normal distribution, researchers can estimate probabilities, calculate confidence intervals, and perform hypothesis tests.
Furthermore, the normal distribution provides a foundation for many statistical techniques and models. Various statistical tests, such as Z-tests and t-tests, assume a normal distribution of the data. Additionally, many machine learning algorithms, such as linear regression and logistic regression, rely on the assumption of normality.
A uniform distribution, also known as a rectangular distribution, is a probability distribution where all outcomes are equally likely. This means that every value in a given range has the same probability of occurring.
In a uniform distribution, the probability density function remains constant within the range of possible values. This results in a rectangular-shaped histogram, where each value has the same height. The area under the curve of a uniform distribution is always equal to 1.
The uniform distribution is commonly used in various fields, such as statistics, finance, and computer science. It is especially useful in scenarios where there is no underlying bias or preference towards any specific outcome.
One example of the uniform distribution is rolling a fair six-sided die. Each side has an equal probability of landing face-up, resulting in a uniform distribution of the six possible outcomes.
To better understand the uniform distribution, let’s consider an example. Suppose you have a bag of colored marbles, with 20 red marbles, 20 blue marbles, and 20 yellow marbles. If you randomly pick a marble from the bag, the probability of selecting any specific color is 1/3 (assuming each marble has an equal chance of being picked).
A table can be used to represent the uniform distribution of the marble colors:
In this example, each color has an equal probability of being picked, resulting in a uniform distribution.
Overall, the uniform distribution provides a simple and equal allocation of probabilities among all possible outcomes. It is commonly used in various fields to model scenarios with no bias or preference, providing a fair and balanced representation of data.
In statistics, skewed distribution refers to a type of probability distribution where the data has a long tail on one side and appears asymmetrical. It deviates from the normal distribution, which has a symmetrical bell-shaped curve. Skewed distribution occurs when there are outliers or extreme values in the data that pull the mean in the direction of the tail.
There are two types of skewed distribution:
- Positive Skewness: Also known as right-skewed distribution, it occurs when the tail extends towards the right side of the distribution. In this case, the majority of the data points are concentrated on the left side, and the mean is greater than the median.
- Negative Skewness: Also known as left-skewed distribution, it occurs when the tail extends towards the left side of the distribution. In this case, the majority of the data points are concentrated on the right side, and the mean is less than the median.
The skewness of a distribution can be quantitatively measured using the skewness coefficient. A positive skewness coefficient indicates a positive skewness, while a negative skewness coefficient indicates a negative skewness.
Skewed distribution can have different practical implications. For example, in finance, stock returns often follow a negative skewed distribution due to the occurrence of market crashes. In such cases, the mean return may be negative, indicating a higher probability of losses. Moreover, skewed distribution can affect the accuracy of certain statistical measures, such as the mean and standard deviation, which are heavily influenced by extreme values.
Understanding skewed distribution is important in data analysis as it helps in identifying patterns, making predictions, and selecting appropriate statistical techniques. When dealing with skewed data, it is often necessary to transform the data or use non-parametric statistical methods that do not assume a normal distribution.
Overall, skewed distribution provides valuable insights into the nature of data and helps researchers and analysts make informed decisions based on the characteristics of the data distribution.
Applications and Examples
Estimates and error distributions have a wide range of applications in various fields, including statistics, economics, finance, and social sciences. They are essential tools for understanding and analyzing data. Here are some examples of how estimates and error distributions are used:
1. Estimating Population Size: One common application of estimates and error distributions is estimating the size of a population. Researchers often use sampling techniques to collect data from a subset of the population and then use statistical methods to estimate the population size. The error distribution helps quantify the uncertainty associated with the estimate.
2. Predictive Modeling: Estimates and error distributions are also crucial in predictive modeling. By analyzing historical data and fitting a distribution to the observed errors, researchers can make predictions about future outcomes. For example, in finance, analysts use estimates and error distributions to predict stock prices or assess investment risks.
3. Quality Control: Estimates and error distributions play a significant role in quality control processes. By collecting data on product or process variables, statisticians can estimate the mean and standard deviation of the population and assess whether the process is in control. Deviations from the expected distribution pattern indicate potential quality issues.
4. Hypothesis Testing: In hypothesis testing, estimates and error distributions are used to assess the significance of the results. Researchers compare the observed data to the expected distribution pattern and calculate the p-value, which measures the likelihood of obtaining the observed results by random chance. This helps determine whether the results are statistically significant.
5. Machine Learning: In machine learning algorithms, estimates and error distributions are used to evaluate the performance of a model. By comparing the predicted outcomes to the actual outcomes, researchers can determine the accuracy of the model and identify any pattern or bias in the errors. This helps improve the model’s predictive ability.
6. Risk Analysis: Estimates and error distributions are extensively employed in risk analysis. By analyzing historical data and estimating the distribution of potential losses, risk analysts can evaluate the likelihood and severity of various risks. This information helps businesses make informed decisions and implement risk mitigation strategies.
Overall, estimates and error distributions are fundamental concepts in data analysis. They provide a framework for understanding uncertainty, making predictions, assessing quality, testing hypotheses, evaluating models, and managing risks. Their applications are widespread, making them essential tools for researchers, analysts, and decision-makers in various fields.
Estimating Population Size
Estimating population size is a crucial aspect of data analysis. It involves using statistical methods to determine the number of individuals in a given population based on sample data. This estimation process plays a vital role in various fields such as market research, ecology, and social sciences.
There are several approaches to estimating population size. One common method is called the capture-recapture method. This technique involves capturing a sample of individuals from a population, marking them in some way, releasing them back into the population, and then capturing another sample at a later time. By comparing the number of marked individuals in the second sample to the total number of individuals captured in the second sample, it is possible to estimate the overall population size.
Another approach is the spatial sampling method. This method involves dividing a study area into smaller sub-areas and then systematically sampling these sub-areas to gather data on the population density within each sub-area. By extrapolating these density estimates to the entire study area, it is possible to estimate the total population size.
Estimating population size also relies on understanding error distributions. Errors can occur during the sampling process, and these errors can introduce bias or variability into the estimates. By studying the error distribution patterns, researchers can account for these errors and improve the accuracy of their population size estimates.
The estimation of population size has numerous practical applications. In market research, it helps businesses determine the potential customer base for a new product or service. In ecology, it assists in understanding wildlife populations and their conservation needs. In social sciences, it aids in studying demographics and making policy decisions.
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The processor’s architecture refers to the way in which a computer’s central processing unit (CPU) is designed and organized. It encompasses the fundamental components, principles, and structures that govern the operation of the CPU. This includes the processor’s instruction set, register design, and memory hierarchy, among other aspects. Understanding the architecture of a processor is crucial for comprehending how a computer performs tasks and interacts with the rest of the system.
The processor’s architecture refers to the design and organization of the components and circuits within the processor. It encompasses the fundamental principles that govern the processor’s operation, including the instruction set architecture (ISA), the processor’s registers, and the logic circuits that perform calculations and execute instructions. The architecture of a processor plays a crucial role in determining its performance, power consumption, and compatibility with software and other components. Understanding the architecture of a processor is essential for developing efficient software, optimizing system performance, and making informed purchasing decisions.
The basics of processor architecture
The central processing unit (CPU)
The central processing unit (CPU) is the primary component of a computer’s processor and is responsible for executing instructions. It is made up of several smaller components that work together to perform calculations and manipulate data.
The control unit
The control unit is responsible for coordinating the flow of data and instructions within the CPU. It receives instructions from the memory and decodes them, determining the actions that need to be taken. It then directs the other components of the CPU to carry out these actions.
The arithmetic logic unit (ALU)
The arithmetic logic unit (ALU) is responsible for performing arithmetic and logical operations. It performs calculations and comparisons, such as addition, subtraction, multiplication, division, and Boolean logic operations like AND, OR, and NOT.
The registers are small amounts of memory located within the CPU. They are used to store data that is being used by the CPU, such as the results of calculations or addresses of memory locations. There are several different types of registers, each with a specific purpose. The accumulator register is used to store the results of arithmetic and logical operations, while the program counter register keeps track of the current instruction being executed. Other registers include the stack pointer, which keeps track of the current position in the stack, and the index register, which is used to access data in memory.
The memory hierarchy
The memory hierarchy is a crucial aspect of a processor’s architecture. It refers to the organization and arrangement of memory components within a computer system. The memory hierarchy includes various types of memory, each with its own unique characteristics and purposes.
Cache memory is a small, high-speed memory that is located close to the processor. It is used to store frequently accessed data and instructions, with the goal of improving the overall performance of the system. Cache memory is designed to be faster than the main memory, which allows the processor to access data quickly and efficiently.
Random Access Memory (RAM)
Random Access Memory (RAM) is a type of volatile memory that is used to store data and instructions that are currently being used by the processor. RAM is referred to as “volatile” because it loses its contents when the power is turned off. Unlike cache memory, RAM is not located near the processor, and access times are slower. However, RAM is used to store data that is actively being used by the processor, and it can be written to and read from by the processor at any time.
Read-Only Memory (ROM)
Read-Only Memory (ROM) is a type of non-volatile memory that is used to store permanent data, such as the computer’s BIOS (Basic Input/Output System). ROM is referred to as “non-volatile” because it retains its contents even when the power is turned off. Unlike RAM, ROM cannot be written to, but it can be read from by the processor. ROM is typically used to store firmware, which is software that is permanently embedded in hardware.
The importance of processor architecture
Processor architecture plays a crucial role in determining the performance of a computer system. It is the design and layout of the processor that dictates how it executes instructions and interacts with other components. In this section, we will discuss the two primary factors that influence the performance of a processor: clock speed and instruction set architecture (ISA).
The clock speed of a processor, often measured in GHz (gigahertz), refers to the number of cycles per second that the processor can perform. In general, a higher clock speed means that the processor can execute more instructions per second, resulting in faster performance. However, clock speed is just one factor that affects performance, and other factors such as the number of cores and the architecture of the processor can also impact performance.
Instruction set architecture (ISA)
The instruction set architecture (ISA) of a processor refers to the set of instructions that the processor can execute. Each processor has its own unique ISA, which defines the types of instructions that it can perform and the order in which they can be executed. The ISA of a processor can have a significant impact on its performance, as it determines the efficiency with which the processor can execute instructions.
In addition to the ISA, the pipeline architecture of a processor can also impact performance. The pipeline architecture refers to the way in which the processor fetches, decodes, and executes instructions. A processor with a deeper pipeline can execute more instructions in parallel, resulting in faster performance. However, a deeper pipeline also requires more transistors, which can increase power consumption and manufacturing costs.
Overall, the performance of a processor is determined by a combination of factors, including clock speed, ISA, and pipeline architecture. Understanding these factors can help system designers and users make informed decisions about the selection and configuration of processors for their systems.
Processor architecture plays a crucial role in determining the power consumption of a computer system. The amount of power consumed by a processor is directly proportional to its performance and complexity. In recent years, there has been a growing concern about the power consumption of computer systems, as it has a significant impact on the environment. Therefore, processor architects have been working to design processors that consume less power while maintaining high performance.
There are two main types of processors based on power consumption: low power processors and high performance processors. Low power processors are designed to consume less power than high performance processors, making them ideal for use in portable devices such as laptops and smartphones. These processors typically have fewer transistors and operate at a lower clock speed than high performance processors.
On the other hand, high performance processors are designed to provide maximum performance, even if it means consuming more power. These processors have more transistors and operate at a higher clock speed than low power processors. High performance processors are commonly used in desktop computers and servers, where raw computing power is required.
However, there is a growing trend towards designing high performance processors that consume less power. This is achieved by using techniques such as power gating, where certain parts of the processor are turned off when they are not in use, and dynamic voltage and frequency scaling, where the voltage and clock speed of the processor are adjusted based on the workload. These techniques allow high performance processors to consume less power without sacrificing performance.
In conclusion, power consumption is an important aspect of processor architecture, and designers must balance performance and power consumption when designing processors. With the growing concern about the environmental impact of computer systems, there is a need for processors that consume less power while maintaining high performance.
Processor architecture refers to the design and organization of a computer processor’s functional units, logic, and control signals. The cost of a processor architecture is a critical factor in the design and development of modern processors.
Cost of production
The cost of production of a processor architecture is a significant factor that affects the overall cost of a processor. The cost of production includes the cost of designing and developing the architecture, as well as the cost of manufacturing the processor.
The cost of production of a processor architecture depends on various factors such as the complexity of the design, the number of transistors used, and the manufacturing process used. The more complex the design and the more transistors used, the higher the cost of production.
In addition, the cost of production also depends on the manufacturing process used. For example, a processor designed using a more advanced manufacturing process such as photolithography will have a higher cost of production compared to a processor designed using a less advanced manufacturing process.
Cost of ownership
The cost of ownership of a processor architecture is another important factor that affects the overall cost of a processor. The cost of ownership includes the cost of maintaining and upgrading the processor over its lifetime.
The cost of ownership of a processor architecture depends on various factors such as the processor’s power consumption, the number of cores, and the processor’s ability to be upgraded. A processor with a high power consumption will have a higher cost of ownership due to the increased energy costs.
Additionally, a processor with a large number of cores will have a higher cost of ownership due to the increased complexity of the architecture and the need for more resources to maintain and upgrade the processor.
In conclusion, the cost of a processor architecture is a critical factor in the design and development of modern processors. The cost of production and the cost of ownership are important factors that affect the overall cost of a processor.
Processor architecture plays a crucial role in determining the compatibility of different computer systems. Compatibility is a key factor in ensuring that various components of a computer system can work together seamlessly. There are two main types of compatibility that are important in processor architecture: backward compatibility and forward compatibility.
Backward compatibility refers to the ability of a newer system to work with older software and hardware. This is an essential feature of processor architecture as it allows users to continue using their existing software and hardware with a new system, without having to upgrade everything at once. For example, a newer processor that is backward compatible with an older motherboard and RAM will be able to run the software that was designed for the older system.
Forward compatibility, on the other hand, refers to the ability of an older system to work with newer software and hardware. This is also an important feature of processor architecture as it allows users to upgrade their system gradually, without having to worry about compatibility issues. For example, a newer operating system that is forward compatible with an older processor will be able to run on the older system, although it may not be optimized for it.
Overall, compatibility is a critical aspect of processor architecture as it ensures that different components of a computer system can work together seamlessly, allowing users to upgrade and improve their systems without having to worry about compatibility issues.
Processor architecture plays a crucial role in ensuring the security of a computer system. It refers to the design and layout of the processor, which determines how it performs various tasks and interacts with other components of the system.
Hardware security features
Hardware security features are physical components built into the processor that provide additional layers of protection against security threats. These features include:
- Secure Boot: This feature ensures that the system boots using only firmware that is trusted by the device manufacturer, providing protection against bootkits and other malware that attempt to infect the boot process.
- Trusted Execution Environment (TEE): TEE is a secure area within the processor that provides a protected environment for sensitive data and code execution. It ensures that data remains confidential and cannot be accessed or tampered with by unauthorized parties.
- Cryptographic acceleration: This feature provides fast and efficient encryption and decryption of data, which is essential for secure communication and storage.
Software security measures
Software security measures are implemented in the form of software programs and protocols that work in conjunction with the processor architecture to provide additional layers of security. These measures include:
- Encryption: Encryption is the process of converting plaintext into ciphertext to prevent unauthorized access to sensitive data.
- Access control: Access control measures ensure that only authorized users have access to certain resources and information within the system.
- Firewalls: Firewalls are software programs that monitor and control incoming and outgoing network traffic, preventing unauthorized access to the system.
Overall, the processor architecture plays a critical role in ensuring the security of a computer system. By incorporating hardware and software security measures, system designers can provide a more secure environment for sensitive data and code execution.
Understanding processor architecture
The role of the processor in a computer system
The processor, also known as the central processing unit (CPU), is the primary component of a computer system that executes instructions and performs various tasks. It is responsible for executing instructions, controlling input/output operations, managing memory, and performing logical and mathematical operations.
The processor manages input/output (I/O) operations by controlling the flow of data between the computer system and external devices such as keyboards, mice, printers, and disk drives. It sends and receives data to and from these devices and converts the data into a format that can be understood by the computer system.
The processor manages the computer system’s memory by controlling the flow of data between the memory and other components of the computer system. It retrieves data from memory and stores data back into memory, and it also manages the allocation of memory to different programs and processes.
Logic and mathematical operations
The processor performs logical and mathematical operations, such as comparison, addition, subtraction, multiplication, and division. It uses binary arithmetic to perform these operations and converts the results into a format that can be used by the computer system. The processor also performs logical operations, such as AND, OR, and NOT, which are used to manipulate data and control the flow of execution in a program.
How processor architecture affects performance
The architecture of a processor refers to the design and layout of its internal components and how they interact with each other. This layout plays a crucial role in determining the overall performance of the processor. The following are some of the key factors that influence the performance of a processor based on its architecture:
Single-core vs multi-core processors
A single-core processor has a single processing unit, while a multi-core processor has multiple processing units. Multi-core processors can perform multiple tasks simultaneously, whereas single-core processors can only perform one task at a time. As a result, multi-core processors tend to be more efficient at handling multiple tasks and can offer better performance for applications that require high levels of concurrency.
Dual-core vs quad-core processors
Dual-core processors have two processing units, while quad-core processors have four processing units. In general, quad-core processors tend to offer better performance than dual-core processors, especially when it comes to multi-tasking and running applications that require a lot of processing power. However, the specific performance gains will depend on the particular workload and the software being used.
32-bit vs 64-bit processors
A 32-bit processor can process 32 bits of data at a time, while a 64-bit processor can process 64 bits of data at a time. This difference can have a significant impact on performance, especially when it comes to handling large amounts of data or running applications that require a lot of memory. In general, 64-bit processors tend to offer better performance than 32-bit processors, especially for applications that require a lot of memory or that need to access large amounts of data. However, the specific performance gains will depend on the particular workload and the software being used.
How processor architecture affects power consumption
Processor architecture refers to the design and layout of a processor, which includes the instructions it can execute, the data structures it can manipulate, and the operations it can perform. The architecture of a processor is a critical factor in determining its power consumption, as it directly affects the amount of energy required to perform various tasks.
In general, the more complex the processor architecture, the higher the power consumption. This is because more complex architectures require more transistors and circuitry to perform the same operations, which increases the amount of energy required to operate the processor. For example, a processor with a higher clock speed and more cores will generally consume more power than a processor with a lower clock speed and fewer cores.
However, it is important to note that power consumption is not the only factor to consider when choosing a processor architecture. Other factors, such as performance, cost, and compatibility, must also be taken into account when selecting a processor for a particular application.
Some of the key aspects of processor architecture that affect power consumption include:
- Instruction set: Different processor architectures have different instruction sets, which determine the types of operations that can be performed by the processor. Some instruction sets are more power-efficient than others, as they require fewer instructions to perform the same task.
- Clock speed: The clock speed of a processor refers to the number of cycles per second that it can perform. A higher clock speed generally means a higher power consumption, as the processor is performing more operations per second.
- Core count: The number of cores in a processor can also affect power consumption. More cores generally mean higher power consumption, as each core requires its own power supply.
- Thermal design power (TDP): TDP is the maximum amount of power that a processor is designed to consume. Processors with a higher TDP will generally consume more power than those with a lower TDP.
In conclusion, the architecture of a processor plays a crucial role in determining its power consumption. Understanding the various aspects of processor architecture and how they affect power consumption can help you make informed decisions when selecting a processor for your applications.
How processor architecture affects cost
Processor architecture refers to the design and layout of the processor, which includes the number and type of cores, the size of the cache, and the way instructions are executed. This architecture directly impacts the cost of the processor, both in terms of production and ownership.
One key factor in the cost of production is the complexity of the architecture. Processors with more cores, larger caches, and more advanced instruction sets require more complex manufacturing processes, which can increase the cost of production. Additionally, the cost of ownership can be affected by the power consumption of the processor, as more power-efficient processors can lead to lower electricity costs over time.
Value for money processors are those that offer a balance of performance and cost-effectiveness. These processors may have a smaller number of cores or a smaller cache, but they are designed to provide reliable performance at a lower cost. For example, a dual-core processor with a smaller cache may be more cost-effective for basic computing tasks, while a quad-core processor with a larger cache may be more suitable for demanding applications such as gaming or video editing.
Ultimately, the choice of processor architecture will depend on the specific needs of the user, including the type of applications they will be running and their budget. Understanding how processor architecture affects cost can help users make informed decisions when selecting a processor for their needs.
How processor architecture affects compatibility
Compatibility with different operating systems
Processor architecture refers to the design and organization of a processor’s logic, which determines how it processes instructions and data. It is an essential aspect of a computer system that affects compatibility with different operating systems and software programs.
In order to understand how processor architecture affects compatibility, it is essential to know the two main types of processor architectures: RISC (Reduced Instruction Set Computing) and CISC (Complex Instruction Set Computing).
RISC processors have a smaller number of instructions, which they execute more quickly, while CISC processors have a larger number of instructions, which can perform more complex tasks. The difference between these two architectures can impact compatibility with different operating systems and software programs.
Compatibility with different software programs
Another way that processor architecture affects compatibility is through the software programs that can run on a computer. Different software programs may be designed to work with specific processor architectures, and if a computer does not have the right architecture, the software may not run correctly or may not run at all.
For example, a software program designed for a RISC-based processor may not work correctly on a CISC-based processor, and vice versa. This is because the two architectures have different instruction sets, and the software may not be able to execute the correct instructions on the wrong architecture.
In conclusion, processor architecture plays a crucial role in determining the compatibility of a computer system with different operating systems and software programs. It is essential to consider the architecture when selecting a processor to ensure that it will work correctly with the other components of the system.
How processor architecture affects security
Processor architecture refers to the design and organization of a processor, including its instruction set, data paths, and control logic. It plays a crucial role in determining the performance, power consumption, and security of a system.
In the context of security, processor architecture can have a significant impact on the level of protection provided by a system. This is because the architecture can either facilitate or hinder the implementation of security features and measures.
One way that processor architecture affects security is through the inclusion of security features within the design of the processor itself. For example, processors may include hardware-based support for encryption and decryption, allowing for more efficient and secure data transfer. Additionally, some processors may have dedicated hardware for performing specific security functions, such as hashing or digital signatures.
Another way that processor architecture affects security is through the interaction between the processor and the operating system or other software running on the system. For instance, certain processor architectures may make it easier or harder to implement virtualization, which can help isolate sensitive applications and data from the rest of the system. Additionally, some architectures may have specific features that can be exploited by attackers, such as the ability to execute code from specific memory locations.
Overall, the impact of processor architecture on security is complex and multifaceted. While some architectures may provide inherent advantages in terms of security, others may be more vulnerable to certain types of attacks. As such, it is important for system designers and developers to carefully consider the security implications of different processor architectures when making design decisions.
1. What is a processor’s architecture?
A processor’s architecture refers to the design and organization of the logic circuits within a computer processor. It defines how the processor carries out instructions and interacts with other components in the system.
2. Why is the processor’s architecture important?
The processor’s architecture is crucial because it determines the performance, power consumption, and capabilities of the processor. It also affects the compatibility of the processor with other system components and the types of software that can run on the system.
3. What are some common processor architectures?
Some common processor architectures include x86, ARM, PowerPC, and MIPS. Each architecture has its own strengths and weaknesses, and different architectures are used in different types of devices, such as smartphones, desktop computers, and servers.
4. How does the processor’s architecture affect software compatibility?
The processor’s architecture affects software compatibility because different software is designed to run on specific architectures. For example, software designed for an x86 processor may not work on a device with an ARM processor, and vice versa. Additionally, some software may be optimized for a specific architecture, which can affect performance on other architectures.
5. Can a processor’s architecture be changed?
In most cases, a processor’s architecture is fixed and cannot be changed. However, some devices, such as smartphones and tablets, may have processors that can be upgraded or replaced with new processors that have different architectures. This can affect the compatibility and performance of the device.
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- 1. Concepts of Alternating Current
- 2. AC and DC
- 3. Disadvantages of DC Compared to AC
- 4. Voltage Waveforms
- 5. Basic AC Generation
- 6. Alternating Current Values
- 7. Ohm’s Law in AC Circuits
1. Concepts of Alternating Current
All of your study thus far has been with direct current (dc), that is, current which does not change direction. However, as you saw in module 1 and will see later in this module, a coil rotating in a magnetic field actually generates a current which regularly changes direction. This current is called ALTERNATING CURRENT or ac.
2. AC and DC
Alternating current is current which constantly changes in amplitude, and which reverses direction at regular intervals. You learned previously that direct current flows only in one direction, and that the amplitude of current is determined by the number of electrons flowing past a point in a circuit in one second. If, for example, a coulomb of electrons moves past a point in a wire in one second and all of the electrons are moving in the same direction, the amplitude of direct current in the wire is one ampere. Similarly, if half a coulomb of electrons moves in one direction past a point in the wire in half a second, then reverses direction and moves past the same point in the opposite direction during the next half-second, a total of one coulomb of electrons passes the point in one second. The amplitude of the alternating current is one ampere. The preceding comparison of dc and ac as illustrated. Notice that one white arrow plus one striped arrow comprise one coulomb.
3. Disadvantages of DC Compared to AC
When commercial use of electricity became wide-spread in the United States, certain disadvantages in using direct current in the home became apparent. If a commercial direct-current system is used, the voltage must be generated at the level (amplitude or value) required by the load. To properly light a 240- volt lamp, for example, the dc generator must deliver 240 volts. If a 120-volt lamp is to be supplied power from the 240-volt generator, a resistor or another 120-volt lamp must be placed in series with the 120-volt lamp to drop the extra 120 volts. When the resistor is used to reduce the voltage, an amount of power equal to that consumed by the lamp is wasted.
Another disadvantage of the direct-current system becomes evident when the direct current (I) from the generating station must be transmitted a long distance over wires to the consumer. When this happens, a large amount of power is lost due to the resistance (R) of the wire. The power loss is equal to I2R. However, this loss can be greatly reduced if the power is transmitted over the lines at a very high voltage level and a low current level. This is not a practical solution to the power loss in the dc system since the load would then have to be operated at a dangerously high voltage. Because of the disadvantages related to transmitting and using direct current, practically all modern commercial electric power companies generate and distribute alternating current (ac).
Unlike direct voltages, alternating voltages can be stepped up or down in amplitude by a device called a TRANSFORMER. (The transformer will be explained later in this module.) Use of the transformer permits efficient transmission of electrical power over long-distance lines. At the electrical power station, the transformer output power is at high voltage and low current levels. At the consumer end of the transmission lines, the voltage is stepped down by a transformer to the value required by the load. Due to its inherent advantages and versatility, alternating current has replaced direct current in all but a few commercial power distribution systems.
O4. What disadvantage of a direct current is due to the resistance of the transmission wires?
4. Voltage Waveforms
You now know that there are two types of current and voltage, that is, direct current and voltage and alternating current and voltage. If a graph is constructed showing the amplitude of a dc voltage across the terminals of a battery with respect to time, it will appear in Figure 1 view A. The dc voltage is shown to have a constant amplitude. Some voltages go through periodic changes in amplitude like those shown in Figure 1 view B. The pattern which results when these changes in amplitude with respect to time are plotted on graph paper is known as a WAVEFORM. Figure 1 view B shows some of the common electrical waveforms. Of those illustrated, the sine wave will be dealt with most often.
5. Basic AC Generation
In a previous chapter, you learned that a current-carrying conductor produces a magnetic field around itself. You also learned how a changing magnetic field produces an emf in a conductor. That is, if a conductor is placed in a magnetic field, and either the field or the conductor moves, an emf is induced in the conductor. This effect is called electromagnetic induction.
The sine wave illustrated in Figure 1 view B is a plot of a current which changes amplitude and direction. Although there are several ways of producing this current, the method based on the principles of electromagnetic induction is by far the easiest and most common method in use.
Figure 2 and Figure 3 show a suspended loop of wire (conductor) being rotated (moved) in a clockwise direction through the magnetic field between the poles of a permanent magnet. For ease of explanation, the loop has been divided into a dark half and light half. Notice in (A) of the figure that the dark half is moving along (parallel to) the lines of force. Consequently, it is cutting NO lines of force. The same is true of the light half, which is moving in the opposite direction. Since the conductors are cutting no lines of force, no emf is induced. As the loop rotates toward the position shown in (B), it cuts more and more lines of force per second (inducing an ever-increasing voltage) because it is cutting more directly across the field (lines of force). At (B), the conductor is shown completing one-quarter of a complete revolution, or 90° , of a complete circle. Because the conductor is now cutting directly across the field, the voltage induced in the conductor is maximum. When the value of induced voltage at various points during the rotation from (A) to (B) is plotted on a graph (and the points connected), a curve appears as shown below.
As the loop continues to be rotated toward the position shown below in (C), it cuts fewer and fewer lines of force. The induced voltage decreases from its peak value. Eventually, the loop is once again moving in a plane parallel to the magnetic field, and no emf is induced in the conductor.
The loop has now been rotated through half a circle (one alternation or 180° ). If the preceding quarter-cycle is plotted, it appears as shown below.
When the same procedure is applied to the second half of rotation (180° through 360° ), the curve appears as shown below. Notice the only difference is in the polarity of the induced voltage. Where previously the polarity was positive, it is now negative.
The sine curve shows the value of induced voltage at each instant of time during rotation of the loop. Notice that this curve contains 360° , or two alternations. TWO ALTERNATIONS represent ONE complete CYCLE of rotation.
Assuming a closed path is provided across the ends of the conductor loop, you can determine the direction of current in the loop by using the LEFT-HAND RULE FOR GENERATORS. Refer to Figure 4. The left-hand rule is applied as follows: First, place your left hand on the illustration with the fingers as shown. Your THUMB will now point in the direction of rotation (relative movement of the wire to the magnetic field); your FOREFINGER will point in the direction of magnetic flux (north to south); and your MIDDLE FINGER (pointing out of the paper) will point in the direction of electron current flow.
By applying the left-hand rule to the dark half of the loop in (B) in Figure 3, you will find that the current flows in the direction indicated by the heavy arrow. Similarly, by using the left-hand rule on the light half of the loop, you will find that current therein flows in the opposite direction. The two induced voltages in the loop add together to form one total emf. It is this emf which causes the current in the loop.
When the loop rotates to the position shown in (D) of Figure 3, the action reverses. The dark half is moving up instead of down, and the light half is moving down instead of up. By applying the left-hand rule once again, you will see that the total induced emf and its resulting current have reversed direction. The voltage builds up to maximum in this new direction, as shown by the sine curve in Figure 3. The loop finally returns to its original position (E), at which point voltage is again zero. The sine curve represents one complete cycle of voltage generated by the rotating loop. All the illustrations used in this chapter show the wire loop moving in a clockwise direction. In actual practice, the loop can be moved clockwise or counterclockwise. Regardless of the direction of movement, the left-hand rule applies.
If the loop is rotated through 360° at a steady rate, and if the strength of the magnetic field is uniform, the voltage produced is a sine wave of voltage, as indicated in Figure 4. Continuous rotation of the loop will produce a series of sine-wave voltage cycles or, in other words, an ac voltage. As mentioned previously, the cycle consists of two complete alternations in a period of time. Recently the HERTZ (Hz) has been designated to indicate one cycle per second. If ONE CYCLE PER SECOND is ONE HERTZ, then 100 cycles per second are equal to 100 hertz, and so on. Throughout the NEETS, the term cycle is used when no specific time element is involved, and the term hertz (Hz) is used when the time element is measured in seconds.
QI7. One cycle is equal to how many degrees of rotation of a conductor in a magnetic field?
If the loop in the figure 1-8 (A) makes one complete revolution each second, the generator produces one complete cycle of ac during each second (1 Hz). Increasing the number of revolutions to two per second will produce two complete cycles of ac per second (2 Hz). The number of complete cycles of alternating current or voltage completed each second is referred to as the FREQUENCY. Frequency is always measured and expressed in hertz.
Alternating-current frequency is an important term to understand since most ac electrical equipments require a specific frequency for proper operation.
An individual cycle of any sine wave represents a definite amount of TIME. Notice that Figure 5 shows 2 cycles of a sine wave which has a frequency of 2 hertz (Hz). Since 2 cycles occur each second, | cycle must require one-half second of time. The time required to complete one cycle of a waveform is called the PERIOD of the wave. In Figure 5, the period is one-half second. The relationship between time (t) and frequency (f) is indicated by the formulas
where t = period mn seconds and t = frequency ti hertz
Each cycle of the sine wave shown in figure 1-10 consists of two identically shaped variations in voltage. The variation which occurs during the time the voltage is positive is called the POSITIVE ALTERNATION. The variation which occurs during the time the voltage is negative is called the NEGATIVE ALTERNATION. In a sine wave, these two alternations are identical in size and shape, but opposite in polarity.
The distance from zero to the maximum value of each alternation is called the AMPLITUDE. The amplitude of the positive alternation and the amplitude of the negative alternation are the same.
The time it takes for a sine wave to complete one cycle is defined as the period of the waveform. The distance traveled by the sine wave during this period is referred to as WAVELENGTH. Wavelength, indicated by the symbol λ (Greek lambda), is the distance along the waveform from one point to the same point on the next cycle. You can observe this relationship by examining Figure 6. The point on the waveform that measurement of wavelength begins is not important as long as the distance is measured to the same point on the next cycle (see Figure 7).
6. Alternating Current Values
In discussing alternating current and voltage, you will often find it necessary to express the current and voltage in terms of MAXIMUM or PEAK values, PEAK-to-PEAK values, EFFECTIVE values, AVERAGE values, or INSTANTANEOUS values. Each of these values has a different meaning and is used to describe a different amount of current or voltage.
6.1. Peak and Peak-To-Peak Values
Refer to Figure 8. Notice it shows the positive alternation of a sine wave (a half-cycle of ac) and a dc waveform that occur simultaneously. Note that the de starts and stops at the same moment as does the positive alternation, and that both waveforms rise to the same maximum value. However, the de values are greater than the corresponding ac values at all points except the point at which the positive alternation passes through its maximum value. At this point the dc and ac values are equal. This point on the sine wave is referred to as the maximum or peak value.
During each complete cycle of ac there are always two maximum or peak values, one for the positive half-cycle and the other for the negative half-cycle. The difference between the peak positive value and the peak negative value is called the peak-to-peak value of the sine wave. This value is twice the maximum or peak value of the sine wave and is sometimes used for measurement of ac voltages. Note the difference between peak and peak-to-peak values in Figure 9. Usually alternating voltage and current are expressed in EFFECTIVE VALUES (a term you will study later) rather than in peak-to-peak values.
6.2. Instantaneous Value
The INSTANTANEOUS value of an alternating voltage or current is the value of voltage or current at one particular instant. The value may be zero if the particular instant is the time in the cycle at which the polarity of the voltage is changing. It may also be the same as the peak value, if the selected instant is the time in the cycle at which the voltage or current stops increasing and starts decreasing. There are actually an infinite number of instantaneous values between zero and the peak value.
6.3. Average Value
The AVERAGE value of an alternating current or voltage is the average of ALL the INSTANTANEOUS values during ONE alternation. Since the voltage increases from zero to peak value and decreases back to zero during one alternation, the average value must be some value between those two limits. You could determine the average value by adding together a series of instantaneous values of the alternation (between 0° and 180° ), and then dividing the sum by the number of instantaneous values used. The computation would show that one alternation of a sine wave has an average value equal to 0.636 times the peak value. The formula for average voltage is
Eavg = 0.636 x Emax
where Eavg is the average voltage of one alternation, and Emax 18 the maximum or peak voltage. Similarly, the formula for average current is
lavg = 0.636 X Imax
where Iavg is the average current in one alternation, and I… 18 the maximum or peak current.
Do not confuse the above definition of an average value with that of the average value of a complete cycle. Because the voltage is positive during one alternation and negative during the other alternation, the average value of the voltage values occurring during the complete cycle is zero.
O30. If Emax, ts 115 volts, what is Eavg? O31. If lavg is 1.272 ampere, what is Imax?
6.4. Effective Value of a Sine Wave
Emax, Eavg, Imax, and Lavg are values used in ac measurements. Another value used is the EFFECTIVE value of ac This is the value of alternating voltage or current that will have the same effect on a resistance as a comparable value of direct voltage or current will have on the same resistance.
In an earlier discussion you were told that when current flows in a resistance, heat is produced. When direct current flows in a resistance, the amount of electrical power converted into heat equals IR watts. However, since an alternating current having a maximum value of 1 ampere does not maintain a constant value, the alternating current will not produce as much heat in the resistance as will a direct current of 1 ampere.
Figure 10 compares the heating effect of 1 ampere of dc to the heating effect of 1 ampere of ac.
Examine views A and B of Figure 10 and notice that the heat (70.7° C) produced by 1 ampere of alternating current (that is, an ac with a maximum value of 1 ampere) is only 70.7 percent of the heat (100° C) produced by | ampere of direct current. Mathematically,
Therefore, for effective value of ac (Ieff) = 0.707 X Imax.
The rate at which heat is produced in a resistance forms a convenient basis for establishing an effective value of alternating current, and is known as the "heating effect" method. An alternating current is said to have an effective value of one ampere when it produces heat in a given resistance at the same rate as does one ampere of direct current.
You can compute the effective value of a sine wave of current to a fair degree of accuracy by taking equally-spaced instantaneous values of current along the curve and extracting the square root of the average of the sum of the squared values.
For this reason, the effective value is often called the "root-mean-square" (rms) value. Thus,
Stated another way, the effective or rms value (Ieff) of a sine wave of current is 0.707 times the maximum value of current (Imax). Thus, Ieff = 0.707 X Imax. When Ieff known, you can find I, by using the formula Imax = 1.414 X Ieff. You might wonder where the constant 1.414 comes from. To find out, examine figure 1-15 again and read the following explanation. Assume that the dc in figure 1-15(A) is maintained at | ampere and the resistor temperature at 100° C. Also assume that the ac in figure 1-15(B) is increased until the temperature of the resistor is 100° C. At this point it is found that a maximum ac value of 1.414 amperes is required in order to have the same heating effect as direct current. Therefore, in the ac circuit the maximum current required is 1.414 times the effective current. It is important for you to remember the above relationship and that the effective value (Ieff) of any sine wave of current is always 0.707 times the maximum value (Imax).
Since alternating current is caused by an alternating voltage, the ratio of the effective value of voltage to the maximum value of voltage is the same as the ratio of the effective value of current to the maximum value of current. Stated another way, the effective or rms value (Eeff) of a sine-wave of voltage is 0.707 times the maximum value of voltage (Emax),
When an alternating current or voltage value is specified in a book or on a diagram, the value is an effective value unless there is a definite statement to the contrary. Remember that all meters, unless marked to the contrary, are calibrated to indicate effective values of current and voltage.
Problem: A circuit is known to have an alternating voltage of 120 volts and a peak or maximum current of 30 amperes. What are the peak voltage and effective current values?
Figure 11 shows the relationship between the various values used to indicate sine-wave amplitude. Review the values in the figure to ensure you understand what each value indicates.
O34. What is the formula for finding the effective value of an alternating current? O35. If the peak value of a sine wave is 1,000 volts, what is the effective (Eeff) value? O36. If Ieff = 4.25 ampere, what is Imax?
6.5. Sine Waves in Phase
When a sine wave of voltage is applied to a resistance, the resulting current is also a sine wave. This follows Ohm’s law which states that current is directly proportional to the applied voltage. Now examine Figure 12. Notice that the sine wave of voltage and the resulting sine wave of current are superimposed on the same time axis. Notice also that as the voltage increases in a positive direction, the current increases along with it, and that when the voltage reverses direction, the current also reverses direction. When two sine waves, such as those represented by Figure 12, are precisely in step with one another, they are said to be IN PHASE. To be in phase, the two sine waves must go through their maximum and minimum points at the same time and in the same direction.
In some circuits, several sine waves can be in phase with each other. Thus, it is possible to have two or more voltage drops in phase with each other and also be in phase with the circuit current.
6.6. Sine Waves Out of Phase
Figure 13 shows voltage wave E1 which is considered to start at 0° (time one). As voltage wave E1 reaches its positive peak, voltage wave E2 starts its rise (time two). Since these voltage waves do not go through their maximum and minimum points at the same instant of time, a PHASE DIFFERENCE exists between the two waves. The two waves are said to be OUT OF PHASE. For the two waves in Figure 13 the phase difference is 90°.
To further describe the phase relationship between two sine waves, the terms LEAD and LAG are used. The amount by which one sine wave leads or lags another sine wave is measured in degrees. Refer again to figure 1-18. Observe that wave E; starts 90° later in time than does wave E;. You can also describe this relationship by saying that wave E, leads wave E> by 90°, or that wave E> lags wave E, by 90° . (Either statement is correct; it is the phase relationship between the two sine waves that is important.)
It is possible for one sine wave to lead or lag another sine wave by any number of degrees, except 0° or 360° . When the latter condition exists, the two waves are said to be in phase. Thus, two sine waves that differ in phase by 45° are actually out of phase with each other, whereas two sine waves that differ in phase by 360° are considered to be in phase with each other.
A phase relationship that is quite common is shown in Figure 14. Notice that the two waves illustrated differ in phase by 180° . Notice also that although the waves pass through their maximum and minimum values at the same time, their instantaneous voltages are always of opposite polarity. If two such waves exist across the same component, and the waves are of equal amplitude, they cancel each other. When they have different amplitudes, the resultant wave has the same polarity as the larger wave and has an amplitude equal to the difference between the amplitudes of the two waves.
To determine the phase difference between two sine waves, locate the points on the time axis where the two waves cross the time axis traveling in the same direction. The number of degrees between the crossing points is the phase difference. The wave that crosses the axis at the later time (to the right on the time axis) is said to lag the other wave.
O39. What is the phase relationship between two voltage waves that differ in phase by 360°?
7. Ohm’s Law in AC Circuits
Many ac circuits contain resistance only. The rules for these circuits are the same rules that apply to dc circuits. Resistors, lamps, and heating elements are examples of resistive elements. When an ac circuit contains only resistance, Ohm’s Law, Kirchhoff’s Law, and the various rules that apply to voltage, current, and power in a dc circuit also apply to the ac circuit. The Ohm’s Law formula for an ac circuit can be stated as
Remember, unless otherwise stated, all ac voltage and current values are given as effective values. The formula for Ohm’s Law can also be stated as
The important thing to keep in mind is: Do Not mix ac values. When you solve for effective values, all values you use in the formula must be effective values. Similarly, when you solve for average values, all values you use must be average values. This point should be clearer after you work the following problem: A series circuit consists of two resistors (R1 = 5 ohms and R2 = 15 ohms) and an alternating voltage source of 120 volts. What is Iavg?
Solution: First sobre for total resistance RT.
The alternating voltage is assumed to be an effective value (since it is not specified to be otherwise). Apply the Ohm’s Law formula.
The problem, however, asked for the average value of current (I 4,2). To convert the effective value of current to the average value of current, you must first determine the peak or maximum value of current, Imax.
You can now find Iavg. Just substitute 8.484 amperes in the Iavg formula and solve for Iavg.
Remember, you can use the Ohm’s Law formulas to solve any purely resistive ac circuit problem. Use the formulas in the same manner as you would to solve a de circuit problem.
O41. A series circuit consists of three resistors (RI = 10Q, R2 = 209 R3 = 15Q)) and an alternating voltage source of 100 volts. What is the effective value of current in the circuit? O42. If the alternating source in Q41 is changed to 200 volts peak-to-peak, what is Iavg? O43. If Eeff 130 volts and Ieff 3 amperes, what is the total resistance (RT) in the circuit?
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Before we define what calibration means, I want you to understand the below scenario:
When you go to the market and buy some meat, you bought 1 kilo and pay for it with your hard-earned money but when you check it at home, it is only ¾ of a kilo, what would you feel?
You just gassed up good for 2 days as your routine but only a day had passed and your meter indicates near-empty – would this not make you mad?
You are baking a cake and the instruction tells you to set the temperature to 50 degrees Celsius for 20 minutes, you followed the steps and specifications but your cake turned into charcoal – will you not be upset?
Now, what do you feel if the above scenario happens to you? Of course, you may get angry, dismayed, or worse complain about the services or products that you received and paid for, all because of the effect of the un-calibrated Instruments.
This is why calibration is important in our daily life not just inside a laboratory or within the manufacturing aspect. There is the involvement of quality, safety, and reliability.
Now let us define What calibration is.
Basics of Calibration
There are definitions of Calibrations by NIST or ISO that you can look for. But to easily understand, below is a simple calibration definition:
Calibration is simply the comparison of Instrument, Measuring and Test Equipment (M&TE), Unit Under Test (UUT), Unit Under Calibration (UUC), a Device Under Test (DUT), or simply a Test Instrument (TI) of unverified accuracy to an instrument or standards with a known (higher) accuracy to detect or eliminate unacceptable variations. It may or may not involve adjustment or repair.
It is making the instruments perform what it displays by referencing or adjusting it based on a Reference Standard.
- to ensure that you get what you have paid for;
- Satisfy your expectations;
- Create win-win situations
In daily operations, Calibration Means:
What is a Reference Standard?
A reference standard is also an Instrument, or equipment, or measuring device with the highest metrological quality or accuracy than the Unit Under Calibration (UUC).
It is where we compare the UUC reading and where measurement values are derived. It is also calibrated, but by a higher-level laboratory with traceability to a higher standard (See traceability below).
The reference standard is also known as the Master Standard. Other terms that I sometimes hear, refer to it as Master Calibrator or simply ‘calibrator’.
Why Calibrate – Reasons for Calibration
There are so many reasons why we need to perform a calibration. Some reasons are:
- For public or consumer protection, like the example above to get the value of the money we spend on a product or services.
- For a technical reason, we need to calibrate because as components age or equipment undergo environmental or mechanical stress, its performance gradually degrades.
This degradation is what we call the ‘drift‘. When this happens, the results or performance generated by certain equipment will be unreliable where design and quality suffer. We cannot eliminate drift, but through the process of calibration, it can be detected and contained.
- There are also practical reasons for implementing calibration. Calibration will eliminate doubts and provide confidence when we encounter the below situations with our instruments:
- when there is a newly installed or purchased instruments
- instruments that are mishandled during transfer (for example: dropped or fell)
- when instrument performance is questionable
- calibration period is overdue
- kept to an unstable environment for too long (exposed to vibrations or too high/low temperatures)
- when a new setting, repair, and/or adjustment is performed
While detecting an inaccuracy is one of the main reasons for calibration, some other reasons are:
- Customer requirements – they want to ensure that the product they buy is within the expected specifications.
- Requirements of a government or statutory regulations – they want to make sure that products produced are safe and reliable for the public
- Audit requirements– as a requirement for achieving a certification like ISO 9001:2015 certification and ISO 17025 accreditations
- Quality and Safety requirements– a reliable and accurate operation through proper use of inspection instruments provides a great deal of confidence for everyone
- Process requirements – to ensure that the product produced is the most accurate and reliable, some operations will not be executed unless the equipment has passed the calibration and verification process, and used in equipment or product qualifications as part of quality control.
Importance of Instrument Calibration
- To establish and demonstrate traceability (I will explain the traceability below). Through calibration, the measurement established by the instrument is the same wherever you are, it means that a 1 Kg of weight in one place is also 1 Kg in other places or wherever it reaches. You can use instruments regardless of the units or parameters they measure on different occasions.
- To determine and ensure the accuracy of instrument readings (through calibration, you can determine how close the actual value is to the true or reference value) – resulting in product quality and safety
- To ensure readings from the instrument are consistent with other measurements. This means that you have the same measurement results regardless of what measuring instrument you use that is compatible in the process.
- To establish the reliability of the instrument making sure that they function in the way they are intended to be – resulting in more confidence in the expected output.
- Provides customer satisfaction by having a product that meets what they have paid for – high quality of a product.
What needs calibration?
- All inspection, measuring, and test equipment that can affect or determine product quality. This means that if you are using the instruments to verify the acceptance of a product whether to pass or fail based on the measured value you have taken, the instrument should be calibrated.
- Equipment which, if out of calibration, would produce unsafe products.
- Equipment that requires calibration because of an agreement. An example is a customer, where before progressing into a contract, they need to ensure that the equipment that will produce their product is calibrated.
- All measuring and testing equipment (standards) affecting the accuracy or validity of calibrations. These are the master standards, go-no-go jigs, check masters, reference materials, and related instruments that we used to verify other instruments or measuring equipment for their accuracy.
What does a calibrated Instrument Look like?
When you have your measuring instruments calibrated, see to it that it has a calibration label where details of its calibration
date and due date are seen, also includes a serial number, certificate number, and person-in-charge of the calibration which depends on the calibration lab. Also, if needed, a ‘void’ seal is placed to protect it from any unauthorized adjustment.
A calibrated Instrument with labels is useless if the calibration certificate is not available, so be sure to keep it safe and readily available once requested.
Be cautious to check calibration certificates once received, not all calibrated instruments are performing the same as you expected, some have a limited use based on the result of the calibration. you must learn how to read or interpret the results in a calibration certificate.
When is Calibration Not Required?
Every measuring instrument needs calibration but not all measuring instruments are required to be calibrated. Below are some of the reasons or criteria to consider before having an instrument calibrated. This may save you some time and money.
- It is not critical in your process ( just to display a certain reading for the purpose of functionality check).
- It functions as an indicator only (for example: high or low and close or open).
- As an accessory only to support the main instrument. For example, a coil of wire is used to amplify current. Current is measured, but the amplification is not that critical, used as an accessory only to amplify a measured current.
- Its accuracy is established by a higher or reference to a higher or more accurate instrument within a group. (for example a set of pressure gauges that are connected in series in which one of them is a more accurate gauge where they are compared or referenced to).
- If the instruments are verified regularly or continuously monitored by a calibrated instrument that is documented in a measurement assurance process. For example, a room thermometer that is verified by a calibrated thermometer regularly.
- If the instrument is a part of a system or integrated into a system where the system is calibrated as a whole. For example, a thermocouple that is permanently connected to the oven (some thermocouples are detached after usage and then transferred to other units).
Please visit this link for more details regarding this topic.
How To Perform Adjustment of Calibration Interval
Every measuring instrument needs calibration and every calibrated instrument needs to be re-calibrated. This means that there is a due date for a re-calibration period that we need to establish.
This recalibration period is a scheduled calibration that is based on the calibration interval that we set. Before we send our instruments for calibration, we need to set in advance our initial interval or our final fixed interval. This will be communicated with the calibration lab.
These initial calibration intervals may be based on the following aspects:
- Manufacturer Requirements – recommended by the manufacturer
- On the frequency of use – the more it is used, the shorter the calibration interval
- Required by the regulatory bodies (for example: required by the government)
- Experience of the user with the same type of instrument
- Based on the criticality of use. – more critical instruments have higher accuracy or very strict tolerance, therefore shorter calibration interval
- Customer Requirements
- Conditions of the environment where it is being used.
- Published Documents
Now that we have an initial calibration interval, we need to set the fixed calibration interval based on the performance of our instruments. We will review the performance by gathering all the calibration records and plot it in a graph to study its stability or the drift encountered by the instruments.
Based on this performance history, we can decide if we need to extend or reduce the calibration interval.
I have shown how to do this in this link >> CALIBRATION INTERVAL: HOW TO INCREASE THE CALIBRATION FREQUENCY OF INSTRUMENTS
Basic Calibration Terms and Principles
What is the Accuracy of an Instrument?
Accuracy is a reference number (usually given in percent (%) error) that shows the degree of closeness to the true value. There is a true value which means that you have a source of known value to compare with.
The closer your measuring instrument reads or measures to the true value, the more accurate your instrument is. Or, in other words, “the smaller the error, the more accurate the instrument”.
Error or Measurement Error – as per JCGM 200:2012, refers to measured quantity value minus a reference quantity value. In simple terms, this is the UUC displayed reading minus the STD reading.
How to calculate the accuracy of measurement?
To find the value of accuracy, you need to calculate the error, and to gauge the “degree of closeness to the true value”, you need to calculate the % error.
To calculate the error and % error, below are the formulas:
Error = Measured value – True value
And for the percent error:
% error = Error/True value x 100
We are performing a calibration to check for the accuracy of instruments, and to determine how close (or far) the reading of our instruments is compared to the reading of the reference standard.
Where can we find the Accuracy of Instruments?
- Original Equipment Manufacturer (OEM) Specifications in manual or brochure
- Publish standards or handbook
- Calibration Certificates
When calibrating, make sure that the reference standard has higher accuracy than the Unit Under Calibration (UUC), usually, a good rule of thumb is to have an accuracy ratio of 4:1.
This means that if your UUC has an accuracy of 1, the reference standard to be used should have at least a 0.25 accuracy, four times more accurate than the UUC (1/0.25 =4). To make it more clear, since most of the time, accuracy is expressed in percentage (% error), the closer the value is to ZERO, the more accurate the instrument.
What is the Tolerance of Instruments?
Tolerance is closely related to accuracy at some point but for clarity, it is the permissible deviation or the maximum error to be expected from a manufactured component and expressed usually in measurement units (examples are psi, volts, meter, etc.).
As per JCGM 106:2012:
Tolerance = difference of upper and lower tolerance limits
Tolerance limit = specified upper or lower bound of permissible values of a property
Tolerance Interval = Intervals of permissible values
A pressure gauge with a full-scale reading of 20 psi with a tolerance limit of +/- 0.5 ( 20 +/-0.5) psi has a tolerance of 1 psi, see the below photo.
After calibration, we should perform a verification to determine if the reading is within the tolerance specified, it may be less accurate but if it is within the specified tolerance limit, it is still acceptable. This is where a pass-or-fail decision can be brought out.
A pass-or-fail decision during verification is best applied by using a “Decision Rule” for a more in-depth assessment of conformity.
The tolerance needed should be determined by the user of the UUC, which is the combinations of many factors includes:
- Process requirements
- The capability of measurement equipment
- Manufacturer’s tolerance specifications (Related to accuracy)
- Published Standards
See more explanation and presentation in this link >> Differences Between Accuracy, Error, Tolerance, and Uncertainty in Calibration Results
What is Precision in Measurement?
Precision is the closeness of repeated measurement to each other. Precision signifies good stability and repeatability of instruments but not accuracy.
A measuring instrument can be highly precise but cannot be accurate. Our goal during calibration for a measuring instrument is to have good accuracy and precision. Precision can be determined without using a reference standard.
How to Determine Precision in Measurements?
Precision is closely related to repeatability, you cannot determine the precision of you cannot have repeated measurements.
In order to calculate and determine ‘precision’, follow the below steps:
1. With the same method of measurement, get a repeated reading. For example, a digital caliper measuring a 10 mm gauge block (or any stable material) for 10 times. See below table.
2. using the Excel worksheet (this will simplify and make the calculation easy) plot all the measured points and calculate the standard deviation. Follow the below instructions to calculate in Excel.
The smaller the value, the more precise the instrument.
Below are the relationships of Accuracy, Precision, and Tolerance to understand better:
What is Stability and How to Determine Stability?
Stability is the ability of the instrument to maintain its output within defined limits over a period of time. A highly stable instrument can be determined by collecting its output data in a fixed interval.
There are many ways to do it but I will share the most basic and simple to do. Just remember that the goal for stability calculation is to determine that the instrument or standard is functioning within specifications on a defined period.
Use this formula:
To determine stability, refer to the control chart below (or the error part on the table). Observe the peak-to-peak value (the highest and the lowest value in the control chart below) then perform the below calculation.
Highest positive error = 0.2
Highest negative error = -0.3
Stability = [0.2 -(-0.3)]/2 = 0.25
Therefore, stability = 0.25 psi
Another method is by determining the standard deviation. If you are using an Excel sheet, this is the simplest to calculate.
Basing it on the table below, just use the formula =STDEV() (see also Excel calculation of precision above) then highlight or choose the ‘UUC Actual Value’. A standard deviation equal to ‘0.192’ will be calculated as the ‘stability’.
Take note that the smaller the value of ‘stability’, the more stable the instrument is. You can also compare this value to the specification of the instrument, which usually can be found on its user manual.
The above calculated ‘stability value’ can also be used during the calculation of measurement uncertainty (source of uncertainty) as one of the component in the uncertainty budget.
How to Monitor Stability?
Below are the things that you can do to monitor the stability of your reference standard.
1. Collect all the past calibration certificates of your reference standards to be evaluated, the more the better. Check the data results.
2. Stability is determined by collecting data on a fixed interval. This could be data from your intermediate check. It can be daily, monthly, or every 3 months.
3. Organize your data on a table, see the below example.
I have collected data through an intermediate check performed every three months on a Test Gauge.
Below are the data.
Determine the error, calculate the mean, and the standard deviation
Plot in the control chart.
As long as the error is within the control limit, we can be sure that the reference standard is very stable and in control.
What is a Drift and How to Determine Drift in Calibration Results?
If a measuring instrument has almost the same measurement output every time you compare its calibration data during re-calibration, we refer to its performance as ‘stable’ or it has good stability as presented above.
But what if the measurement output changes over time where it is reaching the tolerance limit every time we perform a scheduled verification or plotting it in the control chart for performance monitoring? If this is the case, then we can say that our instrument is ‘drifting’, the opposite of a stable reading.
‘Drift’ is the change in output reading of instruments, it is the variation of the performance over time which can be observed by comparing all the calibration results from its calibration history. It is simply the difference between the past result from the present result which can be taken from the UUC’s calibration certificate.
How do we calculate drift? Choose a specific range that you need then summarize it in a table. Subtract the present value with the past value. See the example below:
A drift in our Standards or UUC is normal as long as it is within the acceptable limits and it is under control. Monitoring drift can be the same as the control chart for ‘stability’ that is presented above.
A drift can only determined through the process of calibration. This is one main reason why we need to perform calibration. If we monitor and control drift, then we are confident that our Instruments are performing well.
Traceability in Calibration
As stated, calibration is the comparison of an instrument to a higher or more accurate instrument. These higher accuracy instruments are called the reference standard and are sometimes also known as the calibrator, a master, or a reference.
This reference standard that is used to calibrate your instrument has also a more accurate master standard being used to calibrate which is also linked to a much higher standard until it reaches the main source or SI.
There is an unbroken chain of comparison being linked from top to bottom of the chain. It is passed to local from international standards, the topmost source of traceability in the comparison chain (as shown in the figure above).
This means that the 1 kilogram you used is also 1 kilogram no matter where you go. There is unity in every measurement. Traceability can be determined through its calibration certificate indicating the results and reference standard used to calibrate your instrument.
Why is Traceability in Measurements Necessary?
- For companies engaged in manufacturing and engineering, ensures that parts produced or supplied have the same or acceptable specifications when used by customers anywhere. Compatibility is not an issue.
- Traceability provides confidence to our measurement process because the validity of the measurement results is ensured for its accuracy.
- Traceability has a value, this value can be seen in a calibration certificate as the measurement uncertainty results. With these results, you can determine how accurate the measurement instruments are.
- A requirement by relevant laws and regulations to guarantee product quality.
- A requirement from a contract agreed by two parties (contractual provisions)- a traceable calibration
- Statutory requirements for safety – even though we have different units of measurement, we are confident that compatibility in terms of size, form, or level is not an issue anywhere it goes.
Where can we find the Traceability Information of a Calibrated Instrument?
Since traceability is very important, we should know how to determine or check the traceability information of certain calibrated instruments. We can find it through its calibration certificate, one of the check items in a calibration certificate once it is calibrated by an authorized laboratory.
In a calibration certificate, there is traceability information written usually in the middle part together with the reference standard used or at the bottom and or even both.
This is a requirement so it must not be neglected if the laboratory is a competent one. Moreover, the most important one is the result of its measurement uncertainty, it should be reflected in the data results to ensure that you have a traceable calibration done by an accredited laboratory.
For a deeper explanation with evidence of traceability that you need to know, please visit my other post at this link.
What are the Differences Between Calibration, Verification, and Validation?
Calibration, verification, and validation are the terms that are most confusing if you are not aware of their differences and true meaning when it comes to the measurement process.
To differentiate these terms, below are the main points to remember:
- Calibration is simply the “comparison” of the unknown reading of a UUC to a known reading of a Reference Standard, also known as the Master.
- Verification is a process of “confirming” that a given specification is fulfilled.
- Validation is for “ensuring” the acceptability of the implemented measurement process. Focusing on the final output of the measurement process.
To learn more about their differences including a concrete example, visit my other post at this link>>
We should also understand when to use “calibration or verification” in our measurement process, whether to calibrate to verify or both. check out this link.>> When To Perform Calibration, Verification, or Both?
What Does Measurement Uncertainty Mean?
Calibration is not complete without Measurement uncertainty or Uncertainty of Measurement. This is where the unbroken chain of comparison is being connected or linked.
What does uncertainty mean in calibration? Measurement Uncertainty is the value being displayed to quantify the doubt that exists on a specified measurement result. Since no measurement is exact, there is always an error that is associated with every measurement. To determine the degree, effect, or quantity of this error that exists in every measured parameter, we compute or estimate Measurement Uncertainty.
During uncertainty computation or estimation, we identify all the valid sources of errors that influence our measurement system. It can be from our procedures, instruments, environment, and many more. We evaluate and quantify the value of each error and combine them into a single computed value.
What is the Use of Measurement Uncertainty (MU) in Measurements?
You might just be wondering why Measurement Uncertainty is reported in the calibration certificate. The following are the important uses of measurement uncertainty:
- MU is used for conformity assessments, an important factor in generating a decision rule.
- Use for calculating the TUR (Test Uncertainty Ratio) of Instruments to determine their suitability to be used in a specific measurement process.
- MU is evidence of traceability of an accredited lab or any calibration performed.
- MU uncertainty shows how accurate your instrument is. A smaller uncertainty value means more accurate results.
- If you do not have a basis for your tolerance, MU can be used as your tolerance,
- If you are calculating your own measurement uncertainty, MU in the calibration report is used as the main contributor to be included in the uncertainty budget.
Where Can we find measurement uncertainty? We can find measurement uncertainty results in the calibration certificates usually on the data results page.
Visit my other posts about measurement uncertainty to learn more in this link >> 8 Ways How You Can Use the Measurement Uncertainty Reported in a Calibration Certificate
Difference Between Measurement Uncertainty and Tolerance
Measurement Uncertainty (MU)
Usually defined as the quantification of the doubt. If you measure something, there is always an error (a doubt – no result is perfect) included in the final result since there are no exact measurement results.
Since there are no exact measurement results, what we can do is to determine the range where the true value is located, this range can be determined by adding or subtracting the limits of uncertainty (or the measurement uncertainty result) to the measurement result.
We do not know what the true value is, but because of the measurement uncertainty result, it will show us that the true value lies within the limits of the calculated measurement uncertainty.
“The smaller the measurement uncertainty, the more accurate or exact our measurement results.”
For example, based on the calibration certificate of a pressure switch, it has a measurement result of 10 psi with a calculated measurement uncertainty of +/- 0.3 psi.
As we can see, it has an exact value of 10 psi, but in reality, the exact value is located between the range 9.7 to 10.3 psi’.
It is the maximum error or deviation that is allowed or acceptable as per the design of the user for its manufactured product or components.
If we perform a measurement, the tolerance value will tell us if the measurement we have is acceptable or not.
If you know the tolerance, it will help you answer the questions like:
1. How do you know that your measurement result is within the acceptable range?
2. Is the final product specification pass or fail?
3. Do we need to perform adjustments?
“The bigger the tolerance, the more product or measurement results will pass or be accepted.”
For example, a pressure switch is set to turn on at 10 psi. The process tolerance limit is 1 psi. Therefore, the acceptable range for the switch to turn on is between 9 to 11 psi, beyond this range, we need to perform adjustment and calibration.
See more explanation and presentation in this link >> Differences Between Accuracy, Error, Tolerance, and Uncertainty in Calibration Results
ISO 17025 – Calibration Laboratory Quality Management System
What is ISO 17025?
- International Standard ISO/IEC 17025:2017, General requirements for the competence of testing and calibration laboratories
- As the title implies, it is a standard for laboratory competence, to differ from ISO 9001, which is for certification.
- It is an accreditation standard used by accreditation bodies where a demonstration of a calibration laboratory competency is assessed with regard to its scope and capabilities. Accreditation is simply the formal recognition of a demonstration of that competence.
- It is also a Quality Management System that is comparable to ISO 9001 but it is designed for Calibration Laboratory specifically 3rd Party or External Calibration Labs.
- The usual contents of the quality manual follow the outline of the ISO/IEC 17025 standard.
- It can be divided into two principal parts,
1. Management System Requirements– similar to those specified in ISO 9001:2015, primarily related to the operation and effectiveness of the quality management system within the laboratory.
Some of the requirements include:
- Internal audit
- Document and records control of the management system documentation.
- Handling of complaints
- Control of Non-conforming calibration works
- Implementing corrective actions
- Monitoring improvements
- Maintaining Impartiality and confidentiality
- Management review meeting
- Risk Analysis for Improvements
2. Technical Requirements – include factors that determine the correctness and reliability of the tests and calibrations performed in the laboratory. Some of these factors are personnel competency, facilities and environmental conditions, equipment, calibration methods, reporting of results, and measurement uncertainty calculations.
- These requirements are implementing systems and procedures to be met by testing and calibration labs in their organization and management of their quality system particularly when seeking accreditation.
- Since ISO 17025 is a quality management system specifically for calibration laboratories, it is also a good tool or a guide if you are managing an in-house or internal lab. Following its requirements will help you achieve most of the auditors’ requirements during internal or customer audits.
To learn more about the requirements of ISO 17025:2017, visit my other post at the below link:
ISO/IEC 17025:2017 Requirements: List of Documents Outline and Summary
Learn the basic elements regarding In-House Calibration Management Implementation by visiting this link. >>> ELEMENTS IN IMPLEMENTING IN-HOUSE CALIBRATION.
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What is a Induction Motor?
An induction motor is a type of asynchronous AC motor where power is supplied to the rotating device by means of electromagnetic induction. Induction motor is called asynchronous machines because it will never run at a synchronous speed. Induction motors may be single-phase or three-phase. The single-phase induction motors are usually built-in small sizes (up to 3 H.P). Three phase induction motors are the most commonly used AC motors in the industry because of their simple and rugged construction, low cost, high efficiency, reasonably good power factor, self-starting and low maintenance cost. Almost more than 90% of the mechanical power used in industry is provided by three phase induction motors. Lets see in detail about Working Principle of an Induction Motor – Single & 3 Phase.
How the Induction Motor Works ?
As a general rule, conversion of electrical power into mechanical power takes place in the rotating part of an electric motor. In d.c. motors, the electric power is conducted directly to the armature (i.e. rotating part) through brushes and commutator.
Hence, in this sense, a d.c. motor can be called a conduction motor. However, in a.c. motors, the rotor does not receive electric power by conduction but by induction in exactly the same way as the secondary of a 2-winding transformer receives its power from the primary.
That is why such motors are known as induction motors. In fact, an induction motor can be treated as a rotating transformer i.e., one in which primary winding is stationary but the secondary is free to rotate. Lets see the Working Principle of an Induction Motor further.
Working Principle of an Induction Motor
Principle of an Induction Motor is Electromagnetic induction.
Faradays Laws of Electromagnetic Induction
First Law: This law states that “Whenever a conductor cuts across the magnetic field, an emf is induced in the conductor.” or “Whenever the magnetic flux linking with any circuit (or coil) changes, an emf is induced in the circuit.”
Second Law: This law states that “The magnitude of induced emf in a coil is directly proportional to the rate of change of flux linkages”
Construction of Induction Motor
An induction motor consists essentially of two main parts:
Rotor is classified into two types.
- Squirrel cage Rotor
- Phase wound Rotor or Slip ring Rotor
Induction Motor Types
Starting Methods of Induction Motor
The operation of the squirrel cage induction motor is similar to transformer having short circuited on the secondary side. Due to short circuited on the rotor circuit it will take heavy current when it is directly switched on. Generally, when direct switched, take five to seven times of their full load current.
This initial excessive current is objectionable, because it will produce large line voltage drop. Hence it is not advisable to start directly motors of rating above 5 KW. But the starting torque of an induction motor can be improved by increasing the resistance of the rotor circuit.
This is easily feasible in the case of slip ring induction motor but not in the case of squirrel cage motors. However, in their case, the initial in rush of current is controlled by applying a reduced voltage to the stator during the starting period, full normal voltage being applied when the motor has run up to speed. Speed of squirrel cage induction motor can be controlled from below methods.
Method of Starting of Squirrel Cage Motor:
- Resistors Method
- Star – Delta Method
- Auto transformer Method
Method of Starting of Slip ring Motor:
Rotor Rheostat Method:
These motors are practically always started with full line voltage applied across the stator terminals. The value of starting current is adjusted by introducing a variable resistance in the rotor circuit.
Speed Control of the Induction Motor
The Speed Control of the Induction Motor can be changed under two main headings.
Control from stator side:
- By changing the applied voltage
- By changing the applied frequency
- By changing the no of stator poles.
Control from Rotor side:
- Rotor Rheostatic Control
- Cascade operation
- By injecting emf in the rotor circuit
Single Phase Induction Motors
A single-phase induction motor is very similar to a 3-phase squirrel cage induction motor in construction. Similar to 3-phase motor it consists of two main parts namely stator and rotor.
Working Principle of an Induction Motor– Single Phase:
Why single phase induction motor is not self starting?
When an alternating voltage is applied to the stator winding of single phase motor, an alternating magnetic field(pulsating) is produced. Such a magnetic field acting on squirrel cage rotor cannot produce starting torque needed for motor. Hence single phase induction motors are not self starting. Various method has been developed for obtaining starting torque in these motors. Stator winding is modified or split into two parts, to make itself start.
Types of Single-phase Induction Motors
Based on starting method, single phase induction motors are classified into
- Split Phase Induction Motor
- Capacitor Start Motor
- Capacitor Start Capacitor Run Motor
- Shaded Pole Induction Motor
Three Phase Induction Motor
A 3-phase induction motor consists of two main parts, namely stator and rotor.
1. Stator: It is the stationary part of the motor. It has three main parts, namely.
(i) Outer frame,
(ii) Stator core and
(iii) Stator winding.
(i) Outer frame: It is the outer body of the motor. Its function is to support the stator core and also to protect the inner parts of the machine. For small machines the fame is casted but for large machines it is fabricated. To place the motor on the foundation, feet are provided in the outer frame. The frame of the motor is usually made of cast iron.
(ii) Stator core: When AC supply is given to the motor, an alternating flux is set -up in the stator core. This alternating field produces hysteresis and eddy current loss. To minimize these losses, the core is made of high grade silicon steel stampings. The stampings are assembled under hydraulic pressure and also are keyed to the frame. Each stamping is insulated from the other with a thin varnish layer. The thickness to the stamping usually varies from 0.3 to 0.5 mm. Slots are punched on the inner periphery of the stampings to accommodate stator winding.
(iii) Stator winding: The stator core carries a three phase winding which is usually supplied from a three phase supply system. The six terminals of the winding (two of each phase) are connected in the terminal box of the machine
Rotor: The rotating part of the motor is called rotor. Two types of rotors are used for 3-phase induction motors.
(i) Squirrel cage rotor
(ii) Phase wound or Slip Ring rotor
Working Principle of an Induction Motor– Three Phase
When 3-phase supply is fed to the stator winding of a 3-phase wound rotor induction motor, a resultant rotating magnetic field at constant angular velocity is produced in the stator core. The stator resultant MMF at four different instances along the airgap. Let this field is revolting in an anti-clockwise direction at synchronous speed 𝑛𝑠.
Where, 𝑛𝑠 =120𝑓/𝑃 (𝑟𝑝𝑚)
Where 𝑓 is the frequency of the input electrical power, and 𝑃 is the number of poles of the induction machine. The rotating magnetic field is cut by the stationary rotor conductors and also an emf is induced in the rotor conductors.
As the rotor conductors are short circuited, current flows through them, Furthermore, a resultant field is produced by the rotor current carrying conductors. This field tries to come in line with the stator rotating field, As a result, an electromagnetic torque is developed and rotor starts rotating in same direction as that of stator rotating field.
The rotor then run at a mechanical speed close to and less than the synchronous speed as it tries to attain synchronous speed but never reach. It is because if the rotor revolve at the synchronous speed then the relative speed between rotating stator field and rotor will be zero, therefore, neither emf will be induced in rotor conductors or current nor rotor field and hence no torque will be produced. Thus, an IM can never run at synchronous speed. This is the Working Principle of an Induction Motor -3 Phase.
3 Phase Induction Motor Types
Three phase induction motors are constructed into two major types:
1. Squirrel cage Induction Motors
2. Slip ring Induction Motors
Squirrel cage Induction Motors
It consists of Squirrel cage rotors. The motors in which these rotors are employed are called Squirrel cage induction motors. Because of simple and rugged construction, the most of the induction motors employed in the industry are of this type.
A squirrel cage rotor consists of a laminated cylindrical core having semi-closed circular slots at the outer periphery. Copper or aluminium bar conductors are placed in these slots and short circuited at each end by copper or aluminium rings, called short circuiting rings.
Thus, in these rotors, the rotor winding is permanently short-circuited and also no external resistance can be added in the rotor circuit. Figure 9.3 clearly shows that the slots are not parallel to the shaft but these are skewed.
The skewing provides the following advantages:
- Humming is reduced, that ensures quiet running.
- At different positions of the rotor, smooth and also sufficient torque is obtained.
- It reduces the magnetic locking of the stator and rotor,
- It increases the rotor resistance due to the increased length of the rotor bar conductors.
Slip ring Induction Motors
It consists of Phase wound rotor. The motors in which these rotors are employed are known as phase wound or slip ring induction motors. This rotor is also cylindrical in shape which consists of large number of stampings.
A number of semi-closed slots are punched at its outer periphery. A 3-phase insulated winding is placed in these slots. The rotor is wound for the same number of poles as that of stator. rotor winding is connected in star and also its remaining three terminals are connected to the slip rings.
The rotor core is keyed to the shaft. Similarly, slip-rings are also keyed to the shaft but these are insulated from the shaft. In this case, depending upon the requirement any external resistance can be added in the rotor circuit. In this case also the rotor is skewed.
A mild steel shaft is passed through the centre of the rotor and is fixed to it with key. The purpose of shaft is to transfer mechanical power.
WHAT IS SLIP OF INDUCTION MOTOR
- In practice, the rotor never succeeds in ‘catching up’ with the stator field. If it really did so, then there would be no relative speed between the two, hence no rotor e.m.f., no rotor current and so no torque to maintain rotation. That is why the rotor runs at a speed which is always less than the speed of the stator field. The difference in speeds depends upon the load on the motor.
- The difference between the synchronous speed Ns and the actual speed N of the rotor is known as Slip of Induction Motor.
- Though it may be expressed in so many revolutions/second, yet it is usual to express it as a percentage of the synchronous speed. Actually, the term ‘slip’ is descriptive of the way in which the rotor ‘slips back’ from synchronism.
- Sometimes, Ns − N is called the Slip Speed.
Induction Motor Vs Synchronous Motor
Advantages of Induction Motor:
- It has very simple & extremely rugged, almost unbreakable construction (especially squirrel cage type).
- Its cost is low. It is very reliable.
- It has sufficiently high efficiency. In normal running condition, no brushes are needed, hence frictional losses are reduced. It has a reasonably good power factor.
- It requires minimum of maintenance.
- It starts up from rest and needs no extra starting motor & has not to be synchronized.
Disadvantages of Induction Motor:
- Its speed cannot be varied without sacrificing some of its efficiency.
- Just like a d.c. shunt motor, its speed decreases with increase in load.
- Its starting torque is somewhat inferior to that of a d.c. shunt motor
Induction Motor Applications
The applications of squirrel cage induction motors and slip-ring (phase wound) induction motors are given below:
1. Squirrel cage induction motor application:
These motors are mechanically robust and are operated almost at constant speed. These motors operate at high power factor and also have high over load capacity. However, these motors have low starting torque. (i.e., these motors cannot pick-up heavy loads) and draw heavy current at start. On the bases of these characteristics, these motors are best suited for:
- Printing machinery
- Flour mills
- Saw mills
- Shaft drives of small industries
- Prime-movers with small generators etc.
2. Slip-ring induction motor application:
These motors have all the important characteristics (advantage) of squirrel cage induction motors and at the same time have the ability to pick-up heavy loads at start drawing smaller current from the mains. Accordingly these motors are best suited for;
- Rolling mills
- Lifts & hoists
- Big flour mills
- Large pumps
- Line shafts of heavy industries
- Prime-moves with medium & also large generators.
INDUCTION MOTOR FAQ
What is induction motors?
An induction motor is a type of asynchronous AC motor where power is supplied to the rotating device by means of electromagnetic induction. It is called asynchronous machines because it will never run at a synchronous speed. Induction motors may be single-phase or three-phase.
Why it is called induction motor?
Induction motor works on the principle of electromagnetic induction. These motors are powered at the stator, while the rotor induces current hence it is called as “Induction” motor.
What is speed control of induction motor?
Induction motor speed control is a process of manipulating currents in an induction motor to regulate speed. The speed of an induction motor can be changed under two main headings.
- Control from stator side
- Control from Rotor side
What is rotor and stator?
Stator is the stationary part of the motor. It has three main parts, namely Outer frame, Stator core and Stator winding. The rotating part of the motor is called rotor. There are two types of rotors namely Squirrel cage rotor & Slip ring rotor
What are the two basic types of induction motors?
Induction motors are categorized into two main types: single-phase and three-phase induction motors
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Genes are the fundamental units of heredity, responsible for the production of proteins that carry out key functions in the body. However, not all genes are expressed in every cell or tissue. The intricate regulation of gene expression determines when and where specific genes are activated, leading to the development and maintenance of different cell types and tissues throughout the body.
At any given time, only a subset of genes are actively expressed in a particular cell or tissue. This regulation is crucial for the proper functioning of the body, as different cells have distinct roles and functions. For example, genes that are expressed in muscle cells enable the contraction and movement of the muscles, while genes that are expressed in skin cells determine the production of structural proteins for the skin.
The process of gene expression involves multiple steps, including transcription and translation. During transcription, the DNA sequence of a gene is transcribed into a messenger RNA (mRNA) molecule. This mRNA molecule carries the genetic information to the ribosomes, where it is translated into a specific protein. The regulation of gene expression occurs primarily at the level of transcription, with various factors controlling whether a gene is turned on or off in a specific cell or tissue.
Scientists have made significant progress in understanding the factors that control gene expression. These factors include transcription factors, which bind to specific DNA sequences near a gene and either promote or inhibit its transcription. Additionally, epigenetic modifications, such as DNA methylation and histone modifications, can influence gene expression by altering the accessibility of the DNA to the transcription machinery.
What is Gene Expression?
Gene expression refers to the process by which information from a gene is used to create a functional product, typically a protein. Genes are segments of DNA that contain the instructions for building proteins, which are essential for the structure and function of cells and organisms. The process of gene expression involves two main steps: transcription and translation.
In transcription, the DNA sequence of a gene is copied into a molecule called messenger RNA (mRNA). This mRNA molecule carries the genetic information from the nucleus to the ribosomes in the cytoplasm, where protein synthesis takes place.
In translation, the mRNA molecule is read by the ribosomes, which use the information to assemble amino acids in the correct order to form a protein chain. The sequence of amino acids determines the structure and function of the protein.
Gene expression is a tightly regulated process that allows cells to respond to their environment and carry out specific functions. Different genes are expressed in different cells and tissues, giving rise to the wide variety of cell types and functions in the body. Understanding where genes are expressed is important for understanding how they contribute to the development and function of different tissues and organs.
Importance of Gene Expression
Understanding where genes are expressed in the body is crucial for unraveling the intricate mechanisms underlying human biology. Gene expression refers to the process of turning on specific genes in a cell and allowing them to produce their corresponding proteins. This process plays a fundamental role in the development, function, and maintenance of all living organisms.
Genes are the blueprints for building proteins, which are the workhorses of the cell. They carry out a wide range of functions, including catalyzing chemical reactions, transporting molecules, regulating gene expression, and providing structural support. Therefore, knowing which genes are expressed, and where they are expressed, is essential for understanding the functional capabilities of a cell or tissue.
By studying gene expression, scientists can gain insights into the specific functions of different cell types and tissues in the body. For example, genes that are highly expressed in neurons are likely involved in processes related to brain function, while genes that are expressed in muscle cells are likely involved in muscle contraction and movement.
Regulation of Gene Expression
In addition to understanding the location of gene expression, studying the regulation of gene expression is equally important. The human body consists of trillions of cells, each with the same genetic information. However, not all genes are actively expressed in every cell at all times. Instead, gene expression is tightly regulated, allowing different cells to have distinct functions and characteristics.
Through the study of gene regulation, scientists can uncover the mechanisms that control when and where genes are turned on and off. These regulatory mechanisms involve a complex interplay of DNA sequences, proteins, and other molecules. Disruptions in gene regulation can lead to the development of diseases, such as cancer, where genes that should be off are turned on, or vice versa.
Applications in Medicine
The importance of understanding gene expression extends beyond basic biology and has significant implications in medicine. Identifying genes that are aberrantly expressed in certain diseases can provide targets for therapeutic interventions. For example, drugs can be designed to specifically target and inhibit the expression of genes that are overactive in cancer cells.
Furthermore, gene expression profiling can be used in diagnostics to identify disease subtypes and predict patient outcomes. By analyzing the expression levels of specific genes, doctors can gain insights into the underlying molecular mechanisms driving a patient’s disease and tailor treatment accordingly.
In conclusion, understanding where genes are expressed and how they are regulated is essential for unraveling the complexities of human biology. It provides insights into the functions of different cell types and tissues, offers opportunities for therapeutic interventions, and aids in the diagnosis and treatment of diseases.
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How Does Gene Expression Work?
Gene expression refers to the process by which the information encoded in a gene is used to create a functional product, such as a protein. It is a complex and tightly regulated process that occurs in all living organisms.
At a basic level, gene expression involves two main steps: transcription and translation. Transcription is the process by which the DNA sequence of a gene is copied into a molecule of messenger RNA (mRNA). This mRNA molecule carries the genetic information from the DNA to the site of protein synthesis. The location where genes are expressed in the body is a key factor in gene expression.
During transcription, an enzyme called RNA polymerase binds to the DNA at the start of a gene and unwinds and separates the DNA strands. The RNA polymerase then adds complementary RNA nucleotides to the growing mRNA molecule, using the DNA sequence as a template. This process produces a primary transcript, which undergoes further processing to produce a mature mRNA molecule.
Translation is the process by which the genetic information carried by mRNA is used to create a protein. It takes place in the ribosomes, which are complex structures in the cytoplasm of the cell. The mRNA molecule is read by ribosomes in groups of three nucleotides called codons. Each codon corresponds to a specific amino acid. As the ribosome moves along the mRNA molecule, it adds amino acids to the growing protein chain according to the codons it encounters. This process continues until a stop codon is reached, signaling the end of protein synthesis.
The regulation of gene expression is essential for the proper functioning of cells and tissues. Cells have mechanisms in place to control when and where specific genes are expressed. This regulation can occur at various stages of gene expression, including transcription initiation, mRNA processing, and translation control. Understanding how genes are expressed in different parts of the body can provide valuable insights into the development, function, and diseases of various tissues and organs.
|Stage of Gene Expression
|The process of copying the DNA sequence of a gene into mRNA.
|The process of using the mRNA to synthesize a protein.
|The control of when and where specific genes are expressed.
Types of Gene Expression
Where genes are expressed
Gene expression refers to the process by which information from a gene is used in the synthesis of a protein or functional RNA molecule. Genes can be expressed in a variety of different ways, depending on the specific cell type and the stage of development.
There are two main types of gene expression: constitutive and regulated.
Constitutive gene expression:
In constitutive gene expression, genes are constantly active and produce their products at a relatively constant rate in all cells. These genes are essential for the basic survival and functioning of the organism. Examples of constitutively expressed genes include those involved in cellular metabolism and housekeeping functions.
Regulated gene expression:
In regulated gene expression, genes are only active under specific conditions or in specific cell types. This type of gene expression allows cells to respond to different signals and adapt to changing environments. Regulated gene expression is crucial for the development and specialization of different cell types in multicellular organisms.
Regulation of gene expression can occur at multiple levels, including transcriptional, post-transcriptional, translational, and post-translational regulation. These processes control the amount and timing of gene expression, ensuring that genes are expressed in the right place and at the right time.
Understanding the different types of gene expression is essential for unraveling the complex processes that govern development, physiology, and disease in the human body.
Regulation of Gene Expression
Gene expression refers to the process by which information from a gene is used in the synthesis of a functional gene product. It is a highly regulated process that ensures the correct genes are expressed at the right time and in the right tissues. Understanding the regulation of gene expression is crucial for understanding how cells function and develop.
One of the major levels of gene expression regulation is at the transcriptional level, where genes are transcribed into messenger RNA (mRNA) molecules. Transcriptional regulation involves various factors, such as transcription factors, enhancers, and repressors, that control the initiation and rate of transcription.
Transcription factors are proteins that bind to specific DNA sequences near the gene and either activate or repress transcription. Enhancers are DNA sequences that can enhance the transcription of specific genes when bound by certain transcription factors. Repressors, on the other hand, bind to DNA sequences and inhibit transcription.
Once the mRNA molecules have been transcribed, they undergo various post-transcriptional modifications that can regulate gene expression. These modifications include alternative splicing, where different exons of the mRNA are spliced together in different combinations, and RNA editing, where nucleotides in the mRNA sequence are chemically modified.
RNA stability also plays a role in gene expression regulation. Some mRNA molecules are more stable than others and can persist in the cell for longer periods of time, leading to increased gene expression. On the other hand, some mRNA molecules are targeted for degradation, resulting in decreased gene expression.
After mRNA molecules have undergone post-transcriptional modifications, they are translated into proteins in a process called translation. Translation regulation mechanisms control the efficiency and timing of protein synthesis.
One important mechanism of translation regulation is the binding of small non-coding RNAs, known as microRNAs, to the mRNA molecules. MicroRNAs can either block translation or promote degradation of the mRNA molecules, thus preventing or reducing protein synthesis.
Epigenetic regulation refers to the modifications of DNA and chromatin that do not involve changes in the DNA sequence itself. These modifications can influence gene expression by altering the accessibility of the DNA to transcription factors and other regulatory proteins.
One example of epigenetic regulation is DNA methylation, where methyl groups are added to the DNA molecule. Methylation can silence gene expression by preventing the binding of transcription factors to the DNA. Another example is histone modification, where certain chemical groups are added or removed from the histone proteins that package the DNA. Histone modifications can affect how tightly the DNA is wound around the histones, making it more or less accessible for transcription.
In summary, gene expression is regulated at multiple levels, including transcriptional, post-transcriptional, translational, and epigenetic regulation. These regulatory mechanisms ensure that genes are expressed in the appropriate tissues and at the appropriate times, allowing for the proper development and functioning of organisms.
Factors Affecting Gene Expression
The expression of genes, which refers to the process of turning on or off specific genes, is influenced by a variety of factors. These factors play a crucial role in determining where genes are expressed in the body and the levels of gene expression in different tissues or cell types.
1. Genetic Factors
Genetic factors are one of the primary determinants of gene expression. Each individual inherits a unique set of genetic information, known as their genotype, which influences how and when genes are expressed. Genetic variations, such as mutations or single nucleotide polymorphisms (SNPs), can alter gene expression patterns and contribute to the development of various diseases or traits.
2. Environmental Factors
Environmental factors can also have a significant impact on gene expression. External factors like diet, stress, exposure to toxins, and lifestyle choices can modify the expression of certain genes and influence their activity. For example, a high-fat diet can upregulate genes associated with lipid metabolism, while chronic stress can downregulate genes involved in the immune response.
It is important to note that environmental factors can interact with genetic factors to shape gene expression patterns. For example, certain genetic variants may increase the susceptibility to environmental influences, resulting in different gene expression profiles in individuals exposed to the same environmental factor.
3. Epigenetic Modifications
Epigenetic modifications are chemical tags that can be added to the DNA or histone proteins associated with DNA, and they can affect gene expression without altering the underlying genetic sequence. These modifications, such as DNA methylation or histone acetylation, can either activate or suppress gene expression by modifying the accessibility of genes to transcription factors and other regulatory molecules.
Epigenetic modifications can be influenced by both genetic and environmental factors. They can be stable or reversible and can be inherited through generations, contributing to the regulation of gene expression in different tissues and cell types.
In conclusion, the expression of genes in the body is regulated by a complex interplay of genetic, environmental, and epigenetic factors. Understanding these factors and their interactions is vital for deciphering the mechanisms underlying gene expression and its implications for human health and disease.
Methods to Study Gene Expression
Understanding where and how genes are expressed in the body is crucial for unraveling the complex mechanisms of life. To investigate gene expression, scientists have developed various methods that allow them to peek into the molecular machinery of cells and tissues.
One of the fundamental techniques used to study gene expression is reverse transcription polymerase chain reaction (RT-PCR). This method enables the detection and quantification of mRNA molecules, which are intermediates in the process of gene expression. By analyzing the levels of mRNA, scientists can infer which genes are being actively expressed in a particular cell or tissue.
In recent years, advances in high-throughput sequencing technologies have revolutionized the study of gene expression. RNA sequencing (RNA-seq) allows researchers to analyze the entire transcriptome of a cell or tissue, providing a comprehensive picture of gene activity. This technique can not only reveal the presence and abundance of different mRNA molecules but also uncover novel gene isoforms and identify previously unknown transcripts.
Another powerful method used to study gene expression is in situ hybridization. By labeling specific RNA molecules with fluorescent probes, scientists can visualize the exact location of gene expression within cells and tissues. This technique provides spatial information about gene activity, allowing researchers to map gene expression patterns across different organs and developmental stages.
In addition to these molecular techniques, researchers can also use bioinformatics approaches to study gene expression. This involves analyzing large-scale gene expression datasets to uncover patterns and relationships between genes. By utilizing computational algorithms, scientists can identify co-regulated genes, predict gene functions, and gain insights into the underlying regulatory networks.
Overall, the study of gene expression encompasses a wide range of methods that complement each other and provide a multi-dimensional view of how genes are expressed in the body. By utilizing these techniques, scientists can unravel the intricate processes that govern cell development, tissue specialization, and human health.
Genes and Protein Synthesis
Genes are the units of heredity that are responsible for determining the characteristics of living organisms. They are encoded in the DNA molecules and are expressed in different parts of the body.
Protein synthesis is the process by which genes are turned into functional proteins. This process involves a series of steps, including transcription and translation.
- Transcription: In this step, the DNA sequence of a gene is copied into a molecule called messenger RNA (mRNA). The enzyme RNA polymerase binds to the DNA molecule and synthesizes a complimentary mRNA molecule by adding nucleotides one by one.
- Translation: In this step, the mRNA molecule is used as a template to synthesize a specific protein. The mRNA molecule is read in groups of three nucleotides called codons. Each codon corresponds to a specific amino acid. Ribosomes, the cellular machinery responsible for protein synthesis, read the codons and bring the corresponding amino acids together to form the protein.
Genes are expressed in specific tissues and cell types in the body. The expression of genes can be regulated, meaning that certain genes are turned on or off depending on the needs of the organism. This regulation allows for the development and maintenance of different tissues and organs throughout the body.
Understanding gene expression and protein synthesis is crucial for understanding how genes function and how they contribute to the diversity of living organisms.
Gene Expression and Development
Gene expression plays a crucial role in the development of an organism. It determines where and when genes are expressed, which ultimately influences the formation and function of different body parts.
During development, specific genes are turned on or off in different cells and tissues, allowing for the specialization of cells and the formation of distinct structures. For example, genes involved in muscle development are expressed in muscle cells, while genes involved in brain development are expressed in neuronal cells.
The precise regulation of gene expression during development is essential for the proper growth and differentiation of cells. Misregulation of gene expression can lead to developmental disorders and diseases. For instance, mutations in genes that control limb development can result in limb malformations.
Understanding where genes are expressed in the body during development is a complex task. It requires techniques such as in situ hybridization and whole-mount immunostaining to visualize gene expression patterns in specific tissues and at specific time points. These techniques provide valuable insights into the spatial and temporal dynamics of gene expression during development.
By studying gene expression patterns, researchers can uncover the molecular mechanisms underlying developmental processes. This knowledge is crucial for advancing our understanding of human development and improving the diagnosis and treatment of developmental disorders.
Gene Expression and Disease
Understanding where genes are expressed in the body is crucial for studying the role of gene expression in various diseases. Gene expression refers to the process by which information from a gene is used to create a functional gene product such as a protein. Abnormal gene expression can lead to the development of diseases and understanding the specific tissues or organs where genes are expressed can provide valuable insights into disease mechanisms.
Gene Expression Patterns in Disease
In many diseases, there are specific changes in gene expression patterns. For example, certain genes may be overexpressed, meaning that they are produced in excessive amounts, while others may be underexpressed or completely turned off. These alterations in gene expression can have profound effects on cellular functions and can contribute to disease development and progression.
By studying gene expression patterns in disease, researchers can identify potential biomarkers that can help in the diagnosis and prognosis of diseases. Biomarkers are measurable indicators of disease presence or progression, and understanding where specific genes are expressed can help in identifying these markers. Additionally, gene expression profiling can provide insights into the underlying mechanisms of diseases and can help in the development of targeted therapies.
Identifying Disease-Related Gene Expression
To identify disease-related gene expression, scientists use various techniques such as gene expression microarrays or RNA sequencing. These methods allow researchers to analyze the expression levels of thousands of genes simultaneously. By comparing gene expression profiles between healthy and diseased tissues, researchers can identify genes that are differentially expressed and potentially associated with the disease.
Furthermore, advances in bioinformatics have enabled the integration of gene expression data from different studies and databases, allowing for a comprehensive analysis of gene expression patterns in various diseases. This integrated approach helps in identifying common gene expression signatures across different diseases and can provide insights into shared disease pathways.
Overall, understanding gene expression patterns in disease is crucial for unraveling the complexities of disease biology. By identifying where specific genes are expressed, researchers can gain insights into disease mechanisms and develop targeted treatments for better disease management.
Technologies for Studying Gene Expression
Understanding where genes are expressed in the body is a crucial aspect of studying gene expression. Fortunately, there are several technologies available that help researchers investigate the spatial distribution of gene expression.
One common method is in situ hybridization, which allows scientists to visualize the location of specific RNA molecules within tissues or cells. By using complementary DNA or RNA probes that are labeled with fluorescent or enzymatic markers, researchers can identify where genes are being expressed.
Another technique called RNA sequencing (RNA-seq) has revolutionized the field of gene expression analysis. RNA-seq allows researchers to measure the abundance of RNA molecules in a sample, providing a quantitative assessment of gene expression levels. This technology can also provide information about alternative splicing and novel RNA transcripts.
Microarrays are another widely used technology for studying gene expression. Microarray platforms contain an array of thousands of DNA or RNA probes that can hybridize to target sequences in a sample. By measuring the amount of fluorescence or radioactivity associated with each spot on the microarray, researchers can determine the relative abundance of specific RNA molecules.
Recent advancements in single-cell technologies have also revolutionized the study of gene expression. Single-cell RNA sequencing (scRNA-seq) allows researchers to analyze the gene expression profiles of individual cells. This technology has revealed previously unknown heterogeneity within cell populations and has provided insights into cellular dynamics and development.
Overall, these technologies provide valuable tools for researchers to explore where genes are expressed in the body. By understanding the spatial distribution of gene expression, scientists can gain insights into development, disease progression, and potential therapeutic targets.
Gene Expression Datasets
Gene expression datasets provide valuable information about where genes are expressed in the body. These datasets are generated by various techniques, such as microarray analysis and RNA sequencing, and allow researchers to explore gene expression patterns in different tissues and cell types.
Microarray analysis is a method used to measure the expression levels of thousands of genes simultaneously. In this technique, DNA or RNA molecules are spotted onto a microarray chip, and gene expression levels are detected using fluorescent probes. By comparing the gene expression profiles of different tissues or cells, researchers can identify specific genes that are expressed in a particular location.
RNA sequencing, also known as RNA-Seq, is a method that enables the profiling of all RNA molecules in a given sample. In this technique, RNA molecules are converted into complementary DNA (cDNA) and then sequenced using high-throughput sequencing platforms. By comparing the abundance of different RNA molecules in different tissues or cells, researchers can determine where specific genes are expressed within the body.
These gene expression datasets are often publicly available and can be accessed through various databases and online resources. Researchers can use these datasets to investigate the spatial and temporal patterns of gene expression, identify disease-specific gene expression changes, and gain insights into the functions and regulatory mechanisms of genes.
In conclusion, gene expression datasets have revolutionized our understanding of where genes are expressed in the body. By analyzing these datasets, researchers can unravel the complexity of gene expression patterns and uncover the roles of genes in different tissues and cell types.
Understanding Tissue-Specific Gene Expression
Gene expression is the process by which information from a gene is used to create a functional gene product, such as a protein. Different genes are expressed in different tissues throughout the body, which allows for the specialization and differentiation of cells. Understanding where genes are expressed can provide valuable insight into the development, function, and regulation of different tissues.
Tissue-Specific Gene Expression
Tissue-specific gene expression refers to the phenomenon where certain genes are only expressed in specific tissues or cell types. This means that the gene is activated and its product is produced only in certain cells, while being inactive or producing no product in other cells.
The regulation of tissue-specific gene expression is complex and involves various mechanisms, such as transcription factors, epigenetic modifications, and regulatory elements. These mechanisms work together to ensure that genes are expressed at the right time and in the right place, allowing for the proper development and functioning of different tissues.
By studying tissue-specific gene expression, researchers can gain insights into the molecular mechanisms underlying tissue development, maintenance, and disease. For example, identifying which genes are specifically expressed in a certain tissue can help identify markers for that tissue, which can be useful for diagnostic purposes or the development of targeted therapies.
Methods for Studying Tissue-Specific Gene Expression
There are several methods available for studying tissue-specific gene expression. One commonly used approach is RNA sequencing, which allows researchers to measure the levels of gene expression in different tissues. By comparing the gene expression profiles of different tissues, researchers can identify tissue-specific genes.
Another approach is in situ hybridization, which involves labeling RNA probes that specifically target a gene of interest. These probes are then used to visualize the expression pattern of the gene in tissue sections. This method provides spatial information about gene expression, allowing researchers to identify the specific cells or regions where the gene is expressed.
|Provides information about tissue-specific gene expression
|Does not capture dynamic changes in gene expression over time
|Can be used to identify markers for specific tissues
|Requires careful sample preparation and optimization
Overall, understanding tissue-specific gene expression is crucial for unraveling the complexity of gene regulation and the development of different tissues. It provides valuable insights into the molecular mechanisms underlying tissue specialization and can have important implications for disease research and therapeutic development.
Gene Expression Atlas
The Gene Expression Atlas is a comprehensive database that provides information about where genes are expressed in the body. It allows researchers to explore gene expression patterns across different tissues and cell types.
By analyzing data from various experiments and studies, the Gene Expression Atlas enables scientists to understand which genes are active and where they are active within the body. This information is crucial for understanding the roles and functions of different genes, as well as for studying diseases and developing potential treatments.
Features of the Gene Expression Atlas
- Searchable database: The Gene Expression Atlas allows users to search for specific genes and explore their expression patterns.
- Tissue-specific expression: The database provides information on gene expression in different tissues, allowing researchers to identify genes that are specifically active in certain tissues.
- Cell-type expression: Users can also explore gene expression patterns in different cell types, helping to understand the specific functions of genes in different cell populations.
- Comparative analysis: The Gene Expression Atlas enables researchers to compare gene expression patterns between different tissues and cell types, providing insights into the similarities and differences in gene regulation.
Benefits of the Gene Expression Atlas
- Better understanding of gene function: By knowing where genes are expressed, researchers can gain insights into their roles and functions in specific tissues and cell types.
- Disease research: The Gene Expression Atlas provides valuable information for studying diseases and identifying potential therapeutic targets.
- Drug discovery: By understanding gene expression patterns, scientists can develop more targeted and effective drugs.
- Biomedical research: The database supports a wide range of biomedical research, including genomics, physiology, and developmental biology.
Overall, the Gene Expression Atlas is a valuable resource for understanding gene expression across the body. It helps researchers navigate the complex landscape of gene activity and provides insights into the intricate functioning of cells and tissues.
Gene Expression in Different Organs
Genes are expressed in various organs throughout the body. Each organ has a unique set of genes that are activated and regulated to perform specific functions. Understanding where genes are expressed in different organs is crucial for understanding the development and functioning of the human body.
The brain is one of the most complex organs in terms of gene expression. It contains a wide range of genes that are involved in processes such as neuron development, synaptic transmission, and memory formation. Genes related to neurotransmitters, such as dopamine and serotonin, are particularly expressed in the brain.
The heart is another organ with highly regulated gene expression. Genes involved in cardiac muscle contraction, ion channel function, and cardiac development are predominantly expressed in the heart. Understanding gene expression in the heart is crucial for understanding heart development and function, as well as for studying cardiovascular diseases.
The liver is an organ that plays a vital role in metabolism and detoxification. It has a unique set of genes that are expressed to carry out functions such as protein synthesis, drug metabolism, and bile production. Genes related to enzymes involved in drug metabolism, such as cytochrome P450 enzymes, are highly expressed in the liver.
The pancreas is responsible for producing hormones such as insulin, which regulate blood sugar levels. Genes involved in hormone production and secretion are expressed in the pancreas. Understanding gene expression in the pancreas is essential for understanding diseases such as diabetes.
The lungs are the organs responsible for gas exchange in the body. Genes involved in lung development, oxygen transport, and immune responses are expressed in the lungs. Understanding gene expression in the lungs is crucial for understanding respiratory diseases and conditions.
These are just a few examples of organs where genes are expressed. Gene expression in different organs is tightly regulated and plays a crucial role in organ development, function, and disease. Studying gene expression patterns in different organs can provide valuable insights into the biology of the human body.
Gene Expression in the Nervous System
The nervous system is a complex network of cells and tissues that allows the body to communicate and respond to its environment. Genes play a crucial role in the development, function, and maintenance of the nervous system. Understanding where genes are expressed in the nervous system can provide insights into its structure and function.
Central Nervous System
The central nervous system (CNS) consists of the brain and spinal cord. It is the main control center for the body and coordinates various functions such as movement, sensation, and cognition. Genes are expressed in different regions of the brain and spinal cord, allowing for specialized functions and communication between cells.
Neurons are the building blocks of the nervous system. They are responsible for transmitting electrical signals and information throughout the body. Genes that regulate the development and function of neurons are expressed in specific regions of the CNS, such as the cerebral cortex, hippocampus, and cerebellum.
Glia are supportive cells in the nervous system that provide nutrients and support to neurons. They play a crucial role in maintaining the health and function of the nervous system. Genes involved in glial cell development and function are expressed in various regions of the CNS, including the white matter, grey matter, and ventricles.
Peripheral Nervous System
The peripheral nervous system (PNS) consists of nerves that connect the CNS to various parts of the body. It transmits sensory information to the CNS and sends signals from the CNS to the organs and muscles. Genes expressed in the PNS are involved in regulating the growth and function of peripheral nerves.
Sensory neurons are responsible for transmitting sensory information from the body to the CNS. Genes that control the development and function of sensory neurons are expressed in various sensory organs, such as the eyes, ears, and skin.
Motor neurons transmit signals from the CNS to the muscles, allowing for movement and coordination. Genes that regulate the development and function of motor neurons are expressed in specific regions of the PNS, such as the spinal cord and neuromuscular junctions.
In conclusion, genes are expressed in different regions of the nervous system, allowing for specialized functions and communication between cells. Understanding where genes are expressed in the nervous system provides valuable insights into its structure and function.
Gene Expression in the Cardiovascular System
Genes play a crucial role in the development, function, and maintenance of the cardiovascular system. They determine the proteins and other molecules that are expressed in the heart, blood vessels, and other components of the cardiovascular system. Understanding where genes are expressed in the cardiovascular system helps scientists and researchers better understand the mechanisms behind cardiovascular diseases and conditions.
In the heart, genes are expressed in various cell types, including cardiomyocytes, endothelial cells, and smooth muscle cells. Cardiomyocytes are responsible for the contraction of the heart and are highly specialized cells. Genes involved in the regulation of cardiac muscle development, contraction, and electrical signaling are expressed in cardiomyocytes.
Endothelial cells line the blood vessels and are crucial for maintaining vascular health. Genes involved in the regulation of blood vessel development, endothelial cell function, and vascular repair are expressed in endothelial cells. Dysfunction of these genes can lead to conditions such as atherosclerosis and hypertension.
Smooth muscle cells are found in the walls of blood vessels and are responsible for regulating blood vessel tone and diameter. Genes involved in the regulation of smooth muscle cell contraction and relaxation are expressed in smooth muscle cells. Dysregulation of these genes can lead to conditions such as vasospasm and arterial stiffness.
Additionally, genes involved in the regulation of lipid metabolism, inflammation, and immune response are expressed in various cell types within the cardiovascular system. These genes play important roles in the development of atherosclerosis, the formation of blood clots, and the response to cardiac injury.
Overall, understanding where genes are expressed in the cardiovascular system provides insights into the molecular mechanisms underlying cardiovascular health and disease. It helps identify potential therapeutic targets for the treatment and prevention of cardiovascular diseases.
Gene Expression in the Immune System
The immune system plays a crucial role in protecting the body from pathogens and diseases, and gene expression is a key mechanism that regulates its functioning. Genes are the basic units of heredity and are responsible for creating the proteins that drive all cellular processes. The immune system relies on the coordinated expression of specific genes to mount an effective immune response.
In the immune system, genes are expressed in various cell types, including white blood cells, such as T cells, B cells, and natural killer cells. Each cell type expresses a unique set of genes that contribute to its specialized functions in the immune response. For example, T cells express genes that are involved in recognizing and attacking foreign invaders, while B cells express genes that are responsible for producing antibodies.
Where are Genes Expressed in the Immune System?
Genes are expressed in specific tissues and organs of the immune system. For instance, in the bone marrow, genes are expressed in the hematopoietic stem cells that give rise to different types of blood cells, including immune cells. In the thymus, genes are expressed in T cells as they mature and undergo the selection process to ensure their proper functioning. In the lymph nodes, genes are expressed in immune cells that interact with antigens to initiate an immune response.
Furthermore, genes are also expressed in specialized immune organs, such as the spleen and the tonsils. These organs contain different cell types that express specific genes required for their functions. The spleen, for example, expresses genes that filter the blood and remove old or damaged red blood cells, while the tonsils express genes involved in detecting and responding to pathogens that enter through the mouth and nose.
In summary, gene expression in the immune system is essential for the proper functioning of different cell types and organs involved in the immune response. Genes are expressed in specific tissues and organs, allowing for the coordination of immune processes and the protection of the body against diseases and infections.
Gene Expression in the Digestive System
In the human body, genes are expressed in various tissues and organs to carry out specific functions. One crucial system where gene expression plays a vital role is the digestive system. The digestive system is responsible for breaking down food into smaller molecules, absorbing nutrients, and eliminating waste products.
Within the digestive system, genes are expressed in different organs and tissues such as the mouth, esophagus, stomach, small intestine, and large intestine. Each of these organs has specific gene expression patterns that contribute to their unique functions in the digestive process.
In the mouth, genes are expressed in the salivary glands, which produce saliva containing enzymes that begin the process of breaking down carbohydrates. Additionally, genes are expressed in the taste buds on the tongue, which play a role in detecting different flavors.
In the esophagus, genes are expressed in the smooth muscles responsible for peristalsis, the coordinated contractions that propel food towards the stomach.
In the stomach, genes are expressed in the gastric mucosa, which secretes gastric juices containing enzymes and acid to further break down food. Genes are also expressed in the stomach lining to protect it from the corrosive effects of the gastric juices.
In the small intestine, genes are expressed in the epithelial cells that line the intestinal walls. These genes are responsible for producing enzymes that break down different types of food molecules further. Genes are also expressed in the cells of the intestinal villi, which absorb nutrients into the bloodstream.
In the large intestine, genes are expressed in the cells that line the colon, contributing to the absorption of water and electrolytes and the formation of feces.
Overall, gene expression in the digestive system is essential for the proper functioning of each organ and tissue involved in the complex process of digestion. Understanding the specific genes and their expression patterns in the digestive system can provide valuable insights into digestive disorders and diseases.
Gene Expression in the Respiratory System
The respiratory system plays a crucial role in the exchange of gases, allowing oxygen to enter the bloodstream and carbon dioxide to be expelled from the body. This complex system involves various organs, tissues, and cells, each with their unique gene expression patterns.
Lung Gene Expression
The lungs are the primary organs of the respiratory system and are composed of numerous specialized cells that facilitate efficient gas exchange. Genes expressed in the lungs are responsible for the development, maintenance, and functioning of these cells.
One of the key genes expressed in the lungs is surfactant protein genes, which encode proteins that reduce surface tension in the alveoli, allowing them to remain open and promote efficient gas exchange.
Nasal Gene Expression
The nasal cavity is lined with specialized cells that help filter, warm, and moisten the air we breathe. Genes expressed in the nasal epithelium play a vital role in the production of mucus, cilia movement, and immune responses to pathogens.
One example of genes expressed in the nasal epithelium is the MUC genes, which encode mucins – proteins that form the main component of mucus. These genes are essential for proper mucin production, which helps trap and remove particulate matter and microbes from the air we inhale.
Trachea and Bronchus Gene Expression
The trachea and bronchi are responsible for carrying air to and from the lungs. Genes expressed in these parts of the respiratory system are involved in maintaining the integrity of the airway lining, controlling mucus production, and assisting in coughing or sneezing reflexes.
One critical gene expressed in the trachea and bronchi is the CFTR gene. Mutations in this gene can lead to cystic fibrosis, a condition characterized by the production of thick, sticky mucus that clogs the airways and leads to persistent infections.
- Genes expressed in the respiratory system have unique functions in different organs and tissues.
- Understanding gene expression in the respiratory system can provide insights into respiratory diseases and potential therapeutic targets.
- Further research is needed to fully understand the complex gene regulatory networks that govern respiratory system development and functioning.
In conclusion, gene expression in the respiratory system is highly specialized and critical for its proper functioning. Genes expressed in the lungs, nasal cavity, trachea, and bronchi all contribute to the intricate processes involved in respiration and maintaining respiratory health.
Gene Expression in the Endocrine System
The endocrine system is a network of glands that produce and release hormones into the bloodstream to regulate various bodily functions. Hormones, produced by specific cells within these glands, are responsible for a wide range of activities, including growth, metabolism, reproduction, and response to stress. In order for these hormones to be produced and regulated properly, specific genes must be expressed within the cells of the endocrine system.
Within the endocrine system, there are several hormone-producing glands, each responsible for producing and releasing specific hormones. These glands include the pituitary gland, thyroid gland, adrenal gland, pancreas, and gonads (testes and ovaries). Each gland contains specialized cells that express specific genes to produce the necessary hormones.
Gene Expression in Hormone Production
The genes that are expressed within the hormone-producing cells of the endocrine system play a crucial role in hormone production and regulation. These genes encode for proteins and enzymes that are involved in the synthesis, secretion, and transport of hormones. The expression of these genes is tightly regulated and can be influenced by various factors, such as hormonal signals, environmental cues, and genetic factors.
For example, in the thyroid gland, specific genes are expressed to produce thyroid hormones, such as thyroxine (T4) and triiodothyronine (T3). These hormones play a crucial role in regulating metabolism throughout the body. The expression of genes responsible for thyroid hormone synthesis is regulated by a feedback system involving the hypothalamus and pituitary gland.
In the adrenal gland, genes are expressed to produce hormones such as cortisol and adrenaline. These hormones are involved in the body’s response to stress and regulate various physiological processes related to stress, including blood pressure, immune function, and metabolism.
Regulation of Gene Expression
The expression of genes within the endocrine system is tightly regulated to ensure proper hormone production and regulation. This regulation can occur at various levels, including transcription, translation, and post-translational modification.
Transcriptional regulation involves the control of gene expression at the level of transcription, where the DNA sequence is converted into RNA. Transcription factors, proteins that bind to specific DNA sequences, can activate or inhibit the transcription of target genes. These transcription factors can be influenced by hormonal signals and other signaling pathways.
Post-transcriptional and post-translational modifications, such as mRNA processing and protein modifications, can also regulate gene expression within the endocrine system. These modifications can affect the stability and activity of the mRNA and protein products, ultimately impacting hormone production and function.
|Growth hormone (GH), adrenocorticotropic hormone (ACTH), thyroid-stimulating hormone (TSH)
|Thyroxine (T4), triiodothyronine (T3), calcitonin
|Testosterone, estrogen, progesterone
Gene Expression in the Musculoskeletal System
In the musculoskeletal system, a complex network of genes is expressed to regulate its function and development. These genes play a crucial role in determining the structure, composition, and function of the muscles, bones, and joints in our body.
Muscle Development and Function
Various genes are expressed during muscle development to ensure proper growth and function. MyoD, Myf5, and Pax3 are among the key genes involved in the formation of muscle tissue. They regulate the differentiation of specific cells into muscle fibers and play a crucial role in muscle regeneration and repair.
Additionally, genes such as ACTA1, MYH7, and MYBPC3 are responsible for encoding proteins involved in muscle contraction and force generation. These proteins are crucial for muscle function and enable us to move and perform physical activities.
Bone Formation and Remodeling
Genes such as RUNX2, COL1A1, and BMP2 are expressed during bone development and remodeling. RUNX2 is a transcription factor that plays a central role in bone formation by regulating the differentiation of osteoblasts, the cells responsible for bone synthesis. COL1A1 produces collagen, a major component of the bone matrix, while BMP2 is involved in the induction of bone formation and repair.
Moreover, genes like RANKL and OPG are responsible for maintaining the balance between bone resorption and formation. RANKL promotes bone resorption by activating osteoclasts, while OPG acts as a decoy receptor and inhibits RANKL, preventing excessive bone loss.
These genes collectively contribute to the proper formation, growth, and maintenance of bones, ensuring their strength and integrity.
In conclusion, the musculoskeletal system heavily relies on the expression of specific genes to ensure its proper development, function, and maintenance. Understanding gene expression patterns in this system can provide valuable insights into musculoskeletal disorders and potential therapeutic targets.
Gene Expression in the Reproductive System
The reproductive system is a crucial aspect of an organism’s life cycle, responsible for the production and maintenance of life. Gene expression plays a significant role in the development and function of the reproductive system, allowing for the proper function of reproductive organs and the production of gametes.
Genes expressed in the reproductive system are involved in various processes, such as the development of reproductive organs, regulation of hormonal signaling, and spermatogenesis or oogenesis. These genes are active in specific tissues and cell types within the reproductive system, ensuring the proper function of each component.
Male Reproductive System
In the male reproductive system, gene expression is essential for the development and function of the testes, epididymis, vas deferens, seminal vesicles, and prostate gland. Genes expressed in these tissues regulate the production of sperm, the maturation and storage of spermatozoa, and the secretion of seminal fluid.
For example, the SRY gene, located on the Y chromosome, is specifically expressed in developing testes and is crucial for initiating male sex determination. Other genes, such as those encoding androgen receptors and follicle-stimulating hormone receptors, are essential for the development and function of the male reproductive system.
Female Reproductive System
In the female reproductive system, gene expression plays a crucial role in the development and function of the ovaries, uterus, fallopian tubes, and vagina. Genes expressed in these tissues regulate the development and release of ova, the preparation of the uterine lining for implantation, and the hormone signaling involved in reproductive cycles.
Genes such as FOXL2 and WNT4 are involved in ovarian development and follicle maturation. Other genes, such as those encoding estrogen and progesterone receptors, are critical for the regulation of female reproductive hormone signaling.
Gene Expression Patterns
The expression patterns of genes in the reproductive system can vary depending on the stage of reproductive development and the specific cell type. For example, certain genes may be highly expressed in the testes during embryonic development but become downregulated in adulthood.
Additionally, gene expression patterns can differ between species, contributing to the diversity of reproductive strategies observed in nature. Understanding these gene expression patterns and their regulation allows researchers to gain insights into the molecular mechanisms underlying reproductive processes and the development of reproductive disorders.
|SRY, androgen receptors, follicle-stimulating hormone receptors
|FOXL2, WNT4, estrogen receptors, progesterone receptors
|Estrogen receptors, progesterone receptors
Gene Expression in the Urinary System
The urinary system plays a vital role in maintaining the body’s fluid balance and eliminates waste products from the blood. To carry out these functions, various genes are expressed in different parts of the urinary system.
In the kidneys, genes involved in filtration and reabsorption processes are highly expressed. One such gene is the Aquaporin gene, which codes for proteins that regulate water balance in the body. This gene is found in high levels in the cells of the kidney tubules, where water reabsorption takes place.
In the bladder, genes involved in the contraction and relaxation of muscles are expressed. These genes control the smooth muscle cells in the bladder, allowing it to stretch and contract for the storage and release of urine. One example is the Myosin gene, which codes for proteins that are essential for muscle contraction.
Additionally, genes involved in the production and secretion of hormones related to the urinary system are also expressed. The Renin gene, for example, is expressed in the juxtaglomerular cells of the kidneys. This gene codes for a hormone that regulates blood pressure and fluid balance by controlling the production of another hormone called aldosterone.
Understanding where genes are expressed in the urinary system is crucial for comprehending the functioning and regulation of this essential system in the human body. The coordinated expression of these genes ensures proper kidney function, fluid balance, and elimination of waste products.
Gene Expression in the Integumentary System
The integumentary system is composed of the skin, hair, nails, and glands, and plays a crucial role in protecting the body from external environmental factors. Understanding where genes are expressed in the integumentary system can provide insights into the function and development of these tissues.
Gene Expression in the Skin
The skin is the largest organ of the integumentary system and is responsible for protecting the body from dehydration, temperature fluctuations, and pathogens. Genes involved in the development and maintenance of the skin are predominantly expressed in the epidermis, dermis, and appendages such as hair follicles and sweat glands. For example, genes encoding structural proteins like keratin and collagen are highly expressed in the epidermis, providing strength and flexibility to the skin barrier.
Gene Expression in Hair and Nails
Hair and nails are specialized structures of the integumentary system that serve various functions, including protection and regulation of body temperature. Genes responsible for hair and nail development are predominantly expressed in specialized cells known as hair follicle cells and nail matrix cells. These genes control the growth, pigmentation, and differentiation of these structures, ensuring their proper formation and function.
A variety of genes involved in the production of hair and nail proteins, such as keratins and filaggrin, are expressed in the hair follicles and nail matrix cells, respectively. Additionally, genes involved in the regulation of hair growth and cycle, like the WNT signaling pathway genes, are expressed in hair follicles, enabling the continuous growth and replacement of hair.
Gene Expression in Glands
The integumentary system also includes various glands, such as sweat glands and sebaceous glands, which are responsible for producing and secreting substances that help maintain the health and integrity of the skin. Genes involved in the development and function of these glands are predominantly expressed in the respective glandular cells. For example, genes encoding proteins involved in the production and secretion of sweat are highly expressed in sweat gland cells.
Overall, understanding where genes are expressed in the integumentary system provides valuable insights into the molecular mechanisms underlying the development and function of the skin, hair, nails, and glands. Further research in this area can help unravel the complexities of these tissues and lead to the development of targeted therapies for various diseases and disorders related to the integumentary system.
Gene Expression in the Lymphatic System
The lymphatic system is a crucial component of the body’s immune system, playing a vital role in defending against infections and diseases. Genes in the lymphatic system are expressed in specific locations, where they perform essential functions to ensure the proper functioning of this network.
One of the key areas where genes are expressed in the lymphatic system is the lymph nodes. Lymph nodes are small, bean-shaped structures that are distributed throughout the body and act as filtration centers for lymph, the fluid that carries immune cells. Within the lymph nodes, genes are expressed in various cell types, including lymphocytes, which are the main cellular components of the immune system.
Another important location where gene expression in the lymphatic system occurs is in specialized lymphatic vessels called lymphatic endothelial cells (LECs). LECs line the inner surface of the lymphatic vessels and are involved in the transport of lymph and immune cells. Genes expressed in LECs play a crucial role in maintaining the integrity and function of the lymphatic network.
Additionally, genes are also expressed in other tissues associated with the lymphatic system, such as the spleen and thymus. These tissues have specific functions in immune response and development, respectively. Genes expressed in these tissues are essential for their proper functioning and contribute to overall immune system health.
In summary, genes in the lymphatic system are expressed in various locations, including lymph nodes, lymphatic endothelial cells, and other associated tissues. Understanding where genes are expressed in the lymphatic system provides important insights into the regulation of the immune response and the maintenance of overall health.
Current Research on Gene Expression
Research on gene expression is constantly evolving, providing valuable insights into the intricate mechanisms of how genes are expressed and where they are active within the body.
Scientists have made significant advancements in understanding the factors that influence gene expression. They have identified various regulatory elements, such as promoters and enhancers, that play crucial roles in determining when and where genes are expressed. Through advanced techniques like RNA sequencing, researchers are able to identify and quantify the transcripts produced by individual genes, giving them a more detailed understanding of gene expression patterns.
Recent studies have also focused on understanding the impact of gene expression on different diseases and conditions. By comparing gene expression profiles between healthy and diseased tissues, scientists can identify genes that are specifically upregulated or downregulated in certain diseases. This knowledge can help in developing targeted therapies and diagnostic tools.
Furthermore, researchers are exploring the role of non-coding RNAs in gene expression regulation. These non-coding RNAs have been found to interact with both coding RNAs and DNA sequences, influencing gene expression at various levels. Understanding the complex interactions between different molecules involved in gene expression is a thriving area of research.
In summary, current research on gene expression continues to uncover the vast complexity of this process. Scientists are constantly refining their knowledge of how genes are expressed and where they are active in the body. This research holds great promise in advancing our understanding of diseases and developing new therapeutic strategies.
Future Directions in Gene Expression Research
As researchers continue to explore the fascinating world of gene expression, there are several exciting directions that hold promise for further understanding how genes are expressed and where they are expressed in the body.
- Single-cell gene expression analysis: Current techniques for analyzing gene expression provide an average measurement across a population of cells, but advances in single-cell sequencing technologies are allowing researchers to examine gene expression patterns at the individual cell level. This approach will provide valuable insights into the heterogeneity of gene expression within tissues and organs.
- Temporal and spatial gene expression mapping: Mapping the precise spatiotemporal patterns of gene expression in various tissues and organs is crucial for understanding how genes contribute to development, disease, and normal physiology. Advances in imaging technologies and computational methods will enable researchers to create detailed maps of gene expression throughout the body.
- Investigating the impact of non-coding RNAs: Non-coding RNAs have been found to play important roles in regulating gene expression, but much remains to be discovered about their specific functions and mechanisms of action. Future research will focus on understanding the roles of non-coding RNAs in various cellular processes and their implications for health and disease.
- Integrating multi-omic data: Gene expression is just one layer of the complex regulatory networks within cells. Integrating gene expression data with other omics data, such as epigenetics, proteomics, and metabolomics, will provide a more comprehensive understanding of how genes are expressed and regulated.
- Exploring gene expression dynamics: Gene expression is a dynamic process that can change in response to various stimuli and environmental factors. Future research will aim to unravel the complex dynamics of gene expression and identify the factors that influence gene expression patterns.
These future directions in gene expression research will shed light on the intricacies of gene regulation and provide important insights into human health and disease. By understanding where and how genes are expressed, we can unlock new therapeutic strategies and improve personalized medicine.
Why is understanding gene expression important?
Understanding gene expression is important because it helps us understand how genes function in different tissues and organs of the body. It provides insights into the development, growth, and maintenance of cells, and can help us understand diseases and develop better treatments.
What is gene expression?
Gene expression is the process by which information from a gene is used to create a functional gene product, such as a protein. It involves the conversion of the genetic information stored in DNA into various RNA molecules and ultimately proteins, which carry out specific functions in the body.
How do scientists study gene expression?
Scientists use various techniques to study gene expression. They can analyze gene expression patterns by measuring the levels of RNA molecules in different tissues or cells using techniques like RNA sequencing or microarrays. They can also visualize the location of gene expression within tissues using techniques like in situ hybridization or immunohistochemistry.
What is the significance of tissue-specific gene expression?
Tissue-specific gene expression plays a crucial role in the development and function of different tissues and organs in the body. It allows for the specialization of cells and ensures that they carry out their specific functions. Understanding tissue-specific gene expression can help us understand how different tissues are formed and maintained, and how they can be affected in diseases.
What are some factors that influence gene expression?
Gene expression is influenced by various factors, including genetic factors, environmental factors, and cellular signals. Genetic factors include mutations or variations in the DNA sequence that can affect gene expression. Environmental factors such as diet, stress, or exposure to toxins can also impact gene expression. Additionally, signaling molecules within cells can activate or repress specific genes.
What is gene expression?
Gene expression is the process by which information from a gene is used in the synthesis of a functional gene product, such as a protein or RNA molecule.
Why is understanding gene expression important?
Understanding gene expression is important because it helps us to understand how genes function in different cells and tissues, and how changes in gene expression can contribute to the development of diseases.
What are the techniques used to study gene expression?
There are several techniques used to study gene expression, including DNA microarrays, RNA sequencing, and quantitative polymerase chain reaction (qPCR).
Where are genes expressed in the body?
Genes can be expressed in different tissues and organs throughout the body. Some genes have a ubiquitous expression pattern, meaning they are expressed in nearly all tissues, while others have a more restricted expression pattern and are only expressed in specific cell types or tissues.
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Chapter 13 Special Relativity
- Describe proper length.
- Calculate length contraction.
- Explain why we don’t notice these effects at everyday scales.
Have you ever driven on a road that seems like it goes on forever? If you look ahead, you might say you have about 10 km left to go. Another traveler might say the road ahead looks like it’s about 15 km long. If you both measured the road, however, you would agree. Traveling at everyday speeds, the distance you both measure would be the same. You will read in this section, however, that this is not true at relativistic speeds. Close to the speed of light, distances measured are not the same when measured by different observers.
One thing all observers agree upon is relative speed. Even though clocks measure different elapsed times for the same process, they still agree that relative speed, which is distance divided by elapsed time, is the same. This implies that distance, too, depends on the observer’s relative motion. If two observers see different times, then they must also see different distances for relative speed to be the same to each of them.
The muon discussed in Chapter 28.2 Example 1 illustrates this concept. To an observer on the Earth, the muon travels at for from the time it is produced until it decays. Thus it travels a distance
relative to the Earth. In the muon’s frame of reference, its lifetime is only 2.20μs2.20μs. It has enough time to travel only
The distance between the same two events (production and decay of a muon) depends on who measures it and how they are moving relative to it.
Proper length is the distance between two points measured by an observer who is at rest relative to both of the points.
The Earth-bound observer measures the proper length , because the points at which the muon is produced and decays are stationary relative to the Earth. To the muon, the Earth, air, and clouds are moving, and so the distance it sees is not the proper length.
To develop an equation relating distances measured by different observers, we note that the velocity relative to the Earth-bound observer in our muon example is given by
The time relative to the Earth-bound observer is Δt, since the object being timed is moving relative to this observer. The velocity relative to the moving observer is given by
The moving observer travels with the muon and therefore observes the proper time . The two velocities are identical; thus,
We know that . Substituting this equation into the relationship above gives
Substituting for γ gives an equation relating the distances measured by different observers.
Length contraction is the shortening of the measured length of an object moving relative to the observer’s frame.
If we measure the length of anything moving relative to our frame, we find its length to be smaller than the proper length that would be measured if the object were stationary. For example, in the muon’s reference frame, the distance between the points where it was produced and where it decayed is shorter. Those points are fixed relative to the Earth but moving relative to the muon. Clouds and other objects are also contracted along the direction of motion in the muon’s reference frame.
Example 1: Calculating Length Contraction: The Distance between Stars Contracts when You Travel at High Velocity
Suppose an astronaut, such as the twin discussed in Chapter 28.2 Simultaneity and Time Dilation, travels so fast that . (a) She travels from the Earth to the nearest star system, Alpha Centauri, 4.300 light years (ly) away as measured by an Earth-bound observer. How far apart are the Earth and Alpha Centauri as measured by the astronaut? (b) In terms of , what is her velocity relative to the Earth? You may neglect the motion of the Earth relative to the Sun. (See Figure 3.)
First note that a light year (ly) is a convenient unit of distance on an astronomical scale—it is the distance light travels in a year. For part (a), note that the 4.300 ly distance between the Alpha Centauri and the Earth is the proper distance , because it is measured by an Earth-bound observer to whom both stars are (approximately) stationary. To the astronaut, the Earth and the Alpha Centauri are moving by at the same velocity, and so the distance between them is the contracted length . In part (b), we are given , and so we can find by rearranging the definition of to express in terms of .
Solution for (a)
- Identify the knowns. ;
- Identify the unknown.
- Choose the appropriate equation.
- Rearrange the equation to solve for the unknown.
Solution for (b)
- Identify the known. γ = 30.00
- Identify the unknown. v in terms of c
- Choose the appropriate equation.
- Rearrange the equation to solve for the unkno wn.
Squaring both sides of the equation and rearranging terms gives
Taking the square root, we findv/c = 0.99944
which is rearranged to produce a value for the velocity v = 0.9994c.
First, remember that you should not round off calculations until the final result is obtained, or you could get erroneous results. This is especially true for special relativity calculations, where the differences might only be revealed after several decimal places. The relativistic effect is large here (), and we see that is approaching (not equaling) the speed of light. Since the distance as measured by the astronaut is so much smaller, the astronaut can travel it in much less time in her frame.
People could be sent very large distances (thousands or even millions of light years) and age only a few years on the way if they traveled at extremely high velocities. But, like emigrants of centuries past, they would leave the Earth they know forever. Even if they returned, thousands to millions of years would have passed on the Earth, obliterating most of what now exists. There is also a more serious practical obstacle to traveling at such velocities; immensely greater energies than classical physics predicts would be needed to achieve such high velocities. This will be discussed in Chapter 28.6 Relativistic Energy.
Why don’t we notice length contraction in everyday life? The distance to the grocery shop does not seem to depend on whether we are moving or not. Examining the equation , we see that at low velocities () the lengths are nearly equal, the classical expectation. But length contraction is real, if not commonly experienced. For example, a charged particle, like an electron, traveling at relativistic velocity has electric field lines that are compressed along the direction of motion as seen by a stationary observer. (See Figure 4.) As the electron passes a detector, such as a coil of wire, its field interacts much more briefly, an effect observed at particle accelerators such as the 3 km long Stanford Linear Accelerator (SLAC). In fact, to an electron traveling down the beam pipe at SLAC, the accelerator and the Earth are all moving by and are length contracted. The relativistic effect is so great than the accelerator is only 0.5 m long to the electron. It is actually easier to get the electron beam down the pipe, since the beam does not have to be as precisely aimed to get down a short pipe as it would down one 3 km long. This, again, is an experimental verification of the Special Theory of Relativity.
Check Your Understanding
1: A particle is traveling through the Earth’s atmosphere at a speed of . To an Earth-bound observer, the distance it travels is 2.50 km. How far does the particle travel in the particle’s frame of reference?
- All observers agree upon relative speed.
- Distance depends on an observer’s motion. Proper length is the distance between two points measured by an observer who is at rest relative to both of the points. Earth-bound observers measure proper length when measuring the distance between two points that are stationary relative to the Earth.
- Length contraction is the shortening of the measured length of an object moving relative to the observer’s frame:
1: To whom does an object seem greater in length, an observer moving with the object or an observer moving relative to the object? Which observer measures the object’s proper length?
2: Relativistic effects such as time dilation and length contraction are present for cars and airplanes. Why do these effects seem strange to us?
3: Suppose an astronaut is moving relative to the Earth at a significant fraction of the speed of light. (a) Does he observe the rate of his clocks to have slowed? (b) What change in the rate of Earth-bound clocks does he see? (c) Does his ship seem to him to shorten? (d) What about the distance between stars that lie on lines parallel to his motion? (e) Do he and an Earth-bound observer agree on his velocity relative to the Earth?
Problems & Exercises
1: A spaceship, 200 m long as seen on board, moves by the Earth at . What is its length as measured by an Earth-bound observer?
2: How fast would a 6.0 m-long sports car have to be going past you in order for it to appear only 5.5 m long?
3: (a) How far does the muon in Chapter 28.2 Example 1 travel according to the Earth-bound observer? (b) How far does it travel as viewed by an observer moving with it? Base your calculation on its velocity relative to the Earth and the time it lives (proper time). (c) Verify that these two distances are related through length contraction .
4: (a) How long would the muon in Chapter 28.2 Example 1 have lived as observed on the Earth if its velocity was ? (b) How far would it have traveled as observed on the Earth? (c) What distance is this in the muon’s frame?
5: (a) How long does it take the astronaut in Example 1 to travel 4.30 ly at (as measured by the Earth-bound observer)? (b) How long does it take according to the astronaut? (c) Verify that these two times are related through time dilation with as given.
6: (a) How fast would an athlete need to be running for a 100 metre race to look 100 yards long? Remember that 1 yard is exactly 3 feet or 36 inches (b) Is the answer consistent with the fact that relativistic effects are difficult to observe in ordinary circumstances? Explain.
7: Unreasonable Results
(a) Find the value of for the following situation. An astronaut measures the length of her spaceship to be 25.0 m, while an Earth-bound observer measures it to be 100 m. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
8: Unreasonable Results
A spaceship is heading directly toward the Earth at a velocity of 0.800c. The astronaut on board claims that he can send a canister toward the Earth at 1.20c relative to the Earth. (a) Calculate the velocity the canister must have relative to the spaceship. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
- proper length
- ; the distance between two points measured by an observer who is at rest relative to both of the points; Earth-bound observers measure proper length when measuring the distance between two points that are stationary relative to the Earth
- length contraction
- , the shortening of the measured length of an object moving relative to the observer’s frame:
Check Your Understanding
Problems & Exercises
1: 48.6 m
3: (a) 1.387 km = 1.39 km (b) 0.433 km
Thus, the distances in parts (a) and (b) are related when .
5: (a) 4.303 y (to four digits to show any effect) (b) 0.1434 y
Thus, the two times are related when γ = 30.00.
6: 100 yards is exactly 91.44 metres. The best sprinters in the world can run at about 10 metres per second, but for 100 metres to appear to be 91.44 metres they would have to run at 1.21 x 10 8 m/s. Not reasonable.
7: (a) 0.250 (b) γ must be ≥1 (c) The Earth-bound observer must measure a shorter length, so it is unreasonable to assume a longer length.
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Understanding the Unit Circle Quadrants: A Comprehensive Guide
Author: Noreen Niazi
Last Updated on: September 18, 2023
A useful tool in trigonometry that enables us to comprehend the connection between angles and coordinates on a circle is the unit circle. It is a circle with a radius of 1 that is centered at the coordinate plane’s starting point.
We may quickly calculate the signs of trigonometric functions like sine, cosine, and tangent for every given angle by splitting the unit circle into four quadrants. This in-depth manual covers unit circle quadrants, their coordinates, trigonometric functions, typical errors to avoid, applications in trigonometry issues, practise problems, and advice for mastering these quadrants.
What are unit circle quadrants?
The unit circle is divided into four pieces, which are called quadrants. Beginning at the positive x-axis, these quadrants are numbered anticlockwise. A different set of signs represents each quadrant’s x and y coordinates.
Knowing the unit circle quadrants determines the trigonometric function’s positive or negative values for various angles. Calculations can be made simpler, and trigonometry difficulties can be resolved more quickly by becoming familiar with each quadrant’s properties.
The four quadrants of the unit circle
Where both the x and y values are positive, the first quadrant is located in the upper right corner of the unit circle. Its range is 0 to /2 radians, or 0 to 90 degrees. The sine function, along with the cosine and tangent functions, are all positive in the first quadrant. The cosine and sine of the angle are denoted by the coordinates (x, y), which are the coordinates for the first quadrant.
The upper-left corner of the unit circle, where the x coordinate is negative and the y coordinate is positive, is where the second quadrant is situated. This quadrant ranges from /2 to radians, or 90 to 180 degrees. In the second quadrant, the sine function is positive and the cosine and tangent functions are negative. The second quadrant’s coordinates can be written as (-x, y), where x represents the angle’s sine and y its cosine.
The lower-left corner of the unit circle contains the third quadrant, where both the x and y coordinates are negative. The range of this quadrant is 180–270 degrees, or –3/2 radians. The cosine, tangent, and sine functions are all negative in the third quadrant along with the sine function. (-x, -y), where x is the angle’s cosine and y is its sine, can be used to denote the coordinates in the third quadrant.
In the lower-right corner of the unit circle, where the x coordinate is positive and the y coordinate is negative, is where the fourth quadrant is situated. This quadrant covers the range of 270 to 360 degrees, or 3/2 to 2 radians. The cosine and tangent functions are positive in the fourth quadrant, but the sine function is negative. (x, -y), where x is the angle’s cosine and y is its sine, can be used to denote the coordinates in the fourth quadrant.
Understanding the coordinates in each quadrant
We need to consider how angles relate to the unit circle, the trigonometric functions, and the coordinates in each quadrant. The sine, cosine, and tangent functions are positive in the first quadrant because both the x and y coordinates are positive. The x coordinate turns negative while the y coordinate stays positive as we turn anticlockwise to the second quadrant. The values of the trigonometric functions are impacted by this change in sign. Similarly, the x and y coordinates turn negative in the third and fourth quadrants, changing the signs of the trigonometric functions.
Trigonometric functions in each quadrant
The mnemonic “All Students Take Calculus” can be used to remember the signs of the trigonometric functions in each quadrant. This mnemonic represents:
All: All trigonometric functions are positive in the first quadrant.
Students: Sine is positive in the second quadrant.
Take: Tangent is positive in the third quadrant.
Calculus: Cosine is positive in the fourth quadrant.
When working with unit circle quadrants, you may rapidly discover the signs of the trigonometric functions in each quadrant by memorising this mnemonic. This will save you time and help you avoid making mistakes.
Common mistakes to avoid when working with unit circle quadrants
Although comprehending unit circle quadrants is crucial, working with them can be tricky. Here are some typical errors to avoid:
Forgetting the signs of trigonometric functions
The trigonometric functions in each quadrant’s signs are frequently forgotten, which is a common error. To get accurate results, it’s imperative to keep in mind both the positive and negative sine, cosine, and tangent values in each quadrant.
Confusing coordinates in different quadrants
Another error is mixing up the coordinates for the various quadrants. The x and y coordinates in each quadrant are a special mix of positive and negative values. Incorrect answers can result from combining the coordinates.
Failing to recognize reference angles
Reference angles are created between an angle’s terminal side and the x-axis. It might be difficult to determine the proper quadrant and signs of trigonometric functions if reference angles are not understood.
Applications of unit circle quadrants in trigonometry problems
There are numerous ways to use unit circle quadrants in trigonometry issues. We may calculate the positive or negative values of trigonometric functions for any given angle by studying the signs of these functions in various quadrants. In situations when there are unknown angles and values, this understanding enables us to solve equations, locate those unknown angles, and calculate those values.
Practice exercises for mastering unit circle quadrants
Working through issues involving angles and trigonometric functions is crucial to mastering unit circle quadrants. Here are some exercises to try:
In each quadrant, calculate the sine, cosine, and tangent of an angle of 60 degrees.
With a cosine value of -0.5, find the angle in the third quadrant.
For a 135-degree angle in the second quadrant, get the sine value.
Your comprehension and ability to deal with unit circle quadrants can be enhanced by performing these exercises and other issues of a similar nature.
Tips for memorizing the unit circle quadrants
Memorizing the unit circle quadrants can be challenging, but with the right techniques, it becomes easier. Here are some tips to help you memorize the unit circle quadrants:
Break it down: Divide the knowledge into smaller, more manageable parts rather than trying to memorize it all at once. Before going on to the next quadrant, concentrate on comprehending and memorizing the one before you.
Visualise: Envision yourself traveling around each quadrant of the unit circle as you mentally visualize it. To help you remember, connect each quadrant to its distinctive qualities.
Create flashcards: Make flashcards that include the quadrant numbers, trigonometric function signs, and important coordinates. To improve your memory, periodically review these flashcards.
Routine practice Memorization requires regular practice. To improve your comprehension of unit circle quadrants, solve puzzles, do activities, and apply your learning to actual situations.
Conclusion: The importance of understanding unit circle quadrants in trigonometry
A fundamental idea in trigonometry is the concept of unit circle quadrants. You can improve your ability to solve problems and accuracy in trigonometry by becoming familiar with the coordinates, signs of trigonometric functions, and applications of each quadrant.
You can improve your comprehension of unit circle quadrants by avoiding common mistakes, working through exercises, and using memorising techniques. With these abilities, you’ll be better able to solve difficult trigonometry issues and succeed in your academic and professional pursuits.Now that you have a thorough understanding of unit circle quadrants, apply your knowledge by working through trigonometry problems and investigating practical uses. Your mathematics abilities will be substantially enhanced and you’ll have more opportunities in areas that use trigonometric concepts if you can master unit circle quadrants. Begin your path to trigonometry mastery right now!
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When an object is at rest, it is in a state of equilibrium with no net force acting on it. This means the forces acting on the object are balanced. However, when an unbalanced or net external force acts on an object at rest, it disrupts this equilibrium and causes the object to accelerate in the direction of the net force. This acceleration results in a change in the object’s velocity and allows it to overcome inertia and begin moving.
What is a force?
A force is a push or pull that acts on an object and changes its state of motion. Forces can vary greatly in magnitude and direction. According to Newton’s first law of motion, an object at rest will stay at rest and an object in motion will stay in motion unless acted upon by an unbalanced force.
Some examples of forces include:
– Friction: A force that resists the relative motion between two surfaces in contact. Acts to slow down moving objects.
– Tension: Force that acts through a rope, string, cable or wire when it is pulled tight by forces acting from each end.
– Normal force: Force exerted by a surface on an object to prevent it from sinking into the surface. Acts perpendicular to the surface.
– Gravity: Downward force exerted by earth’s mass that pulls objects toward it. Gravity gives weight to objects.
– Air resistance: Frictional force exerted by air pushing against a moving object. Tends to slow down objects moving through air.
– Applied force: External force applied to an object by direct contact, such as a person pushing or pulling it.
Requirements for an unbalanced force
In order for an unbalanced or net force to act on an object at rest, two key requirements must be met:
Requirement 1: The force must be external
The net force must come from outside the object rather than internal forces. When all the internal forces within an object are balanced, the object maintains its state of rest or constant velocity motion.
An external force is needed to disrupt this equilibrium. Some examples of external forces include gravity, magnetism, applied force from contact, air resistance, tension or friction from another object. Internal forces such as the structural forces holding an object together do not cause acceleration.
Requirement 2: The force must be non-zero
If the external force equals zero, there is no net force on the object. A non-zero net external force is required to accelerate a stationary object. This means the vector sum of all external forces acting on the object cannot equal zero.
There must be an unbalance between the various forces acting horizontally and vertically on the object in order for any acceleration to take place. Even a small net external force will cause the object to accelerate.
Effects of Unbalanced Force on an Object at Rest
When a non-zero net external force acts on an object at rest, two main effects are observed:
Effect 1: The object accelerates
Newton’s second law of motion states that the net external force on an object is equal to its mass multiplied by its acceleration.
Fnet = ma
Fnet = net external force (N)
m = mass of object (kg)
a = acceleration (m/s2)
This means an unbalanced external force causes an object at rest to accelerate in the direction of the net force. The greater the net force, the greater the magnitude of acceleration produced.
Effect 2: Work is done on the object
Work is done on an object when an applied force moves it through a displacement. Since the unbalanced force causes the stationary object to accelerate and move, it is doing work on the object.
Work = Force x Displacement
W = Fd
W = Work done by force (J)
F = Force applied (N)
d = Displacement of object (m)
The work done on the object by the unbalanced force results in a transfer of energy to the object. This energy goes into overcoming its inertia, causing it to accelerate from rest.
Examples of Unbalanced Forces Causing Motion
Pushing a box at rest
When a person exerts a push on a stationary box with their hands, this applied force acts horizontally on the box. As there are no other significant horizontal forces, the applied force is unbalanced. This results in the box accelerating in the direction it was pushed.
When an object is held at rest above the ground and then released, the gravitational force acting downwards is unbalanced. This results in a downwards acceleration and the object starts falling vertically towards the ground.
Magnet moving a paperclip
When a magnet is brought close to a stationary paperclip, the magnetic force exerted on the paperclip is unbalanced. This causes the paperclip to accelerate towards the magnet. The paperclip moves with increasing velocity until is sticks to the magnet.
Static and kinetic friction
When a force is applied to an object at rest on a flat surface, static friction comes into play. As long as static friction is equal and opposite to the applied force, the object remains stationary. However, when the applied force exceeds the maximum possible static friction force, the object accelerates due to the unbalanced force. This unbalanced force then becomes the kinetic friction as the object slides across the surface.
Analysis Methods for Unbalanced Force Problems
To analyze scenarios involving unbalanced forces on objects at rest, some useful methods include:
Free body diagrams
Drawing a free body diagram isolates the object and shows all forces acting on it. This allows identification of any unbalanced force.
Applying Newton’s laws of motion
Newton’s first and second laws can determine whether a net external force exists, and predict the motion that will result.
Determining the net force quantitatively
All the forces are resolved into vertical and horizontal components using trigonometry. The components along each axis are summed to find the net force.
Using force tables
Tabulating all the forces with their direction, magnitude and components clarifies which direction has an unbalanced force.
Applying friction equations
The maximum static friction force can be calculated and compared to applied force to determine if motion will occur.
How to Calculate the Acceleration from an Unbalanced Force
The acceleration of an object at rest produced by an unbalanced force can be calculated using:
Step 1) Draw a free body diagram
Isolate the object and draw all external forces acting on it. Assign appropriate symbols.
Step 2) Apply Newton’s 2nd law
Fnet = ma
Where Fnet is the vector sum of all external forces.
Step 3) Calculate net force
Add vector components of all forces acting along the movement direction.
Or find the magnitude and direction of the resulting net force vector.
Step 4) Determine mass of object
Obtain mass m in kg.
Step 5) Calculate acceleration a
Divide net force by mass:
a = Fnet/m
Acceleration direction is same as net force direction.
Step 6) Plug in values and solve
Substitute the known values and calculate the acceleration of the object in m/s2.
For the object above with mass 2 kg:
Fnet = Fapp – Ffriction
= 18 N – 6 N
= 12 N
a = Fnet/m = 12 N/2 kg = 6 m/s2
Therefore, with an unbalanced force of 12 N, the object’s acceleration is 6 m/s2.
Effects of Varying Force Magnitude
According to Newton’s second law, doubling the net force on an object at rest doubles its acceleration. Some general effects of increasing unbalanced force magnitude:
– Greater net force causes greater acceleration
– Heavier objects accelerate slower from the same force
– Net force directly proportional to mass x acceleration
– For a given mass, larger force decreases time to reach a velocity
– Force is a vector so direction affects motion produced
This relationship allows predicting motion based on the force magnitude and direction.
Table showing effect of different net forces on a 5 kg object initially at rest:
|Net Force (N)
|Time to reach 10 m/s (s)
Factors Opposing the Unbalanced Force
While an unbalanced force causes an object at rest to accelerate, there are some factors that hinder this motion by opposing the external net force:
Frictional forces between an object and the surface it rests on creates resistance to the applied force. Static friction initially holds the object in place until exceeding the maximum static friction force. Kinetic friction then slows down its motion.
The frictional drag force exerted by air pushes against objects moving through the air. Air resistance tends to decrease acceleration from an applied force.
Forces transmitted through cables, ropes and strings act in the opposite direction of an applied force. The tension provides resistance to motion.
This resistance to change in motion makes it harder to accelerate stationary objects. More force is required to overcome inertia and accelerate larger mass objects.
If the applied force deforms an object, some energy is absorbed in deforming rather than accelerating it. This decreases the acceleration produced.
By accounting for these factors impeding the unbalanced force, more accurate motion prediction is possible. Minimizing these opposing forces allows the object to accelerate faster.
Understanding the effect of unbalanced forces has useful applications in:
Calculating friction and air resistance allows vehicle design for faster acceleration. Aerodynamic design minimizes drag.
The motion of athletes and sports projectiles considers unbalanced forces and oppositional factors. This improves performance.
Engineers analyze unbalanced force effects when designing moving structures and mechanisms to improve functionality.
Measuring the acceleration of objects from known unbalanced forces allows determination of their mass and verification of physics theories.
Amusement park rides
The motions of rollercoasters involve unbalanced forces. Ride designs apply physics concepts for safety and thrill factor.
An unbalanced external force applied to an object at rest overcomes inertia and produces acceleration in the direction of the net force. This disrupts the object’s equilibrium and imparts motion. Greater net force causes greater magnitude of acceleration, with various factors like friction reducing the realized motion. Understanding the precise effects of unbalanced forces allows predicting and controlling an object’s motion. This has important use across science, sports and engineering.
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An artificial neural network (ANN) is a machine learning model inspired by the structure and function of the human brain's interconnected network of neurons. It consists of interconnected nodes called artificial neurons, organized into layers. Information flows through the network, with each neuron processing input signals and producing an output signal that influences other neurons in the network.
A multi-layer perceptron (MLP) is a type of artificial neural network consisting of multiple layers of neurons. The neurons in the MLP typically use nonlinear activation functions, allowing the network to learn complex patterns in data. MLPs are significant in machine learning because they can learn nonlinear relationships in data, making them powerful models for tasks such as classification, regression, and pattern recognition. In this tutorial, we shall dive deeper into the basics of MLP and understand its inner workings.
Basics of Neural Networks
Neural networks or artificial neural networks are fundamental tools in machine learning, powering many state-of-the-art algorithms and applications across various domains, including computer vision, natural language processing, robotics, and more.
A neural network consists of interconnected nodes, called neurons, organized into layers. Each neuron receives input signals, performs a computation on them using an activation function, and produces an output signal that may be passed to other neurons in the network. An activation function determines the output of a neuron given its input. These functions introduce nonlinearity into the network, enabling it to learn complex patterns in data.
The network is typically organized into layers, starting with the input layer, where data is introduced. Followed by hidden layers where computations are performed and finally, the output layer where predictions or decisions are made.
Neurons in adjacent layers are connected by weighted connections, which transmit signals from one layer to the next. The strength of these connections, represented by weights, determines how much influence one neuron's output has on another neuron's input. During the training process, the network learns to adjust its weights based on examples provided in a training dataset. Additionally, each neuron typically has an associated bias, which allows the neuron to adjust its output threshold.
Neural networks are trained using techniques called feedforward propagation and backpropagation. During feedforward propagation, input data is passed through the network layer by layer, with each layer performing a computation based on the inputs it receives and passing the result to the next layer.
Backpropagation is an algorithm used to train neural networks by iteratively adjusting the network's weights and biases in order to minimize the loss function. A loss function (also known as a cost function or objective function) is a measure of how well the model's predictions match the true target values in the training data. The loss function quantifies the difference between the predicted output of the model and the actual output, providing a signal that guides the optimization process during training.
The goal of training a neural network is to minimize this loss function by adjusting the weights and biases. The adjustments are guided by an optimization algorithm, such as gradient descent. We shall revisit some of these topics in more detail later on in this tutorial.
Types of Neural Network
Picture credit: Keras Tutorial: Deep Learning in Python
The ANN depicted on the right of the image is a simple neural network called ‘perceptron’. It consists of a single layer, which is the input layer, with multiple neurons with their own weights; there are no hidden layers. The perceptron algorithm learns the weights for the input signals in order to draw a linear decision boundary.
However, to solve more complicated, non-linear problems related to image processing, computer vision, and natural language processing tasks, we work with deep neural networks.
Check out Datacamp’s Introduction to Deep Neural Networks tutorial to learn more about deep neural networks and how to construct one from scratch utilizing TensorFlow and Keras in Python. If you would prefer to use R language instead, Datacamp’s Building Neural Network (NN) Models in R has you covered.
There are several types of ANN, each designed for specific tasks and architectural requirements. Let's briefly discuss some of the most common types before diving deeper into MLPs next.
Feedforward Neural Networks (FNN)
These are the simplest form of ANNs, where information flows in one direction, from input to output. There are no cycles or loops in the network architecture. Multilayer perceptrons (MLP) are a type of feedforward neural network.
Recurrent Neural Networks (RNN)
In RNNs, connections between nodes form directed cycles, allowing information to persist over time. This makes them suitable for tasks involving sequential data, such as time series prediction, natural language processing, and speech recognition.
Convolutional Neural Networks (CNN)
CNNs are designed to effectively process grid-like data, such as images. They consist of layers of convolutional filters that learn hierarchical representations of features within the input data. CNNs are widely used in tasks like image classification, object detection, and image segmentation.
Long Short-Term Memory Networks (LSTM) and Gated Recurrent Units (GRU)
These are specialized types of recurrent neural networks designed to address the vanishing gradient problem in traditional RNN. LSTMs and GRUs incorporate gated mechanisms to better capture long-range dependencies in sequential data, making them particularly effective for tasks like speech recognition, machine translation, and sentiment analysis.
It is designed for unsupervised learning and consists of an encoder network that compresses the input data into a lower-dimensional latent space, and a decoder network that reconstructs the original input from the latent representation. Autoencoders are often used for dimensionality reduction, data denoising, and generative modeling.
Generative Adversarial Networks (GAN)
GANs consist of two neural networks, a generator and a discriminator, trained simultaneously in a competitive setting. The generator learns to generate synthetic data samples that are indistinguishable from real data, while the discriminator learns to distinguish between real and fake samples. GANs have been widely used for generating realistic images, videos, and other types of data.
A multilayer perceptron is a type of feedforward neural network consisting of fully connected neurons with a nonlinear kind of activation function. It is widely used to distinguish data that is not linearly separable.
MLPs have been widely used in various fields, including image recognition, natural language processing, and speech recognition, among others. Their flexibility in architecture and ability to approximate any function under certain conditions make them a fundamental building block in deep learning and neural network research. Let's take a deeper dive into some of its key concepts.
The input layer consists of nodes or neurons that receive the initial input data. Each neuron represents a feature or dimension of the input data. The number of neurons in the input layer is determined by the dimensionality of the input data.
Between the input and output layers, there can be one or more layers of neurons. Each neuron in a hidden layer receives inputs from all neurons in the previous layer (either the input layer or another hidden layer) and produces an output that is passed to the next layer. The number of hidden layers and the number of neurons in each hidden layer are hyperparameters that need to be determined during the model design phase.
This layer consists of neurons that produce the final output of the network. The number of neurons in the output layer depends on the nature of the task. In binary classification, there may be either one or two neurons depending on the activation function and representing the probability of belonging to one class; while in multi-class classification tasks, there can be multiple neurons in the output layer.
Neurons in adjacent layers are fully connected to each other. Each connection has an associated weight, which determines the strength of the connection. These weights are learned during the training process.
In addition to the input and hidden neurons, each layer (except the input layer) usually includes a bias neuron that provides a constant input to the neurons in the next layer. The bias neuron has its own weight associated with each connection, which is also learned during training.
The bias neuron effectively shifts the activation function of the neurons in the subsequent layer, allowing the network to learn an offset or bias in the decision boundary. By adjusting the weights connected to the bias neuron, the MLP can learn to control the threshold for activation and better fit the training data.
Note: It is important to note that in the context of MLPs,
bias can refer to two related but distinct concepts: bias as a general term in machine learning and the bias neuron (defined above). In general machine learning, bias refers to the error introduced by approximating a real-world problem with a simplified model. Bias measures how well the model can capture the underlying patterns in the data. A high bias indicates that the model is too simplistic and may underfit the data, while a low bias suggests that the model is capturing the underlying patterns well.
Typically, each neuron in the hidden layers and the output layer applies an activation function to its weighted sum of inputs. Common activation functions include sigmoid, tanh, ReLU (Rectified Linear Unit), and softmax. These functions introduce nonlinearity into the network, allowing it to learn complex patterns in the data.
Training with Backpropagation
MLPs are trained using the backpropagation algorithm, which computes gradients of a loss function with respect to the model's parameters and updates the parameters iteratively to minimize the loss.
Workings of a Multilayer Perceptron: Layer by Layer
Example of a MLP having two hidden layers
In a multilayer perceptron, neurons process information in a step-by-step manner, performing computations that involve weighted sums and nonlinear transformations. Let's walk layer by layer to see the magic that goes within.
- The input layer of an MLP receives input data, which could be features extracted from the input samples in a dataset. Each neuron in the input layer represents one feature.
- Neurons in the input layer do not perform any computations; they simply pass the input values to the neurons in the first hidden layer.
- The hidden layers of an MLP consist of interconnected neurons that perform computations on the input data.
- Each neuron in a hidden layer receives input from all neurons in the previous layer. The inputs are multiplied by corresponding weights, denoted as
w. The weights determine how much influence the input from one neuron has on the output of another.
- In addition to weights, each neuron in the hidden layer has an associated bias, denoted as
b. The bias provides an additional input to the neuron, allowing it to adjust its output threshold. Like weights, biases are learned during training.
- For each neuron in a hidden layer or the output layer, the weighted sum of its inputs is computed. This involves multiplying each input by its corresponding weight, summing up these products, and adding the bias:
n is the total number of input connections,
wi is the weight for the i-th input, and
xi is the i-th input value.
- The weighted sum is then passed through an activation function, denoted as
f. The activation function introduces nonlinearity into the network, allowing it to learn and represent complex relationships in the data. The activation function determines the output range of the neuron and its behavior in response to different input values. The choice of activation function depends on the nature of the task and the desired properties of the network.
- The output layer of an MLP produces the final predictions or outputs of the network. The number of neurons in the output layer depends on the task being performed (e.g., binary classification, multi-class classification, regression).
- Each neuron in the output layer receives input from the neurons in the last hidden layer and applies an activation function. This activation function is usually different from those used in the hidden layers and produces the final output value or prediction.
During the training process, the network learns to adjust the weights associated with each neuron's inputs to minimize the discrepancy between the predicted outputs and the true target values in the training data. By adjusting the weights and learning the appropriate activation functions, the network learns to approximate complex patterns and relationships in the data, enabling it to make accurate predictions on new, unseen samples.
This adjustment is guided by an optimization algorithm, such as stochastic gradient descent (SGD), which computes the gradients of a loss function with respect to the weights and updates the weights iteratively.
Let’s take a closer look at how SGD works.
Stochastic Gradient Descent (SGD)
- Initialization: SGD starts with an initial set of model parameters (weights and biases) randomly or using some predefined method.
- Iterative Optimization: The aim of this step is to find the minimum of a loss function, by iteratively moving in the direction of the steepest decrease in the function's value.
For each iteration (or epoch) of training:
- Shuffle the training data to ensure that the model doesn't learn from the same patterns in the same order every time.
- Split the training data into mini-batches (small subsets of data).
- For each mini-batch:
- Compute the gradient of the loss function with respect to the model parameters using only the data points in the mini-batch. This gradient estimation is a stochastic approximation of the true gradient.
- Update the model parameters by taking a step in the opposite direction of the gradient, scaled by a learning rate:
Θt+1 = θt - n * ⛛ J (θt)
θtrepresents the model parameters at iteration
t. This parameter can be the weight
⛛ J (θt) is the gradient of the loss function
Jwith respect to the parameters
nis the learning rate, which controls the size of the steps taken during optimization
- Direction of Descent: The gradient of the loss function indicates the direction of the steepest ascent. To minimize the loss function, gradient descent moves in the opposite direction, towards the steepest descent.
- Learning Rate: The step size taken in each iteration of gradient descent is determined by a parameter called the learning rate, denoted above as
n. This parameter controls the size of the steps taken towards the minimum. If the learning rate is too small, convergence may be slow; if it is too large, the algorithm may oscillate or diverge.
- Convergence: Repeat the process for a fixed number of iterations or until a convergence criterion is met (e.g., the change in loss function is below a certain threshold).
Stochastic gradient descent updates the model parameters more frequently using smaller subsets of data, making it computationally efficient, especially for large datasets. The randomness introduced by SGD can have a regularization effect, preventing the model from overfitting to the training data. It is also well-suited for online learning scenarios where new data becomes available incrementally, as it can update the model quickly with each new data point or mini-batch.
However, SGD can also have some challenges, such as increased noise due to the stochastic nature of the gradient estimation and the need to tune hyperparameters like the learning rate. Various extensions and adaptations of SGD, such as mini-batch stochastic gradient descent, momentum, and adaptive learning rate methods like AdaGrad, RMSProp, and Adam, have been developed to address these challenges and improve convergence and performance.
You have seen the working of the multilayer perceptron layers and learned about stochastic gradient descent; to put it all together, there is one last topic to dive into: backpropagation.
Backpropagation is short for “backward propagation of errors.” In the context of backpropagation, SGD involves updating the network's parameters iteratively based on the gradients computed during each batch of training data. Instead of computing the gradients using the entire training dataset (which can be computationally expensive for large datasets), SGD computes the gradients using small random subsets of the data called mini-batches. Here’s an overview of how backpropagation algorithm works:
- Forward pass: During the forward pass, input data is fed into the neural network, and the network's output is computed layer by layer. Each neuron computes a weighted sum of its inputs, applies an activation function to the result, and passes the output to the neurons in the next layer.
- Loss computation: After the forward pass, the network's output is compared to the true target values, and a loss function is computed to measure the discrepancy between the predicted output and the actual output.
- Backward Pass (Gradient Calculation): In the backward pass, the gradients of the loss function with respect to the network's parameters (weights and biases) are computed using the chain rule of calculus. The gradients represent the rate of change of the loss function with respect to each parameter and provide information about how to adjust the parameters to decrease the loss.
- Parameter update: Once the gradients have been computed, the network's parameters are updated in the opposite direction of the gradients in order to minimize the loss function. This update is typically performed using an optimization algorithm such as stochastic gradient descent (SGD), that we discussed earlier.
- Iterative Process: Steps 1-4 are repeated iteratively for a fixed number of epochs or until convergence criteria are met. During each iteration, the network's parameters are adjusted based on the gradients computed in the backward pass, gradually reducing the loss and improving the model's performance.
Data Preparation for Multilayer Perceptron
Preparing data for training an MLP involves cleaning, preprocessing, scaling, splitting, formatting, and maybe even augmenting the data. Based on the activation functions used and the scale of the input features, the data might need to be standardized or normalized. Experimenting with different preprocessing techniques and evaluating their impact on model performance is often necessary to determine the most suitable approach for a particular dataset and task.
- Data Cleaning and Preprocessing
- Handle missing values: Remove or impute missing values in the dataset.
- Encode categorical variables: Convert categorical variables into numerical representations, such as one-hot encoding.
- Feature Scaling
- Standardization or normalization: Rescale the features to a similar scale to ensure that the optimization process converges efficiently.
- Standardization (Z-score normalization): Subtract the mean and divide by the standard deviation of each feature. It centers the data around zero and scales it to have unit variance.
- Normalization (Min-Max scaling): Scale the features to a fixed range, typically between 0 and 1, by subtracting the minimum value and dividing by the range (max-min).
To learn more about feature scaling, check out Datacamp’s Feature Engineering for Machine Learning in Python course.
- Train-Validation-Test Split
- Split the dataset into training, validation, and test sets. The training set is used to train the model, the validation set is used to tune hyperparameters and monitor model performance, and the test set is used to evaluate the final model's performance on unseen data.
- Data Formatting
- Ensure that the data is in the appropriate format for training. This may involve reshaping the data or converting it to the required data type (e.g., converting categorical variables to numeric).
- Optional Data Augmentation
- For tasks such as image classification, data augmentation techniques such as rotation, flipping, and scaling may be applied to increase the diversity of the training data and improve model generalization.
- Normalization and Activation Functions
- The choice between standardization and normalization may depend on the activation functions used in the MLP. Activation functions like sigmoid and tanh are sensitive to the scale of the input data and may benefit from standardization. On the other hand, activation functions like ReLU are less sensitive to the scale and may not require standardization.
General Guidelines for Implementing Multilayer Perceptron
Implementing a MLP involves several steps, from data preprocessing to model training and evaluation. Selecting the number of layers and neurons for a MLP involves balancing model complexity, training time, and generalization performance. There is no one-size-fits-all answer, as the optimal architecture depends on factors such as the complexity of the task, the amount of available data, and computational resources. However, here are some general guidelines to consider when implementing MLP:
1. Model architecture
- Begin with a simple architecture and gradually increase complexity as needed. Start with a single hidden layer and a small number of neurons, and then experiment with adding more layers and neurons if necessary.
2. Task Complexity
- For simple tasks with relatively low complexity, such as binary classification or regression on small datasets, a shallow architecture with fewer layers and neurons may suffice.
- For more complex tasks, such as multi-class classification or regression on high-dimensional data, deeper architectures with more layers and neurons may be necessary to capture intricate patterns in the data.
3. Data Preprocessing
- Clean and preprocess your data, including handling missing values, encoding categorical variables, and scaling numerical features.
- Split your data into training, validation, and test sets to evaluate the model's performance.
- Initialize the weights and biases of your MLP appropriately. Common initialization techniques include random initialization with small weights or using techniques like Xavier or He initialization.
- Ultimately, the best approach is to experiment with different architectures, varying the number of layers and neurons, and evaluate their performance empirically.
- Use techniques such as cross-validation and hyperparameter tuning to systematically explore different architectures and find the one that performs best on the task at hand.
- Train your MLP using the training data and monitor its performance on the validation set.
- Experiment with different batch sizes, number of epochs, and other hyperparameters to find the optimal training settings.
- Visualize training progress using metrics such as loss and accuracy to diagnose issues and track convergence.
7. Optimization Algorithm
- Experiment with different learning rates and consider using techniques like learning rate schedules or adaptive learning rates.
8. Avoid Overfitting
- Be cautious not to overfit the model to the training data by introducing unnecessary complexity.
- Use techniques such as regularization (e.g., L1, L2 regularization), dropout, and early stopping to prevent overfitting and improve generalization performance.
- Tune the regularization strength based on the model's performance on the validation set.
9. Model Evaluation
- Monitor the model's performance on a separate validation set during training to assess how changes in architecture affect performance.
- Evaluate the trained model on the test set to assess its generalization performance.
- Use metrics such as accuracy, loss, and validation error to evaluate the model's performance and guide architectural decisions.
10. Iterate and Experiment
- Experiment with different architectures, hyperparameters, and optimization strategies to improve the model's performance.
- Iterate on your implementation based on insights gained from training and evaluation results.
Multilayer perceptrons represent a fundamental and versatile class of artificial neural networks that have significantly contributed to the advancement of machine learning and artificial intelligence. Through their interconnected layers of neurons and nonlinear activation functions, MLPs are capable of learning complex patterns and relationships in data, making them well-suited for a wide range of tasks. The history of MLPs reflects a journey of exploration, discovery, and innovation, from the early perceptron models to the modern deep learning architectures that power many state-of-the-art systems today.
In this article, you’ve learned the basics of artificial neural networks, focused on multilayer perceptrons, learned about stochastic gradient descent and backpropagation. If you are interested in getting hands-on experience and using deep learning techniques to solve real-world challenges, such as predicting housing prices, building neural networks to model images and text - we highly recommend following Datacamp’s Keras toolbox track.
Working with Keras, you’ll learn about neural networks, deep learning model workflows, and how to optimize your models. Datacamp also has a Keras cheat sheet that can come in handy!
I wear multiple hats: Software Developer, Programmer, Data Scientist, Business Intelligence Developer, Product Owner
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1. What is Surface Tension
Surface Tension Definition: Surface tension is the force acting along the surface of a liquid, causing the liquid to behave like a stretched elastic skin. We can also define it as the force per unit length acting on the surface at right angles to one side of a line drawn on the surface. Hence, it’s the property of a liquid in which the surface acts as though it’s covered with elastic skin.
The formula for surface tension is γ = F / L. Where γ = surface tension, F is the force, and L is the length. The si unit of surface tension is Newton per meter.
Additionally, we can say that it is a result of the cohesive forces between the molecules of the liquid. Consequently, because of this effect, they will refuse to separate from each other. Surface tension arises when the surface of a liquid interacts with another phase, which can be another liquid or even a solid. The defining characteristic of this phenomenon is the tendency of liquids to acquire the least possible surface area. This behaviour results in the liquid’s surface acting as an elastic sheet, a feature that sets the stage for numerous intriguing observations.
2. Surface Tension Formula
The formula for calculating surface tension is in force per unit length. Hence we can write the formula as
Surface Tension (S) = Force (F) / Length (L)
S = F / L
We need to also know that the S.I unit of the surface tension is in Newton per meter (Nm-1). Another unit for this force is dynes per centimeter (dyn/cm). We also have erg/cm2, and Joules per meter square (J/m2)
We can equally write surface tension equation as
S = (1/2) (F / L)
3. Surface Tension In Our Dialy Lives
Many of the everyday observations, show that the surface of a liquid behaves as it were a stretched elastic skin. For example, close a tap of a water but not very tight. Now observe the drops of the water from the tap. You will see the water forming slowly as it comes out from the tap. Therefore, as the water is coming out from the tap. It will appear to be making a bubble (in the form of balloon or elastic skin or bag). Hence, the elastic bag supports the weight of the water until the spherical drop of water falls on the floor.
This force is due to the attraction between the same molecules (cohesive force). Therefore, it makes the surface of the liquid to behave like a stretched elastic sheet. Thanks to this type of a force, insects can comfortably float (walk) on a water. When insects walk on a water. The surface they currently occupy will behave like an elastic skin. You will see the stretch on the surface.
a. Everyday Observations: Glass of Water and Broken Thermometer:
Surface tension becomes particularly evident when you observe how liquid behaves in everyday scenarios. Filling a glass with water to its brim allows for the addition of a few more drops before it overflows, defying gravity and our expectations. Similarly, a broken thermometer releases a small pool of mercury that exhibits peculiar characteristics due to surface tension. The captivating nature of these observations lies in the fact that they provide tangible evidence of an otherwise imperceptible force at work.
b. Significance of Surface Tension in Nature:
Surface tension is a fundamental force that influences numerous natural phenomena. It’s responsible for the behavior of liquid droplets, the floating of small objects on the surface of water, and even the formation of bubbles. Understanding surface tension allows us to appreciate the intricacies of our world and can be applied in diverse fields, from biology to materials science.
4. Surface Tension Definition: Explanation
Surface tension is a physical phenomenon that results from the cohesive forces between the molecules of a liquid at the interface between the liquid and another medium, such as air or another liquid. It is a fundamental physical phenomenon that plays a crucial role in many aspects of our lives. These roles ranges from the behavior of fluids to the way our bodies function.
Therefore, these cohesive forces make the surface of the liquid to contract and form a thin, elastic layer. The layer is known as the surface film or surface skin. We should also know that this layer behaves as if it were a stretched membrane. Hence, when you place an object on it, it is capable of supporting that object. It can also resist deformation from external forces.
The magnitude of the surface tension of a liquid depends on various factors. These factors depends on
- The nature of the liquid,
- Temperature, and
- The presence of impurities or dissolved substances.
The study of surface tension has applications in many fields, including physics, chemistry, materials science, and engineering. For example, it is a fundamental factor in determining the behavior of fluids in:
- Capillary tubes,
- The formation of drops and bubbles
- Wetting of surfaces, and
- Stability of emulsions and foams.
5. Causes and Fundamentals of Surface Tension
a. Attraction of Liquid Particles:
At the core of surface tension lies the attraction between liquid particles. These particles exhibit a cohesive force, drawing them together within the liquid. Along the surface, these particles are pulled toward the rest of the liquid, creating the tension we observe. This cohesive force is instrumental in minimizing the surface area.
b. Role of Interactions with Solid, Liquid, or Gas:
Surface tension is not solely dependent on the forces of attraction between liquid particles. It also hinges on the forces of attraction with solid, liquid, or gas substances in contact with the liquid. This multifaceted interplay adds complexity to the surface tension phenomenon, making it an intriguing field of study that extends beyond the liquid phase.
c. The Concept of Surface Energy:
The energy responsible for the surface tension phenomenon is akin to the work required to remove the surface layer of molecules in a unit area. This concept is fundamental to understanding surface tension’s impact and how it contributes to the overall behaviour of liquids.
6. Surface Tension Definition: Measurement
We use a tensiometer to measure the surface tension of a liquid. We have different types of tensiometers which are:
- Drop volume tensiometer
- Force tensiometer
- Spinning drop tensiometer
- Bubble pressure tensiometer
This brings us to three methods for measuring surface tension:
- Wilhelmy plate method
- Du nouy ring method
- Optical method
7. Measurement and Units of Surface Tension
a. Surface Tension Units: Dynes/cm and Newton per Meter (N/m):
Surface tension is typically quantified in two main units: dynes/cm and Newton per Meter (N/m). Dynes/cm represent the force in dynes required to break a film of length 1 cm, providing a practical measurement of surface tension. Newton per Meter is the SI unit, expressing surface tension as the force required to break a film of length 1 meter. These units are essential for understanding and comparing surface tension values across different liquids.
b. Mathematical Expression: Tension (T) = Force (F) / Length (L):
Mathematically, surface tension can be expressed through a straightforward formula: Tension (T) equals the force (F) per unit length (L). This equation provides a clear and quantifiable relationship between the fundamental parameters of surface tension.
c. Dimensional Formula of Surface Tension (MT-2):
The dimensional formula of surface tension is MT-2. This succinct expression encapsulates the fundamental dimensions of mass (M) and time (T), providing a comprehensive understanding of surface tension’s physical characteristics.
8. Examples of Surface Tension in Nature
a. Water Striders: Walking on Water:
In biology, water striders, small insects with negligible weight, appear to defy gravity as they effortlessly walk on the water’s surface. This remarkable feat is a testament to the power of surface tension, which supports their diminutive weight and prevents them from sinking into the water. Understanding this behaviour allows us to appreciate the intricate adaptations of creatures in their natural habitats.
b. Rainproof Tents: Bridging Water’s Pores:
The concept of surface tension extends beyond the realm of insects. In everyday life, rainproof tents rely on the surface tension of water to bridge the pores in their material. By forming a barrier against water infiltration, this application of surface tension showcases its practical importance in materials science.
c. Clinical Tests for Jaundice:
Surface tension finds applications in clinical diagnostics, where it is employed in tests for conditions like jaundice. The behavior of blood components in the presence of reagents with specific surface tension properties aids in diagnosing health issues, highlighting surface tension’s role in medical science.
d. Cleaning with Soaps and Detergents:
The use of soaps and detergents for cleaning is a common practice that capitalizes on surface tension. These cleaning agents lower the surface tension of water, allowing it to penetrate fabrics and surfaces more effectively. By reducing surface tension, soaps and detergents enable thorough cleaning processes and illustrate the practical implications of this force.
e. Round Bubbles: Wall Tension and Liquid Droplet Shapes:
Surface tension is instrumental in the formation of round bubbles. It provides the necessary wall tension that shapes these delicate structures. The role of surface tension in the formation and stability of bubbles demonstrates its importance in everyday phenomena, from dishwashing to the visual arts.
9. The Science Behind Surface Tension
a. Intermolecular Forces – Van der Waals Force:
The science of surface tension delves into the intricacies of intermolecular forces, including the Van der Waals force. These forces play a pivotal role in drawing liquid particles together, creating the cohesive strength that underlies surface tension. Understanding the nature of intermolecular forces is central to comprehending the driving factors behind surface tension.
b. Exploring the Relationship Between Forces and Length:
Surface tension hinges on the relationship between forces and length. As we explore this connection, we gain insight into how different forces act on liquid surfaces, creating tension along specific lengths. This exploration is a key element in unraveling the mysteries of surface tension.
10. How Does Surface Tension Work?
Surface tension is due to intermolecular forces, which are the attractive forces between molecules. These forces are stronger between like molecules than between unlike molecules. Thus, they create a cohesive force at the surface of a liquid.
At the surface of a liquid, the molecules are more strongly attracted to each other than they are to the air above the surface. Therefore, it makes the surface of the liquid to behave like a stretched elastic sheet. Consequently, this behavior is the reason why it resists penetration by objects and why it can form droplets.
11. Molecular Explanation of Surface Tension
The cohesive forces between the molecules of a liquid are responsible for a stretched elastic skin of the liquid. These forces are due to the presence of intermolecular forces. The intermolecular forces responsible are van der Waals forces and hydrogen bonds. They cause the molecules to come together. This attraction creates a net inward force on the surface molecules. Subsequently, it pulls them towards the bulk of the liquid. Hence, it makes the surface to act as if it is a stretched elastic membrane.
The strength of the cohesive forces between the molecules of the liquid determines the magnitude of the stretched elastic skin of a liquid, with stronger forces resulting in higher surface tension.
12. Methods of Surface Tension Measurement
a. Spinning Drop Method:
The spinning drop method is one of the techniques used to measure surface tension. It involves the rotation of a drop of liquid to determine its surface tension properties, providing valuable insights into the nature of the liquid’s surface.
b. Pendant Drop Method:
In the pendant drop method, a drop of liquid is suspended from a solid object. By studying the shape of the pendant drop, researchers can gather data on surface tension and use it for various scientific applications.
c. Du Noüy–Padday Method:
The Du Noüy–Padday method relies on the use of a precision instrument known as a Du Noüy ring. This method offers a straightforward approach to measuring surface tension, making it a valuable tool in laboratories.
d. Du Noüy Ring Method:
The Du Noüy ring method, a variant of the Du Noüy–Padday method, involves using a carefully designed ring to study the surface tension of a liquid. Its simplicity and accuracy contribute to its widespread use in scientific research.
e. Wilhelmy Plate Method:
The Wilhelmy plate method employs a thin flat plate that is immersed in a liquid to measure surface tension. By assessing the liquid’s interaction with the plate, researchers can gain valuable data on surface tension properties.
f. Pendant Drop Method:
The pendant drop method, as previously mentioned, involves suspending a drop of liquid from a solid object. This technique offers a practical and versatile approach to surface tension measurement.
g. Stalagmometric Method:
The stalagmometric method focuses on the use of capillary tubes to measure surface tension. By observing the meniscus in these tubes, researchers can determine surface tension values with precision.
h. Capillary Rise Method:
Capillary rise is another phenomenon closely related to surface tension. This method explores the rise of liquids in narrow tubes, shedding light on surface tension and its impact on capillary action.
i. Bubble Pressure Method:
The bubble pressure method assesses the pressure within gas bubbles formed within a liquid. This method provides valuable insights into the surface tension properties of the surrounding liquid.
j. Resonant Oscillations of Liquid Drops:
Resonant oscillations of liquid drops offer a unique perspective on surface tension. By studying the frequency of these oscillations, researchers can glean information about the liquid’s surface properties.
k. Sessile Drop Method:
The sessile drop method involves analyzing a drop of liquid on a solid surface. By observing the shape and behaviour of the drop, scientists can infer surface tension characteristics.
13. Surface Tension Definition: Effects
Surface tension has several effects on the behavior of liquids. For instance, it causes liquids to form droplets. These are spherical in shape due to the minimization of the surface area. Therefore, the spherical shape of droplets is due to the balance between the stretched elastic skin of the liquid and the hydrostatic pressure.
It also affects the wetting of a solid surface by a liquid. When the cohesive forces between the liquid molecules are stronger than the adhesive forces between the liquid and the solid. The liquid forms a droplet on the surface of the solid. On the other hand, if the adhesive forces are stronger. The liquid spreads over the surface, forming a thin film.
We use surfactants (molecules) to lower the surface tension of a liquid. They are amphiphilic molecules, which means they have both hydrophobic and hydrophilic regions. The hydrophobic region is attracted to non-polar molecules, while the hydrophilic region is attracted to polar molecules.
When you add it to a liquid. Surfactant molecules align themselves at the surface, with their hydrophobic regions in the liquid and their hydrophilic regions in the air. This alignment reduces the cohesive forces between the liquid molecules.
15. Surface Curvature and Pressure
We can describe the curvature of the surface by the radius of curvature. It is the radius of the circle that best fits the surface at a given point. Laplace equation describes the relationship between the surface tension, curvature, and pressure.
The surface curvature of a liquid interface refers to the curvature of the liquid’s surface at a given point. A liquid interface can have different curvatures at different points, and the curvature can change depending on the surrounding environment.
Surface pressure, on the other hand, refers to the force per unit length that acts on a liquid surface. This pressure is caused by the stretched elastic skin of the liquid and is directed perpendicular to the facet. It is directly proportional to the surface tension and inversely proportional to the radius of curvature of the surface.
Mathematically, the relationship between surface tension, surface pressure, and surface curvature can be expressed using the Laplace-Young equation:
ΔP = S(1/R1 + 1/R2)
S = Surface tension
ΔP is the pressure difference across the interface
R1 and R2 are the principal radii of curvature at the interface.
The Laplace-Young equation states that the pressure difference across a curved interface is proportional to the surface tension and the sum of the inverse radii of curvature.
This equation explains why small bubbles have higher internal pressure than large bubbles. This is because the radius of curvature of a small bubble is smaller than that of a large bubble. Hence, the pressure difference across the interface causes the bubble to expand or contract until the internal and external pressures are equal.
16. Force Due to Surface Tension
When we consider a small segment of the liquid surface with length L. The cohesive forces between the molecules of the liquid cause a net inward force on this segment. This force is directed tangentially along the surface and is perpendicular to the length of the segment.
The formula for force due to surface tension is F = γL,
This coefficient (γ) represents the amount of energy required to increase the surface area of the liquid by one unit.
If the surface area of the liquid changes by a small amount dA. Then the work done is given by
dW = FdA.
This work represents the energy required to increase the surface area of the liquid by dA, and is equal to γdA.
This force plays an important role in many natural and technological processes. We have roles like the formation of droplets, the behavior of bubbles, and the adhesion of materials. Understanding its mathematical expression helps in designing and optimizing these processes.
17. How to Calculate Surface Tension
a. Practical Calculation Example:
Calculating surface tension is a practical application of the mathematical expression Tension (T) = Force (F) / Length (L). By working through a real-world example, we can see how these components come together to quantify the surface tension of a given liquid.
b. Force and Length Parameters:
Surface tension calculations rely on two fundamental parameters: force (F) and length (L). These parameters serve as the building blocks for understanding and quantifying surface tension in a tangible way.
18. Wider Applications of Surface Tension
The study of a stretched elastic skin of a liquid has many practical applications in daily activities. One of these applications is capillary action (Capillarity). This is because it plays a key role in the transport of water in plants. It is also important in the absorption of liquids in paper towels and sponges.
Another example we can capitalize on is the formation and behavior of bubbles. The spherical shape of bubbles helps the surface of the bubble to contract and minimize its surface area.
Wetting and spreading are other important applications we need to consider. When a liquid comes into contact with a solid surface, it will either wet or not wet the surface. This is depending on the relative strengths of the adhesive and cohesive forces involved.
19. The Importance of Surface Tension
This type of force plays a crucial role in many aspects of our lives. It is what allows plants to transport water and nutrients through their stems and leaves. It enables insects to walk on water. When we look into medicine, we apply it to create microfluidic devices for drug delivery and to study the behavior of cells and proteins.
It is also responsible for the behavior of surfactants. As we earlier explained, they are molecules that reduce the surface tension of a liquid. Surfactants are found in many household products. These products can be detergents and soaps. We use them to enable more penetration in surfaces more easily.
20. Surface Tension Definition: Effects in Our Daily Lives
Surface tension plays a role in many everyday activities, from washing dishes to blowing bubbles. Soap and other cleaning agents can penetrate surfaces more easily and helps to remove dirt and grease.
Additionally, it allows droplets to form on surfaces. We can see these droplets in the dew that forms on grass in the morning. It is what allows inkjet printers to create sharp, precise images. It also enables the formation of bubbles in carbonated beverages.
21. Surface Tension Definition: Applications in Nature
Surface tension is essential to the functioning of many natural systems. In plants, it allows water and nutrients to be transported from the roots to the leaves through the stem. A quite obvious example is the one we mentioned earlier, where Insects are able to walk on water. Some species even use it to trap prey.
When we look into the behavior of marine organisms. We can see that the ocean allows the formation of waves. Therefore, it helps certain species to float on the surface of the water.
22. Applications in Science and Technology
It is an important factor in many scientific and technological applications. In materials science, we use it to study the behavior of materials at the nanoscale. In energy harvesting, we use it to maximize the efficiency of solar cells by optimizing the surface tension of the materials we use.
We also use it in the development of microfluidic devices. Which are used in a variety of applications, from drug delivery to lab-on-a-chip technology. Deeper research can help to create precise channels for the transport of fluids and particles at the micrometer scale.
23. Future Research
Research into the properties of this topic is ongoing, and there are many potential applications for this knowledge. One area of research is the development of new surfactants and other materials that can be used in a variety of industrial and biological processes.
Another area of research is in microfluidics. This is the study of fluids at the microscale. Hence, it can be used to control the behavior of fluids in microfluidic devices. This is because they have applications in fields such as medical diagnostics and drug delivery.
24. Frequently Asked Questions (FAQs)
Q: What is the difference between surface tension and viscosity?
A: Surface tension is the cohesive force that exists between molecules at the surface of a liquid, while viscosity is a measure of a liquid’s resistance to flow.
Q: How is surface tension measured?
A: It is measured in units of force per unit length. The unit is Newtons per meter or dynes per centimeter.
Q: What are some examples of how surface tension is used in everyday life?
A: We use it for washing dishes, blowing bubbles, and even in the formation of dew on grass.
Q: How does surface tension impact the behavior of insects on water?
A: It allow insects to walk on its surface by providing a supportive layer of cohesive molecules.
You may also like to read:
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Introduction to Computer Information Systems/The System Unit
Data and Program Representation[edit | edit source]
Digital data and numerical data[edit | edit source]
Most computers are digital computers which use a specific language to communicate within itself in order to process information. If there are programs running in the background or a person is typing up a word document for example, the computer needs to be able to interpret the data that is being put into it by the human as well as communicate to working components within itself. This language that digital computers use is called binary code and is a very basic form of language composed of only two figures; 1 and 0. Whereas the English language is composed of 26 figures which we commonly call the alphabet, computers use a language composed of only two figures, hence its name "binary code". These 1's and 0's are referred to as "bits" - which are known as the smallest unit of data that a binary computer can recognize. They are found through every action, memory, storage, or computation that is done through a computer, such as creating a document, opening a web browser, or downloading media. In order to comply with more actions memory or storage, bits must compound together to form a larger unit referred to as "bytes".
Bytes are commonly used when referring to the size of the information being provided. For example, a song that is downloaded may contain several kilobytes or perhaps even a few megabytes if it is a whole c.d. and not just a single track. Likewise, pictures and all other documents in general are stored on the computer based on their size or amount of bytes they contain. The amount of information that can be stored onto a computer is also shown or displayed in bytes as is the amount left on a computer after certain programs or documents have been stored. Since bytes can be extremely long, we have come up with prefixes that signify how large they are. These prefixes increase by three units of ten so that a Kilobyte represents around 1,000 bytes, a Megabyte represents around one million bytes (1,000,000 bytes), a Gigabyte represents around one billion bytes (1,000,000,000 bytes), etc. Computers components have become so small that we can now store larger and larger amounts of data bytes in the same size computers resulting in the use of other larger prefixes such as Tera, Peta, Exa, Zetta, and Yotta. Below is a chart outlining the name of the prefix used and powers of ten they symbolize.
Digital Data Representation, otherwise known as how the computer interprets data, is a key concept to understanding computer data processing, as well as overall functioning. Data is represented by particular coding systems. The computer recognizes coding systems- rather than letters or phrases that the user of a computer views. The actual process of the computer understanding coding systems is called digital data representation. A digital computer operates by understanding two different states, on or off. This means that the data is represented by numbers- 0’s and 1’s, and is known as a binary computer. The binary code is a very basic coding system for computers to comprehend. An advantage to digital data computing lies behind the binary coding systems. Although the binary code has become decreasingly popular in the professional, recreational fields due to an increase in technology, they still provide a use in programming. Digital data creates a simple way to duplicate and transfer information accurately from computer to computer, which is why it is still used today. The terminology for the smallest unit of data is a bit, which consists of a single numeric value,0 or 1. Bytes, on the other hand, consist of groupings of multiple groupings of bits. Bytes allow the computer hardware to work more quickly and efficiently.
(from the SI page on Wikipedia):
Representing data in a way that can be understood by a digital computer is called Digital Representation and Binary Code is the most commonly used form of this. Binary Code is a Numerical Representation of data that uses only 1 and 0 to represent every possible number. Mathematics uses 10 symbols ranging from 1 TO 0 and include 2, 3, 4, 5, 6, 7, 8, and 9 as well. This Numerical Representation of data is called the decimal numbering system because it uses ten symbols. As shown on the chart, the prefix deci symbolizes ten. In both systems, the position of each digit determines to which power that number is raised. In the decimal system each digit is raised by ten so that the first column equals 1 (10^1) or ten raised to the first power, the second column equals 10 (10^2) or ten raised to the second power, the third column equals 100 (10^3) or ten raised to the third power and so on. However, since Binary Code only operates with two symbols, each digit is a power of two instead of ten. In binary the first column equals 1 (2^0) or two raised to the zero power, the second column equals 2 (2^1) or two raised to the first power, the third column equals 4 (2^2) or two raised to the second power, the fourth column equals 8 (2^3) or two raised to the third power, and so forth. Because the Binary system takes advantage of so few symbols, the result is that more positions for digits are used to express the same number than in decimal form, leaving long lines of information for even the simplest expressions.
Coding systems[edit | edit source]
There are a few different coding systems, EBCDIC, ASCII and Unicode. EBCDIC (extended binary coded decimal interchange code) was created for use in mainframes, developed by IBM. The code uses a unique combination of 0’s and 1’s, 8-bits in length, which allows for 256 different combinations. ASCII ( American standard code for information interchange) was created for a more personal use. ASCII uses a 7 bit code, though there is an extended code which adds an extra bit, which nearly doubles the amount of unique characters the code can represent. However Unicode is a much longer string of code, between 8 and 32 bits. With over one million different possibilities, every language can be represented with this code, every mathematical symbol can be represented, every punctuation mark, and every symbol or sign from any culture.
Unicode is universal. With using 0’s and 1’s to represent different data, it has become fit for any language used all over the world. This code is replacing ASCII (American Standard Code for Information Interchange) because the characters in this code can be transformed into Unicode, a much more practical system for data. ASCII is known as the alphabet code, and its numbering codes range from 0 all the way to 127 considered to be a 7 bit code. Alphabets vary from language to langue, but 0’s and 1’s can be understood worldwide. The problem with Unicode is that it is not compatible with each computer system used today. Windows 95/98 does not have the ability to run Unicode while other Windows such as NT and 2000 are closer to being able to. There is a program Sun Microsystem’s Java Software Development Kit which allows you to convert files in ASCII format into Unicode. While Unicode is a huge improvement for coding systems today, it cannot process all symbols that are possible, leaving room for new systems to one day take its place.
Graphics Data[edit | edit source]
One type of multimedia data is graphics data. These data are of still images, and can be stored in the form of a bitmap image file. A bitmap image is a type of graphic that contains pixels, or picture elements, that are arranged in a grid-like pattern. Each pixel is made up of a specific group of numbers which corresponds to the color, and the color’s intensity. Although there are a few other key factors when determining the detail quality of an image, pixels play an important role. An image with many pixels allows there to be more potential of higher quality in that image. However, this doesn’t mean that more pixels in an image definitely results in a higher quality picture. When shopping for digital cameras consumers must be aware of the amount of megapixels, or pixels by the million, the cameras in front of them have. Today, an average person wishing to take decent and basic everyday pictures will be satisfied with about an 8 megapixel camera. In fact, many new smartphone cameras use 16 megapixels, like the HTC Titan 2, a popular smartphone released in April, 2012. Someone with different intentions of using images, perhaps for making high definition prints, will require a camera with more megapixels. This would allow for their prints to be large, but with appropriate and exceptional quality.
Audio Data[edit | edit source]
Audio Data is very similar to graphics data in that it is understood in pieces. Instead of using pixels, however, audio data uses samples. Audio data is usually recorded with an input device such as a microphone or a MIDI controller. Samples are then taken from the recording thousands of times every second and when they are played back in the same order, they create the original audio file. Because there are so many samples within each sound file, files are often compressed into formats such as MP3 or MP4 so that they take up less storage space. This makes them easier to download, send over the internet, or even store on your MP3 player.
Video Data[edit | edit source]
Video data is also similar to graphic and audio data, but instead of using pixels or samples, video data is recorded with the use of frames. Frames are still images that are taken numerous times per second and that when played simultaneously, create a video (most films are recorded using twenty-four frames per second). Similar to audio data, because video data contains so much information, the files can be compressed, making it possible for full length movies containing thousands of frames to be stored on optical discs.
The System Unit - The Motherboard and CPU[edit | edit source]
Motherboard[edit | edit source]
"The motherboard can be thought of as the "back bone" of the computer." This quote is from the article Motherboard. Inside the system unit contains the motherboard. The motherboard is the "glue" of the computer. It connects the CPU, memory, hard drive, optical drives, video card, and sound card together. The front of the motherboard are peripheral card slots. The slots contain different types of cards which are connected to the motherboard. The left side of the motherboard contain ports. The ports connect to the monitor, printer, keyboard, mouse, speakers, phone line, and network cables.
Like many of the components of computers, motherboards have not always been as advanced as they are today. Motherboards on early PCs did not have many integrated parts located directly on the board. Instead, most of the devices, such as display adapters and hard disk controllers, were connected through expansion slots. As technology advanced, more and more devices were built in directly to the board itself. At first, this began to create problems as manufacturers began to find that if one of the devices on the motherboard was faulty or in some way damaged, that the entire motherboard must be replaced. This led manufactures to change the design in a way that allowed them to remove faulty parts easily and replace them, especially parts that are growing and changing so quickly, such as the RAM or CPU. Today, a motherboard comes equipped with many parts working in conjunction with each other. One can find anything, from back up batteries, keyboard and mouse connectors, to cache memory chips, in close proximity to the CPU. The computer is able to do tasks faster as its components continue to be closer to one another. The advancement of technology has allowed for these parts to become smaller and more powerful, allowing more surface area on the motherboard to fit more devices. It is common today to find even audio and video components built into it as well. With technology moving as fast as it is, one may wonder what a motherboard will be capable of containing in the near future.
RepRap Motherboard v1.1
Real-time clock on a motherboard
Expansion Cards[edit | edit source]
An expansion card, also known as an expansion board, adapter card, or accessory board, is a printed circuit board that can be inserted into an expansion slot on the motherboard to add functionality to a computer system. The three most common expansion cards are the audio card, graphics card, and network card. Each type of expansion card has a self-explanatory name and all serve the same purpose of adding functionality to the computer. The audio card is responsible for producing sound that is then transferred to speakers or headphones. Commonly audio cards are built onto the motherboard, however, they can be purchased separately. The graphics card turns the data produced by a CPU to an image that is able to be seen on a computer's display. Along with the audio card, graphics cards are commonly built onto the motherboard, yet graphics card that produce higher resolution images can be bought separately. Lastly, the network card is an expansion card that connects the computer to a computer network. This allows for a computer to exchange data with the computer network through a commonly used number of protocols called IEEE 802.11, popularly known as wireless LAN or Wi-Fi.
CPU[edit | edit source]
The central processing unit, also known as the CPU, is responsible for executing a sequence of instructions called a program. The computer needs the CPU in order to function correctly. It is known as the brains of the computer where the calculations occur. The microprocessor and the processor are two other names for the central processing unit. The Central processing unit attaches to a CPU socket on the motherboard. A multi-core CPU contains more than one processor chips. This specific type of CPU is efficient because it allows computers to work on more than one task at a time because the singular processor can run multiple instructions on the different cores at the same time. Also, these multi core CPU's experience less over heating than the original CPU which causes much less problems to the computer.
Intel i7 940
AMD Dual Core
History of the CPU[edit | edit source]
The first CPU ever made was the Intel 4004, which was designed by Federico Faggin. After ten months of Faggin and his colleagues working on the chip, it was released by Intel Corporation in January 1971. Even though this first generation, 4-bit microprocessor could only add and subtract, it was a major breakthrough in technology. The amazing quality was that all of the processing was done on one chip, as opposed to prior computers which had a collection of chips wired together. This invention lead to the first portable electronic calculator.
While technology has advanced quite a bit since 1971, old technology is not as “out-of-date” as one thinks. There are still CPU chips made in the 1970’s and 1980’s that are still being used today. Personal computers, such as PC’s and Mac’s, use faster, more up-to-date CPU’s because the users run many programs at the same time. However, the more simple computers embedded in cars, printers, and microwaves can still use the older forms of microprocessors. For example, one famous CPU was the MOS 6502, made in 1975, and it was still being used in many appliances up until 2009. Control processing units are the key component in any computer, and thus sometimes the simpler styles work best.
The System Unit - Memory, Buses, Ports[edit | edit source]
Memory[edit | edit source]
Memory identifies data storage that comes in the form of chips and is used to store data and programs on a temporary or permanent basis. There are two main types of memory storage which are random- access memory (RAM) and read-only memory (ROM). Inside the system unit, ROM is attached to the motherboard. Random-access memory can read data from RAM and write data into RAM in the same amount of time. RAM capacity is measured in bytes. It is volatile which means that it loses the information/data stored on it when the power is turned off. In order to retrieve an important file at a later date, one needs to store it on a separate, non-volatile, storage medium (such as a flash drive or hard-drive) so that, even though the information is erased from RAM, it is stored elsewhere. RAM has different slots where it stores data and keeps track of addresses. Read-only memory cannot be written to and is non-volatile which means it keeps its contents regardless of whether the power is turned off or not. Flash memory (solid-state) is starting to replace ROM. It is also a non-volatile memory chip that is used for storage on devices, like mobile phones, tablets, digital cameras, etc. This type of memory can often be found in the form of flash drives, SD cards, and Solid-State hard drives. The reason for this is so that the data can be quickly updated over time while taking up a smaller amount of physical space in comparison to its precursors. Flash memory is also more resistant to outside forces, such as electro-magnetic fields or shock, than other memory alternatives such as traditional hard-drives.
Cache memory and Registers are special types of volatile memory that allows a computer to perform certain tasks much more quickly. The cache memory is a high speed circuitry that can either be built right into the CPU or very close to the CPU. Registers are built into the CPU to store intermediary results during processing. A good analogy from HowStuffWorks compares the computer to a librarian, data to books, and cache to a backpack. Suppose somebody walks into a library and asks the librarian for a copy of the book Moby Dick. The librarian goes back into the room full of books, grabs that book, and gives it to the reader. Later that day, the reader returns, having finished the book, and gives it back to the librarian, who returns it to the same storage room. Then, a second reader walks in asking for the same book, Moby Dick. The librarian has to get up and go all the way back to the room in order to get the book he was just handling, which is a waste of time. Instead, suppose the librarian had a backpack that could store up to 10 books. When the first person returns Moby Dick, the librarian puts it into his backpack instead (after making sure the backpack doesn't have 10 books in it already.) Then, when the second person comes in requesting that same book, the librarian can just check his bag, get the book out, and hand it to the second person without having to walk all the way back into the other room. Cache memory functions like that backpack, it stores previously accessed data in a specific area with a limited amount of memory so that the processor can get this data much more quickly.
Ports[edit | edit source]
Ports are on the outside of the system unit and they are used to connect hardware devices. There are physical ports and virtual ports. A physical port is a physical connection to a computer where data is transferred. It is when something is physically plugged into the computer or some other device. Virtual ports allow software applications to share hardware resources without having to physically connect to each other or to interfere with one another. Parallel ports are most often used with a keyboard, printer or mouse, but these are more commonly known as legacy ports instead. Each port has a certain connector to plug it into the computer. Different type of ports would be power connectors, VGA monitor port, USB ports, Firewire port, HDMI port, Network port, audio ports, and empty slots. The connectors would be Monitor (VGA, HDMI), USB, Firewire, network, and audio connector. Each port also has a different purpose and connector. Almost all PCs come with a serial RS-232C port or a RS-4222 port and they are used for connecting a modem, mouse, or keyboard. They also have parallel ports that are used to connect printers. These are also considered USB ports because they are physical ports and which standardize communications between computers and peripheral. USB ports were created in the mid 1990’s; USB stands for Universal Serial Bus. There are also network ports used to connect a computer to a network. Ethernet was developed in the 1980s and it is a system for connecting a number of computer systems to form a local area network (LAN).
A serial port is used to connect modems to personal computers. The term “serial” signifies that data sent in one direction always travels over a single wire within the cable. The last main kind of port is the FireWire, which are used to connect FireWire devices to the computer via a FireWire connector. These are used with mostly digital video cameras and other multimedia devices.
Thunderbolt port[edit | edit source]
A Thunderbolt port connects peripheral devices through that cable. These ports allow you to connect more devices to your computer and are very fast. Thunderbolt ports support hardware controller I/0 protocols with the use of a single cable. I/O technology is input and output, and is a device that transfers the data to the computer peripherally (a CD-ROM would be an example of an I/O technology). This port supports full bandwidth for both directions of the port, thus allowing the user to be faster and more efficient with the connected devices. This type of technology allows people to plug in as many devices as they can use on their computer while not slowing any of those devices down. The thunderbolt port is also small, so it is easy to travel with as well.
Power supply unit[edit | edit source]
Computers need power and there are two main functions the power supply unit, also commonly referred to as the PSU, is responsible for. The first is to convert the type of electrical power available at the wall outlet such as 110 V 60 Hz AC (alternating current) or 230 V 50 Hz AC to the type the computer circuits can use. The other crucial task is to deliver low voltages to each device due their requirements. The converting currents could be represented either by built in PSU (desktops, servers, mainframes) or by the separate power supply adapters for computers with rechargeable batteries inside (laptops, tablets). Three main voltages are used to power computer : +3.3 V, +5 V, and +12 V DC, Usually, the +3.3 or +5 voltages are being used by logic circuits and some digital electronic components (motherboard, adapter cards, and disk drive logic boards) while the motors (disk drive motors and any fans) use the +12 V power. The power supply must provide a good, steady supply of DC power for the proper system operation. Devices that run on voltages other than these must be powered by onboard voltage regulators. For example the CPUs operate 1.5 V and 2 V and require very stable power with high power consumption.
Ethernet Cable in Theatre[edit | edit source]
A commonly used cable today is Ethernet cable. You are probably most familiar with its use involving the internet in your home, mostly going from your modem, to another computer of to a Wi-Fi router. However, the use of Ethernet cable has been instrumental in the changing would of technical theatre. Before its introduction, the most common computer cables used in theatre were DMX and XLR, for lighting and sound respectively. The issue with this is that each cable can only carry the information for one device, be that a microphone or light. In addition, if these cables are stored improperly, they can corrupt the information being transmitted. Ethernet is much smaller, and can transmit far more data. Also, there is less of a danger regarding storing cable. Ethernet, combined with new operating system and equipment, has made things far more efficient. For example, an analog board must have one XLR cable go to each microphone, so if you wanted to run 40 microphones, you must have 40 channels available on your soundboard. Also, the size of a cable with 40 smaller lines inside it can reach a one-inch diameter, and can weigh several hundred pounds. Now, a digital soundboard can control up to 100 microphones on a single Ethernet cable.
How the CPU Works[edit | edit source]
CPU Architecture and Components[edit | edit source]
As previously discussed on this page, the CPU is a complex piece of the computer made up of many parts. The way these parts all fit together inside the CPU is different in each processor but they mainly contain the same parts from device to device. The most abundant part in the CPU would be the transistor. Modern CPU's typically hold several hundred million transistors with some of the more high-end computers holding over a billion, and for good reason. Calculations in a computer can be performed thanks to the combination of transistors turning off or on. Besides these transistors, there are several parts that make up the CPU. Some of these include the arithmetic/logic unit (ALU) and floating point unit (FPU), the control unit, and the prefetch unit. The ALU is the part of the CPU that deals with the mathematics involving whole numbers and any functions done with those numbers. The FPU takes care of the mathematics with other numbers like fractions, or numbers with decimal places. These two parts work hand in hand, using arithmetic and logical processes, to allow you to perform basically any function you perform on your computer. The control unit takes charge in controlling where and when information is transferred to and from the CPU. When information leaves the control unit, it is usually sent to the ALU/FPU where it can be converted into a process. The prefetch unit, as its name implies, fetches data before it is needed. It uses a sequence of processes to guess what information will be needed next, and have it readily available before the time it needed. Other components of the CPU include the cache, the decode unit, and the bus interface unit. The cache serves as high-speed memory for instructions that the CPU would like to access faster, in other words instructions that the CPU would rather avoid retrieving from RAM or the hard drive. The decode unit, just as it sounds, decodes instructions. Once the prefetch unit fetches data, the data goes through the decode unit so the instructions can be understood by the control unit. The bus interface unit allows communication between the core and other CPU components. Think of it as literally a bus, taking information from one place and transporting it somewhere else.
The Internal Clock[edit | edit source]
Every computer actually has two different clocks. One is the virtual or system clock that runs and is displayed whenever the computer is on and running. The other is a real-time clock or hardware clock that runs continuously, and is responsible for tracking the correct time and day. This device does not count time in days and hours for example. Instead it just runs a counter at times per second. As far as the century goes, it is the job of the BIOS, the Basic Input-Output System, to track this and save it in the non-volatile memory of the hardware clock. These two clocks run independently on each other. The system clock is physically a small quartz crystal that can be found on the motherboard. It also helps synchronize all computer functions by sending out signals- or cycles- on a regular basis to all parts, much like a person’s heartbeat. Hertz is the unit of measure used to count the number of cycles per second. For example, one megahertz is one million ticks of the system clock. This clock is very important to the CPU because the higher the CPU clock speed, the more instructions per second it could process. Since the entire system is tied to the speed of the system clock, increasing the system clock speed is usually more important than increasing the processor speed.
PCs in the past only had one unified system clock with a single clock, which drove the processor, memory, and input/output bus. However, as technology advanced, the need for a higher speed, and thus multiple clocks, arose. Therefore, a typical modern PC now has multiple clocks, all running at different speeds to enable any data to “travel” around the PC. Furthermore, two CPUs with the same clock speed will not necessarily perform equally. For instance, if an old microprocessor required 20 cycles to perform a simple arithmetic equation, a newer microprocessor can perform the same calculation in a single clock tick. Therefore, even if both processors had the same clock speed, the newer processor would be a lot faster than the old.
As mentioned previously, a CPU serves as a great example for the synchronization that the system clock performs. To synchronize, most CPUs start an operation on either the falling edge, when the clock goes from one to zero, or the rising edge, when the clock goes from zero to one. All devices, such as a CPU, synchronized with the system clocks run at either the system clock speed or at a fraction of the system clock speed; therefore, the CPU is unable to perform tasks any faster than the clock. For example, during each system clock tick, a CPU clock speed of 2 GHz allows the CPU clock to “tick” 10 times, executing one or more pieces of microcode. This ability to process multiple pieces of microcode at one time is known as superscalar.
The Machine Cycle[edit | edit source]
A machine cycle is a term often used when discussing the clock. It has four main parts- fetch, decode, execute, and store. The machine cycle occurs whenever a CPU processes a single piece of microcode. The fetch operation requires the program instruction to be fetched from either the cache or RAM, respectively. Next, the instructions are decoded so that the ALU or FPU can understand it, known as the decode operation. Then, the execute operation occurs when the instructions are carried out. Finally, the data or result from the ALU or FPU operations is stored in the CPU’s registers for later retrieval, known as the store operation. A fifth possible step in the cycle is the register write back operation, which occurs in certain CPUs. The RISC CPU, which stands for reduced instruction set computer processing unit, is an example that uses the fifth step of the machine cycle. Machine cycles can only process a single piece of microcode, which forces simple instructions, like addition or multiplication, to require more than one machine cycle. In order to make computers faster, a system known as pipelining has been created. Originally, one machine cycle would have to finish processing a single instruction before another instruction could be carried out through a second machine cycle. With pipelining, as soon as an instruction passes through one operation of the machine cycle, a second instruction can start that operation. For example, after one instruction is fetched and moves on to decoding, the CPU can fetch a second instruction. This invention allows for multiple machine cycles to be carried out at the same time, which boosts the performance of the computer. Also, because of how fast the CPU can work with pipelining, it can be measured in millions of instructions per second.
Typical CPU Components (continued)[edit | edit source]
To round up the simplified inventory of a CPU's guts, we have the decode unit, the registers and internal cache memory, and the bus interface unit. Of the remaining three sections of a CPU, the decode unit is easiest to understand because its job immediately follows the job of the prefetch unit. After the prefetch unit collects the data, the decode unit decodes the data into a language that is easier for the ALU/FPU to understand. It does that by consulting a ROM memory that exists inside the CPU, called microcode. The registers are used during processing; they're groups of high-speed memory located within the CPU that can be accessed by the ALU and FPU, or for other assorted optimization purposes. While the registers provide the fastest speed of memory, their space is extremely limited. In the cases where the small register space isn't good enough, there are the caches to save the day. The cache is used by the CPU for memory which is being accessed repeatedly, speeding up the access time and having a slightly larger storage than the register. The bus interface unit does exactly what it sounds like; it buses the data back and forth, connecting the core of the CPU to interact with other components.
Another aspect of the CPU is improving processing performance. In the past most CPUs designed for desktop computers had only one single core, so the only way to improve performance was to increase the speed of the CPU; however, increasing the speed also caused the CPU to heat up. So now a days CPU have multiple cores in order to increase the performance. The new iPhone XS, for example, will have six CPU cores. In an article by Stephen Shankland from CNET on September 12, 2018, he explains how the new Apple iPhone XS CPU will be able to perform faster. The new Apple iPhone is going to have a new A12 Bioinic chip. It is going to have more transistors, which if you recall, are small devices made of semiconductor material that acts like a switch to open and close electrical circuits. This new A12 chip will have about 7 billion transistors according to the article Mr. Shankland wrote. Mr. Shankland states in his article that that the new A12 will be 15 percent faster than 2017’s iPhone X, and consume 40 percent less power. As of now, this information is coming from graphs and information that Apple has shared. The thing to know and realize is that companies are constantly striving to improve performance and reworking the architecture of the CPU can improve the performance.
Improving the Performance of Your Computer[edit | edit source]
Add More Memory and Buy a Separate Hard Drive[edit | edit source]
When it comes to technology, there is no question that newer is better. New systems are able to process faster, store more, and run more applications at one time. However, it is obviously not within everybody’s means to just run out and purchase the newest technology the minute it hits the market. Technology is expensive, and therefore it is important to know your options. For example, if you have a computer that is a couple of years old, it is not unreasonable to assume that the hard drive and memory on the system are starting to slow down. However, what many people may not know is that buying a new computer is not the only solution to the problem. You can add memory to your old system simply by purchasing a new memory card and installing it into the computer hardware. By doing this, you are saving money and buying yourself a little bit more time with the computer. Another way to speed up your computer with out having to invest in a whole new one is by buying a second hard drive. When the original hard drive starts to fill up, one can simply purchase either an internal or external hard drive for the computer and drastically increase the operating speed.
Upgrade To A Solid-State Drive[edit | edit source]
Since solid-state drives (SSDs) are hard drives that use flash memory technology instead of hard disk platters they have no moving parts. They also no longer make noise, consume less power thus generating less heat, and are much faster than hard drives. Since they are much faster than hard drives, the performance of the computer would also be improved. Running programs, opening files, saving things to the disk, even browsing the web will be much faster. Also with a mechanical hard drive, physical heads have to move around to read data from the disk while in a solid-state drive data can be read and written on any location thus there is no penalty in performance. Not only are solid-state drives faster but they have also become less expensive that upgrading to them is much more affordable and reasonable. Even further, installing solid-state drives is not too difficult or complex. It is basically the same as installing regular hard drives. Also if the decision of upgrading to solid-state drives seems a little too final, it is possible to just add a solid-state drive alongside the hard drive. Thus not only having more space, but also having the ability to keep the old mechanical drive.
Upgrade Your Internet Connection[edit | edit source]
If your system seems to be running poorly while using the internet, you may have to upgrade your internet connection. Upgrading your internet connection may become more costly but there is a significant change in the processor. Your first step would be to discuss any upgrades or check if the provider needs to be enhanced in any way. Then find a browser that is suitable for your connection type. With that being stated, you can change the settings on the router in order to speed up the internet connection. In order to prevent your internet connection becoming slower, it’s highly suggested to have a password in order to access the internet. In addition, every computer owner should provide maintenance to their computer in order to prevent viruses or any bugs the computer may receive but it also prevents an internet connection from being slow. In order to do so, keep up with upgrading and cleaning the computer because the more the computer is trying to maintain, the slower the internet connection may become.
System Maintenance[edit | edit source]
In order for computers to operate at their maximum efficiency, users must be aware of the importance of system maintenance because, over a period of time, one may notice a reduction in system performance. This can be attributed to a number of common factors that lead to the degradation in performance. One major reason is hard drive fragmentation. As more programs are installed onto the hard drive, the pieces of the files that are on those programs take longer to be located. The longer pieces of the program become shorter and fragmented, leading to a longer waiting period for the user as the computer searches for these scattered pieces. Related to this, although not nearly as detrimental to system performance as fragmentation, is the cluttering of pieces and references to uninstalled programs in the operating system. For Windows users, this occurs in the Registry. After the user uninstalls a program, there are references to that program left behind in the Registry that can possibly impact performance. However, performance is not necessarily the issue here. For example if the user is going to update the system by switching from an Nvidia graphics card to an AMD one, it might be a good idea to not only uninstall all drivers and related programs but also to clean the Registry of any references to the Nvidia drivers and software (in order to avoid possible conflicts when the AMD card is installed). This will ensure a “clean” install of both the hardware and software components. A free registry cleaner utility one can use is CCleaner.
Temporary files (e.g. from web browsers and installation programs) can take up valuable storage space if they are not removed after extended periods of times. Also, users should be aware of the programs they are installing and decide which specific programs are to run at startup. Too many programs can slow down the initial startup time of the computer because it must launch program after program. Only those programs that are necessary should be included, and to check for this, click Start (in the lower-left Windows icon) and enter the command msconfig in the search tab. This will open the System Configuration window. Programs that run at startup are listed under the Startup tab. Here the user can enable or disable programs, which can affect startup time.
Another important factor in determining system performance is the corruption of system files by malware. Viruses, worms, trojans, spyware, and other forms of malware can infect a system by various means, so it is important for the user to be aware and defensive. Anti-virus programs and other security software provide protection from malware, so it is recommended that a user has some sort of program installed and regularly scans the system for any traces.
Lastly, dust can accumulate in and on heatsink fans (e.g. processor and graphics card), case fans, ports, power supplies, and motherboards. Every internal component can accumulate dust, and this can be a major issue for system integrity because dust acts as an insulator by trapping heat. Fans with too much dust do not operate efficiently because the fins do not spin quickly, which further exacerbates the heating problem. Not only that, but dust can also cause electrical shorting of the circuits, which can irreversibly damage components. To clean the computer, power off the system, which includes turning off the power supply. It should not be connected to any source. Then open the case and use a can of compressed air to blow out the dust wherever it may be. The goal is to rid the case of any remnants of dust. Following this and the other tips listed above will help ensure reliable performance and a longer lifespan for the computer.
Future Trends[edit | edit source]
The challenge of making computers faster and more efficient has brought new ideas to the table of technology. One such idea is nanotechnology, which uses microscopic components only nanometers in length. Carbon nanotubes are already being used in technology today in products such as lithium ion batteries because of their great performance conducting electricity. Other nanotechnology includes nanoparticles and nanosensors. Another idea that has received increased recent attention is quantum computing. These computer’s go beyond regular computers’ binary system using qubits, which can be either a 1, a 0, or both simultaneously. Although these computers are only able to perform seemingly simple tasks like sudoku puzzles as of recently, their potential is outrageous for tasks such as encryption. Optical computing is another form of future technology which uses light waves to transfer data. Since the in fared beams do not interfere with each other, optical computers can be much smaller and more efficient that electronic computers. In fact, once optical computers have been mastered the computers will be able to process information at the speed of light using very little power at all. In years to come, the extraordinary power of supercomputers is predicted to be available in more common computers using technology like terascale computing to process at incredible speeds.
Review Definitions[edit | edit source]
Application Software: Programs that enable users to perform specific tasks on a computer, such as writing letters or playing games.
Computer: A programmable, electronic device that accepts data input, performs processing operations on that data, and outputs and stores the results.
Data: Raw, unorganized facts.
Information: Data that has been processed into a meaningful form.
Computer Network: A collection of computers and other hardware devices that are connected together to share hardware, software, and data, as well as to communicate electronically with one another.
Hardware: The physical parts of a computer system, such as the keyboard, monitor, printer, and so forth.
Internet Appliance: A specialized network computer designed primarily for Internet access and/or e-mail exchange.
Operating System: The main component of system software that enables a computer to operate, manage its activities and the resources under its control, run application programs, and interface with the user.
Output: The process of presenting the results of processing; can also refer to the results themselves.
Software: The instructions, also called computer programs, that are used to tell a computer what it should do.
Storage: The operation of saving data, programs, or output for future use.
URL: An Internet address (usually beginning with http://) that uniquely identifies a Web page.
Web browser: A program used to view Web pages.
World Wide Web (WWW): The collection of Web pages available through the Internet.
Web server: A computer that is continually connected to the Internet and hosts Web pages that are accessible through the Internet.
Review Questions[edit | edit source]
1) What is the key element of the CPU?
2) What are the connectors located on the exterior of the system unit that are used to connect external hardware devices?
3) What is an electronic path over which data travels?
4) _________ are locations on the motherboard into which _________ can be inserted to connect those cards to the motherboard.
5) What is used to store the essential parts of the operating system while the computer is running?
6) The ______________________ consists of a variety of circuitry and components that are packaged together and connected directly to the motherboard
7) A _________ is a thin board containing computer chips and other electronic components.
8) The main circuit board inside the system unit is called the ___________ .
9) Before a computer can execute any program instruction, such as requesting input from the user, moving a file from one storage device to another, or opening a new window on the screen, it must convert the instruction into a binary code known as ____________.
10) In order to synchronize all of a computer's operations, a __________ is used.
Review Answers[edit | edit source]
1) Transistor 2) Ports 3) Bus 4) Expansion slots, Expansion cards 5) RAM 6) Central Processing Unit 7) Circuit Board 8) Motherboard 9) Machine Language 10) System Clock
References[edit | edit source]
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https://en.wikibooks.org/wiki/Introduction_to_Computer_Information_Systems/The_System_Unit
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Degaussing is a process that reduces or eliminates the magnetism from an object or magnetic storage media. According to Wikipedia, degaussing works by applying an alternating magnetic field to an object, gradually reducing the magnetic field to zero. The term originated from a device known as a “degausseur” that was patented in 1884 by Charles Eugene Lancelot Brown to demagnetize the steel hulls of ships.
Degaussing became commonly used in the computing industry in the 1950s to erase data from magnetic storage like hard disk drives and tapes. It works by randomizing the magnetic domains on the drive platter, effectively removing any previously stored data. When applied to a hard drive, degaussing renders all the data unreadable by eliminating the magnetic signature holding the encoded bits. The process essentially demagnetizes the entire drive surface.
Compared to simply deleting files or reformatting a drive, degaussing is a more effective data sanitization method that helps prevent recovery of erased data. However, degaussing renders the drive permanently unusable. The process damages the platters and drive components through strong magnetic forces, making the hard drive unusable.
How Degaussing Works
Degaussing works by exposing magnetic storage media, like hard drives, to an alternating magnetic field. This field randomizes the orientation of magnetic domains on the drive’s platter surface, effectively erasing any previously stored data (Ontrack).
Specifically, degaussing machines generate a strong alternating magnetic field using electromagnets. As this alternating field is applied to a hard drive, it randomizes the magnetic orientation of each bit, disrupting the previously organized pattern that comprised the drive’s data. This renders the original data unreadable and overwritten (Shredstation).
The powerful alternating electromagnetic currents inside a degausser are able to penetrate casing and coatings to reach the internal platters. This allows degaussing to completly scramble the magnetic domains to erase data, without requiring the drive to be opened or removed from the computer (Ontrack).
Effect on Hard Drives
Degaussing has a significant impact on hard drives. It randomizes all of the data stored on the drive by exposing it to a powerful, alternating magnetic field. This essentially scrambles the magnetic orientation of the bits on the drive platters into a completely random pattern.
Once a hard drive has been degaussed, the data is rendered unrecoverable. Even using advanced forensic data recovery techniques, it is virtually impossible to reconstruct the original data after degaussing. This is because the process essentially destroys the underlying magnetic structure that the data was recorded onto.
Degaussing is considered more thorough and secure than simply formatting or deleting files on a hard drive. Formatting only removes the file system structure and does not touch the actual contents of the drive. Degaussing physically randomizes the magnetic fields, making recovery impossible. This is why degaussing is recommended for permanently destroying sensitive data on hard drives.
According to Drivesavers, “Data stored on a degaussed drive is irrevocably destroyed and has no hope of return or recovery.” (https://drivesaversdatarecovery.com/everything-you-need-to-know-about-hard-drive-degaussing/)
Advantages of Degaussing
There are several key advantages to using degaussing to erase data from hard drives:
Degaussing is one of the most secure methods for data destruction. According to Mediaduplicationsystems.com, “Degaussing a hard drive is the most secure way to erase data as it permanently destroys all data on the drive by randomizing the magnetic fields on the disk” (source). This makes retrieving or reconstructing data from a degaussed drive practically impossible.
The degaussing process is also much more environmentally friendly compared to physically destroying hard drives. As noted by AOZhouClick, “degaussing allows organizations to reuse, recycle, sell or donate hard drives after erasing them” (source). It is a non-destructive method.
Finally, degaussing can be a relatively quick process taking just seconds or minutes to completely erase data from a hard drive. The entire degaussing procedure is automated and efficient compared to manual data destruction techniques.
Limitations of Degaussing
While degaussing can be an effective method for sanitizing traditional hard disk drives, the technology does have some limitations to be aware of:
Degaussing may not fully erase data from solid-state drives (SSDs). SSDs store data in flash memory chips rather than on magnetic platters like traditional hard drives. The electromagnetic degaussing fields can disrupt but not completely erase the data stored on SSDs (https://datarecovery.com/rd/what-is-hard-drive-degaussing/).
Degaussing cannot target and erase specific data or files. The process simply eliminates all data stored on the drive by disrupting the magnetic field. You cannot selectively degauss certain files or directories (https://www.bitraser.com/article/data-destruction-techniques.php).
Specialized and expensive degaussing equipment is required. Degaussing machines generate powerful magnetic fields and are not standard IT equipment. Purchasing degaussing hardware represents a significant upfront investment (https://www.bitraser.com/article/data-destruction-techniques.php).
Degaussing vs Other Methods
Degaussing differs from other common data destruction methods like formatting and physical destruction in some key ways:
Compared to formatting, degaussing more completely removes data from a hard drive. Formatting simply removes address tables and pointers to data, but does not erase the data itself. Degaussing magnetically erases data from the drive platters at a low level to make recovery very difficult, if not impossible. This source provides more details on the differences.
Unlike physical destruction through shredding or crushing, degaussing allows the hard drive hardware to be reused after data erasure. So degaussing is preferable when the goal is to securely wipe sensitive data but also recover value from the hardware.
Degaussing is best suited for quickly erasing large volumes of data and hard drives. It’s preferable over other methods when reusable hardware and fast, secure data destruction are top priorities.
Standards and Regulations
There are standards and regulations regarding the degaussing of hard drives, especially when handling sensitive or classified data. Government agencies such as the Department of Defense (DoD) and National Security Agency (NSA) have established requirements for proper degaussing methods and equipment. Some key standards include:
The NSA Information Systems Security Organization has specified the NSA/CSS Evaluated Products List for High Security Crosscut Paper Disintegrators as NSA/CSS Specification 02-01 for degaussing and destroying paper documents (1).
DoD standard 5220.22-M provides requirements for clearing, sanitizing and destroying data storage devices including degaussing. Degaussers must meet a minimum level of 10,000 Gauss (2).
NIST Special Publication 800-36 provides guidelines on data sanitization including degaussing and specifies field strengths needed to sanitize various media (3).
To comply with these standards, organizations must utilize degaussers that are NSA-approved and meet the minimum Gauss rating. Using certified equipment from reputable vendors ensures proper sanitization and compliance with regulations like HIPAA, GDPR, and others requiring secure data destruction. Proper degaussing documentation should be maintained as evidence of due diligence.
When to Use Degaussing
Degaussing is most commonly used for secure data destruction when recycling or disposing of old hard drives. It provides a way to completely erase confidential data so it cannot be recovered. According to Verity Systems, degaussing is the most secure method for erasing hard drives. It is an ideal solution when you need to protect sensitive information before getting rid of a hard drive.
Specifically, you may want to degauss hard drives when:
- Permanently erasing data from old hard drives before disposal
- Sanitizing drives as part of an IT asset disposition or e-waste recycling program
- Removing confidential data prior to selling or donating used hard drives
- Destroying sensitive information from drives that are no longer needed
- Wiping data from hard drives of decommissioned computers and servers
Degaussing ensures data cannot be recovered from the drive even using advanced forensic methods. It provides a high level of security when permanently destroying data. This makes it an ideal choice over simply formatting or deleting files when faced with securely wiping confidential hard drive contents.
Degaussing Process Step-by-Step
To properly degauss a hard drive, you should take the following steps:
1. Prepare the hard drive by removing it from any enclosure or external casing. The degaussing field needs to be able to reach the disk platters directly.
2. Only use a certified degaussing tool that complies with industry standards like NSA/CSS EPL-1M (per PartitionWizard). Handheld degaussing wands are not powerful enough for modern high-density drives.
3. Run the degaussing cycle at least 3 times to ensure complete erasure. The high magnetic field will realign the magnetic domains to random patterns.
4. To confirm all data has been destroyed, run a utility like DiskWipe or Darik’s Boot and Nuke (DBAN) on the drive after degaussing. Then, do a full reformatting of the hard drive.
Following this proper procedure will magnetically sanitize the drive and ensures no data can be recovered, even using advanced forensic methods.
In summary, degaussing is an effective method for completely erasing data from hard drives by exposing them to a strong magnetic field. This magnetic field randomizes the orientation of magnetic domains on the drive, rendering previous data unrecoverable. Degaussing has advantages over physical destruction and software erasure, as it is fast, efficient, and meets regulatory standards like HIPAA for safe data destruction.
Looking ahead, degaussing will continue to play an important role in secure data wiping, especially for organizations with large volumes of drives to purge. As storage technologies evolve, degaussers will need to adapt to effectively erase new types of drives. Solid state drives may present challenges in the future, requiring adaptations or alternative erasure methods.
For readers interested in exploring degaussing further, associations like the National Association for Information Destruction provide resources on degausser selection, standards compliance, and best practices. Manufacturers also offer detailed product information to help organizations choose the right degaussing equipment to meet their data security needs.
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By the end of this section, you will be able to:
- Identify the mathematical relationships between the various properties of gases
- Use the ideal gas law, and related gas laws, to compute the values of various gas properties under specified conditions
During the seventeenth and especially eighteenth centuries, driven both by a desire to understand nature and a quest to make balloons in which they could fly (Figure 9.9), a number of scientists established the relationships between the macroscopic physical properties of gases, that is, pressure, volume, temperature, and amount of gas. Although their measurements were not precise by today’s standards, they were able to determine the mathematical relationships between pairs of these variables (e.g., pressure and temperature, pressure and volume) that hold for an ideal gas—a hypothetical construct that real gases approximate under certain conditions. Eventually, these individual laws were combined into a single equation—the ideal gas law—that relates gas quantities for gases and is quite accurate for low pressures and moderate temperatures. We will consider the key developments in individual relationships (for pedagogical reasons not quite in historical order), then put them together in the ideal gas law.
Pressure and Temperature: Amontons’s Law
Imagine filling a rigid container attached to a pressure gauge with gas and then sealing the container so that no gas may escape. If the container is cooled, the gas inside likewise gets colder and its pressure is observed to decrease. Since the container is rigid and tightly sealed, both the volume and number of moles of gas remain constant. If we heat the sphere, the gas inside gets hotter (Figure 9.10) and the pressure increases.
This relationship between temperature and pressure is observed for any sample of gas confined to a constant volume. An example of experimental pressure-temperature data is shown for a sample of air under these conditions in Figure 9.11. We find that temperature and pressure are linearly related, and if the temperature is on the kelvin scale, then P and T are directly proportional (again, when volume and moles of gas are held constant); if the temperature on the kelvin scale increases by a certain factor, the gas pressure increases by the same factor.
Guillaume Amontons was the first to empirically establish the relationship between the pressure and the temperature of a gas (~1700), and Joseph Louis Gay-Lussac determined the relationship more precisely (~1800). Because of this, the P-T relationship for gases is known as either Amontons’s law or Gay-Lussac’s law. Under either name, it states that the pressure of a given amount of gas is directly proportional to its temperature on the kelvin scale when the volume is held constant. Mathematically, this can be written:
where ∝ means “is proportional to,” and k is a proportionality constant that depends on the identity, amount, and volume of the gas.
For a confined, constant volume of gas, the ratio is therefore constant (i.e., ). If the gas is initially in “Condition 1” (with P = P1 and T = T1), and then changes to “Condition 2” (with P = P2 and T = T2), we have that and which reduces to This equation is useful for pressure-temperature calculations for a confined gas at constant volume. Note that temperatures must be on the kelvin scale for any gas law calculations (0 on the kelvin scale and the lowest possible temperature is called absolute zero). (Also note that there are at least three ways we can describe how the pressure of a gas changes as its temperature changes: We can use a table of values, a graph, or a mathematical equation.)
Predicting Change in Pressure with TemperatureA can of hair spray is used until it is empty except for the propellant, isobutane gas.
(a) On the can is the warning “Store only at temperatures below 120 °F (48.8 °C). Do not incinerate.” Why?
(b) The gas in the can is initially at 24 °C and 360 kPa, and the can has a volume of 350 mL. If the can is left in a car that reaches 50 °C on a hot day, what is the new pressure in the can?
Solution(a) The can contains an amount of isobutane gas at a constant volume, so if the temperature is increased by heating, the pressure will increase proportionately. High temperature could lead to high pressure, causing the can to burst. (Also, isobutane is combustible, so incineration could cause the can to explode.)
(b) We are looking for a pressure change due to a temperature change at constant volume, so we will use Amontons’s/Gay-Lussac’s law. Taking P1 and T1 as the initial values, T2 as the temperature where the pressure is unknown and P2 as the unknown pressure, and converting °C to K, we have:
Rearranging and solving gives:
Check Your LearningA sample of nitrogen, N2, occupies 45.0 mL at 27 °C and 600 torr. What pressure will it have if cooled to –73 °C while the volume remains constant?
Volume and Temperature: Charles’s Law
If we fill a balloon with air and seal it, the balloon contains a specific amount of air at atmospheric pressure, let’s say 1 atm. If we put the balloon in a refrigerator, the gas inside gets cold and the balloon shrinks (although both the amount of gas and its pressure remain constant). If we make the balloon very cold, it will shrink a great deal, and it expands again when it warms up.
These examples of the effect of temperature on the volume of a given amount of a confined gas at constant pressure are true in general: The volume increases as the temperature increases, and decreases as the temperature decreases. Volume-temperature data for a 1-mole sample of methane gas at 1 atm are listed and graphed in Figure 9.12.
The relationship between the volume and temperature of a given amount of gas at constant pressure is known as Charles’s law in recognition of the French scientist and balloon flight pioneer Jacques Alexandre César Charles. Charles’s law states that the volume of a given amount of gas is directly proportional to its temperature on the kelvin scale when the pressure is held constant.
Mathematically, this can be written as:
with k being a proportionality constant that depends on the amount and pressure of the gas.
For a confined, constant pressure gas sample, is constant (i.e., the ratio = k), and as seen with the P-T relationship, this leads to another form of Charles’s law:
Predicting Change in Volume with TemperatureA sample of carbon dioxide, CO2, occupies 0.300 L at 10 °C and 750 torr. What volume will the gas have at 30 °C and 750 torr?
SolutionBecause we are looking for the volume change caused by a temperature change at constant pressure, this is a job for Charles’s law. Taking V1 and T1 as the initial values, T2 as the temperature at which the volume is unknown and V2 as the unknown volume, and converting °C into K we have:
Rearranging and solving gives:
This answer supports our expectation from Charles’s law, namely, that raising the gas temperature (from 283 K to 303 K) at a constant pressure will yield an increase in its volume (from 0.300 L to 0.321 L).
Check Your LearningA sample of oxygen, O2, occupies 32.2 mL at 30 °C and 452 torr. What volume will it occupy at –70 °C and the same pressure?
Measuring Temperature with a Volume ChangeTemperature is sometimes measured with a gas thermometer by observing the change in the volume of the gas as the temperature changes at constant pressure. The hydrogen in a particular hydrogen gas thermometer has a volume of 150.0 cm3 when immersed in a mixture of ice and water (0.00 °C). When immersed in boiling liquid ammonia, the volume of the hydrogen, at the same pressure, is 131.7 cm3. Find the temperature of boiling ammonia on the kelvin and Celsius scales.
SolutionWhen immersed in an ice-water bath at 0.00 °C (T1), the thermometer’s gas volume is 150.0 cm3 (V1). When immersed in boiling liquid ammonia (T2), the thermometer’s gas volume is 131.7 cm3. The relation between volume and temperature at constant pressure is provided by Charles’s Law:
Subtracting 273.15 from 239.8 K, we find that the temperature of the boiling ammonia on the Celsius scale is –33.4 °C.
Check Your LearningWhat is the volume of a sample of ethane at 467 K and 1.1 atm if it occupies 405 mL at 298 K and 1.1 atm?
Volume and Pressure: Boyle’s Law
If we partially fill an airtight syringe with air, the syringe contains a specific amount of air at constant temperature, say 25 °C. If we slowly push in the plunger while keeping temperature constant, the gas in the syringe is compressed into a smaller volume and its pressure increases; if we pull out the plunger, the volume increases and the pressure decreases. This example of the effect of volume on the pressure of a given amount of a confined gas is true in general. Decreasing the volume of a contained gas will increase its pressure, and increasing its volume will decrease its pressure. In fact, if the volume increases by a certain factor, the pressure decreases by the same factor, and vice versa. Volume-pressure data for an air sample at room temperature are graphed in Figure 9.13.
Unlike the P-T and V-T relationships, pressure and volume are not directly proportional to each other. Instead, P and V exhibit inverse proportionality: Increasing the pressure results in a decrease of the volume of the gas. Mathematically this can be written:
with k being a constant. Graphically, this relationship is shown by the straight line that results when plotting the inverse of the pressure versus the volume (V), or the inverse of volume versus the pressure (P). Graphs with curved lines are difficult to read accurately at low or high values of the variables, and they are more difficult to use in fitting theoretical equations and parameters to experimental data. For those reasons, scientists often try to find a way to “linearize” their data. If we plot P versus V, we obtain a hyperbola (see Figure 9.14).
The relationship between the volume and pressure of a given amount of gas at constant temperature was first published by the English natural philosopher Robert Boyle over 300 years ago. It is summarized in the statement now known as Boyle’s law: The volume of a given amount of gas held at constant temperature is inversely proportional to the pressure under which it is measured.
Volume of a Gas SampleThe sample of gas in Figure 9.13 has a volume of 15.0 mL at a pressure of 13.0 psi. Determine the pressure of the gas at a volume of 7.5 mL, using:
(a) the P-V graph in Figure 9.13
(b) the vs. V graph in Figure 9.13
(c) the Boyle’s law equation
Comment on the likely accuracy of each method.
Solution(a) Estimating from the P-V graph gives a value for P somewhere around 27 psi.
(b) Estimating from the versus V graph give a value of about 26 psi.
(c) From Boyle’s law, we know that the product of pressure and volume (PV) for a given sample of gas at a constant temperature is always equal to the same value. Therefore we have P1V1 = k and P2V2 = k which means that P1V1 = P2V2.
Using P1 and V1 as the known values 13.0 psi and 15.0 mL, P2 as the pressure at which the volume is unknown, and V2 as the unknown volume, we have:
It was more difficult to estimate well from the P-V graph, so (a) is likely more inaccurate than (b) or (c). The calculation will be as accurate as the equation and measurements allow.
Check Your LearningThe sample of gas in Figure 9.13 has a volume of 30.0 mL at a pressure of 6.5 psi. Determine the volume of the gas at a pressure of 11.0 psi, using:
(a) the P-V graph in Figure 9.13
(b) the vs. V graph in Figure 9.13
(c) the Boyle’s law equation
Comment on the likely accuracy of each method.
(a) about 17–18 mL; (b) ~18 mL; (c) 17.7 mL; it was more difficult to estimate well from the P-V graph, so (a) is likely more inaccurate than (b); the calculation will be as accurate as the equation and measurements allow
Breathing and Boyle’s Law
What do you do about 20 times per minute for your whole life, without break, and often without even being aware of it? The answer, of course, is respiration, or breathing. How does it work? It turns out that the gas laws apply here. Your lungs take in gas that your body needs (oxygen) and get rid of waste gas (carbon dioxide). Lungs are made of spongy, stretchy tissue that expands and contracts while you breathe. When you inhale, your diaphragm and intercostal muscles (the muscles between your ribs) contract, expanding your chest cavity and making your lung volume larger. The increase in volume leads to a decrease in pressure (Boyle’s law). This causes air to flow into the lungs (from high pressure to low pressure). When you exhale, the process reverses: Your diaphragm and rib muscles relax, your chest cavity contracts, and your lung volume decreases, causing the pressure to increase (Boyle’s law again), and air flows out of the lungs (from high pressure to low pressure). You then breathe in and out again, and again, repeating this Boyle’s law cycle for the rest of your life (Figure 9.15).
Moles of Gas and Volume: Avogadro’s Law
The Italian scientist Amedeo Avogadro advanced a hypothesis in 1811 to account for the behavior of gases, stating that equal volumes of all gases, measured under the same conditions of temperature and pressure, contain the same number of molecules. Over time, this relationship was supported by many experimental observations as expressed by Avogadro’s law: For a confined gas, the volume (V) and number of moles (n) are directly proportional if the pressure and temperature both remain constant.
In equation form, this is written as:
Mathematical relationships can also be determined for the other variable pairs, such as P versus n, and n versus T.
The Ideal Gas Law
To this point, four separate laws have been discussed that relate pressure, volume, temperature, and the number of moles of the gas:
- Boyle’s law: PV = constant at constant T and n
- Amontons’s law: = constant at constant V and n
- Charles’s law: = constant at constant P and n
- Avogadro’s law: = constant at constant P and T
Combining these four laws yields the ideal gas law, a relation between the pressure, volume, temperature, and number of moles of a gas:
where P is the pressure of a gas, V is its volume, n is the number of moles of the gas, T is its temperature on the kelvin scale, and R is a constant called the ideal gas constant or the universal gas constant. The units used to express pressure, volume, and temperature will determine the proper form of the gas constant as required by dimensional analysis, the most commonly encountered values being 0.08206 L atm mol–1 K–1 and 8.314 kPa L mol–1 K–1.
Gases whose properties of P, V, and T are accurately described by the ideal gas law (or the other gas laws) are said to exhibit ideal behavior or to approximate the traits of an ideal gas. An ideal gas is a hypothetical construct that may be used along with kinetic molecular theory to effectively explain the gas laws as will be described in a later module of this chapter. Although all the calculations presented in this module assume ideal behavior, this assumption is only reasonable for gases under conditions of relatively low pressure and high temperature. In the final module of this chapter, a modified gas law will be introduced that accounts for the non-ideal behavior observed for many gases at relatively high pressures and low temperatures.
The ideal gas equation contains five terms, the gas constant R and the variable properties P, V, n, and T. Specifying any four of these terms will permit use of the ideal gas law to calculate the fifth term as demonstrated in the following example exercises.
Using the Ideal Gas LawMethane, CH4, is being considered for use as an alternative automotive fuel to replace gasoline. One gallon of gasoline could be replaced by 655 g of CH4. What is the volume of this much methane at 25 °C and 745 torr?
SolutionWe must rearrange PV = nRT to solve for V:
If we choose to use R = 0.08206 L atm mol–1 K–1, then the amount must be in moles, temperature must be in kelvin, and pressure must be in atm.
Converting into the “right” units:
It would require 1020 L (269 gal) of gaseous methane at about 1 atm of pressure to replace 1 gal of gasoline. It requires a large container to hold enough methane at 1 atm to replace several gallons of gasoline.
Check Your LearningCalculate the pressure in bar of 2520 moles of hydrogen gas stored at 27 °C in the 180-L storage tank of a modern hydrogen-powered car.
If the number of moles of an ideal gas are kept constant under two different sets of conditions, a useful mathematical relationship called the combined gas law is obtained: using units of atm, L, and K. Both sets of conditions are equal to the product of n R (where n = the number of moles of the gas and R is the ideal gas law constant).
Using the Combined Gas LawWhen filled with air, a typical scuba tank with a volume of 13.2 L has a pressure of 153 atm (Figure 9.16). If the water temperature is 27 °C, how many liters of air will such a tank provide to a diver’s lungs at a depth of approximately 70 feet in the ocean where the pressure is 3.13 atm?
SolutionLetting 1 represent the air in the scuba tank and 2 represent the air in the lungs, and noting that body temperature (the temperature the air will be in the lungs) is 37 °C, we have:
Solving for V2:
(Note: Be advised that this particular example is one in which the assumption of ideal gas behavior is not very reasonable, since it involves gases at relatively high pressures and low temperatures. Despite this limitation, the calculated volume can be viewed as a good “ballpark” estimate.)
Check Your LearningA sample of ammonia is found to occupy 0.250 L under laboratory conditions of 27 °C and 0.850 atm. Find the volume of this sample at 0 °C and 1.00 atm.
The Interdependence between Ocean Depth and Pressure in Scuba Diving
Whether scuba diving at the Great Barrier Reef in Australia (shown in Figure 9.17) or in the Caribbean, divers must understand how pressure affects a number of issues related to their comfort and safety.
Pressure increases with ocean depth, and the pressure changes most rapidly as divers reach the surface. The pressure a diver experiences is the sum of all pressures above the diver (from the water and the air). Most pressure measurements are given in units of atmospheres, expressed as “atmospheres absolute” or ATA in the diving community: Every 33 feet of salt water represents 1 ATA of pressure in addition to 1 ATA of pressure from the atmosphere at sea level. As a diver descends, the increase in pressure causes the body’s air pockets in the ears and lungs to compress; on the ascent, the decrease in pressure causes these air pockets to expand, potentially rupturing eardrums or bursting the lungs. Divers must therefore undergo equalization by adding air to body airspaces on the descent by breathing normally and adding air to the mask by breathing out of the nose or adding air to the ears and sinuses by equalization techniques; the corollary is also true on ascent, divers must release air from the body to maintain equalization. Buoyancy, or the ability to control whether a diver sinks or floats, is controlled by the buoyancy compensator (BCD). If a diver is ascending, the air in their BCD expands because of lower pressure according to Boyle’s law (decreasing the pressure of gases increases the volume). The expanding air increases the buoyancy of the diver, and they begin to ascend. The diver must vent air from the BCD or risk an uncontrolled ascent that could rupture the lungs. In descending, the increased pressure causes the air in the BCD to compress and the diver sinks much more quickly; the diver must add air to the BCD or risk an uncontrolled descent, facing much higher pressures near the ocean floor. The pressure also impacts how long a diver can stay underwater before ascending. The deeper a diver dives, the more compressed the air that is breathed because of increased pressure: If a diver dives 33 feet, the pressure is 2 ATA and the air would be compressed to one-half of its original volume. The diver uses up available air twice as fast as at the surface.
Standard Conditions of Temperature and Pressure
We have seen that the volume of a given quantity of gas and the number of molecules (moles) in a given volume of gas vary with changes in pressure and temperature. Chemists sometimes make comparisons against a standard temperature and pressure (STP) for reporting properties of gases: 273.15 K and 1 atm (101.325 kPa).1 At STP, one mole of an ideal gas has a volume of about 22.4 L—this is referred to as the standard molar volume (Figure 9.18).
- 1The IUPAC definition of standard pressure was changed from 1 atm to 1 bar (100 kPa) in 1982, but the prior definition remains in use by many literature resources and will be used in this text.
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STEP 3 Develop Strategies for Success
6 General Strategies
IN THIS CHAPTER
Summary: This chapter contains general strategies useful for the entire AP Physics 2 exam—multiple-choice and free-response sections. First, let’s talk about AP Physics 1 and what you need to remember. Second, I’ll discuss the tools you have at your disposal (calculator, a table of information, and equation sheet) and how to use them. Next, I’ll investigate what those equations you are given mean, how to relate them to a graph, and how to use a graph to find information. Finally, we’ll work on ranking task skills.
You should dust off that 5 Steps to a 5 AP Physics 1 book you had last year. The skills you learned in AP Physics 1 are going to be needed.
Sure you can have a calculator, but it won’t help you for most of the exam. Only use it when you actually need it.
The table of information/equation sheet is good to have in a pinch, but it won’t save you if you don’t know what it all means.
Each equation tells a story of a relationship. Graphs are a picture of these relationships. Learn to see the relationships.
There are three ways to get information from a graph: (1) read it, (2) find the slope, and (3) calculate the area under the graph.
Ranking task questions show up in both multiple-choice and free-response questions. Some require conceptual analysis and others have numbers.
What Do I Need to Remember from AP Physics 1?
The short answer is everything. The prior skills you learned in AP Physics 1 are expected knowledge on the AP Physics 2 exam. Don’t panic. You won’t be asked any questions about blocks on an incline attached to a pulley. Only the content in AP Physics 2 is tested. However, the information you learned about forces, energy, momentum, motion, graphing, free body diagrams, and all the rest is assumed to be still accessible in your brain. Physics is cumulative. There won’t be any roller coasters going around a track, but there will be charged particles that experience forces, accelerate, and convert potential energy into kinetic energy. All the skills you learned last year will help you this year.
So what do you do if all that past information is fuzzy? Ask your teacher to review the concepts and dust off that 5 Steps to a 5 AP Physics 1 book you had last year.
Tools You Can Use
You can use a calculator on both sections of the AP exam. Most calculators are acceptable—scientific calculators, programmable calculators, graphing calculators. However, you cannot use a calculator with a QWERTY keyboard, and you’ll be restricted from using any calculators that make noise. You also cannot share a calculator with anyone during the exam.
The real question, though, is whether a calculator will really help you. The short answer is “Yes”: You will be asked a few questions on the exam that require you to do messy calculations. The longer answer, though, is “Yes, but it won’t help very much.”
The majority of the questions on the exam, both multiple choice and free response, don’t have any numbers at all.
There are questions that have numbers but don’t want a numerical answer. For example:
A convex lens of focal length f = 0.2 m is used to examine a small coin lying on a table. During the examination the lens is held a distance of 0.3 m above the coin and is moved slowly to a distance of 0.1 m above the coin. During this process, what happens to the image of the coin?
(A) The image continually increases in size.
(B) The image continually decreases in size.
(C) The image gets smaller at first and then bigger in size.
(D) The image flips over.
The numbers in these problems are only there to set the problem up. (The correct answer is D).
Then there are questions with numerical answers but using a calculator is counterproductive. For example:
A cylinder with a movable piston contains a gas at pressure P = 1 × 105 Pa, volume V = 20 cm3, and temperature T = 273 K. The piston is moved downward in a slow, steady fashion, allowing heat to escape the gas and the temperature to remain constant. If the final volume of the gas is 5 cm3, what will be the resulting pressure?
(A) 0.25 × 105 Pa
(B) 2 × 105 Pa
(C) 4 × 105 Pa
(D) 8 × 105 Pa
Using your calculator to solve this one will take too much time. You can do this one in your head: PV = nRT, nRT is constant. So, if the volume is four times smaller, the pressure has to be four times greater! Correct answer (C) 4 × 105 Pa. In fact, many times the numerical calculations are simple or involve ratios that don’t require a calculator.
Here is the big takeaway—use your calculator only when it is absolutely necessary.
Special Note for Students in AP Physics 2 Classes
Many, if not most, of your assignments in class involve numerical problems. What can you do? Start by trying to solve every problem without a calculator first. Be resourceful. Draw a diagram, sketch a graph, use equations with symbols only, etc. Second, work the conceptual problems from your textbook and ask your teacher for the key. Practice the skills that will make you successful on the AP exam.
The Table of Information and the Equation Sheet
The other tools you can use are the table of information and equation sheet. You will be given a copy of these sheets in your exam booklet. It’s a handy reference because it lists all the constants, math formulas, and the equations that you’re expected to know for the exam.
However, the equation sheet can also be dangerous. Too often, students interpret the equation sheet as an invitation to stop thinking: “Hey, they tell me everything I need to know, so I can just plug-and-chug through the rest of the exam!” Nothing could be further from the truth.
First of all, you’ve already memorized the equations on the sheet. It might be reassuring to look up an equation during the AP exam, just to make sure that you’ve remembered it correctly. And maybe you’ve forgotten a particular equation, but seeing it on the sheet will jog your memory. This is exactly what the equation sheet is for, and in this sense, it’s pretty nice to have around. But beware of the following:
• Don’t look up an equation unless you know exactly what you’re looking for. It might sound obvious, but if you don’t know what you’re looking for, you won’t find it.
• Don’t go fishing. If part of a free-response question asks you to find an object’s velocity, and you’re not sure how to do that, don’t just rush to the equations sheet and search for every equation with a “V” in it.
If your teacher has not issued you the official AP Physics 2 table of information and equation sheet, download one from the College Board at https://apstudents.collegeboard.org/courses/ap-physics-2-algebra-based/assessment. Exam day shouldn’t be the first time you see these tools.
Get to Know the Relationships
Now that you have an official AP Physics 2 equation sheet, let’s talk about what the jumble of symbols tell us. Take a look under the “FLUID MECHANICS AND THERMAL PHYSICS” heading. See the equation PV = nRT? What does it tell us? It shows us how all these individual quantities are related and what their relationship is. Rearranging the equation for P we get: . T is in the numerator, which means that if T doubles, and all the other variables on the right stay the same, P must also double. Pressure is directly proportional to temperature: P ∝ T. See graph #2 below. If V doubles, and all the other variables stay the same, P will be cut in half. Pressure is inversely proportional to volume: . On a graph, an inverse relationship looks like #4. What other relationships are we likely to see? Shown below are the six most frequent relationships in AP Physics 2.
Kinetic energy . Kinetic energy is directly proportional to the velocity squared, K ∝ v2. It will have a graph like #3. If you double the velocity, the kinetic energy quadruples: 4K ∝ (2v)2.
Rearrange the kinetic energy equation to solve for . Velocity is proportional to the square root of the kinetic energy ; see graph #5. If you double the kinetic energy, the velocity goes up by a factor of .
Electric force: . The electric force is inversely proportional to the radius squared , but it is directly related to the charge FE ∝ q. See graphs #2 and #4. This means that if you double one charge and also double the radius, the force will be cut in half: .
There are always questions on the exam that can be solved this way. Learning how to work with these relationships is crucial to doing well on the exam because they don’t require a calculator and save you time. Put your calculator away and practice this skill all year long.
What Information Can We Get from a Graph?
Gathering information from a graph is another highly prized skill on the AP exam. Let’s spend some time making sure you have it down cold. The good thing is there are only three things you can do with a graph: read it, find the slope, or find the area. But, before you can do that, you need to examine the graph. Look at the x-axis and y-axis. What do they represent? What are the variables? What are the units? Which physics relationships (equations) relate to this graph?
1. Read the Graph
Look at this graph. The data is not in a perfect straight line. This is common on the AP exam, as it includes real data like you would get in an actual lab. If you get data like this, sketch a line or curve that seems to best fit the data.
This data seems straight, so draw a “best-fit” line through the data that splits it down the middle. Your line may not touch any of the data points. That’s OK. Once you have your best-fit line, forget about the data points and concentrate only on the line you have drawn. The AP exam may ask you to extrapolate beyond the existing data or interpolate between points. For example: at a current of 2 amps the power is approximately 6 watts.
2. Find the Slope
In math class you calculated the slope of lines, but most of the time it didn’t have a physical meaning. In physics, the slope usually represents something real. Take a look at the axis on the graph. Power is on the y-axis and current on the x-axis. Ask yourself if there is a physics relationship between power and current. P = IΔV seems to fit the bill. In math class the equation of a line is y = mx + b. Now line up the physics equation with the math equation to find out what the slope’s physics meaning is. Turns out that the slope is the potential difference!
This procedure of matching up the physical equation with the math equation of a line will help you find the physics meaning of the slope every time.
Now that we know the slope represents the potential difference, we need to calculate it. Slope is rise over run. Pick two convenient points. I used points (1 A, 3 W) and (3 A, 9 W). Thus, the slope is (9 W — 3 W) / (3 A — 1 A) = 3 W/A = 3 V.
CAUTION! Never choose a plotted point unless it actually falls on your best-fit line. This will give you the wrong slope. Notice that one of the chosen points was an actual data point (1 A, 3 W) because it was on the best-fit line. That’s OK. The other point (3 A, 9 W) was not a data point.
Let’s practice. The graph (shown above) has pressure as a function of depth. (Watch out for the units on the axis: kPa.) This looks like a fluids problem. The equation P = P0 + ρgh seems to fit. Now let’s match the physics equation with the math equation.
So the slope is the density times the acceleration due to gravity. Using the circled points, we get a slope of 10 kPa/m. This time there is a y-intercept, which is the pressure on top of the fluid, P0 = 100 kPa.
Sometimes the axes are strange. Take a look at the graph, which shows the inverse of current as a function of resistance. Don’t fret! What is the physics relationship between current and resistance? . Now match up this equation with y = mx + b:
The slope is the inverse of the potential difference. That’s kind of strange but if that is the graph they give you, just go with it. The slope = 0.006 (1/A)/Ω. Taking the inverse, we get the potential difference of 167 V.
One last hard one! The graph, shown above, is the inverse of the image distance as a function of the inverse of the object distance. What a mess, but who cares? We can handle it. Image and object distances imply optics: . Now match the equation up:
The slope turns out to equal −1 and does not have any physical meaning this time. The y-intercept is the inverse of the focal length.
Occasionally the AP exam asks that you to take a graph that is curved and produce a graph that has a straight line. This is called linearization. It is the reverse process of above. Take the equation and match it up with y = mx + b to see what you should graph. Take this equation: n1 sin θ1 = n2 sin θ2. Let’s put the θ1 on the x-axis and θ2 on the y-axis. Match the equation. Plot what it tells you to plot, and you get a straight line from your data. Piece of cake.
3. Find the Area
This time we look to see if multiplying the x- and y-axis variables will produce anything meaningful. If so, the space under the graph has a physical meaning. Take a look at the graph above—pressure times a changing volume. That looks like something to do with gases: W = —PΔV. The area under the graph is the work. Calculate the area of a graph just like the area of a geometric shape. Keep in mind that the “units” of our graph area will be Pa · m not a geometric unit like meters squared.
The area of this graph is (6 × 105 Pa)(9 × 10—3 m − 2 × 10—3 m) = 4200 Pa · m = 4200 J. Since our equation was W = —PΔV, our final answer is negative: W = —4200 J of work.
Ranking Task Skills
Ranking tasks are an interesting type of question that can show up in both the multiple-choice and free-response portions of the exam. Here is an example:
A battery of potential difference ε is connected to the circuit pictured above. The circuit consists of three resistors and four ammeters. Rank the readings on the ammeters from greatest to least.
They are not asking for any numbers, and in most cases, trying to use numbers to solve the problem is much more time-consuming than using conceptual reasoning and semi-quantitative reasoning. In this example, we see that ammeters A1 and A4 have the same current because they are in a same single pathway. This is the main pathway that feeds the rest of the circuit. This main current splits before passing through the lower two resistors. The 20-Ω resistor will have less current passing through it than the 10-Ω resistor. Thus, the ranking from greatest to least is A1 = A4 > A3 > A2. No numerical calculations were needed, just physics reasoning.
On a free-response question, make sure you write your answer in a clear way that cannot be misunderstood and designate any that are equal. For example: Greatest (A1 = A4) > A3 > A2 Least. On a multiple-choice question, look to save time. As soon as you figure out the ranking of any pair, look for any answer choices that don’t have that pairing and cross them out. For example, look at the answer choices below:
(A) A1 > A3 > A2 > A4
(B) A1 > A3 = A2 > A4
(C) A1 > A4 > A3 > A2
(D) A1 = A4 > A3 > A2
When you determine that A1 = A4, a quick look at the answer choices shows that only answer choice (D) will work.
There are students who are more comfortable with numerical thinking. If that is the case with you, you can choose a number for ε and work out the currents. But this will almost always take longer.
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Executive functionNeed help?
Executive function is a set of skills that stems from the coordination of three cognitive processes: cognitive flexibility, working memory, and inhibitory control.
- Cognitive flexibility is the ability to pay attention and switch attention from one task to another.
- Working memory enables us to mentally hold and process information.
- Inhibitory control allows us to stop an impulse and display a more appropriate response.
These skills help us plan, focus, remember instructions, and complete tasks.
Use the following reading to learn more about the various components of executive function and how these can be strengthened. As you read, think about how the different processes that make up executive function support children with learning, and with social and emotional competence.
What is executive function?
Some children have an easier time paying attention than others. Some children follow directions well, but others do not. Some children are more likely to hit others when they feel frustrated, rather than stopping and using their words instead. Many of the differences we see in young children’s behaviour relate to their executive function. Executive function is a set of skills that stems from the coordination of three cognitive processes: cognitive flexibility, working memory, and inhibitory control. These skills help us plan, focus, remember instructions and complete tasks. Executive function is important throughout life and starts to develop early.
Cognitive flexibility is the ability to pay attention and switch attention from one task to another. For example, children use cognitive flexibility when they focus on one activity, such as building with blocks, but then switch to another activity, putting the blocks away and joining their peers for a story.
Working memory enables us to mentally hold and process information. Young children use working memory when they have to remember and follow one or more instructions, such as when working on an art project and then putting their materials away.
Inhibitory control allows us to stop an impulse and display a more appropriate response. We see this often in young children when they have to take turns in sharing a desirable toy (for example, asking ‘Can I have a turn?’ rather than grabbing the toy). In young children, the three aspects of executive function work together and can be seen in many different ways, such as when a child has to listen and follow directions, ignore distractions, and wait in line.
How executive function develops
Executive function begins to develop early in life. Babies who experience warm and supportive interactions with important adults in their lives are more likely to feel safe and secure. This helps children develop positive relationships with parents and adults, giving children the confidence they need to explore their world and develop independence and problem-solving skills. Secure relationships also lead to strong social emotional development and executive function skills in young children. Children who develop executive function skills early in life are more likely to show self-control, especially as they get older and make the transition to more structured learning environments.
Executive function skills are important
Early education teachers often report that children’s executive function skills are foundational for success in educational settings and social situations. More than two decades of research have shown that these skills are important for many aspects of our lives, including:
- Mental and physical health across the lifespan
- Effective social communication
- Short and long-term success in school
- University completion
In fact, executive function has been a stronger predictor of early academic achievement than IQ.
Although executive function is a key predictor of many outcomes, a significant number of young children struggle with these skills. This is especially evident when children make the transition from early childhood settings to formal educational settings such as primary school, which are often more structured than ECE settings. Many young children easily transition to primary school but a significant number of children experience difficulty. Teachers report that young children struggle most with challenging behaviours that relate to aspects of executive function like being able to focus and pay attention, persisting with tasks, and demonstrating self-control in academic and social situations. This is concerning because we know that these skills help children navigate classroom settings. In fact, children who struggle with executive function are more likely to dislike school and become disengaged, which can place them at risk long-term.
Strengthening executive function skills
Based on evidence showing us how important executive function is for children’s school success, an essential question to consider is how to support development of these skills in young children. Executive function skills are particularly malleable in early childhood, and intervention research has shown that these skills can be taught, practised, and improved. This is especially evident for children who struggle with executive function skills. For example, children aged 3-5 who participated in an intervention aimed at helping children practise executive function skills with music and movement games (called Red Light, Purple Light!) demonstrated improvement in their executive function skills and early academic achievement compared with children in a control group. Providing children with opportunities to practise executive function skills in fun and engaging ways has been shown to help children improve these skills and then demonstrate them in a variety of settings, including home and school.
Strategies to improve executive function skills
Parents, teachers and other adults serve an important role in helping children develop executive function skills. As noted, positive early relationships lay the foundation for executive function skills by helping children feel safe, secure and ready to explore and problem-solve. Parents and teachers do many things that encourage the development of children’s executive function, even if they do not know it! Below we include several strategies that teachers can use to support these skills in early childhood settings.
- Take time to build relationships with children. This can be hard when there are many children in a group and when individual children may need extra support! However, taking time to build positive teacher-child relationships provides children with a strong foundation for social emotional skills and learning. Children who have strong relationships with their teachers make greater gains in school readiness and positive behaviour over the year.
- Model what strong executive function skills look like. Children look to adults as a guide for their own behaviour, and one way teachers can support executive function in the early childhood settings is by talking aloud. For example, teachers can narrate their actions as they walk through the space and clean up: ‘We need to clean up the toys at activity centres, so I’m going to start with the art centre first and then clean the dramatic play centre. Then we will be ready to go outside to play!’ By modelling positive behaviour, children can see how adults use executive function in their daily lives to be organised and planful.
- Set up the space to promote executive function skills. Teachers can organise learning and play spaces in ways that encourage children to practise executive function skills. In order to support executive function, it helps to plan and focus activities that can build upon one other. For example, teachers can allow children the opportunity to move between relatively unstructured activities, like dramatic play, and more complex activities like a multi-step art activity where children need to remember and follow directions while ignoring distractions to stay on task. Children need both types of activities to practise executive function skills and then process what they are learning through play. Teachers can also give children materials and activities that require them to practise executive function skills. For example, teachers can promote focus and attention (which are important parts of executive function) by having children practise their fine motor skills in a maths game that involves them having to use tweezers to sort small manipulatives into categories (such as colour).
- Use games as a teaching tool. Children develop strong executive function when they practise these skills in different contexts and settings. This means that it is important to practise executive function skills outside of challenging moments. One fun and simple way to incorporate executive function into everyday activities is to use music and movement games and add steps to make them more complex over time. For example, interventions such as Red Light, Purple Light! include games that become more challenging over time. In one game, the Freeze game, children dance to music and then freeze when the music stops. After children practise the basic rules of the game, more complex rules are added. Children are then asked to dance quickly to fast music, slowly to slow music and freeze when the music stops. To add another level of complexity, children are then asked to do the opposite (which can be tricky!) and dance quickly to slow music and slowly to fast music. In another game, called the Sleeping Game, the teacher sings a short lullaby and the children pretend to go to sleep when they hear the song. The children then ‘wake up’ when the teacher says: ‘and when they woke up, they were kangaroos hopping around the room’. Children move around the room pretending to be the animal or action named by the teacher. The teacher then uses the lullaby as a cue for children to pretend to sleep again. As children learn new executive function activities and games, teachers can also allow opportunities for children to lead the group in game play (for example, a child names the animal or action during the sleeping game). These are just a few examples of ways that teachers can embed aspects of executive function into their everyday activities. Typical activities can be easily modified to more explicitly support children’s executive function skills as well.
- Engage families in supporting executive function at home. Children’s first teachers are their parents and other important adults in their life. Engaging families in activities surrounding the development of executive function skills can provide the extra support children need to succeed. Teachers can play games like the ones mentioned above with children as part of family events or open days. This helps parents see some of the ways that they can promote executive function skills at home. Teachers can also share information with families about the importance of executive function skills, encourage parents and other adults to model these skills themselves, and send home flyers with examples of activities that families can do at home to help children practise executive function skills.
Children’s executive function skills include their ability to focus and pay attention, remember instructions and demonstrate self-control. These skills are important aspects of early learning and development that help children regulate their behaviour and they are correlated with social and academic success. Executive function skills develop early in life and are supported through warm and secure relationships. The early childhood years are a sensitive period of development when these skills are especially malleable. Teachers can do many things to build positive teacher-child relationships and promote executive function skills in early childhood settings, including adapting existing activities to help children practise these skills. Including families in these efforts can also help support children’s executive function at home and in other important contexts of their lives.
To read the fully referenced version of this research review by Dr Megan McClelland and Dr Shauna Tominey, click here.
In this video, Dr Dione Healey talks about how executive function fits into the broader skill of self-regulation, and discusses a programme she has developed called ENGAGE, which is designed to support children to develop their executive function and emotional regulation skills.
About Dr Dione Healey
Dr Dione Healey is an Associate Professor at the University of Otago and the developer of the ENGAGE programme. She is a clinical psychologist with a PhD in Psychology. She trained at the University of Canterbury before heading to the USA to do postdoctoral work with Distinguished Professor Jeffrey Halperin at Queens College of the City University of New York. She now works in the Department of Psychology at the University of Otago and has extensive clinical and research experience in the area of self-regulation in young children. Dione is the lead developer of the ENGAGE programme and is the Director of the ENGAGE research and development programme based in her research laboratory at the University of Otago.
Self-regulation is such a core skill that is strongly associated with all sorts of outcomes in everyday functioning: social emotional functioning, academic functioning, learning. There is research, one of the main ones being the findings from the Dunedin Longitudinal Study, where they found that poorer self-regulation skills very early in childhood, at age three, were predictive of a wide array of adverse adult outcomes. So, they really spanned quite a spectrum: poorer employment and work functioning, poorer mental health, higher rates of criminality, more relationship difficulties, higher rates of unemployment. So, that shows that this is a really core, important skill for functioning across the lifespan, and it is really important to be fostering it early on. It’s developing around that early child – by age three, four, five, children are starting to develop the ability to self-regulate, and we’re constantly developing it, right throughout our lifespans.
What is the relationship between self-regulation, emotional regulation, and executive function?
The three terms are often used quite inter-changeably, and they overlap quite significantly. So, people do often get confused, and in a sense, they’re kind of all describing a similar, or the same type of thing. I would see self-regulation as the umbrella term that encompasses executive functioning and emotion regulation within it. So, I see self-regulation as consisting of behaviour regulation, so being able to regulate your behaviour and inhibit your responses – stop yourself from responding in certain ways. Cognitive regulation: your ability to think, focus, concentrate, remember information. Then, emotion regulation: your ability to manage your emotions in different situations. When people typically talk about executive functions, they’re talking more about the behavioural cognitive aspect of regulation, so you think of it as being associated to the prefrontal cortex functioning. So, you’ll often think about executive functioning as the brain’s kind of management system, so the planning, the organising, the inhibiting, the remembering aspects of self-regulation. Then, the emotional regulation piece is sort of the other part of it, which is about managing your emotions.
How are self-regulation skills important for social emotional competence and learning?
All three of those aspects of self-regulation are really important for your social functioning, and also later on in your learning and your academic functioning, because we need all of those skills to varying degrees in varying situations, really, across life and functioning. So, if we think about social functioning: for example, if you’re having a conversation with another child – two children are having a conversation – there are a lot of complex skills involved in that. You have to be able to attend to what the child is saying, understand what they’re saying, remember what they’ve said so that you can respond to what they’ve said, and sort of flow in the conversation, but also, having that inhibitory control, so that you’re able to stop yourself from just butting in or talking over them, but waiting for your turn, until you can response in that reciprocal way within the conversation. So, that would be one example of where self-regulation is important in social functioning.
Other aspects are, of course, that emotional piece, so being able to manage frustrations, if you’re playing a game with other children, and you’re not winning, or they’re not playing the game the way you wanted to them to play the game, so being able to negotiate, compromise, manage your emotions, if you’re feeling frustrated that it’s not going the way you would like, necessarily. When you’re learning information, you have to be able to remember the information that you’ve learned, but also, typically learning new skills, be them everyday basic skills, or more formal academic learning later on down the piece, you need to be able to focus, concentrate, manage your emotions, because learning a new skill typically requires quite a lot of persistence, and lots of attempts at doing it – not necessarily getting it right the first time, so being able to manage the emotion that’s associated, and also being able to then persist and keep going, even though you might be finding it hard or distressing.
You create the ENGAGE programme to help young children develop self-regulation. Tell us about the programme and it works.
Essentially, the idea around ENGAGE is that it’s a framework for teaching self-regulation through play. So, the idea is that we’re teaching these skills across the three areas of self-regulation – behavioural, cognitive and emotional – via playing varying games in a structured way. So, it’s easiest to just kind of give an example of how it works, to try and describe it. So, essentially with ENGAGE, there’s varying steps to it. The first idea is to think of a skill that you want to teach, so, for example, you want to help a child learn to regulate their behaviour – help them being able to slow down or calm down when they’re being overly-active. Then, you need to think of a game that involves that skill. So, an example we often use is an animal speeds game, where essentially, they do different activities and tasks across three different speeds, and again, you can choose animals or any kind of superheroes or characters, or anything that children relate to as being fast, moderate, and slow. So, we’ll often use a cheetah for being really fast, a giraffe for being moderate, and a tortoise for being slow. Then, you can do varying activities at these different speeds, but before you start the game, it’s also really important to kind of connect the game to the child’s everyday functioning, so that there is a relationship, because that’s important for later on when we use the programme. So, you’ll often introduce the game by saying, we’re going to do this animal speeds game, where you going to do things at different speeds, and then, get them to think about times where they do things at different speeds, and where there are times where it’s good or you’re allowed to be fast and wild and jump all over the place, or whatever you want – however you want to describe it. So, maybe at the park, it’s fine to run around and climb, and swing off the monkey bars, but when is a good time to be a moderate speed? Maybe when you’re at the centre, and you’re outside playing, but you don’t want to get too overactive, or if you’re going with a walk with your mum, or some sort of example for them. Then, think about tortoise-mode: when it’s important to be really slow, calm, and methodical. So, again, maybe when you’re inside at the centre, rather than when you’re playing outside, or you might be very slow at times where you don’t really want to go somewhere, like if your mum says it’s time to leave the park, and you don’t really want to go, you might go into tortoise-mode, and you might need to speed up a bit into giraffe-mode. So, you try to relate it to their everyday functioning, so the children kind of understand the game, and what it involves.
Then, you play the game. So, you play the game – think of some different activities. What’s really important when you introduce the game, or start the game, is to try and aim it at a level that will be slightly challenging for the child, and this is a lot harder when you’re working with a group of children, of course, because there’ll be a variation within the centre, and the children, but trying to aim it sort of generally at a level that will be slightly challenging, at least for the majority of the children in the group, and then building the skill up. So, it’s kind of having that approach that comes out of the Vygotskian approach from children development around scaffolding, and building skills up over time. So, we aim at slightly challenging, and then find ways to make the game more complicated over time. So, with the animal speeds, you might make the activity that they’re doing at different speeds more complicated, or you might make them switch speeds more rapidly – something like that. Then, when you get to a point where you think you’ve kind of maxed out the skills within this game, you might have to find another game that uses similar skills, to continue to build the skill over time.
Then, the last part will be using the game as a reference when they are functioning in everyday life, so when there are times when they might be outside in the centre, and they’re being overly active, or over-excited, you might say to them, ‘Johnny, it’s time to go into giraffe-mode’, or ‘go into giraffe-mode’. Then, they will be able to associate that with the game, and should more easily be able to switch down into regulating their behaviour down. I’ve had a lot of positive feedback with this game, from parents and teachers, that it does work really well when this was something that a child really struggled with before they started doing the ENGAGE games, and now they just have to call out this ‘giraffe-mode’, for example, and the child straight away knows what that means and is able to regulate themselves.
The thinking around cognitive regulation, that’s focusing a lot more on memory, planning, organisation, so again, we have a range of games there, but a common one we would do is just doing a puzzle, which for a lot of children is actually quite challenging. Some kids love them but there’s a huge group of young children that are not particularly fans of doing puzzles, and again we use that approach of building up the skill over time. So for children that really don’t like doing a puzzle, what we’ve done in the past is just have a basic puzzle and actually, they only need to do two or three pieces each day, so we’re teaching them the strategy about starting at the corners, looking at the picture, trying to plan how you would approach the puzzle, and then getting them to do a few pieces each day, and then over time building it up, so doing more pieces per day to get to the point where they can do a whole puzzle in one sitting, and then doing more complicated puzzles over time.
Thinking about emotion regulation, we’ve got varying games that are focusing on learning to manage your emotion and calm down, but also around noticing emotions and identifying emotions, so being able to know what you’re feeling and then how to respond to that. So, we have deep breathing type exercises – just learning to slow down your breathing, for example. Or we have varying muscle relaxation exercises, so we talk about tensing and relaxing muscles, and we’ll talk to the children around how when you get upset and frustrated, you might feel quite tense and your muscles feel really hard, so we’ll tell them to tense themselves up, so sometimes we’ll say ‘screw your face up like a scary monster’, so you’re trying to get them to really tense different muscles. And then you might flop around like a tree that’s blowing in the wind, so relaxing your muscles and feeling that feeling, and then relating that to when they’re feeling stressed and tense, how they can relax through using their muscle relaxation, for example.
Dione’s conceptualisation of self-regulation clearly shows how executive function skills are essential to the different kinds of regulation – behavioural regulation, cognitive regulation, and emotional regulation. Basically, these three forms of regulation are focused on self-control in different areas – being able to control your behaviour so that it is appropriate for a given situation, being able to control your cognitive processes so that you can focus and think clearly and in an organised way, and being able to control the expression of emotion. Dione explains that executive function is associated with the pre-frontal lobe and is the part of self-regulation which focuses on the cognitive and behavioural aspects such as inhibitory control, working memory, planning, and organising. Dione offers a useful metaphor for executive function – it is like the brain’s management system, deciding where attention will be directed, what information will be held in mind, which thinking tasks will be undertaken, and in which order.
Dione provides great examples about the way in which all three parts of self-regulation (the two aspects of executive function and emotional regulation) are crucial to being successful in everyday activities such as having a conversation with a friend, playing a game, or learning new information. She also cites the research about the longitudinal outcomes for children with poor self-regulation skills, demonstrating the long-term importance of supporting children’s self-management and emotional regulation skills.
Finally, Dione gives some examples of games or activities that can be used to support children’s developing executive function skills, such as movement games and puzzles. Some really important points to take away from this video are to do with the way that games and activities to practise executive function skills should be framed and presented to children. It is essential to help children to connect the skills they are practising in the game with the everyday use of these skills, and then to refer back to and reinforce the skill learnt in the game in everyday situations. You can easily imagine the usefulness of a prompt like ‘giraffe mode’ or ‘tortoise mode’ to support children to regulate their behaviour in different situations.
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A cellular network or mobile network is a telecommunications network where the link to and from end nodes is wireless and the network is distributed over land areas called cells, each served by at least one fixed-location transceiver (typically three cell sites or base transceiver stations). These base stations provide the cell with the network coverage which can be used for transmission of voice, data, and other types of content. A cell typically uses a different set of frequencies from neighboring cells, to avoid interference and provide guaranteed service quality within each cell.
When joined together, these cells provide radio coverage over a wide geographic area. This enables numerous portable transceivers (e.g., mobile phones, tablets and laptops equipped with mobile broadband modems, pagers, etc.) to communicate with each other and with fixed transceivers and telephones anywhere in the network, via base stations, even if some of the transceivers are moving through more than one cell during transmission.
Cellular networks offer a number of desirable features:
Major telecommunications providers have deployed voice and data cellular networks over most of the inhabited land area of Earth. This allows mobile phones and mobile computing devices to be connected to the public switched telephone network and public Internet access. Private cellular networks can be used for research or for large organizations and fleets, such as dispatch for local public safety agencies or a taxicab company.
In a cellular radio system, a land area to be supplied with radio service is divided into cells in a pattern dependent on terrain and reception characteristics. These cell patterns roughly take the form of regular shapes, such as hexagons, squares, or circles although hexagonal cells are conventional. Each of these cells is assigned with multiple frequencies (f1 – f6) which have corresponding radio base stations. The group of frequencies can be reused in other cells, provided that the same frequencies are not reused in adjacent cells, which would cause co-channel interference.
The increased capacity in a cellular network, compared with a network with a single transmitter, comes from the mobile communication switching system developed by Amos Joel of Bell Labs that permitted multiple callers in a given area to use the same frequency by switching calls to the nearest available cellular tower having that frequency available. This strategy is viable because a given radio frequency can be reused in a different area for an unrelated transmission. In contrast, a single transmitter can only handle one transmission for a given frequency. Inevitably, there is some level of interference from the signal from the other cells which use the same frequency. Consequently, there must be at least one cell gap between cells which reuse the same frequency in a standard frequency-division multiple access (FDMA) system.
Consider the case of a taxi company, where each radio has a manually operated channel selector knob to tune to different frequencies. As drivers move around, they change from channel to channel. The drivers are aware of which frequency approximately covers some area. When they do not receive a signal from the transmitter, they try other channels until finding one that works. The taxi drivers only speak one at a time when invited by the base station operator. This is a form of time-division multiple access (TDMA).
See also: History of mobile phones
The history of cellular phone technology began on December 11, 1947 with an internal memo written by Douglas H. Ring, a Bell Labs engineer in which he proposed development of a cellular telephone system by AT&T.
The first commercial cellular network, the 1G generation, was launched in Japan by Nippon Telegraph and Telephone (NTT) in 1979, initially in the metropolitan area of Tokyo. Within five years, the NTT network had been expanded to cover the whole population of Japan and became the first nationwide 1G network. It was an analog wireless network. The Bell System had developed cellular technology since 1947, and had cellular networks in operation in Chicago and Dallas prior to 1979, but commercial service was delayed by the breakup of the Bell System, with cellular assets transferred to the Regional Bell Operating Companies.
The wireless revolution began in the early 1990s, leading to the transition from analog to digital networks. This was enabled by advances in MOSFET technology. The MOSFET, originally invented by Mohamed M. Atalla and Dawon Kahng at Bell Labs in 1959, was adapted for cellular networks by the early 1990s, with the wide adoption of power MOSFET, LDMOS (RF amplifier), and RF CMOS (RF circuit) devices leading to the development and proliferation of digital wireless mobile networks.
The first commercial digital cellular network, the 2G generation, was launched in 1991. This sparked competition in the sector as the new operators challenged the incumbent 1G analog network operators.
To distinguish signals from several different transmitters, frequency-division multiple access (FDMA, used by analog and D-AMPS systems), time-division multiple access (TDMA, used by GSM) and code-division multiple access (CDMA, first used for PCS, and the basis of 3G) were developed.
With FDMA, the transmitting and receiving frequencies used by different users in each cell are different from each other. Each cellular call was assigned a pair of frequencies (one for base to mobile, the other for mobile to base) to provide full-duplex operation. The original AMPS systems had 666 channel pairs, 333 each for the CLEC "A" system and ILEC "B" system. The number of channels was expanded to 416 pairs per carrier, but ultimately the number of RF channels limits the number of calls that a cell site could handle. FDMA is a familiar technology to telephone companies, which used frequency-division multiplexing to add channels to their point-to-point wireline plants before time-division multiplexing rendered FDM obsolete.
With TDMA, the transmitting and receiving time slots used by different users in each cell are different from each other. TDMA typically uses digital signaling to store and forward bursts of voice data that are fit into time slices for transmission, and expanded at the receiving end to produce a somewhat normal-sounding voice at the receiver. TDMA must introduce latency (time delay) into the audio signal. As long as the latency time is short enough that the delayed audio is not heard as an echo, it is not problematic. TDMA is a familiar technology for telephone companies, which used time-division multiplexing to add channels to their point-to-point wireline plants before packet switching rendered FDM obsolete.
The principle of CDMA is based on spread spectrum technology developed for military use during World War II and improved during the Cold War into direct-sequence spread spectrum that was used for early CDMA cellular systems and Wi-Fi. DSSS allows multiple simultaneous phone conversations to take place on a single wideband RF channel, without needing to channelize them in time or frequency. Although more sophisticated than older multiple access schemes (and unfamiliar to legacy telephone companies because it was not developed by Bell Labs), CDMA has scaled well to become the basis for 3G cellular radio systems.
Other available methods of multiplexing such as MIMO, a more sophisticated version of antenna diversity, combined with active beamforming provides much greater spatial multiplexing ability compared to original AMPS cells, that typically only addressed one to three unique spaces. Massive MIMO deployment allows much greater channel reuse, thus increasing the number of subscribers per cell site, greater data throughput per user, or some combination thereof. Quadrature Amplitude Modulation (QAM) modems offer an increasing number of bits per symbol, allowing more users per megahertz of bandwidth (and decibels of SNR), greater data throughput per user, or some combination thereof.
The key characteristic of a cellular network is the ability to reuse frequencies to increase both coverage and capacity. As described above, adjacent cells must use different frequencies, however, there is no problem with two cells sufficiently far apart operating on the same frequency, provided the masts and cellular network users' equipment do not transmit with too much power.
The elements that determine frequency reuse are the reuse distance and the reuse factor. The reuse distance, D is calculated as
where R is the cell radius and N is the number of cells per cluster. Cells may vary in radius from 1 to 30 kilometres (0.62 to 18.64 mi). The boundaries of the cells can also overlap between adjacent cells and large cells can be divided into smaller cells.
The frequency reuse factor is the rate at which the same frequency can be used in the network. It is 1/K (or K according to some books) where K is the number of cells which cannot use the same frequencies for transmission. Common values for the frequency reuse factor are 1/3, 1/4, 1/7, 1/9 and 1/12 (or 3, 4, 7, 9 and 12, depending on notation).
In case of N sector antennas on the same base station site, each with different direction, the base station site can serve N different sectors. N is typically 3. A reuse pattern of N/K denotes a further division in frequency among N sector antennas per site. Some current and historical reuse patterns are 3/7 (North American AMPS), 6/4 (Motorola NAMPS), and 3/4 (GSM).
If the total available bandwidth is B, each cell can only use a number of frequency channels corresponding to a bandwidth of B/K, and each sector can use a bandwidth of B/NK.
Code-division multiple access-based systems use a wider frequency band to achieve the same rate of transmission as FDMA, but this is compensated for by the ability to use a frequency reuse factor of 1, for example using a reuse pattern of 1/1. In other words, adjacent base station sites use the same frequencies, and the different base stations and users are separated by codes rather than frequencies. While N is shown as 1 in this example, that does not mean the CDMA cell has only one sector, but rather that the entire cell bandwidth is also available to each sector individually.
Recently also orthogonal frequency-division multiple access based systems such as LTE are being deployed with a frequency reuse of 1. Since such systems do not spread the signal across the frequency band, inter-cell radio resource management is important to coordinate resource allocation between different cell sites and to limit the inter-cell interference. There are various means of inter-cell interference coordination (ICIC) already defined in the standard. Coordinated scheduling, multi-site MIMO or multi-site beamforming are other examples for inter-cell radio resource management that might be standardized in the future.
Cell towers frequently use a directional signal to improve reception in higher-traffic areas. In the United States, the Federal Communications Commission (FCC) limits omnidirectional cell tower signals to 100 watts of power. If the tower has directional antennas, the FCC allows the cell operator to emit up to 500 watts of effective radiated power (ERP).
Although the original cell towers created an even, omnidirectional signal, were at the centers of the cells and were omnidirectional, a cellular map can be redrawn with the cellular telephone towers located at the corners of the hexagons where three cells converge. Each tower has three sets of directional antennas aimed in three different directions with 120 degrees for each cell (totaling 360 degrees) and receiving/transmitting into three different cells at different frequencies. This provides a minimum of three channels, and three towers for each cell and greatly increases the chances of receiving a usable signal from at least one direction.
The numbers in the illustration are channel numbers, which repeat every 3 cells. Large cells can be subdivided into smaller cells for high volume areas.
Cell phone companies also use this directional signal to improve reception along highways and inside buildings like stadiums and arenas.
Practically every cellular system has some kind of broadcast mechanism. This can be used directly for distributing information to multiple mobiles. Commonly, for example in mobile telephony systems, the most important use of broadcast information is to set up channels for one-to-one communication between the mobile transceiver and the base station. This is called paging. The three different paging procedures generally adopted are sequential, parallel and selective paging.
The details of the process of paging vary somewhat from network to network, but normally we know a limited number of cells where the phone is located (this group of cells is called a Location Area in the GSM or UMTS system, or Routing Area if a data packet session is involved; in LTE, cells are grouped into Tracking Areas). Paging takes place by sending the broadcast message to all of those cells. Paging messages can be used for information transfer. This happens in pagers, in CDMA systems for sending SMS messages, and in the UMTS system where it allows for low downlink latency in packet-based connections.
In a primitive taxi system, when the taxi moved away from a first tower and closer to a second tower, the taxi driver manually switched from one frequency to another as needed. If communication was interrupted due to a loss of a signal, the taxi driver asked the base station operator to repeat the message on a different frequency.
In a cellular system, as the distributed mobile transceivers move from cell to cell during an ongoing continuous communication, switching from one cell frequency to a different cell frequency is done electronically without interruption and without a base station operator or manual switching. This is called the handover or handoff. Typically, a new channel is automatically selected for the mobile unit on the new base station which will serve it. The mobile unit then automatically switches from the current channel to the new channel and communication continues.
The exact details of the mobile system's move from one base station to the other vary considerably from system to system (see the example below for how a mobile phone network manages handover).
The most common example of a cellular network is a mobile phone (cell phone) network. A mobile phone is a portable telephone which receives or makes calls through a cell site (base station) or transmitting tower. Radio waves are used to transfer signals to and from the cell phone.
Modern mobile phone networks use cells because radio frequencies are a limited, shared resource. Cell-sites and handsets change frequency under computer control and use low power transmitters so that the usually limited number of radio frequencies can be simultaneously used by many callers with less interference.
A cellular network is used by the mobile phone operator to achieve both coverage and capacity for their subscribers. Large geographic areas are split into smaller cells to avoid line-of-sight signal loss and to support a large number of active phones in that area. All of the cell sites are connected to telephone exchanges (or switches), which in turn connect to the public telephone network.
In cities, each cell site may have a range of up to approximately 1⁄2 mile (0.80 km), while in rural areas, the range could be as much as 5 miles (8.0 km). It is possible that in clear open areas, a user may receive signals from a cell site 25 miles (40 km) away. In rural areas with low-band coverage and tall towers, basic voice and messaging service may reach 50 miles (80 km), with limitations on bandwidth and number of simultaneous calls.
Since almost all mobile phones use cellular technology, including GSM, CDMA, and AMPS (analog), the term "cell phone" is in some regions, notably the US, used interchangeably with "mobile phone". However, satellite phones are mobile phones that do not communicate directly with a ground-based cellular tower but may do so indirectly by way of a satellite.
There are a number of different digital cellular technologies, including: Global System for Mobile Communications (GSM), General Packet Radio Service (GPRS), cdmaOne, CDMA2000, Evolution-Data Optimized (EV-DO), Enhanced Data Rates for GSM Evolution (EDGE), Universal Mobile Telecommunications System (UMTS), Digital Enhanced Cordless Telecommunications (DECT), Digital AMPS (IS-136/TDMA), and Integrated Digital Enhanced Network (iDEN). The transition from existing analog to the digital standard followed a very different path in Europe and the US. As a consequence, multiple digital standards surfaced in the US, while Europe and many countries converged towards the GSM standard.
A simple view of the cellular mobile-radio network consists of the following:
This network is the foundation of the GSM system network. There are many functions that are performed by this network in order to make sure customers get the desired service including mobility management, registration, call set-up, and handover.
Any phone connects to the network via an RBS (Radio Base Station) at a corner of the corresponding cell which in turn connects to the Mobile switching center (MSC). The MSC provides a connection to the public switched telephone network (PSTN). The link from a phone to the RBS is called an uplink while the other way is termed downlink.
Radio channels effectively use the transmission medium through the use of the following multiplexing and access schemes: frequency-division multiple access (FDMA), time-division multiple access (TDMA), code-division multiple access (CDMA), and space-division multiple access (SDMA).
Main article: Small cell
Small cells, which have a smaller coverage area than base stations, are categorised as follows:
Main article: Handover
As the phone user moves from one cell area to another cell while a call is in progress, the mobile station will search for a new channel to attach to in order not to drop the call. Once a new channel is found, the network will command the mobile unit to switch to the new channel and at the same time switch the call onto the new channel.
With CDMA, multiple CDMA handsets share a specific radio channel. The signals are separated by using a pseudonoise code (PN code) that is specific to each phone. As the user moves from one cell to another, the handset sets up radio links with multiple cell sites (or sectors of the same site) simultaneously. This is known as "soft handoff" because, unlike with traditional cellular technology, there is no one defined point where the phone switches to the new cell.
In IS-95 inter-frequency handovers and older analog systems such as NMT it will typically be impossible to test the target channel directly while communicating. In this case, other techniques have to be used such as pilot beacons in IS-95. This means that there is almost always a brief break in the communication while searching for the new channel followed by the risk of an unexpected return to the old channel.
If there is no ongoing communication or the communication can be interrupted, it is possible for the mobile unit to spontaneously move from one cell to another and then notify the base station with the strongest signal.
Main article: Cellular frequencies
The effect of frequency on cell coverage means that different frequencies serve better for different uses. Low frequencies, such as 450 MHz NMT, serve very well for countryside coverage. GSM 900 (900 MHz) is suitable for light urban coverage. GSM 1800 (1.8 GHz) starts to be limited by structural walls. UMTS, at 2.1 GHz is quite similar in coverage to GSM 1800.
Higher frequencies are a disadvantage when it comes to coverage, but it is a decided advantage when it comes to capacity. Picocells, covering e.g. one floor of a building, become possible, and the same frequency can be used for cells which are practically neighbors.
Cell service area may also vary due to interference from transmitting systems, both within and around that cell. This is true especially in CDMA based systems. The receiver requires a certain signal-to-noise ratio, and the transmitter should not send with too high transmission power in view to not cause interference with other transmitters. As the receiver moves away from the transmitter, the power received decreases, so the power control algorithm of the transmitter increases the power it transmits to restore the level of received power. As the interference (noise) rises above the received power from the transmitter, and the power of the transmitter cannot be increased anymore, the signal becomes corrupted and eventually unusable. In CDMA-based systems, the effect of interference from other mobile transmitters in the same cell on coverage area is very marked and has a special name, cell breathing.
One can see examples of cell coverage by studying some of the coverage maps provided by real operators on their web sites or by looking at independently crowdsourced maps such as Opensignal or CellMapper. In certain cases they may mark the site of the transmitter; in others, it can be calculated by working out the point of strongest coverage.
A cellular repeater is used to extend cell coverage into larger areas. They range from wideband repeaters for consumer use in homes and offices to smart or digital repeaters for industrial needs.
The following table shows the dependency of the coverage area of one cell on the frequency of a CDMA2000 network:
|Cell radius (km)
|Cell area (km2)
|Relative cell count
Lists and technical information:
Starting with EVDO the following techniques can also be used to improve performance:
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A lack of symmetry is called skewness for data distribution. In other words, a departure from symmetry is called skewness.
A distribution is simply a collection of data, or scores, on a variable. Usually, these scores are arranged in order from smallest to largest and then they can be presented graphically.Page 6, Statistics in Plain English, Third Edition, 2010.
Let’s look at pictures of a Symmetric curve:
The measure of skewness gives the direction and the magnitude of the lack of symmetry.
If the distribution is not symmetric, the frequencies will not be uniformly distributed about the center of the distribution. We will look at pictures of asymmetric distributions shortly.
In mathematics, a figure is called symmetric if there exists a point in it through which if a perpendicular is drawn on the X-axis, it divides the figure into two congruent parts i.e. identical in all respect or one part can be superimposed on the other i.e mirror images of each other.
In Statistics, a distribution is called symmetric if the mean, median, and mode coincide. Otherwise, the distribution becomes asymmetric.
Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point.
Let’s look at pictures of asymmetric distributions:
If the right tail is longer, we get a positively skewed distribution for which mean > median > mode:
If the left tail is longer, we get a negatively skewed distribution for which mean < median < mode:
Skewness gives the direction of variability.
Measures of skewness help us to know to what degree and in which direction (positive or negative) the frequency distribution has a departure from symmetry.
Although positive or negative skewness can be detected graphically depending on whether the right tail or the left tail is longer, we don’t get an idea of the magnitude.
The following are the absolute measures of skewness:
1. Skewness (Sk) = Mean – Median
2. Skewness (Sk) = Mean – Mode
3. Skewness (Sk) = (Q3 – Q2) – (Q2 – Q1)
For comparison to series, we do not calculate these absolute measures. We calculate the relative measures which are called the coefficient of skewness. The coefficient of skewness is pure numbers independent of units of measurement.
Relative Measures of Skewness
Karl Pearson’s Coefficient of Skewness
This method is most frequently used for measuring skewness. The formula for measuring the coefficient of skewness is given by:
Sk = (Mean-Mode) / standard deviation
The value of this coefficient would be zero in a symmetrical distribution. If the mean is greater than the mode, the coefficient of skewness would be positive otherwise negative. The value of Karl Pearson’s coefficient of skewness usually lies between 1 for moderately skewed distribution.
If the value of mean, median, and mode are the same in any distribution, then the skewness does not exist in that distribution. Larger the difference in these values, the larger the skewness.
If sum of the frequencies are equal on both sides of the mode then skewness does not exist.
If the distance of the first quartile and third quartile are the same from the median then a skewness does not exist. Similarly, if deciles (first and ninth) and percentiles (first and ninety-nine) are at an equal distance from the median. Then there is no asymmetry.
If a graph of data becomes a normal curve and when it is folded in middle and one part overlaps fully with the other one then there is no asymmetry.
Skewness refers to the extent to which the data is asymmetrical or skewed to one side. It helps to identify whether a distribution is symmetric or skewed. Hope this article had helped in shedding some light on “skewness for a data distribution”.
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- What are quartiles, deciles and percentiles in statistics?
- Standard deviation and variance in statistics
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- Kurtosis for a data distribution
- Interpretation of Covariance and Correlation
- Lorenz Curve and Gini Coefficient Explained
- Normalization vs Standardization
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- What do you mean by Weight of Evidence (WoE) and Information Value (IV)?
- Statistics Interview Questions 101
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Contour plots in Excel are a powerful way to visualize 3-dimensional data in a 2-dimensional space. They allow you to see how a particular variable changes across two dimensions, with the third dimension represented by contour lines or colours. While contour plots may seem complex, they are easy to create in Excel and can help you gain insights from your data that may be difficult to see in a table or chart.
Table of Contents
In this article, we’ll walk you through everything you need to know to create and customize contour plots in Excel. We’ll cover the basics of contour plots, how to create them using Excel’s built-in tools, and how to customize them to make them more visually appealing and informative. So, let’s dive in!
What are Contour Plots in Excel?
Contour plots in Excel are an essential tool for visualizing data in 3D. This type of graph visually represents the relationship between three variables, with each variable represented by a different axis. Contour plots can identify patterns in data that may not be immediately apparent from a simple 2D graph. This article will explore the steps required to create a contour plot in Excel.
Understanding the Basics of Contour Plots
Before we dive into the specifics of creating and customizing contour plots in Excel, it’s important to understand the basics of how they work. Here are some key things to keep in mind:
- A contour plot is a graphical representation of a 3-dimensional surface on a 2-dimensional plane.
- The surface is divided into a series of contours representing lines of constant values for a particular variable.
- Contour plots are useful for visualizing how variable changes across two dimensions, such as time and temperature or height and weight.
- Contour plots can be created using Excel’s built-in tools, which we’ll cover in the next section.
Creating Plots in Excel
Now that you understand the basics of contour plots let’s walk through the steps to create one in Excel. Here’s what you’ll need to do:
- Organize your data. Your data should be organized in three columns: one for the x-axis variable, one for the y-axis variable, and one for the z-axis variable. For example, if you’re plotting temperature and pressure over time, your columns might look something like this:
- Select your data. Highlight all three columns of data.
- Insert a 2D contour plot. Go to the Insert tab on the Excel ribbon, select the Charts dropdown, and choose the 2D contour plot option.
- Customize your plot. Once your plot is created, you can customize it in a number of ways. For example:
- You can change the colour scale to make the contour lines more visually appealing.
- You can add a title and axis labels to make your plot more informative.
- You can adjust the contour levels to show more or less detail in your data.
Customizing Plots in Excel
While Excel’s built-in contour plot tool is a great starting point, you may want to customize your plot further to make it more visually appealing or informative. Here are some tips for doing so:
- Change the colour scheme. By default, Excel uses a rainbow colour scheme for contour plots, but you may want to choose a different colour scheme that better suits your data or your audience.
- Adjust the contour levels. By default, Excel chooses the contour levels for you, but you can manually adjust them to show more or less detail in your data. This can help you emphasize important trends or patterns in your data.
- Add labels and annotations. You can add labels and annotations to your contour plot to highlight important features or provide additional context for your audience.
- Change the chart type. If you want to display your data differently, change the chart type. For example, you could create a 3D surface chart or a heat map instead of a contour plot.
Only 4 Easy Steps: Creating Plots in Excel
Step 1: Prepare your data
The first step in creating a contour plot in Excel is to ensure that your data is in the correct format. The data should be arranged in columns or rows, with each column or row representing a different variable. In addition, the data should be sorted in ascending order based on the value of the first variable.
Step 2: Insert a 3D Scatter Chart
Once your data is prepared, the next step is to insert a 3D scatter chart. This can be done by selecting the data and then clicking on the Insert tab. From there, select the Scatter Chart option and then choose the 3D Scatter option. This will create a basic 3D scatter chart, which we will modify in the next step.
Step 3: Add Contour Lines
To add contour lines to our chart, we must select the chart and click the Chart Elements button. From there, select the Contour option and choose the desired contour level. Excel will automatically generate contour lines for our chart, which can be customized further by adjusting the contour level or adding colour and shading.
Step 4: Format the Chart
The final step in creating a contour plot in Excel is to format the chart. This can be done by selecting the chart and then clicking on the Format tab. From there, you can adjust the chart title, axis labels, and other formatting options to create a professional-looking chart.
FAQs about Contour Plots in Excel
What types of data are best suited for contour plots?
Contour plots are best suited for data that varies across two dimensions and can be represented by a continuous range of values. For example, temperature and pressure over time or height and weight across a population.
How do I choose the right contour levels for my data?
Choosing the right contour levels depends on the nature of your data and what you want to emphasize. If you want to show fine detail, you may want to use more contour levels. If you want to emphasize broader trends or patterns, you may want to use fewer contour levels.
Can I add multiple data sets to a single contour plot?
You can add multiple data sets to a single contour plot by selecting the data from each set and creating a new contour plot. You can then overlay the plots on top of each other to see how the variables interact.
Contour plots are a powerful way to visualize complex data in an informative and visually appealing way. While they may seem complex at first, they are easy to create and customize in Excel. Following the steps outlined in this guide, you can create contour plots to help you gain insights from your data and impress your colleagues with your data visualization skills. So, experiment with Excel contour plots today and see what insights you can uncover!
Hello, I’m Cansu, a professional dedicated to creating Excel tutorials, specifically catering to the needs of B2B professionals. With a passion for data analysis and a deep understanding of Microsoft Excel, I have built a reputation for providing comprehensive and user-friendly tutorials that empower businesses to harness the full potential of this powerful software.
I have always been fascinated by the intricate world of numbers and the ability of Excel to transform raw data into meaningful insights. Throughout my career, I have honed my data manipulation, visualization, and automation skills, enabling me to streamline complex processes and drive efficiency in various industries.
As a B2B specialist, I recognize the unique challenges that professionals face when managing and analyzing large volumes of data. With this understanding, I create tutorials tailored to businesses’ specific needs, offering practical solutions to enhance productivity, improve decision-making, and optimize workflows.
My tutorials cover various topics, including advanced formulas and functions, data modeling, pivot tables, macros, and data visualization techniques. I strive to explain complex concepts in a clear and accessible manner, ensuring that even those with limited Excel experience can grasp the concepts and apply them effectively in their work.
In addition to my tutorial work, I actively engage with the Excel community through workshops, webinars, and online forums. I believe in the power of knowledge sharing and collaborative learning, and I am committed to helping professionals unlock their full potential by mastering Excel.
With a strong track record of success and a growing community of satisfied learners, I continue to expand my repertoire of Excel tutorials, keeping up with the latest advancements and features in the software. I aim to empower businesses with the skills and tools they need to thrive in today’s data-driven world.
Suppose you are a B2B professional looking to enhance your Excel skills or a business seeking to improve data management practices. In that case, I invite you to join me on this journey of exploration and mastery. Let’s unlock the true potential of Excel together!
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https://www.projectcubicle.com/mastering-contour-plots-in-excel-a-comprehensive-guide/
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By the end of this section, you will be able to do the following:
- Explain gravitational potential energy in terms of work done against gravity
- Show that the gravitational potential energy of an object of mass m at height h on Earth is given by PEg = mgh
- Show how knowledge of potential energy as a function of position can be used to simplify calculations and explain physical phenomena
The information presented in this section supports the following AP® learning objectives and science practices:
- 2.E.1.1 The student is able to construct or interpret visual representations of the isolines of equal gravitational potential energy per unit mass, and identify each line as a gravitational equipotential.
- 4.C.1.1 The student is able to calculate the total energy of a system and justify the mathematical routines used in the calculation of component types of energy within the system whose sum is the total energy. (S.P. 1.4, 2.1, 2.2)
- 5.B.1.1 The student is able to set up a representation or model showing that a single object can only have kinetic energy and use information about that object to calculate its kinetic energy. (S.P. 1.4, 2.2)
- 5.B.1.2 The student is able to translate between a representation of a single object, which can only have kinetic energy, and a system that includes the object, which may have both kinetic and potential energies. (S.P. 1.5)
Work Done Against Gravity
Work Done Against Gravity
Climbing stairs and lifting objects is work in both the scientific and everyday sense—it is work done against the gravitational force. When there is work, there is a transformation of energy. The work done against the gravitational force goes into an important form of stored energy that we will explore in this section.
Let us calculate the work done in lifting an object of mass through a height , such as in Figure 7.5. If the object is lifted straight up at constant speed, then the force needed to lift it is equal to its weight . The work done on the mass is then . We define this to be gravitational potential energy put into, or gained by, the object-Earth system. This energy is associated with the state of separation between two objects that attract each other by the gravitational force. For convenience, we refer to this as the gained by the object, recognizing that this is energy stored in the gravitational field of Earth. Why do we use the word system? Potential energy is a property of a system rather than of a single object—due to its physical position. An object’s gravitational potential is due to its position relative to the surroundings within the Earth-object system. The force applied to the object is an external force, from outside the system. When it does positive work it increases the gravitational potential energy of the system. Since gravitational potential energy depends on relative position, we need a reference level at which to set the potential energy equal to 0. We usually choose this point to be Earth’s surface, but this point is arbitrary; what is important is the difference in gravitational potential energy, because this difference is what relates to the work done. The difference in gravitational potential energy of an object, in the Earth-object system, between two rungs of a ladder will be the same for the first two rungs as for the last two rungs.
Converting Between Potential Energy and Kinetic Energy
Converting Between Potential Energy and Kinetic Energy
Gravitational potential energy may be converted to other forms of energy, such as kinetic energy. If we release the mass, gravitational force will do an amount of work equal to on it, thereby increasing its kinetic energy by that same amount, by the work-energy theorem. We will find it more useful to consider just the conversion of to without explicitly considering the intermediate step of work (see Example 7.7). This shortcut makes it is easier to solve problems using energy, when possible, rather than explicitly using forces.
More precisely, we define the change in gravitational potential energy to be
where, for simplicity, we denote the change in height by rather than the usual . Note that is positive when the final height is greater than the initial height, and vice versa. For example, if a 0.500-kg mass hung from a cuckoo clock is raised 1.00 m, then its change in gravitational potential energy is
Note that the units of gravitational potential energy turn out to be joules, the same as for work and other forms of energy. As the clock runs, the mass is lowered. We can think of the mass as gradually giving up its 4.90 J of gravitational potential energy, without directly considering the force of gravity that does the work.
Using Potential Energy to Simplify Calculations
Using Potential Energy to Simplify Calculations
The equation applies for any path that has a change in height , not just when the mass is lifted straight up (see Figure 7.6). It is much easier to calculate , a simple multiplication, than it is to calculate the work done along a complicated path. The idea of gravitational potential energy has the double advantage that it is very broadly applicable and it makes calculations easier. From now on, we will consider that any change in vertical position of a mass is accompanied by a change in gravitational potential energy , and we will avoid the equivalent but more difficult task of calculating work done by or against the gravitational force.
Example 7.6 The Force to Stop Falling
A 60.0-kg person jumps onto the floor from a height of 3.00 m. If he lands stiffly, with his knee joints compressing by 0.500 cm, calculate the force on the knee joints.
This person’s energy is brought to zero in this situation by the work done on him by the floor as he stops. The initial is transformed into as he falls. The work done by the floor reduces this kinetic energy to zero.
The work done on the person by the floor as he stops is given by
with a minus sign because the displacement while stopping and the force from floor are in opposite directions The floor removes energy from the system, so it does negative work.
The kinetic energy the person has upon reaching the floor is the amount of potential energy lost by falling through height
The distance that the person’s knees bend is much smaller than the height of the fall, so the additional change in gravitational potential energy during the knee bend is ignored.
The work done on the person by the floor stops the person and brings the person’s kinetic energy to zero
Combining this equation with the expression for gives
Recalling that is negative because the person fell down, the force on the knee joints is given by
Such a large force, 500 times more than the person's weight, over the short impact time is enough to break bones. A much better way to cushion the shock is by bending the legs or rolling on the ground, increasing the time over which the force acts. A bending motion of 0.5 m this way yields a force 100 times smaller than in the example. A kangaroo's hopping shows this method in action. The kangaroo is the only large animal to use hopping for locomotion, but the shock in hopping is cushioned by bending its hind legs in each jump (see Figure 7.7).
Example 7.7 Finding the Speed of a Roller Coaster from Its Height
(a) What is the final speed of the roller coaster shown in Figure 7.8 if it starts from rest at the top of the 20.0 m hill and work done by frictional forces is negligible? (b) What is its final speed, again assuming negligible friction, if its initial speed is 5.00 m/s?
The roller coaster loses potential energy as it goes downhill. We neglect friction, so that the remaining force exerted by the track is the normal force, which is perpendicular to the direction of motion and does no work. The net work on the roller coaster is then done by gravity alone. The loss of gravitational potential energy from moving downward through a distance equals the gain in kinetic energy. This can be written in equation form as . Using the equations for and , we can solve for the final speed , which is the desired quantity.
Solution for (a)
Here, the initial kinetic energy is zero, so that . The equation for change in potential energy states that . Since is negative in this case, we will rewrite this as to show the minus sign clearly. Thus,
Solving for , we find that mass cancels and that
Substituting known values,
Solution for (b)
Again . In this case there is initial kinetic energy, so . Thus,
This means that the final kinetic energy is the sum of the initial kinetic energy plus the gravitational potential energy. Mass again cancels, and
This equation is very similar to the kinematics equation , but it is more general—the kinematics equation is valid only for constant acceleration, whereas our equation above is valid for any path regardless of whether the object moves with a constant acceleration. Now, substituting known values gives
Discussion and Implications
First, note that mass cancels. This is quite consistent with observations made in Falling Objects that all objects fall at the same rate if friction is negligible. Second, only the speed of the roller coaster is considered; there is no information about its direction at any point. This reveals another general truth. When friction is negligible, the speed of a falling body depends only on its initial speed and height, and not on its mass or the path taken. For example, the roller coaster will have the same final speed whether it falls 20.0 m straight down or takes a more complicated path like the one in the figure. Third, and perhaps unexpectedly, the final speed in part (b) is greater than in part (a), but by far less than 5.00 m/s. Finally, note that speed can be found at any height along the way by simply using the appropriate value of h at the point of interest. While changes in the potential and kinetic energies depend only on h, changes in the potential and kinetic energies, expressed in terms of other quantities like time t or horizontal distance x, depend on constraints defined by how the roller coaster is constructed. The height h, for example, can be considered a function of x that essentially describes the design of the roller coaster.
We have seen that work done by or against the gravitational force depends only on the starting and ending points, and not on the path between, allowing us to define the simplifying concept of gravitational potential energy. We can do the same thing for a few other forces, and we will see that this leads to a formal definition of the law of conservation of energy.
Making Connections: Take-Home Investigation—Converting Potential to Kinetic Energy
You can study the conversion of gravitational potential energy into kinetic energy in this experiment. On a smooth, level surface, use a ruler of the kind that has a groove running along its length and a book to make an incline (see Figure 7.9). Place a marble at the 10-cm position on the ruler and let it roll down the ruler. When it hits the level surface, measure the time it takes to roll one meter. Now place the marble at the 20-cm and the 30-cm positions and again measure the times it takes to roll 1 m on the level surface. Find the velocity of the marble on the level surface for all three positions. Plot velocity squared versus the distance traveled by the marble. What is the shape of each plot? If the shape is a straight line, the plot shows that the marble’s kinetic energy at the bottom is proportional to its potential energy at the release point.
Newton’s Universal Law of Gravitation and Gravitational Potential Energy
Near the surface of Earth, where the gravitational force on an object of mass m is given by , there is an associated gravitational potential energy, , where h is the height above some reference value (e.g., sea level), and the potential is defined to be zero at that reference height (). In chapter 6 we learned that the magnitude of the gravitational force between two bodies having masses m and M, with a distance r between their centers of mass, is given by the equation . Again, in this case, an associated gravitational potential energy can be determined by
where the potential energy approaches zero as r approaches infinity.
Conservation of energy principles can again be used to solve many problems of practical interest. Suppose you want to launch an object from Earth’s surface with just enough energy to escape Earth’s gravitational influence. At Earth’s surface, the total energy will be , where M and R are Earth’s mass and radius, respectively. The magnitude of the object’s velocity v will drop toward zero as r approaches infinity, leading to a final energy of . Setting the two equal to each other and solving for v gives
Substituting in values for G, M, and R gives m/s, which is the escape velocity for objects launched from Earth.
The potential energy is proportional to the test mass m. To have a physical quantity that is independent of test mass, we define the gravitational potential to be the potential energy per unit mass. Near the surface of Earth, the gravitational potential is given by
The more general form for the potential due to an object of mass M, derived from Newton’s universal law of gravitation, is
where r is the distance from the object’s center of mass. Since the gravitational potential is a scalar quantity, the potential described as a function of location in three-dimensional space corresponds to a scalar field.
When we are interested in the influence of multiple masses on a test mass, the gravitational potential at any given point is simply the sum of the gravitational potentials of each individual object. Suppose that two point objects of mass M are located along the x-axis at . The gravitational potential at any point in the x-y-plane is given by
There are a couple of ways to visualize a function of the sort represented by the preceding equation. You can, for example, look at a 2-D plot for a specific x- or y-value. For example, if we want to look at the potential only along the x-axis, we can set , which results in
Enter this formula into a real or online graphing calculator. The origin is shown at the top of the graph, the locations of the two objects at are noted along the x-axis, and the gravitational potential is plotted in arbitrary units. Keeping in mind that the potential represents the potential energy per unit mass of a test object, you can envision what would happen to such a test object located at some particular point along the x-axis. An object placed anywhere to the left of the origin will fall into the potential well (i.e., be drawn to the object at ). An object placed to the right of the origin will be drawn to the object at . An object located precisely at will be in a state of unstable equilibrium.
Note also that details of the curve in the above plot provide information about the location and relative magnitude of the two masses. Even without knowing the function describing the potential, the location of the potential wells in the plot make clear the locations of the objects. The left-right symmetry of the plot also indicates that the masses of the two objects are equal.
Another way to visualize the potential is to draw a contour plot of the potential in a given plane. Figure 7.10 shows the gravitational potential energy of three objects.
The gravitational potential is constant along each of the lines, which are known as isolines. The potentials are in arbitrary units, with the outermost red line corresponding to a negative potential with relatively small magnitude. The innermost green lines correspond to negative potentials with relatively large magnitudes. The remaining lines correspond to equally spaced intermediate values of the potential. The locations of the three objects are clear from the contour plot, and the symmetry across the y-axis shows that their masses are not equal. Like the contour lines on a topographic map, the relative spacing between adjacent isolines represents how rapidly the potential changes with location. This information provides insight into the direction and magnitude of the gravitational force a test mass would experience at any particular point in the x-y-plane.
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https://www.texasgateway.org/resource/73-gravitational-potential-energy?book=79096&binder_id=78541
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In the realm of mathematics education, innovative tools and methods play a crucial role in enhancing the learning experience for students. One such powerful tool that has proven to be instrumental in understanding trigonometry is the Unit Circle Chart. In this comprehensive guide, we will explore how the use of Unit Circle Charts can impact both the learning and teaching of mathematics.
What Is A Unit Circle Chart?
A Unit Circle Chart is a visual representation of the relationships between angles and the trigonometric functions—sine, cosine, and tangent. It is a fundamental tool in trigonometry that provides a concise and organized way to grasp complex concepts.
Components of a Unit Circle Chart
Radians and Degrees
The Unit Circle Chart typically displays angles in both radians and degrees, allowing students to seamlessly transition between the two measurement systems.
The chart includes values for sine, cosine, and tangent corresponding to different angles, offering a quick reference for solving trigonometric equations.
How Unit Circle Charts Enhance Mathematical Understanding?
Visual Learning Aids
The visual nature of Unit Circle Charts makes them an excellent aid for visual learners. By representing abstract mathematical concepts in a graphical format, students can better comprehend the relationships between angles and trigonometric functions.
Memorization and Retention
Sin Cos Tan Unit Circle Chart: A Mnemonic Device
The Sin Cos Tan Unit Circle Chart serves as a mnemonic device, aiding students in memorizing the values of sine, cosine, and tangent for common angles. This mnemonic strategy enhances long-term retention.
Trigonometry Unit Circle Chart: Simplifying Complex Concepts
The Trigonometry Unit Circle Chart simplifies the understanding of complex trigonometric concepts, such as the periodic nature of sine and cosine functions.
Integrating Unit Circle Charts Into Mathematics Instruction
Interactive Learning Activities
Utilizing Technology: Interactive Unit Circle Apps
Teachers can incorporate technology, such as interactive Unit Circle apps, into their lessons to engage students actively. These tools allow students to explore the Unit Circle dynamically.
Classroom Demonstrations: Bringing the Unit Circle to Life
Engaging classroom demonstrations with physical Unit Circle charts help make abstract concepts more tangible, fostering a deeper understanding among students.
Incorporating Unit Circle Charts in Lesson Plans
Unit Circle Chart in Radians: A Seamless Transition
Integrating the Unit Circle Chart in radians into lesson plans ensures that students become proficient in both radians and degrees, a crucial skill for advanced mathematics.
Unit Circle Chart Values: Emphasizing Practical Application
Teachers can emphasize the practical application of Unit Circle Chart values in solving real-world problems, linking mathematical concepts to everyday scenarios.
FAQs: Addressing Common Queries on Unit Circle Charts
What Is The Significance Of The Unit Circle In Trigonometry?
The Unit Circle serves as a fundamental tool in trigonometry, providing a visual representation of the relationships between angles and trigonometric functions. It simplifies complex concepts and aids in problem-solving.
How Does The Sin Cos Tan Unit Circle Chart Help With Memorization?
The Sin Cos Tan Unit Circle Chart acts as a mnemonic device, offering a systematic way to memorize the values of sine, cosine, and tangent for common angles. This aids in quick recall during mathematical problem-solving.
Are There Online Resources For Interactive Unit Circle Learning?
Yes, several online platforms offer interactive Unit Circle tools and apps that allow students to explore and interact with the chart dynamically, enhancing their understanding of trigonometry.
Can The Unit Circle Chart Be Used In Advanced Mathematics Courses?
Yes, The Unit Circle Chart is a versatile tool that finds applications in advanced mathematics courses, especially in fields like calculus and physics. Its principles remain foundational in higher-level studies.
Conclusion: Empowering Mathematics Education with Unit Circle Charts
In conclusion, the integration of Unit Circle Charts into mathematics learning and teaching has a profound impact on student understanding and engagement. By leveraging the visual and mnemonic aspects of these charts, educators can transform complex trigonometric concepts into accessible and memorable lessons. As we continue to explore innovative tools in mathematics education, the Unit Circle Chart stands out as a timeless and indispensable resource.
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https://techvizer.com/how-we-can-impact-maths-learning-teaching/
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Let us now look at how domains and zones are related.
At the beginning of this tutorial series, we discussed domains and how DNS is used to resolve domain names to IP addresses. Let us now take a closer look at the domain’s structure.
DNS Name Space
The Domain Name System (DNS) is a hierarchical and distributed naming system. A tree data structure is used to represent the domain name space.
A namespace is a context in which all object names must be unambiguously resolvable. The internet, for example, is a single DNS name space in which all network devices with a DNS name can be resolved to a specific address (for example, www.microsoft.com resolves to 188.8.131.52).
A namespace can be hierarchical or flat. A flat namespace does not scale, well, because it can only expand so far before all available names are exhausted. When a name is used more than once in a namespace, the namespace violates the requirement of being unambiguously resolvable.
A hierarchical namespace is divided into areas known as sub-namespaces. Within the overall namespace, each area has its own sub-namespace.As a consequence, in order to have an unambiguously resolvable name within the namespace hierarchy, each object must have a unique name only within its sub-namespace. As a direct consequence, hierarchical namespaces can scale to extremely large networks—as more objects are added to the overall name space, you must find unique names for them within only the sub-namespace to which they belong. That is why your host machine can be called www along with millions other hosts.
DNS namespaces are all hierarchical. Domains are the sub-namespaces in the DNS hierarchical namespace. A relatively distinguished name is the unique name of a computer within a domain.
Because a fully-qualified domain name (FQDN) can be fully resolved to a unique object within the entire DNS hierarchy, computers with the same relative distinguished name can exist in different sub-namespaces (domains) of the namespace hierarchy.
You could, for example, have server1 in the widgets.yourdomain.com domain (the widgets.yourdomain.com namespace) and server1 in the gadgets.widgets.yourdomain.com namespace. They can be resolved to different FQDNs because they are in different sub-namespaces in the hierarchical namespace—server1.widgets.yourdomain.com and server1.gadgets.widgets.yourdomain.com.
Don’t be concerned if it doesn’t make sense to you. It will become clear to you in a few minutes.
The DNS can be thought of as an upside down tree.
DNS domains, like the UNIX file system, are organized as a series of descending branches, similar to tree roots. Each branch represents a domain, and each sub branch represents a subdomain. Domain and subdomain are relative terms. A given domain is a subdomain in the hierarchy to the domains above it and a parent domain to the subdomains below it.
In the above figure, for example, .edu is a parent domain to the nsu, and mit domains. Alternatively, you could say that those are subdomains of the .com domain.
In the below figure, .com is a parent domain to the acme, apex and buss domains. Alternatively, you could say that those are subdomains of the .com domain. The acme domain, in turn, is the parent of three subdomains (boss, toys, and sec).
Each node in the tree has a text label (without dots) of up to 63 characters. The root is given a null (zero-length) label. The full domain name of any node in the tree is the label sequence from that node to the root. Domain names are always read from the node to the root (“up” the tree), with dots separating the names.
If the root node’s label actually appears in a node’s domain name, the name looks as though it ends in a dot, as in “www.google.com.” . (It actually ends with a dot – the separator – and the null label for the root). When the label of the root node appears alone, it is written as a single dot, “.”, for convenience. As a consequence, some software interprets a trailing dot in a domain name to indicate that it is absolute. An absolute domain name is written relative to the root and specifies a node’s position in the hierarchy unambiguously. A fully qualified domain name, or FQDN, is another term for an absolute domain name.
Names that lack trailing dots are sometimes interpreted as relative to a domain other than the root, just as directory names that lack a leading slash are frequently interpreted as relative to the current directory.
DNS requires sibling nodes or nodes that are children of the same parent, to have distinct labels. This constraint ensures that a domain name only identifies a single node in the tree. The restriction isn’t really a restriction because the labels only have to be unique among the children, not among all the nodes in the tree.
The same rule applies to the UNIX filesystem: two sibling directories cannot have the same name. You can’t have two /usr/bin directories in the same namespace, just like you can’t have two hobbes.pa.ca.us nodes ( figure below). However, you can have both a hobbes.pa.ca.us and a hobbes.lg.ca.us node, just as you can have a /bin directory and a /usr/bin directory.
Domains And Domain Names
People frequently mix up domain names and domains. Though they are frequently confused, they are technically two distinct concepts. A domain is merely a branch(subtree) of the domain name space. A domain’s domain name is the same as the domain name of the node at the domain’s very top. As shown in Figure 2.3, the top of the purdue.edu domain is a node named purdue.edu.
Any domain name in the subtree is considered a part of the domain. A domain name can be in many domains because it can be in many subtrees. As shown in Figure 2.5, the domain name pa.ca.us is part of the ca.us domain as well as the us domain.
In essence, a domain is a subtree of the domain name space. And the domain name is the name of the top node of the subtree.
But where are all the hosts if a domain is simply made up of domain names and other domains? Domains are just groups of hosts, aren’t they?
The hosts, as represented by domain names, are present. Keep in mind that domain names are simply indexes in the DNS database. “Hosts” are domain names that point to information about specific hosts. And a domain includes all of the hosts whose domain names are contained within the domain. The hosts are linked logically, usually by geography or organizational affiliation, rather than by network, address, or hardware type. You could have ten different hosts, each on a different network and possibly even in a different country, but all in the same domain. This is how google can have hundreds of thousands of different hosts all in the same domain(google.com).
Individual hosts are generally represented by domain names at the tree’s leaves, which may point to network addresses, hardware information, and mail routing information. Domain names in the tree’s interior can both name hosts and point to domain information. Interior domain names do not have to be one or the other. They can represent both the domain to which they belong and a specific host on the network. For example, google.com is the domain name of Google as well as the domain name of a host that hosts Google’s primary web server.
The above diagram depicts an interior node with both host and structural data. Let us go over this again. A domain can have multiple subtrees, known as subdomains. In DNS documentation, the terms domain and subdomain are frequently used interchangeably or nearly so. Subdomain is only used as a relative term in this context: a domain is a subdomain of another domain if the root of the subdomain is within the domain.
Comparing their domain names is a simple way to determine whether one domain is a subdomain of another. The domain name of a subdomain is followed by the domain name of its parent domain. For example, because shop.mycompany.com ends with mycompany.com, it must be a subdomain of mycompany.com. Similarly, mycompany.com is a subdomain of com.
Domains are frequently referred to by level, in addition to being referred to in relative terms as subdomains of other domains. The terms top-level domain and second-level domain may be heard on mailing lists and in Usenet newsgroups. These terms simply refer to the position of a domain in the domain name space:
A top-level domain is a child of the root.
A first-level domain is a child of the root (i.e., a top-level domain).
A second-level domain is a child of a first-level domain, and so on.
Is the difference between a domain and a domain name now clear to you?
If not, please allow us to elaborate. Recognizing the distinction is advantageous to you.
A network domain is an administrative grouping of multiple private computer networks or local hosts that are all part of the same infrastructure. An example might help you understand.
As you can see, Hogwarts has several towers. We designed and built networks in all of the towers. Because there are so many users in the Slytherin Dungeon, we’ve set up two networks, say, network A and network B.
Half of the Slytherin students use Network A, 192.168.10.0/24. The VLAN identifier for this network is VLAN 10. The other half of Slytherin students connect to Network B, 192.168.20.0/24. VLAN 20 is the VLAN identifier for this network. Do not be concerned about the VLAN identifier.
The dark tower now has its own network as well. Dark Tower’s entire staff connects to Network C, 192.168.0.0/24. VLAN 11 is the VLAN identifier for this.
The router Router1 acts as the gateway for all three networks, and the entire infrastructure is physically connected via ethernet. Networks B and C are connected via Router1 and have full access to one another.
Network A is completely separate from the other two and has no access to them. As a direct consequence, Networks B and C are in the same network domain, whereas Network A is in its own network domain, albeit alone.
Let us now talk about the domain name. A domain name is used to identify a domain. What are you going to call domains A and B? A is a distinct domain from B. B is made up of two distinct networks. You can call them anything. You can call Domain A ‘voodoo’ and Domain B ‘blah’. It’s completely up to you.
A domain name is a string of characters that identifies a domain’s administrative autonomy, authority, or control on the Internet. Domain names are frequently used to identify Internet-based services such as websites, email services, and so on. In 2017, 330.6 million domain names were registered. Domain names are used in a variety of networking contexts as well as for application-specific naming and addressing. A domain name, in general, identifies a network domain or an Internet Protocol (IP) resource, such as a personal computer or a server computer.
The distinction between domain and domain name is analogous to the distinction between a street name and the actual street. It’s a thing with a name. It’s a digital thing in the case of a domain…
Or the distinction between you and your name. A car and its manufacturer. One is a thing, and the other is its name.
What about the DNS? What is the function of DNS?
Within the Domain Name System, domains that must be accessible from the public Internet can be assigned a globally unique name (DNS).
Domain names are formed by the Domain Name System’s rules and procedures for networks that are publicly accessible. Any name registered in the DNS is a domain name. Domain names are organized into subordinate levels (subdomains) of the nameless DNS root domain. The top-level domains (TLDs), which include the popular domains com, info, net, edu, and org, and the country code top-level domains, are the first-level set of domain names (ccTLDs). The second-level and third-level domain names in the DNS hierarchy are typically open for reservation by end-users who wish to connect local area networks to the Internet, create other publicly accessible Internet resources, or run web sites.
A second- or third-level domain name is typically registered by a domain name registrar, who sells its services to the general public.
A fully qualified domain name (FQDN) is a domain name that is fully specified with all labels in the DNS hierarchy, with no parts left out. A FQDN is traditionally terminated with a dot (.) to denote the top of the DNS tree. Labels in the Domain Name System are case-insensitive and can thus be written in any desired capitalization method, but in technical contexts, domain names are typically written in lowercase.
Remember our dig command output?
; <<>> DiG 9.16.35 <<>> ns.google.com
;; global options: +cmd
;; Got answer:
;; ->>HEADER<<- opcode: QUERY, status: NOERROR, id: 24785
;; flags: qr rd ra; QUERY: 1, ANSWER: 1, AUTHORITY: 0, ADDITIONAL: 1
;; OPT PSEUDOSECTION:
; EDNS: version: 0, flags:; udp: 4096
; COOKIE: bb7d6c724eb39b63ebcfb71f63b0580020c97b5fc3b155da (good)
;; QUESTION SECTION:
;ns.google.com. IN A
;; ANSWER SECTION:
ns.google.com. 0 IN A 184.108.40.206
;; Query time: 145 msec
;; SERVER: 172.31.6.10#53(172.31.6.10)
;; WHEN: Sat Dec 31 21:40:49 Bangladesh Standard Time 2022
;; MSG SIZE rcvd: 86
It is now time to explain what the output actually means and how you can use this information.
The Domain Name System defines a database of network resource information elements. The information elements are classified and organized using a list of DNS record types and resource records (RRs).
Resource records, or RRs, contain information about domain names. Records are classified into classes, each of which corresponds to a specific type of network or software. There are currently classes for internet (any TCP/IP-based internet), Chaosnet-based networks, and Hesiod-based networks. (Chaosnet is an ancient network with mostly historical significance).
Each record has a type (name and number), an expiration time (time to live), a class, and data that is unique to that type. A resource record set (RRset) is a collection of resource records of the same type that have no special ordering. When queried, DNS resolvers return the entire set, but servers may use round-robin ordering to achieve load balancing. Domain Name System Security Extensions (DNSSEC), on the other hand, work on the entire set of resource records in canonical order.
All records sent over an Internet Protocol network use the common format specified in RFC 1035.
By far the most popular is the internet class. (We’re not sure if anyone still uses the Chaosnet class, and the Hesiod class is mostly used at MIT). The internet class is the focus of this tutorial.
Within a class, records can be of various types, which correspond to the various types of data that can be stored in the domain name space. Different classes define different record types, though some are shared by multiple classes. Almost every class, for example, defines an address type. Each record type in a given class defines a specific record syntax that must be followed by all resource records of that class and type.
Don’t be concerned if this information appears shady. It will become clear in no time.
So far, we’ve discussed the theoretical structure of the domain name space, what kind of data it stores, and even hinted at the types of names you might find in it with our (sometimes fictitious) examples. However, this will not assist you in decoding the domain names you encounter on a daily basis on the Internet.
The Domain Name System does not impose many rules on domain name labels and does not assign any particular meaning to the labels at any level. You can choose your own semantics for your domain names when you manage a part of the domain name space. Nobody would object if you named your subdomains A through Z (though they might strongly recommend against it).
The existing Internet domain name space, on the other hand, has some self-imposed structure. Domain names, particularly in the upper-level domains, adhere to certain conventions (not rules, really, as they can be and have been broken). These customs keep domain names from appearing completely random. Understanding these traditions is extremely useful when attempting to decipher a domain name.
You’ll probably find it much easier to understand most domain names now that you know what most top-level domains represent and how their namespaces are structured. Let’s practice dissecting a few:
You’ve already gotten a head start on this one because we’ve already told you that google.com is the domain of Google. Google employees are in charge of managing this domain. (If you didn’t already know, you could have deduced that the name belongs to a commercial organization because it’s in the top-level com domain). Google.com’s corporate section’s subdomain is corporate. Finally, lithium is the name of a specific host in the domain – one of about a hundred or so, if they have one for each element.
This example is a little more difficult, but not by much. The hp.com domain is almost certainly owned by the Hewlett-Packard Company (in fact, we mentioned this earlier, too). Their corporate headquarters is undoubtedly their corp subdomain. And Winnie is probably just a silly name that someone made up for a host.
You’ll need to apply your knowledge of the US domain here. ca.us is obviously the domain of California, but mpk is anyone’s guess. Unless you know your San Francisco Bay Area geography, it would be difficult to tell that this is Menlo Park’s domain. (And no, this is not the same Menlo Park where Edison lived; that is in New Jersey.)
We’ve included this example to avoid the misconception that all domain names have four labels. gryffindortower is a subdomain of hogwarts.com. commonroom is gryffindor’s site. And daphne is a host in the commonroom.
We hope that all of your questions about domains, domain names, and DNS have been answered. Let’s talk about zones and zone files now.
Zones have already been discussed in an earlier section. We discussed zones and how they help us. Let’s talk about zones some more. To fully comprehend the concept and significance of zones, you must first comprehend delegation.
What is DNS Zone Delegation?
Remember how one of the primary goals of the Domain Name System design was to decentralize administration? Delegation is used to accomplish this. Domain delegation is similar to task delegation at work. A manager may divide a large project into smaller tasks and assign responsibility for each to different employees.
Similarly, an organization that manages a domain can subdivide it. Each of these subdomains can be delegated to a different organization. This means that an organization is responsible for the upkeep of all data in that subdomain. It has complete control over the data and can even divide and delegate its subdomain. The parent domain retains only pointers to the subdomain’s data sources in order to refer queries there. The domain hogwarts.edu, for example, is delegated to Hogwarts network administrators, as shown in the below diagram.
Not all organizations delegate their entire domain, and not all managers delegate their entire workload. A domain can have multiple delegated subdomains as well as hosts that do not belong in the subdomains. For example, the Acme Corporation (which provides most of the gadgets for a certain coyote) has a division in Rockaway and its headquarters in Kalamazoo, so it could have a rockaway.acme.com subdomain and a kalamazoo.acme.com subdomain. However, the few hosts in the Acme sales offices spread across the United States would fit better under acme.com than either subdomain.
Later, we’ll go over how to create and delegate subdomains. For the time being, it is sufficient to understand that the term delegation refers to the assignment of responsibility for a subdomain to another organization.
You’d be surprised at how many subdomains a business manages. Click on the link to view the Microsoft.com Subdomains.
This section may feel like we’re going over old ground. Perhaps we are. In fact, we will be expanding on a previously discussed topic in this section.
Name servers are programs that store information about the domain name space. Name servers typically have complete information about a portion of the domain name space (a zone) that they load from a file or another name server. The name server is then said to have zone authority for that zone. Name servers may also be authoritative for multiple zones.
As you can see, domains, domain names, DNS, and name servers all serve different functions. They all work together to make your internet surfing experience a bit easier.
The distinction between a zone and a domain is significant but subtle. All top-level domains, and many domains at the second level and lower, such as hogwarts.edu and google.com, are broken into smaller, more manageable units by delegation.
These units are called zones. The edu domain is divided into many zones, as shown in the below diagram , including the hogwarts.edu zone, the mit.edu zone, and the nwu.edu zone. There is also an edu zone at the top of the domain. It’s natural for the people in charge of edu to split up the edu domain; otherwise, they’d have to manage the hogwarts.edu subdomain themselves. Delegating hogwarts.edu to Hogwarts makes far more sense. What remains for those in charge of edu? The edu zone, which would contain mostly delegation information for subdomains of edu.
The above diagram depicts The edu domain broken into zones.
The hogwarts.edu subdomain is, in turn, broken up into multiple zones by delegation. Delegated subdomains include gt, sd, rt, ht, and others. Each of these subdomains is delegated to a set of name servers, some of which are also authoritative for hogwarts.edu. However, the zones remain distinct and may have a completely different set of authoritative name servers.
A DNS zone is a section of the DNS namespace managed by a specific organization or administrator. A DNS zone is an administrative space that enables more granular control of DNS components like authoritative nameservers.
A common misperception is to associate a DNS zone to a domain name or a single DNS server. A DNS zone can, in fact, contain multiple subdomains, and multiple zones can coexist on the same server. DNS zones are not required to be physically separated from one another; zones are solely used for control delegation.
A zone and a domain can have the same domain name but different nodes. The zone, in particular, lacks nodes in delegated subdomains. The top-level domain ca (for Canada) has subdomains ab.ca, on.ca, and qc.ca for the provinces of Alberta, Ontario, and Quebec, respectively. Name servers in each province may be delegated authority over the ab.ca, on.ca, and qc.ca subdomains. The domain ca includes all of the data in ca as well as all of the data in ab.ca, on.ca, and qc.ca. However, the zone ca only contains the data in ca (see Figure 2-10), which is most likely pointers to the delegated subdomains. And the zones ab.ca, on.ca, and qc.ca are distinct from the ca zone.
Consider the example of quidditch. Headmaster Dumbledore is in charge of everything at the school. And he delegated house maintenance to four house masters. As a consequence, all of the houses are subdomains of Hogwarts. And house masters are in charge of various tasks such as managing a quidditch team. Headmaster Dumbledore no longer has a list of each house’s players. It is not his responsibility to manage the house quidditch team. He had already assigned the task to the housemasters. The list(zone file in the house zone) of players is kept by the house masters(authority). They have access to the players’ information. And Dumbledore has a list (a zone file in the school zone) of all the house masters.
The domain hogwarts includes all of the data in Hogwarts as well as all of the data in gryffindor.hogwarts, slytherin.hogwarts, ravenclaw.hogwarts and hufflepuff.hogwarts. However, the zone hogwarts only contains the data in hogwarts, which is most likely pointers to the delegated subdomains. And the zones gryffindor.hogwarts, slytherin.hogwarts, ravenclaw.hogwarts and hufflepuff.hogwarts are distinct from the hogwarts zone.
However, if a subdomain of the domain is not delegated, the zone contains the domain names and data in the subdomain. As a result, the bc.ca and sk.ca (British Columbia and Saskatchewan) subdomains of the ca domain may exist but are not delegated. (Perhaps the provincial authorities in British Columbia and Saskatchewan aren’t yet ready to manage their own zones, but the authorities in charge of the top-level ca zone want to maintain namespace consistency and implement subdomains for all Canadian provinces right away.) As shown in Figure 2-11, the zone ca has a ragged bottom edge and contains bc.ca and sk.ca but not the other ca subdomains.
It is now clear why name servers load zones rather than domains: a domain may contain more information than the name server requires. A domain may contain data that has been delegated to other name servers. Because a zone is defined by delegation, it never contains delegated data.
Consider what would happen if a root name server loaded the root domain rather than the root zone: it would load the entire namespace!
However, if you’re just starting out, your domain will most likely not have any subdomains. In this case, because there is no delegation, your domain and zone contain the same data. It’s like one ring to rule them all, Or in your case one administrator to manage them all.
Even if you don’t need to delegate parts of your domain just yet, it’s useful to understand how the process of delegating a subdomain works. In general, delegation entails delegating responsibility for a portion of your domain to another organization. What actually occurs is the delegation of authority for your subdomains to various name servers. (Note that we said “name servers,” not just “name server”).
Instead of containing information in the subdomain you’ve delegated, the data in your zone includes pointers to the name servers that are authoritative for that subdomain. If one of your name servers is asked for data in the subdomain, it can now respond with a list of the appropriate name servers to contact.
That’s all for DNS.
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Lab Assignment 3: Newton’s Laws
Newton’s laws of motion are a central component of our understanding of physics. As we discussed in Module 5, Newton’s laws can be summarized as follows:
- 1. Inertia – An object tends to resist changes in its motion.
- 2. Relationship between the mass of an object, the net applied force, and the resulting acceleration – F = m a.
- 3. Action-reaction pairs – Forces come in pairs.
In this lab, you will perform experiments to explore each of the laws of motion.
This activity is based on Lab 5 of the eScience Lab kit. Although you should read all of the content in Lab 5, we will be performing a targeted subset of the eScience experiments.
Our lab consists of three main components. These components are described in detail in the eScience manual (pages 55-61). Here is a quick overview:
- • In the first part of the lab, you will use a bowl full of water to understand the concept of inertia. (eScience Experiment 1)
- • In the second part of the lab, you will recreate a classic physics experiment, the Atwood Machine. This system consists of a pulley holding a string with two unequal masses. Experimenting with an Atwood Machine is an excellent way to understand Newton’s second law of motion. (eScience Experiment 2, Procedure 1)
- • In the final part of the lab you will create a balloon-powered vehicle to elucidate Newton’s third law of motion. (eScience Experiment 4)
Note: Record all of your data in the tables that are provided in this document.
Take detailed notes as you perform the experiment and fill out the sections below. This document serves as your lab report. Please include detailed descriptions of your experimental methods and observations.
Newton’s First Law – Water in a Bowl
- • I recommend that you perform this experiment outdoors as there most likely will be some spillage of water.
Newton’s Second Law – The Atwood Machine
- • Prior to determining the mass of the washers, make sure to zero your spring scale. To zero your spring scale, hold it vertically with no mass attach and turn the top screw until the scale reads 0 grams. Refer to the following picture:
- • You may want to use the hooks on the pulley to hang your Atwood machine. I placed mine on a hanger:
Newton’s Third Law – Balloon-Powered Vehicle
- • Here is a picture of my balloon-powered vehicle:
- • To add mass, I taped washers to the straw.
Experiment 1 – inertia – Newton’s first law of motion.
See page 119 of Physics by James Walker, 5th edition, for a statement of Newton’s First Law of Motion.
1. Fill the container with a couple of inches of water.
2. Find an open space outside to walk around in with the container of water in your hands.
3. Perform the following activities and record your observations of each motion in Table 1:
a. Start with the water at rest (e.g., on top of a table). Grab the container and quickly accelerate it.
b. Walk with constant speed in a straight line for 15 feet.
c. After walking a straight line at constant speed, make an abrupt right-hand turn. Repeat with a left-hand turn.
d. After walking a straight line at constant speed, stop abruptly.
Experiment 2 – mass and acceleration – Newton’s 2nd Law of Motion
See page 122 of Physics by James Walker, 5th edition, for a statement of Newton’s Second Law of Motion.
A diagram, equations, and free body diagram for the Atwood Machine (a pulley with hanging masses) are shown on page 176 of Physics by James Walker, 5th edition.
You will use the metal washers to make the masses. You can tie the washers to the string or use a hanger, such as a paper clip. If you do use a hanger you will have to include its mass into the total mass, mass of washers + mass of hanger.
You will use 15 washers to make the larger mass and 5 washers to make the smaller mass.
Use enough string to allow a mass to fall to floor when starting from near the pulley. The other mass is going from the floor to near the pulley.
With the masses hanging from the pulley, the greater mass near the top, measure the distance the mass will fall to the floor. Time the fall of that mass.
Calculate the acceleration of the falling mass:
y = ½ a t2 : we assume no initial velocity when you started timing, that is you just let it drop and started timing when you let go. From the above we solve for a (acceleration):
Note: This assumes you do not have a constant velocity. A constant velocity could occur with a significant pulley friction. Having masses with a large difference in value helps reduce the effect of pulley friction.
Record 10 trials.
Data table for the Atwood Machine experiment (Experiment 2, Procedure 1):
Height = __ meters
Mass of 10 washers = __ grams
Mass of 5 washers = __ grams
M1 = __ grams (lighter mass)
M2 = __ grams (heavier mass)
Fall time (sec)
Calculated acceleration (m/s2)
Experiment 3 – Balloon-Powered Vehicle – Newton’s third law of motion.
A statement of Newton’s Third Law of Motion is on page 129 of Physics by James Walker, 5th edition.
This will be easier with an assistant if available. Blow up the balloon similar to that shown in the picture above but do NOT tie the balloon. Attach a straw by taping to the balloon. Thread a string of about 10 ft. length through the balloon. Attach the string to two chairs and separate them until the string is tight.
Release the balloon and observe its motion. Does it appear to accelerate?
When released can you feel the air rushing out the orifice (nozzle if you like).
Tape a washer to the balloon and repeat the experiment noting any observed difference in the balloons motion.
ANALYSIS and DISCUSSION
Based on your experimental results, please answer the following questions:
Explain how your observations of the water demonstrate Newton’s law of inertia.
Draw a free body diagram of your containers of water from the situation in part d (After walking in a straight line at constant speed, stop abruptly). In your free body diagram, draw arrows for the force of gravity, the normal force (your hand pushing up on the container), and the stopping force (your hand decelerating the container as you stop.)
What is the direction of the water’s acceleration.
Describe two instances where you feel inertial forces in a car.
Draw a FBD for M1 and M2 in your Atwood machine. Draw force arrows for the force due to gravity acting on both masses and the force of tension
Copy the Atwood Machine acceleration equation from the text.
Using the masses M1 and M2, use the above expression to calculate the acceleration of the system. Make sure to show your calculation for the acceleration. How does this value compare to your experimentally measured acceleration? What factors may cause discrepancies between the two values?
Explain what caused the balloon to move in terms of Newton’s Third Law.
What is the force pair in this experiment? Draw a free body diagram to represent the (unbalanced) forces on the balloon/straw combination.
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You must first learn the concept of a function to understand the difference between a function’s differential and derivative.
One of the fundamental ideas in mathematics is the concept of a function, which describes the connection between a set of inputs and a set of potential outputs, where each input connects to one output. The independent variable is one variable, and the dependent variable is another.
In mathematics, variables are any physically altering entities. The difference between the differential and derivative is that if there is a change in the rate of one variable for another variable, we refer to it as a derivative.
On the other hand, a differential equation is a corresponding equation that expresses the relationship between variables mathematically.
Let us learn the basic definitions and difference between the differential and derivative:
Table of Contents
What is Derivative?
The speed at which a function changes at a specific instant in time is called a derivative.
For instance, a line’s slope, which determines its rate of change, is constant along the entire line. The derivative informs us what the slope of a parabola is at a specific point because the slope of a parabola fluctuates.
Another way to depict a derivative is to use the slope of a line that is drawn tangent to a curve at a certain point. Taking a derivative is the process of differentiating. Differential equations and derivatives are frequently used together.
Finding derivatives is done through the differentiation process. They are used to denote a tangent line’s slope. Derivatives quantify the steepness of the slope of a function over a specified time interval.
Derivative of an Integral
The outcome of differentiating an integral’s result is the derivative of an integral. Since differentiation of an integral should provide the original function itself, integration is the process of locating the “anti” derivative.
When the integral’s lower limit is a constant and its upper limit is just a variable, its derivative is the function itself. In other words, where “a” is a constant, d/dx ∫ax f(t) dt = f(x).
What is Differential?
Differential equations are a calculus branch that depicts the slightest variation in some variable quantities. Derivatives and their functions are part of differential equations.
Calculus’ fundamental divisions include the differential and integral branches. The environment we live in is full of constantly changing, interconnected quantities.
The symbol for a differential, an infinitesimally small amount, is often dy/dx.
Because a derivative reflects the slope of a function on an infinitesimally short interval comparable to a single point, it is frequently thought of as a quotient of differentials, such as dy/dx, dy/dx.
Difference Between the Differential and Derivative
The main between the differential and derivative given below:
The primary difference between the differential and derivative is that a derivative refers to the rate of a function’s change, but a differential refers to the actual change in the function.
Another way to define derivatives is to consider the ratio of the function and variable differentials.
Another difference between the differential and derivative is that a differential is a change in a variable (dx). In contrast, a derivative is a change in a function (dy/dx) (dy/dx) (dx).
The derivative is a ratio of differentials because a function represents the relationship between two variables. Differentials indicate the actual value change through a linear map, whereas derivatives represent the same change through a slope map.
Differential Vs. Derivative
In terms of relationships, the terms differential and derivative are closely related to one another. Variables are changing objects in mathematics, and the rate at which one variable changes in relation to another is known as a derivative.
Differential equations describe the relationship between these variables and their derivatives. Finding a derivative is the process of differentiation.
The comparison between differential vs. derivative is that the differential of a function is the actual change in the function, whereas the derivative is the rate at which the output value changes in relation to the input value.
Representation of Differential Vs. Derivative
Differentials can be represented as dx, dy, and so on, where dx represents a small change in x, dy represents a small change in y. The differential dy can be expressed as follows when contrasting changes in related values where y is a function of x:
A function’s slope at any given point is its derivative and can be represented as d/dx.
For example, we can write the derivative of sin(x) as:
d/dx sin(x) = sin(x)’ = cos(x)
Relationship Between Derivative and Differential
Differentiation is a technique for calculating a derivative or the rate at which the output y of a function changes as a function of the changing variable x.
Simply put, a derivative is a rate at which y changes about x.
This relationship represents y = f(x), which denotes that y is a function of x.
The function whose value generates the slope of the function f(x) where it is specified and f(x) is differentiable is said to be the derivative of f(x). It speaks of the graph’s slope at a specific location.
What is the Difference Between the Differential and Derivative?
Highlighting the main difference between the differential and derivative in the following table:
|The rate of change of the variables in a differential equation is represented by derivatives.
|Differentials indicate the smallest variations in variables’ quantities.
|The slope of the graph at a particular point is calculated.
|It is calculated to find the linear difference.
|Simple derivatives merely indicate how quickly the dependent variable is changing in relation to the independent variable.
|Derivatives are a tool used in the solution of differential equations. Differential equations have derivatives as well.
|It is known how different variables relate functionally.
|There are no known functional relationships between the variables.
|There are many derivative degrees and different representational formulae. The formula for a derivative that is most frequently employed is: d/dx
|Numerous formulas can be used to express differential equations. One of the most used formulas is: dy/dx = f. (x)
A differential and a derivative differ in terms of the functions they execute and the values they each represent. The difference between the differential and derivative is that differentials describe minor variations in varying amounts, such as the area of a body.
It makes it possible to compute the equation’s dependent and independent variable relationships.
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Genetic algorithms are a class of search algorithms inspired by the process of natural selection and evolution. They are widely used to solve optimization problems in various fields such as engineering, finance, and computer science. The core idea behind genetic algorithms is to mimic the evolutionary process by continuously evolving a population of candidate solutions to a problem.
In a genetic algorithm, each candidate solution, often referred to as an individual, is represented as a string of bits or numbers, called chromosomes. These chromosomes encode the parameters or features that define a solution to the problem at hand. The process of evolution involves several key steps, including selection, crossover, and mutation.
Selection is the process of identifying the fittest individuals in the population based on their fitness values. Fitness is a measure of how well an individual solves the problem, and it is typically evaluated using a fitness function. Crossover involves combining the genetic material of two individuals to create offspring. This process simulates the genetic recombination that occurs during sexual reproduction in nature. Mutation introduces small random changes in the chromosomes of the offspring to introduce diversity and prevent premature convergence to suboptimal solutions.
MATLAB, a popular software environment for numerical computation and data analysis, provides a convenient platform for implementing and experimenting with genetic algorithms. Its extensive library of functions for vector and matrix manipulation, optimization, and plotting makes it an ideal tool for tackling complex optimization problems. By leveraging the power of MATLAB, researchers and practitioners can easily develop and test new genetic algorithms for a wide range of applications.
Understanding Genetic Algorithm
Genetic algorithm is a search algorithm inspired by the process of natural selection and genetic evolution. It is used in various optimization problems to find the optimal solution.
The first step in genetic algorithm is the selection of individuals for the next generation. This process is based on the fitness of each individual, which represents how well it solves the optimization problem. The individuals with higher fitness are more likely to be selected for reproduction.
Crossover and Mutation
After the selection process, the selected individuals undergo crossover and mutation to create new individuals. Crossover involves exchanging genetic material between two parent individuals to create offspring. Mutation involves introducing small random changes in the genetic material of an individual. These processes help introduce diversity in the population and explore different regions of the search space.
In genetic algorithm, the population is evolved over multiple generations. The individuals with higher fitness are more likely to survive and pass their genetic material to the next generation. This process continues until a satisfactory solution is found or a termination condition is met.
In MATLAB, genetic algorithm can be implemented using the
ga function. This function takes an objective function, constraints, and other parameters as inputs, and returns the optimal solution.
In conclusion, genetic algorithm is a powerful optimization technique that mimics the process of natural selection and genetic evolution. It is widely used in various fields to find optimal solutions for complex problems.
Advantages of Genetic Algorithm
The genetic algorithm is a powerful optimization algorithm that is widely used in various fields. It has several advantages over other optimization algorithms:
1. Fitness: The genetic algorithm incorporates a fitness function that evaluates the quality of each potential solution. This allows the algorithm to focus on finding the best solutions to the optimization problem.
2. Mutation: Unlike other algorithms that rely solely on selection and crossover, the genetic algorithm includes a mutation operator. This helps introduce diversity in the population, allowing for exploration of new and potentially better solutions.
3. Selection: The genetic algorithm employs a selection mechanism that favors better-performing individuals in the population. This ensures that the overall quality of the population improves over time.
4. Optimization: The genetic algorithm is well-suited for optimization problems, where the goal is to find the best solution among a large set of possible solutions. It can handle both single-objective and multi-objective optimization problems.
5. Matlab Implementation: Implementing a genetic algorithm in MATLAB is relatively easy, thanks to the availability of built-in functions and tools for genetic algorithm optimization. This makes it convenient for researchers and practitioners to use this algorithm in their projects.
6. Genetic Evolution: The genetic algorithm is inspired by the process of natural evolution. It mimics the concepts of reproduction, mutation, and natural selection to evolve solutions over generations. This makes it a powerful and intuitive algorithm for optimization problems.
These advantages make the genetic algorithm a popular choice for solving optimization problems in various domains.
Applications of Genetic Algorithm
Genetic algorithms (GAs) are powerful optimization techniques inspired by the principles of natural evolution. These algorithms simulate the process of survival of the fittest, where solutions with the highest fitness are more likely to survive and reproduce.
GAs have been successfully applied to a wide range of optimization problems in various fields. Some of the common applications of genetic algorithms include:
- Function Optimization: Genetic algorithms can be used to find the global or local optimum of a given function, even when the function is complex or has multiple peaks. The algorithm starts with an initial population of solutions and uses selection, crossover, and mutation operations to evolve the population towards better solutions.
- Machine Learning: Genetic algorithms can be employed in the training and optimization of machine learning models. For example, they can be used to optimize the hyperparameters of a neural network or to evolve decision trees.
- Routing and Scheduling: Genetic algorithms can be used to find optimal routes for vehicles or to schedule tasks in a way that minimizes total cost or maximizes efficiency. These algorithms can consider various constraints and objective functions to find the best possible solutions.
- Image and Signal Processing: Genetic algorithms can be used for image restoration, feature selection, or image segmentation tasks. They can also be applied to signal processing problems, such as finding the optimal filters or feature extraction methods.
- Data Mining and Clustering: Genetic algorithms can be utilized to discover hidden patterns in large datasets or to cluster data points based on similarity. These algorithms can handle high-dimensional data and can find globally optimal solutions.
Implementing genetic algorithms in MATLAB provides a convenient and efficient environment for solving optimization problems. The MATLAB Genetic Algorithm Toolbox provides various built-in functions for population initialization, fitness evaluation, selection, crossover, and mutation. This allows researchers and practitioners to easily implement and customize genetic algorithms for their specific applications.
In conclusion, genetic algorithms have proven to be effective in solving a wide range of optimization problems. They can be applied to problems in various fields, including function optimization, machine learning, routing and scheduling, image and signal processing, and data mining. MATLAB provides a powerful platform for implementing and experimenting with genetic algorithms to find optimal solutions.
Implementing Genetic Algorithm in MATLAB
Genetic algorithms are optimization techniques inspired by the process of natural selection. They are used to solve complex optimization problems by mimicking the process of biological evolution. One popular implementation of genetic algorithms is in MATLAB, a programming language and software platform commonly used in scientific research and engineering.
In a genetic algorithm, a population of candidate solutions is evolved over multiple generations. Each candidate solution, also known as an individual, is represented as a set of genes that encode a potential solution to the optimization problem. The process of evolution involves several key steps: selection, crossover, and mutation.
In the selection step, individuals with higher fitness, which represents their suitability as a solution, are more likely to be chosen as parents for the next generation. This mimics the natural selection process, where individuals with higher reproductive success are more likely to pass on their genes.
The crossover step involves combining the genes of two parent individuals to create offspring. This is achieved by randomly selecting a crossover point and swapping the genes between the parents. The resulting offspring inherit some characteristics from each parent, potentially creating a better solution than either parent alone.
The mutation step introduces random changes to the genes of the offspring. This adds diversity to the population and helps explore different areas of the solution space. Without mutation, the genetic algorithm may get stuck in local optima and fail to find the global optimum.
By repeating the steps of selection, crossover, and mutation over multiple generations, the genetic algorithm converges towards an optimal solution to the optimization problem.
Implementing genetic algorithms in MATLAB is straightforward due to its powerful matrix manipulation capabilities and extensive library of mathematical functions. MATLAB provides functions for generating initial populations, evaluating fitness, performing crossover and mutation, and tracking the evolution process.
Using MATLAB, researchers and engineers can easily apply genetic algorithms to a wide range of optimization problems, such as parameter tuning, system design, and pattern recognition. By fine-tuning the parameters and fitness function, they can achieve efficient and effective solutions.
In conclusion, implementing genetic algorithms in MATLAB allows researchers and engineers to leverage the power of genetic evolution for solving complex optimization problems. With its rich set of features and ease of use, MATLAB provides a reliable platform for developing and implementing genetic algorithms.
Choosing Fitness Function
In the optimization process of a genetic algorithm, the fitness function plays a crucial role. It is the measure of how well a particular solution performs in solving the given problem. The fitness function evaluates the quality of each individual in the population based on its ability to meet the desired objectives.
When implementing a genetic algorithm in MATLAB for optimization problems, choosing an appropriate fitness function is essential. The fitness function should be designed to quantify the objective goals of the optimization problem and guide the evolution of the population towards better solutions.
The fitness function typically takes the candidate solution as input and returns a value that represents its fitness. This value is used to assess the solution’s suitability for survival and reproduction in the evolutionary process. Solutions with higher fitness values are more likely to be selected for reproduction and crossover, while those with lower fitness values are more likely to be mutated or eliminated.
In MATLAB, the fitness function can be implemented as a separate function or as an anonymous function within the genetic algorithm code. It should be designed to evaluate the performance of a solution based on the problem’s constraints and objectives.
Factors to consider when designing the fitness function include the problem’s specific requirements, performance metrics, and the trade-offs between different objectives. The fitness function may involve mathematical calculations, simulations, or evaluations of the solution’s performance against specific criteria.
It is important to note that the fitness function should be carefully chosen to capture the desired optimization goals without bias towards certain solutions. A well-designed fitness function enables the genetic algorithm to explore the solution space effectively and converge towards a near-optimal solution.
Overall, choosing an appropriate fitness function is a critical step in implementing a genetic algorithm in MATLAB for optimization problems. The fitness function guides the evolution of the population, influencing the selection, mutation, and crossover processes to improve the quality of the solutions. By selecting and designing the fitness function effectively, the genetic algorithm can efficiently search for optimal or near-optimal solutions to complex optimization problems.
Selecting Appropriate Selection Method
The selection phase plays a crucial role in the optimization process of genetic algorithms. It determines which individuals are chosen to undergo genetic operations such as crossover and mutation, ultimately influencing the evolutionary search for an optimal solution in a given problem space. In MATLAB, various selection methods are available, providing different approaches to balance exploration and exploitation during the optimization process.
One commonly used selection method in MATLAB is tournament selection. This method involves randomly selecting a subset of individuals as potential parents and then selecting the best individual from this subset as a parent for the next generation. The size of the subset and the number of individuals to be selected can be controlled to influence the selection pressure. Tournament selection is advantageous as it does not require high computational power and allows for diverse solutions to be explored.
An alternative selection method is roulette wheel selection, also known as fitness proportionate selection. This method assigns a probability of selection to each individual in the population based on its fitness value. The individuals with higher fitness values are more likely to be selected as parents. Roulette wheel selection is advantageous as it allows for a more natural selection process, favoring individuals with higher fitness values and improving convergence towards optimal solutions.
One of the advantages of tournament selection is the ability to control the selection pressure by adjusting the size of the subset and the number of individuals to be selected. A larger subset size and a smaller number of individuals selected will result in higher selection pressure, favoring the fittest individuals and potentially converging towards optimal solutions more quickly. On the other hand, a smaller subset size and a larger number of individuals selected will result in lower selection pressure, allowing for more exploration of the search space and potentially finding diverse solutions.
Roulette Wheel Selection:
Roulette wheel selection assigns a probability of selection to each individual based on its fitness value. The higher the fitness value, the higher the probability of selection. This method allows for a more natural selection process, as individuals with higher fitness values are more likely to be selected as parents. However, care should be taken to avoid premature convergence, where only a small subset of the population is selected as parents, potentially limiting exploration of the search space. To counter this, techniques such as scaling fitness values or implementing elitism can be used.
In conclusion, the selection method employed in a genetic algorithm implemented in MATLAB should be carefully chosen based on the problem at hand. Tournament selection provides control over selection pressure and allows for exploration of diverse solutions, while roulette wheel selection favor individuals with higher fitness values and improves convergence towards optimal solutions. Depending on the characteristics of the problem and the desired behavior of the optimization process, either selection method can be used effectively in the implementation of a genetic algorithm.
Deciding on Crossover Strategy
When implementing a genetic algorithm for optimization problems, one of the key decisions is choosing the appropriate crossover strategy. Crossover is a genetic operator that combines the genetic material of two parent individuals to create new offspring individuals. It helps to maintain diversity in the population and allows for the exploration of different solutions in the search space.
In the context of optimization, the selection of the appropriate crossover strategy depends on the characteristics of the problem at hand and the desired properties of the solution. There are several commonly used crossover strategies in genetic algorithms:
One-point crossover is a simple and widely used crossover strategy. In this approach, a random point is selected on the chromosomes of the parents and the genetic material beyond that point is swapped between the parents. This creates two offspring individuals with a recombined set of genes.
Two-point crossover is similar to one-point crossover, but instead of one point, two random points are selected on the chromosomes of the parents. The genetic material between these two points is swapped between the parents, creating two offspring with a mix of genes from both parents.
Uniform crossover is a more flexible crossover strategy. In this approach, each gene in the offspring is randomly selected from either parent with a certain probability. This allows for a greater exploration of the search space and can be particularly useful when the optimal solution is not easily represented by specific gene combinations.
It is important to note that the choice of crossover strategy should be considered in conjunction with the selection and mutation strategies. The selection strategy determines which individuals are chosen as parents for crossover, while the mutation strategy introduces random changes to the offspring. A balanced combination of these components is crucial for the success of the genetic algorithm in finding optimal solutions to the optimization problem.
In MATLAB, there are various functions and libraries available for implementing genetic algorithms, such as the Global Optimization Toolbox. These resources provide tools for defining the fitness function, specifying the crossover strategy, mutation strategy, and other parameters, and running the genetic algorithm to find the optimal solution.
Determining Mutation Rate
Mutation is a key component of the genetic algorithm (GA) in the evolution of solutions for optimization problems. It introduces diversity into the population by randomly altering the genetic material, allowing the algorithm to explore new areas of the search space and potentially find a better solution.
The mutation rate determines the probability of a mutation occurring in each individual during the evolution process. If the mutation rate is too low, the algorithm may get stuck in a local optima, as there is not enough exploration happening. On the other hand, if the mutation rate is too high, the algorithm may lose the beneficial solutions it has already found.
Determining the optimal mutation rate for a specific problem is a challenging task, as it depends on the nature of the problem, the size of the search space, and the characteristics of the initial population. However, there are some general guidelines that can help in selecting an appropriate mutation rate.
1. Problem Complexity
The complexity of the optimization problem is one of the factors that influences the mutation rate. If the problem has multiple local optima or a rugged landscape, a higher mutation rate is usually beneficial to escape from local optima and explore different regions of the search space.
2. Fitness Landscape
The shape of the fitness landscape, which represents the relationship between solution fitness and the corresponding genetic material, can also provide insights into the appropriate mutation rate. If the landscape is flat or has a lot of plateaus, a higher mutation rate might be needed to avoid getting stuck in suboptimal solutions.
3. Genetic Operators
The mutation rate should be balanced with other genetic operators, such as crossover and selection. If the crossover rate is high, the mutation rate could be set lower, as the crossover already introduces diversity by combining the genetic material of two individuals. On the other hand, if the selection pressure is high, a higher mutation rate might be necessary to maintain sufficient exploration.
It is important to note that the optimal mutation rate may vary for different problem instances or even at different stages of the evolution process. Therefore, it is recommended to experiment with different mutation rates and observe their effects on the algorithm’s convergence and solution quality.
Finally, it is worth mentioning that determining the optimal mutation rate is not a straightforward process and often requires empirical testing and fine-tuning. The success of the genetic algorithm heavily relies on finding a good balance between exploration and exploitation, and the mutation rate plays a crucial role in achieving this balance.
Setting Population Size
The population size is an important parameter in genetic algorithms, as it determines the number of individuals that will be tested and evolved in each generation. A larger population size allows for more exploration of the search space, but it also increases the computational time required for each generation.
When implementing a genetic algorithm in MATLAB for optimization problems, it is critical to carefully select the population size to balance the tradeoff between exploration and computational efficiency.
Factors to consider when setting the population size:
1. Search space complexity: The size and complexity of the search space can impact the choice of the population size. If the optimization problem has a large and complex search space, a larger population size may be necessary to adequately explore the solution space.
2. Computation time: The population size directly affects the computation time required for each generation. For complex problems with long evaluation functions, a smaller population size may be preferred to minimize the computational burden.
3. Genetic operators: The genetic operators, such as crossover and mutation, also impact the choice of population size. If the genetic operators are highly effective at generating diversity and exploring the search space, a smaller population size may suffice. On the other hand, if the genetic operators are less effective, a larger population size may be necessary to compensate.
Table: Population size recommendations for different scenarios
|Population Size Recommendation
|Simple optimization problem with a small search space
|Complex optimization problem with a large search space
|Optimization problem with highly effective genetic operators
|Optimization problem with less effective genetic operators
It is important to note that these recommendations are not absolute and may vary depending on the specific problem and algorithm implementation. Experimentation and tuning of the population size may be necessary to find the optimal value for a given problem.
Setting the population size in a genetic algorithm is a crucial step in achieving optimal optimization performance. Careful consideration of factors such as search space complexity, computation time, and the effectiveness of genetic operators will help in determining the most appropriate population size for a specific problem.
Controlling Generation Limit
Controlling the generation limit is an important aspect of implementing a genetic algorithm in MATLAB for optimization problems. The generation limit determines the number of iterations or generations the algorithm will go through in search of an optimal solution.
Setting the generation limit appropriately is crucial for achieving the desired balance between exploration and exploitation in the search space. If the limit is set too low, the algorithm may not have enough iterations to adequately explore the search space and find the optimal solution. On the other hand, setting the limit too high may result in excessive calculations and unnecessary computation time.
The generation limit can be controlled by specifying a maximum number of iterations or using a stopping criterion based on the convergence of the fitness values. The convergence criterion involves monitoring the fitness values of the population over successive generations. If the fitness values become stable, indicating that the algorithm has reached a near-optimal solution, the algorithm can be terminated.
One common approach to controlling the generation limit is to combine the convergence criterion with a maximum number of iterations. This ensures that the algorithm terminates if the convergence criterion is not met within the specified number of iterations. This approach provides a balance between exploring the search space and avoiding excessive computation time.
In MATLAB, the generation limit can be implemented using a loop structure. The loop iterates until the convergence criterion is met or the maximum number of iterations is reached. Within each iteration, the genetic algorithm performs the crossover, selection, and mutation operations to evolve the population towards better fitness values. The fitness values are evaluated using the objective function of the optimization problem.
To track the progress of the genetic algorithm, it is useful to keep a record of the best fitness value and the corresponding solution for each generation. This information can be stored in a table, allowing for further analysis and comparison of different algorithm settings or parameter values.
|Best Fitness Value
|[1, 0, 1, 0, 1]
|[1, 1, 0, 1, 0]
|[0, 1, 1, 0, 0]
By controlling the generation limit effectively, the genetic algorithm in MATLAB can efficiently solve optimization problems by iteratively evolving the population through crossover, selection, and mutation operations. The convergence criterion and maximum number of iterations provide the necessary control to strike a balance between exploration and exploitation in the evolutionary process.
In the field of optimization, it is common to encounter problems that have certain constraints that need to be satisfied. Constraints can be seen as additional requirements or limitations that a solution must meet. As a result, handling constraints becomes an essential part of the genetic algorithm process.
When dealing with optimization problems with constraints, the fitness function needs to incorporate the constraints in order to ensure that the generated solutions adhere to the specified limitations. This can be achieved by penalizing solutions that violate the constraints or by adjusting the fitness value accordingly.
The first step in handling constraints is to evaluate the feasibility of a solution. A solution is considered feasible if it satisfies all the constraints. If a solution is not feasible, it is deemed infeasible and its fitness is adjusted accordingly to reflect its violation of the constraints.
The next step is to modify the selection, crossover, and mutation operators to ensure that the generated offspring solutions also satisfy the constraints. This can be achieved by implementing techniques such as constraint handling mechanisms, where the constraints are explicitly taken into account during the evolution process.
One common technique is to assign a penalty to infeasible solutions during selection, crossover, and mutation. This penalty can be used to decrease the chances of infeasible solutions being selected or to bias the crossover and mutation operators towards feasible solutions.
Additionally, incorporating constraints during selection can be achieved by using fitness scaling techniques. These techniques adjust the fitness values of the solutions based on their feasibility, giving more weight to feasible solutions and penalizing infeasible ones.
In conclusion, handling constraints in optimization problems is crucial for the success of a genetic algorithm. By incorporating the constraints in the fitness function and modifying the genetic operators, it is possible to ensure that the generated solutions satisfy the necessary limitations and produce optimal results.
Optimizing Convergence Speed
Convergence speed is a crucial factor in any optimization algorithm, including genetic algorithms. In MATLAB, there are several techniques that can be employed to optimize the convergence speed of a genetic algorithm.
Firstly, the selection mechanism plays a significant role in determining the convergence speed. Selection is the process of choosing individuals from the current population for reproduction based on their fitness values. By using a suitable selection mechanism, such as tournament selection or roulette wheel selection, the algorithm can focus on the most promising individuals and discard less fit ones. This helps to speed up the convergence process.
Another technique to optimize convergence speed is to carefully design the fitness function. The fitness function evaluates the quality of each individual in the population. By defining a fitness function that closely reflects the optimization problem’s objectives, the genetic algorithm can quickly identify promising solutions. This can be achieved by considering the problem-specific requirements and constraints when designing the fitness function.
Crossover is another crucial aspect that can affect the convergence speed of a genetic algorithm. Crossover is the process of combining genetic information from two parent individuals to produce offspring individuals. By choosing an appropriate crossover method, such as one-point crossover or uniform crossover, the algorithm can efficiently explore the search space and produce diverse offspring. This diversification helps in discovering new promising solutions and speeding up convergence.
Lastly, mutation, which is the process of introducing random changes in individuals’ genetic material, can also impact convergence speed. By employing a suitable mutation rate and mutation operator, the algorithm can explore different regions of the search space. This exploration capability helps in escaping local optima and converging to better solutions faster.
In summary, to optimize convergence speed in MATLAB’s implementation of the genetic algorithm, careful consideration should be given to the selection mechanism, fitness function, crossover method, and mutation strategy. By fine-tuning these aspects, the algorithm can converge more quickly and efficiently towards optimal solutions for the given optimization problem.
Testing Genetic Algorithm with Benchmark Problems
Once the genetic algorithm is implemented and the necessary functions for selection, crossover, mutation, and evolution are defined, it is important to test the algorithm on benchmark optimization problems. These benchmark problems provide a standardized set of test cases that allow for the evaluation of the performance and effectiveness of the genetic algorithm.
The selection process in a genetic algorithm involves choosing individuals from the current population based on their fitness. Various techniques can be used, such as tournament selection or roulette wheel selection, to ensure that fitter individuals have a higher likelihood of being selected for reproduction.
Crossover is a fundamental operation in genetic algorithms where the genetic information from two parent individuals is combined to create offspring. Different crossover techniques, such as one-point crossover or uniform crossover, can be used to explore different parts of the search space and potentially discover better solutions.
The evolution of the population through selection and crossover allows the genetic algorithm to gradually improve the fitness of the individuals over generations. This process mimics the natural evolution of species.
Mutation introduces random changes in the genetic information of individuals. This randomness helps prevent the algorithm from getting stuck in local optima and encourages exploration of the search space. By occasionally introducing small changes in individuals, the genetic algorithm can potentially find better solutions that were not present in the initial population.
The fitness function is a crucial component of the genetic algorithm as it determines how well each individual performs in the optimization problem. The fitness function maps the solution space to a scalar value, indicating the quality of a given solution. The aim of the genetic algorithm is to find the solution with the highest fitness value.
By testing the genetic algorithm on benchmark problems, it is possible to assess its performance in terms of convergence speed, solution quality, and robustness. Benchmark problems provide a standardized way of comparing different algorithms and evaluating their strengths and weaknesses.
Testing the genetic algorithm on benchmark problems is an essential step in assessing its performance. The algorithm’s ability to handle various optimization problems and produce high-quality solutions is critical for its applicability in real-world scenarios. By understanding the strengths and weaknesses of the algorithm, researchers can further refine its implementation for specific optimization problems.
Comparing Genetic Algorithm with Other Optimization Techniques
In the field of optimization, various techniques have been developed to solve complex problems and find the best possible solution. One popular technique is the Genetic Algorithm (GA), inspired by the process of natural evolution.
The key idea behind the Genetic Algorithm is to mimic the process of natural selection to search for the optimal solution. The algorithm works by maintaining a population of potential solutions and iteratively applying genetic operators such as selection, crossover, and mutation to evolve the population.
Compared to other optimization techniques, the Genetic Algorithm offers several advantages. Firstly, it can handle large search spaces and does not require the function being optimized to be differentiable. This makes it suitable for a wide range of problems where other algorithms may struggle.
Another advantage of the Genetic Algorithm is its ability to find global optima, rather than getting stuck in local optima. This is achieved by maintaining diversity within the population and exploring different regions of the search space.
Additionally, the Genetic Algorithm is highly parallelizable, which means it can take advantage of modern computing architectures to speed up the optimization process. This is especially useful for large-scale problems that require extensive computations.
Comparison with other techniques
When compared to traditional optimization techniques such as gradient descent or simulated annealing, the Genetic Algorithm has shown better performance in certain scenarios. For example, when dealing with combinatorial optimization problems or problems with discrete or binary variables, the Genetic Algorithm often outperforms other techniques.
Moreover, the Genetic Algorithm is known for its ability to handle complex, multimodal functions with multiple peaks and valleys in the search space. This is an area where gradient-based techniques may struggle, as they tend to converge to local optima and miss the global optimum.
However, it is important to note that the Genetic Algorithm may not always be the best choice for every optimization problem. In some cases, other techniques such as gradient descent or particle swarm optimization may provide faster convergence or better solutions.
In conclusion, the Genetic Algorithm is a powerful optimization technique that offers advantages such as handling large search spaces, finding global optima, and being highly parallelizable. While it outperforms other techniques in certain scenarios, the choice of optimization algorithm should depend on the specific problem at hand.
Modifying Genetic Algorithm for Specific Problems
Genetic algorithms are powerful optimization techniques inspired by the principles of evolution. They are commonly used to solve a wide range of optimization problems, including those that involve finding the optimal values for a set of parameters or decision variables. In MATLAB, the genetic algorithm toolbox provides a convenient way to implement and customize genetic algorithms for specific problem domains.
1. Evolution and Selection
The core idea behind genetic algorithms is to simulate the process of natural evolution. A population of potential solutions, known as individuals, is evolved over a number of generations. This evolution is driven by a fitness function that evaluates the quality of each individual in the population. In each generation, selection operators are used to choose individuals with higher fitness values for reproduction, while individuals with lower fitness values are less likely to be selected.
In some cases, the default selection operators provided by the genetic algorithm toolbox may not be suitable for specific problem domains. In such cases, it is important to modify the selection operators to ensure that individuals with the desired characteristics are favored for reproduction. This can be achieved by using custom fitness functions that incorporate domain-specific knowledge and constraints.
2. Crossover and Mutation
Crossover and mutation are two key operators in genetic algorithms that introduce genetic diversity into the population. Crossover involves combining the genetic material of two parent individuals to generate new offspring individuals. Mutation involves randomly modifying the genetic material of individuals to explore new areas of the solution space.
While the default crossover and mutation operators provided by the genetic algorithm toolbox are generally applicable to a wide range of problems, they may need to be modified for specific problem domains. For example, if the problem has a specific structure or constraints, it may be necessary to design custom crossover and mutation operators to ensure the generated offspring individuals are feasible and conform to the problem requirements.
In MATLAB, it is relatively straightforward to define custom crossover and mutation functions using the built-in capabilities of the language. This allows for flexibility in adapting the genetic algorithm to specific problem requirements.
Overall, modifying the genetic algorithm for specific problems involves customizing the evolution, selection, crossover, and mutation operators to better suit the problem domain. It requires a deep understanding of the problem and the constraints involved, as well as familiarity with the available tools and techniques in MATLAB.
Combining Genetic Algorithm with Other Metaheuristic Algorithms
In the field of optimization, metaheuristic algorithms such as genetic algorithms have gained significant popularity due to their efficiency and effectiveness in finding optimal solutions. However, no single algorithm can guarantee the best results for all optimization problems. Therefore, combining genetic algorithm with other metaheuristic algorithms can yield even better results.
When combining genetic algorithm with other metaheuristic algorithms, it is important to consider the strengths of each algorithm and leverage them to improve the overall optimization process. One common approach is to use a multi-objective optimization technique, which allows for the simultaneous optimization of multiple objectives. This can be achieved by combining the fitness function of the genetic algorithm with the fitness functions of other metaheuristic algorithms, such as simulated annealing or particle swarm optimization.
1. Crossover and Selection with Other Metaheuristic Algorithms
The crossover and selection operators are key components of the genetic algorithm that contribute to the exploration and exploitation of the search space. By combining these operators with those of other metaheuristic algorithms, the search algorithm can benefit from their respective strengths.
For example, the crossover operator of genetic algorithm can be combined with the movement operators of particle swarm optimization to create a new hybrid operator that combines the best features of both algorithms. Similarly, selection operators, such as tournament selection or roulette wheel selection, can be combined with the diversification strategies of other metaheuristic algorithms to create a more powerful selection mechanism.
2. Genetic Mutation and Other Metaheuristic Algorithms
Genetic mutation is another important operation in genetic algorithm that introduces random changes in the search space. When combined with other metaheuristic algorithms, it can enhance the exploration capabilities of the overall algorithm.
For instance, the mutation operator of genetic algorithm can be combined with the neighborhood search technique of simulated annealing to create a new mutation operator that balances exploration and exploitation. This hybrid mutation operator can guide the search process towards the promising regions of the search space while avoiding premature convergence.
|Efficient exploration of large search spaces
|Effective at escaping local optima
|Particle Swarm Optimization
|Fast convergence to global optima
Table: Strengths of Genetic Algorithm and Other Metaheuristic Algorithms
By combining the strengths of genetic algorithm with those of other metaheuristics algorithms, it is possible to achieve a more robust and efficient optimization process. The search algorithm can leverage the exploration capabilities of genetic algorithm and the exploitation properties of other algorithms to find high-quality solutions to complex optimization problems.
Parallelizing Genetic Algorithm for Faster Performance
Genetic algorithms are commonly used for solving optimization problems by mimicking the process of evolution. The algorithm works by maintaining a population of potential solutions and repeatedly applying genetic operators such as mutation, crossover, and selection to evolve new generations.
However, as the complexity of optimization problems increases, the time required to find the optimal solution can also increase significantly. To address this issue, parallelization techniques can be applied to speed up the performance of genetic algorithms.
Parallelizing a genetic algorithm involves dividing the population into multiple subpopulations and running the genetic operators on each subpopulation simultaneously. This allows for parallel execution of the fitness evaluation, selection, and evolution steps, resulting in faster convergence to the optimal solution.
By distributing the computation across multiple processors or threads, parallel genetic algorithms can take advantage of the available computing resources to explore the search space more efficiently. This can greatly reduce the overall runtime of the algorithm and enable the exploration of larger problem spaces.
However, parallelization introduces additional challenges, such as coordinating the communication and synchronization between the different subpopulations. Strategies such as master-slave architectures or island models can be used to manage the interaction between the parallel subpopulations and ensure the proper exchange of genetic information.
The effectiveness of parallelization in a genetic algorithm depends on several factors, such as the problem size, the number of available processors or threads, and the nature of the optimization problem. In some cases, parallelization may not provide significant performance gains if the computational overhead of coordinating the parallel execution outweighs the benefits.
In conclusion, parallelizing a genetic algorithm can lead to faster performance and improved optimization results. However, it is important to carefully consider the specific characteristics of the optimization problem and the available computing resources to determine whether parallelization is a suitable approach.
Implementing Genetic Algorithm on Distributed Systems
Genetic algorithms (GA) are widely used for solving optimization problems in various fields. They are inspired by the process of natural selection and mimic the principles of genetic evolution to find the optimal solution.
In a typical GA, a population of potential solutions, represented as chromosomes, undergoes three main operations: crossover, mutation, and fitness evaluation. These operations gradually improve the population over generations, leading to an optimal solution.
When dealing with complex optimization problems, the computational requirements for running a GA can be significant. This is where distributed systems come into play. By leveraging the power of multiple computers or processors, the performance of a GA can be greatly enhanced.
Distributed Genetic Algorithm
In a distributed genetic algorithm, the population and its associated operations are distributed across multiple nodes or machines. Each node performs a subset of the overall tasks, such as evaluating fitness, generating offspring through crossover and mutation, and sharing the best individuals.
The distributed nature of the algorithm allows for parallel processing, which can significantly reduce the execution time for large-scale optimization problems. Additionally, it provides fault tolerance by distributing the workload, ensuring that the algorithm can continue running even if a node fails.
Implementing Genetic Algorithm on MATLAB
MATLAB is a powerful software environment commonly used for implementing and analyzing genetic algorithms. Its extensive library of functions and toolboxes makes it an ideal choice for developing distributed genetic algorithms.
To implement a distributed genetic algorithm in MATLAB, the following steps can be followed:
- Partition the population across multiple nodes.
- Parallelize the fitness evaluation, crossover, and mutation operations using parallel computing techniques available in MATLAB.
- Synchronize the population and share the best individuals between nodes periodically.
- Implement termination criteria to stop the algorithm when a satisfactory solution is found or a maximum number of generations is reached.
By distributing the workload and leveraging the parallel processing capabilities of MATLAB, the performance of the genetic algorithm can be greatly enhanced, enabling the solution of complex optimization problems in a shorter time.
Implementing a genetic algorithm on distributed systems offers several advantages, including improved performance, fault tolerance, and scalability. By distributing the workload across multiple nodes and leveraging parallel processing capabilities, the algorithm can efficiently solve optimization problems.
When using MATLAB for implementing the algorithm, the extensive library of functions and toolboxes available in MATLAB can be utilized to parallelize and optimize the operations. This combination provides a powerful tool for solving complex optimization problems.
– Goldberg, D. E. (1989). Genetic algorithms in search, optimization, and machine learning. Addison-Wesley Longman Publishing Co., Inc.
– Davis, L. (1991). Handbook of genetic algorithms. Van Nostrand Reinhold.
Handling Large-scale Optimization Problems
Optimization problems in various fields, such as engineering, economics, and biology, often involve a large number of variables and constraints. Dealing with such large-scale problems can be challenging due to the computational complexity and the time required to obtain optimal solutions. However, with the help of genetic algorithms implemented in MATLAB, it is possible to tackle these problems efficiently.
Genetic Algorithm in MATLAB
A genetic algorithm is a search heuristic inspired by the process of natural selection. It mimics the evolution of populations over generations to find optimal solutions to complex optimization problems. In MATLAB, the Genetic Algorithm and Direct Search Toolbox provides a powerful framework for implementing genetic algorithms and solving large-scale optimization problems.
The genetic algorithm works by creating a population of potential solutions represented as individuals. Each individual is evaluated based on a fitness function, which quantifies how well it solves the optimization problem. The algorithm then applies genetic operators, like crossover and mutation, to generate new offspring. The offspring inherit characteristics from their parents, and the process continues iteratively until a satisfactory solution is found.
Approaches for Large-scale Problems
When dealing with large-scale optimization problems, it is essential to consider strategies to improve the efficiency of the genetic algorithm. One approach is to use parallel computing techniques to exploit the computational power of multiple processors or cores. MATLAB provides functionality to implement parallel computing, which can significantly reduce the execution time for large-scale problems.
Another approach is to use advanced selection methods that incorporate a balance between exploration and exploitation. While traditional selection methods, such as tournament or roulette wheel selection, might work well for small-scale problems, they may not be as effective for large-scale problems. Advanced selection methods, such as rank-based or fitness-scaling selection, can help ensure a diverse population and prevent premature convergence.
In addition, fine-tuning the genetic operators, including crossover and mutation, is crucial when dealing with large-scale problems. Adjusting the parameters such as crossover probability and mutation rate can have a significant impact on the algorithm’s performance and convergence. Experimenting with different operator settings and performing sensitivity analyses can help find the optimal combination for solving large-scale optimization problems.
In conclusion, by leveraging the capabilities of MATLAB’s Genetic Algorithm and Direct Search Toolbox and implementing strategies tailored for large-scale problems, it is possible to effectively tackle complex optimization problems. With careful selection of genetic operators, efficient parallel computing techniques, and advanced selection methods, the genetic algorithm can be a powerful tool for handling large-scale optimization problems in various domains.
Optimizing Genetic Algorithm Parameters
Genetic algorithms are a popular method for solving optimization problems. When using genetic algorithms, it is important to select the right parameters to achieve the best performance and accuracy. In this article, we will discuss some key parameters that can be optimized to improve the performance of a genetic algorithm.
Crossover is the process of combining the genetic material of two parent individuals to produce offspring. The selection of the crossover parameter determines how many bits or genes from each parent are exchanged. In some cases, a high crossover rate can result in faster convergence but might lead to loss of diversity. Conversely, a low crossover rate may preserve diversity but can slow down the convergence process. Optimizing the crossover rate is crucial to strike a balance between exploration and exploitation of the search space.
Mutation introduces random changes into the genetic material of individuals. It helps maintain diversity and prevent premature convergence. The mutation rate is an important parameter that determines the probability of a gene being mutated. A high mutation rate can increase exploration but may slow down convergence, while a low mutation rate can lead to premature convergence. It is essential to find the optimal mutation rate that balances exploration and exploitation.
Genetic operators, including crossover and mutation, play a critical role in the evolution process. There are various types of crossover and mutation operators available, and their selection can significantly impact the optimization performance. Experimenting with different genetic operators and their combinations can help identify the most suitable ones for the problem at hand.
The fitness function defines the objective or fitness measure for each individual in the population. It quantifies the quality of the solution and guides the evolution process. Optimizing the fitness function is essential to ensure the algorithm focuses on the most relevant aspects of the problem. A well-designed fitness function can lead to faster convergence and better solutions.
Optimizing genetic algorithm parameters is not a trivial task and often requires an iterative process. It involves experimenting with different parameter values, evaluating the algorithm’s performance, and fine-tuning the parameters based on the results. MATLAB provides powerful tools for implementing and optimizing genetic algorithms, making it an excellent choice for researchers and practitioners in the field of optimization.
Integrating Genetic Algorithm with MATLAB Toolbox
When it comes to solving optimization problems, genetic algorithms provide an efficient and effective approach. These algorithms are inspired by the process of evolution in nature, where genetic information is combined through crossover and mutation to improve the fitness of individuals. The integration of genetic algorithms with the MATLAB Toolbox makes it even easier to implement and solve complex optimization problems.
The MATLAB Toolbox provides a set of functions and tools specifically designed for genetic algorithm optimization. These functions allow users to define their optimization problem, set the parameters for the genetic algorithm, and run multiple iterations to find the best solution. The genetic algorithm implementation in MATLAB Toolbox follows a standardized procedure, making it simple and straightforward to use.
|Runs the genetic algorithm optimization
|Defines the fitness function to be optimized
|Determines how the crossover operation is performed
|Determines how the mutation operation is performed
|Determines how the selection operation is performed
Using these functions, users can easily customize the genetic algorithm implementation according to their specific problem requirements. The fitness function defines the objective function that needs to be optimized, while the crossover, mutation, and selection functions determine how the genetic information is combined and selected at each iteration.
The genetic algorithm in MATLAB Toolbox also allows users to set various parameters, such as the population size, number of generations, and crossover/mutation rates. These parameters can be adjusted to achieve the desired balance between exploration and exploitation, ensuring that the genetic algorithm effectively explores the search space while converging towards the optimal solution.
Overall, integrating genetic algorithms with the MATLAB Toolbox provides a powerful tool for solving optimization problems. The standardized implementation and customizable functions make it easy for users to define and solve their optimization problems efficiently. Whether it is finding the optimal solution to a complex engineering problem or optimizing a financial portfolio, the genetic algorithm implementation in MATLAB Toolbox offers a versatile and effective approach.
What is a genetic algorithm?
A genetic algorithm is a search heuristic inspired by the process of natural selection. It is used to find approximate solutions to optimization and search problems.
What is a genetic algorithm?
A genetic algorithm is a type of optimization algorithm inspired by the process of natural selection. It uses concepts from genetics and evolution to find the best solution to an optimization problem.
How does a genetic algorithm work?
A genetic algorithm works by creating a population of individuals, where each individual represents a potential solution to the problem. These individuals then go through a series of operations such as selection, crossover, and mutation to produce a new generation of individuals. The process is repeated until a satisfactory solution is found.
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A histogram represents the frequency distribution of continuous data, while a bar graph compares different categories. Histograms use adjacent bars to show the distribution of numerical data, whereas bar graphs have gaps between bars indicating discrete variables.
Understanding data presentations is essential in statistics and data analysis. Histograms and bar graphs are fundamental visualization tools that serve different purposes. Histograms help visualize the underlying frequency distribution of a dataset, particularly useful for showing the shape and spread of continuous data.
On the other side of the spectrum, bar graphs enable comparison among discrete categories, conveying information about counts or proportions. They are versatile for displaying any data where distinct groups – such as survey responses, sales by quarter, or population by country – can be compared. Visual learners often find these charts helpful as they provide a clear snapshot of data that supports better decision-making. These tools transform numbers into visual stories, making data trends and patterns easier to grasp.
Key Differences Between Histogram And Bar Graph:
Understanding the differences between histograms and bar graphs is vital for anyone delving into the world of data visualization. While both are graphical representations of data, they each convey information in distinctly different ways. Let’s explore the key differences that set these two types of charts apart.
Definition Of Histogram
A histogram is a type of chart that depicts the distribution of numerical data. It is used primarily for continuous data where the bins represent ranges of data, and the height of each bar reflects the frequency of data within each range. Key characteristics of a histogram include:
- Continuous Data: Histograms are ideal for showcasing data that flow on a continuum and have different intervals or bins.
- No Gaps: Bars in a histogram touch each other to signify the continuous nature of the data.
- Variable Width: Each bar can have a different width depending on the range it represents.
Definition Of Bar Graph
In contrast, a bar graph is used to display categorical data with rectangular bars representing the values. Bar graphs emphasize the comparison between discrete categories. Distinct features of bar graphs are:
- Categorical Data: Suited for data that is segmented into separate categories (e.g., survey responses, population by country).
- Gaps Between Bars: Unlike histograms, bar graphs have gaps between bars to highlight that each category is distinct and independent.
- Uniform Width: Bars typically have the same width, as each category is equally significant in the comparison.
|No (Gaps Present)
What Is A Bar Graph?
A bar graph is a visual representation of data that uses bars to compare different categories of information. Each bar’s length or height represents the value it holds, making it easy to compare the sizes of different groups.
Understanding The Purpose And Application
Bar graphs serve a critical role in the display of categorical data. Unlike histograms, which are used for continuous data, bar graphs are ideal for discrete data. This distinction means that bar graphs are best suited for data that represent non-numerical categories, such as survey responses, different species of plants, or types of snacks.
These graphs are particularly effective in the business and education fields, where they aid in making data-driven decisions and illustrating comparisons clearly. For instance, a company might use a bar graph to compare the sales figures of various products, or a teacher might use one to show the number of students who prefer different kinds of books.
When it comes to application, bar graphs can be depicted in two orientations:
- Vertical (Column Graph): Bars run vertically from the bottom up.
- Horizontal (Bar Chart): Bars run horizontally from left to right.
The choice of orientation depends on the specific requirements of the data presentation and the preferences of the presenter or audience.
|Use Case Example
|Comparing monthly sales across different regions.
|Showing the percentage of responses for each option in a survey.
|Illustrating test score distributions across various subjects.
In the digital world, bar graphs are not only a staple in printed reports and presentations, but also a common feature in interactive dashboards and analytics platforms, where users can often click on bars to drill down into the data for more detail. Such interactivity allows for a more in-depth analysis and user engagement.
Regardless of the format—static or interactive—the primary goal of a bar graph remains consistent: it is a tool for communicating information efficiently and effectively, enabling viewers to grasp complex data at a glance.
Applications Of A Bar Graph
The Applications of a Bar Graph provide a window into the versatility of this data visualization tool. From the realms of business to education and even in everyday data interpretation, bar graphs play a key role in presenting information in an easily digestible format. Their straightforward structure makes them an ideal choice for comparing discrete categories or showcasing the frequency of outcomes within a dataset.
While both bar graphs and histograms might seem similar at first glance, they serve distinct purposes and have unique characteristics. Understanding these differences is key to using each visual tool effectively in various applications.
- Bar Graphs: Often utilized to display and compare the quantity or frequency of different categories. These categories are independent and non-numeric.
- Histograms: Specifically designed to show the distribution of variables, often displaying the frequency of data within certain ranges for continuous, numerical intervals.
Bar graphs stand out in scenarios where clear, crisp categorization of data is vital. The space between bars in a bar graph symbolizes the distinction between the categories, reinforcing their discrete nature. This feature is crucial in helping viewers differentiate between the segments being compared.
Let’s delve deeper into the variety of settings where bar graphs excel:List of Applications
- Business Analytics: Bar graphs effectively display sales data, customer demographics, and resource allocation, making them indispensable in corporate presentations and reports.
- Education: Teachers frequently adapt bar graphs to illustrate differences in test scores, student attendance, or class performance for a visual and comparative analysis.
- Healthcare Data: Hospitals and clinics might use bar graphs to track patient admissions, treatment outcomes, or disease incidence, promoting an understanding of critical health trends.
- Public Opinion and Survey Data: In fields such as market research or psychology, bar graphs offer a straightforward representation of survey results or public opinion polls to simplify complex datasets.
Each of these applications benefits from the clear communicative power of bar graphs, allowing for more informed decision-making and accessible presentations of data.
In this exploration of bar graphs and their distinct applications, remember that the choice between a bar graph and a histogram hinges on the nature of the data at hand and the message intended to be conveyed.
Data Representation Variances
Data representation variances play a critical role in the way we interpret and analyze information. Choosing the right graphical representation can drastically affect the insights that one draws from a dataset. Two common types of data visuals are histograms and bar graphs. While they may look similar at first glance, their use and the level of detail they communicate can vary substantially. Understanding the differences between these two forms of data presentation is key to effectively conveying the nuances within a dataset. Let’s delve into these variations and explore how they impact data analysis.Analysis of Data Presentation
Analysis Of Data Presentation
The choice between a histogram and a bar graph is determined by the nature of the data and the specifics of the information one aims to present. Analyzing data presentation thus requires a close look at the type and granularity of the data. Here are some of the fundamental contrasts:
- Histograms are typically used for continuous data, allowing individuals to see the distribution of numerical values within different intervals, or ‘bins’.
- Bar graphs, on the other hand, are best suited for categorical data. They display comparisons among discrete categories or groups.
By understanding the context and the type of data, one can choose the appropriate graphic that aligns with the objectives of the data presentation.Data representation details in Histograms and Bar Graphs
|Vertical or Horizontal
|Vertical or Horizontal
|X-axis shows ranges, Y-axis shows frequency
|X-axis shows categories, Y-axis shows values
With histograms, the adjacent bars touch each other to signify the continuity of data, whereas bar graphs have space between bars to emphasize the distinct categories.
Comparison Of Categorical Versus Continuous Data
The analysis of data through visual representations can significantly enhance comprehension and facilitate better decision-making. Two such methods are histograms and bar graphs, each designed to display different kinds of data. A primary distinction lies in their treatment of categorical versus continuous data. This segment delves into how these two types of graphs bring diverse insights to the surface, depending on whether they are handling discrete categories or a range of numerical values.
An understanding of categorical and continuous data is foundational to choosing between a histogram or a bar graph. Categorical data are variables that are grouped into categories and are often qualitative in nature. Contrastingly, continuous data emerge from measurements and are quantitative, presenting an infinite number of possibilities within a range.
Bar graphs shine when it comes to categorical data:
- They represent discrete groups, like different brands, countries, or years.
- Each bar stands alone, separated by a gap, underscoring their independence.
On the flip side, histograms are the go-to for continuous data:
- They depict intervals of values, known as bins, without gaps between bars, illustrating the data’s continuum.
- Frequency of data within a certain range is readily visible.
Significance In Decision-making
When interpreting information for strategic choices, the histogram and bar graph can inform in distinct ways. Applying histograms to visualize continuous data can aid in spotting trends and distributions, such as the most common customer spending range. Bar graphs, meanwhile, excel in showcasing comparisons between different groups, like sales performance by region or customer satisfaction ratings by product.
Achieving in-depth insights requires the correct application of each graph type concerning the data at hand. A histogram’s ability to show frequency distributions can unveil insights into variation and central tendency, essential for statistical analysis and quality control. In contrast, the clarity of a bar graph in presenting categorical data facilitates the identification of standout categories or anomalies that warrant further investigation or action.
Frequently Asked Questions Of Difference Between Histogram And Bar Graph
What Defines A Histogram?
A histogram is a graphical representation of frequency distribution. It uses bars to depict the frequency of data points within consecutive numerical intervals, with each bar’s height indicating the number of cases in each interval.
How Does A Bar Graph Differ?
Bar graphs represent categorical data with rectangular bars, where the length of each bar is proportional to the value it represents. Unlike histograms, bar graphs handle data in separate categories without inherent order.
When Should You Use A Histogram?
Use a histogram to analyze the distribution of a continuous data set. It’s particularly effective when you want to see the shape of data distribution, such as determining skewness, bimodality, or central tendency.
Can Histograms Have Gaps Between Bars?
Histograms should not have gaps between the bars as it represents continuous data. The bars are drawn adjacent to each other to show that the data is from a continuous range where intervals are directly next to one another.
Understanding the nuances between histograms and bar graphs empowers you to present data effectively. Histograms offer insight into distribution, whereas bar graphs compare different categories. Both tools are indispensable for statistical analysis, and selecting the appropriate one hinges on your data’s nature and your communication goals.
Embrace their unique strengths to enhance your data storytelling capabilities.
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Chapter 1 Introduction to C Language
- What is C language?
- History of C Language
- Features of C Program Language
- What Are the Disadvantage of C Programming Language ?
- Analysis of the Main Application Fields of C Program Language
- Why Learn C Program Language?
- Downloading and Installing the GCC Compiler for C Language Under Windows Environment
- Why Choose the Code Block Compiler?
- Hello World! Your first C program
- Detailed Explanation of the Compilation and Linking Process of C Program Language
- How Does Preprocessor Work in C ?
- A List of C Library Functions and Header Files
Chapter 2 Basic of C Language
- The Program Structure of C Program Language
- Basic Syntax of C Program Language
- The Semicolon (;) is Used to Terminate Statements
- C Language Identifier
- C Program Language Token
- Introduction of printf() function
- A Placeholder in C Program
- Escape Sequences in C
Chapter 3 Data Types of C Language
- Data types in C Program
- Memory and variables in C language
- The Relationship Between Variables and Memory
- Conversion between decimal and binary
- How to Convert Decimal to Hexadecimal?
- How to Convert Hexadecimal to Binary
- Relationship Between Memory and Data Types
- Storage of integers in the computer, character data type and character variables
- Literals in C Programming
- Character Literals and Escape Sequences in C Programming
- How does Sign-Magnitude and Two’s Complement Work ?
- Integer Data Type, Integer Variables, Integer Variable Overflow.
- The Storage of Floating-Point Variable, Float Type and Float Variables
- Why do integers stored in computers in the form of complement?
- What is complement? Expression of binary complement (optional course)
- Real number type, floating-point variables, storage of floating-point variables
- Conversion of decimal fractions to binary formats
- Difference between integer data types and floating-point data types
- Binary representation of negative numbers
- Rounding error of C language real-type data (optional)
- The Binary Representation of Positive and Negative Numbers
- Complete table of C language escape characters and ASCII codes
- Constants in C language
- Difference between variable definition and variable declaration in C language
Chapter 4 Operators of C Programming
An operator is a symbol that operates on a value or a variable. For example: + is an operator to perform addition. C has a wide range of operators to perform various operations.
- C Program Operators
- C Programming : Arithmetic Operators
- Increment and Decrement Operators
- C Programming : Assignment Operators
- C Programming: The sizeof Operator
- C Programming: Relational Operators
- C Programming: Logical Operators
- C Program : Bitwise Operators
- C Operator Precedence and Associativity
- Explicit Type Conversion (Type Casting) and Implicit Type Conversion (Automatic Type Conversion)
- C Programming: The sizeof Operator
Chapter 5 Input/Output (I/O)
- printf( ) function in C Program
- C Input Function : scanf() in C Program
- What is the Difference Between printf() and scanf() ?
- Input/Output Format Specifiers in C Program
Chapter 6 Conditional Statements
- Basic structure of C Program statements
- Branch statements in C Program
- IF statement and IF ELSE statement in C language
- Ternary Conditional Operator in C Program
- if-else-if Ladder Statements and Nested if-else Statements in C Program
- Issues to note when using IF statement
- Switch statement and nested switch statement in C language
Chapter 6: C Language Loop Statements
- Loop Statements in C Program
- While loop statement in C
- Do-while loop statement in C
- For loop statement
- for Loop Statement and Loop Nesting in C Program
- goto Statement in C
- break and continue Statements in C
- Using infinite series to calculate the value of π in C
- How to determine if a number is prime in C
Chapter 7: Strings and Arrays in C Language
- What are strings in C?
- Input/output functions for characters and strings in C
- Functions for manipulating strings in C
- How to convert a string to a number in C
- Arrays and their applications in C
- Two-dimensional and multidimensional arrays in C
Chapter 8: Pointers in C Language
- What are pointers in C?
- How do pointers work?
- Pointer declaration and initialization in C
- Pointer types in C
- Address-of and indirection operators, as well as operator precedence for pointers in C
- Pointer operations between variables in C
- Pointers and arrays in C
- Pointers and strings in C
Chapter 9: Functions and Function Calls in C Language
- Overview of function applications
- Function definition in C
- User-defined functions in C
- Parameters and formal and actual parameters in C
- How to call a function in C
- Nested function calls in C
- Recursive functions in C
- Functions with pointers as arguments in C
- Functions with array names as arguments in C
- Function pointers in C
- Pointer functions in C
- Parameters of the main function
Chapter 10: Preprocessor Commands and Macro Definitions in C Language
- Overview of preprocessor commands in C
- What are macros in C? What are parameterized macros in C?
- Differences between parameterized macros and functions in C
- Predefined macros and file inclusion in C
- Conditional compilation in C
- Summary of preprocessor commands and macro definitions
Chapter 11: Structures and Unions in C Language
- Structures in C
- Three ways to declare structure variables in C
- Assignment and initialization of structure variables in C
- Definition and manipulation of structure arrays in C
- Explanation and usage of structure pointer variables in C
- Unions in C
- Differences and advantages and disadvantages between structures and unions in C
Chapter 12: Storage Classes and Memory Management in C Language
- Local and global variables in C
- Dynamic storage and static storage in C
- Storage classes auto, extern, static, and register in C
- How to call a function in another file in C
- Dynamic memory allocation in C
- Dynamic arrays in C
- Implementation of linked lists in C
- Enumerated types in C
- Typedef type definition specifier in C
- Differences between typedef and #define in C
Chapter 13: File Management in C Language
- Overview of files in C
- Opening, reading, and writing files in C
- File management functions in C
Chapter 14: Bitwise Operators in C Language
- Bitwise operators in C
Chapter 15: Programming Exercises in C Language
- Programming exercises in C language
- Arrays and pointer exercises
- Decision and loop statement exercises in C
- Functions exercises in C
C language is a general-purpose, procedural programming language. In 1972, Dennis Ritchie designed and developed the C language at Bell Telephone Laboratories for the purpose of porting and developing the UNIX operating system.
Students use C to learn programming, but its role goes far beyond that. It is not only an academic language. It is not the simplest language because C is a very low-level programming language.
Today, C is widely used in embedded devices and drives the majority of Internet servers built with Linux. The Linux kernel is written in C, which also means that C drives the kernel of all Android devices. It can be said that at this moment, a large part of the world is running on C code, which is amazing.
C brought an easy-to-implement language, and its compiler could be easily ported to different machines.
- C does not support garbage collection, which means we must manage memory ourselves. Managing memory is a complex task that requires extreme care to prevent defects, but C has become the ideal language for embedded device programming, such as Arduino.
- C does not hide the complexity and capabilities of the lower-level machine. Once you know what you can do, you can have tremendous power.
- C programming is considered the foundation of other programming languages, which is why it is called the mother tongue.
C language can be defined in the following ways:
- The foundation of all modern programming languages
- System programming language
- Procedural programming language
- Structured programming language
- Intermediate programming language.
The Difference Between C Program and C++ Program in Tablet Format
|C is a procedural programming language.
|C++ is an object-oriented programming language.
|C programs use functions to organize code.
|C++ programs use classes to organize code.
|C does not support function overloading.
|C++ supports function overloading.
|C does not support namespaces.
|C++ supports namespaces.
|C does not support exception handling.
|C++ supports exception handling.
|C programs use structures to group related data.
|C++ programs use classes to group related data and functions.
|C programs have simpler syntax and fewer keywords.
|C++ programs have more complex syntax and more keywords.
|C does not support templates.
|C++ supports templates.
|C programs are typically faster and more efficient.
|C++ programs are typically slower and less efficient due to the additional overhead of object-oriented features.
the Difference Between C Program and Python Program
|C is a compiled language, which means that the code needs to be compiled before it can be executed.
|Python is an interpreted language, which means that the code can be executed directly without needing to be compiled.
|C is a statically typed language, which means that the data type of a variable must be declared before it can be used.
|Python is a dynamically typed language, which means that the data type of a variable is determined at runtime.
|C is a low-level language, which means that it gives the programmer direct access to the computer’s hardware.
|Python is a high-level language, which means that it provides a lot of built-in functionality that makes programming easier and more efficient.
|C programs typically require more lines of code to achieve the same result as a Python program.
|Python programs are typically shorter and more concise than C programs.
|C programs can be more efficient than Python programs because C code can be optimized to take advantage of the computer’s hardware.
|Python programs can be less efficient than C programs because they run in a virtual machine and are subject to interpretation overhead.
|C has a more limited standard library than Python, which means that developers need to write more code from scratch.
|Python has a comprehensive standard library that provides a wide range of built-in functionality.
|C is often used for system-level programming, such as operating systems, device drivers, and embedded systems.
|Python is often used for web development, data analysis, scientific computing, and artificial intelligence.
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The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or graphically from a plotted curve. (Lines are classified as straight curves.) Algebraically, a linear equation typically takes the form y = mx + b, where m and b are constants, x is the independent variable, y is the dependent variable. In a statistical context, a linear equation is written in the form y = a + bx, where a and b are the constants. This form is used to help readers distinguish the statistical context from the algebraic context. In the equation y = a + bx, the constant b, called a coefficient, represents the slope. The constant a is called the y-intercept.
The slope of a line is a value that describes the rate of change between the independent and dependent variables. The slope tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average. The y-intercept is used to describe the dependent variable when the independent variable equals zero.
Scatter plots are particularly helpful graphs when we want to see if there is a linear relationship among data points. They indicate both the direction of the relationship between the x variables and the y variables, and the strength of the relationship. We calculate the strength of the relationship between an independent variable and a dependent variable using linear regression.
A regression line, or a line of best fit, can be drawn on a scatter plot and used to predict outcomes for the x and y variables in a given data set or sample data. There are several ways to find a regression line, but usually the least-squares regression line is used because it creates a uniform line. Residuals, also called “errors,” measure the distance from the actual value of y and the estimated value of y. The Sum of Squared Errors, when set to its minimum, calculates the points on the line of best fit. Regression lines can be used to predict values within the given set of data, but should not be used to make predictions for values outside the set of data.
The correlation coefficient r measures the strength of the linear association between x and y. The variable r has to be between –1 and +1. When r is positive, the x and y will tend to increase and decrease together. When r is negative, x will increase and y will decrease, or the opposite, x will decrease and y will increase. The coefficient of determination r2, is equal to the square of the correlation coefficient. When expressed as a percent, r2 represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression line.
Linear regression is a procedure for fitting a straight line of the form ŷ = a + bx to data. The conditions for regression are:
- Linear In the population, there is a linear relationship that models the average value of y for different values of x.
- Independent The residuals are assumed to be independent.
- Normal The y values are distributed normally for any value of x.
- Equal variance The standard deviation of the y values is equal for each x value.
- Random The data are produced from a well-designed random sample or randomized experiment.
The slope b and intercept a of the least-squares line estimate the slope β and intercept α of the population (true) regression line. To estimate the population standard deviation of y, σ, use the standard deviation of the residuals, s. . The variable ρ (rho) is the population correlation coefficient. To test the null hypothesis H0: ρ = hypothesized value, use a linear regression t-test. The most common null hypothesis is H0: ρ = 0 which indicates there is no linear relationship between x and y in the population. The TI-83, 83+, 84, 84+ calculator function LinRegTTest can perform this test (STATS TESTS LinRegTTest).
After determining the presence of a strong correlation coefficient and calculating the line of best fit, you can use the least squares regression line to make predictions about your data.
To determine if a point is an outlier, do one of the following:
- Input the following equations into the TI 83, 83+,84, 84+:
where s is the standard deviation of the residuals
If any point is above y2 or below y3 then the point is considered to be an outlier.
- Use the residuals and compare their absolute values to 2s where s is the standard deviation of the residuals. If the absolute value of any residual is greater than or equal to 2s, then the corresponding point is an outlier.
- Note: The calculator function LinRegTTest (STATS TESTS LinRegTTest) calculates s.
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Table of content
- Introduction: Unlock the Hidden Power of Sigma Symbols
- Understanding Sigma Symbols
- Types of Sigma Symbols
- Tips for Using Sigma Symbols
- Tricks for Maximizing the Use of Sigma Symbols
- Code Examples for Sigma Symbols
- Common Mistakes When Using Sigma Symbols
- Conclusion: Mastering Sigma Symbols
Introduction: Unlock the Hidden Power of Sigma Symbols
Sigma symbols are an integral part of the Python programming language, and mastering them can unlock a world of possibilities for data analysts, mathematicians, and anyone who needs to perform complex calculations. Whether you are a programming novice or an experienced coder, understanding how to use sigma symbols effectively can take your Python skills to the next level.
In this article, we will explore the power of sigma symbols in Python, providing tips, tricks, and code examples that will help you maximize their potential. We will cover key concepts such as sigma notation, series, and sequences, as well as more advanced topics like numerical integration and approximation. By the end of this article, you will have a deeper understanding of how to use sigma symbols in Python code, and how they can be leveraged to solve complex problems with ease.
So if you're ready to take your Python programming skills to the next level, let's dive in and explore the hidden power of sigma symbols!
Understanding Sigma Symbols
In Python programming, the sigma symbol represents a summation operation where a sequence of values are summed up. is an essential aspect of coding and unlocking their power can lead to more efficient and concise code.
The sigma symbol, often written as Σ, is used in Python code to perform a summation operation on a set of values. A typical use case would be to calculate the sum of a sequence of numbers. The "sigma" or summation function is written as "sum" in Python, and it takes an iterable as an argument containing the values to be summed up.
For example, we can calculate the sum of a list of numbers using the "sum" function in Python as follows:
numbers = [2, 4, 6, 8, 10]
sum_numbers = sum(numbers)
This code will output the sum of the numbers in the "numbers" list, which is 30.
The sigma symbol can also be used in combination with conditional statements like "if" statements to perform more complex calculations. For instance, we can sum up only the even numbers in a list using the "if" statement as follows:
numbers = [2, 3, 4, 5, 6, 7, 8, 9, 10]
even_numbers = [num for num in numbers if num % 2 == 0]
sum_even_numbers = sum(even_numbers)
This code will first create a new list "even_numbers" containing only the even numbers from the "numbers" list using a list comprehension and the "if" statement. It will then use the "sum" function to calculate the sum of the even numbers in the list, which will be 20.
Overall, understanding the use of sigma symbols in Python code can help developers write more efficient and concise code, leading to better performance and readability.
Types of Sigma Symbols
There are two main in Python: uppercase Sigma and lowercase sigma. The uppercase Sigma, denoted by Σ, is used to represent a sum of values. For instance, if we want to add up all the values in a list, we can use the Σ symbol to write a concise and readable code snippet. The lowercase sigma, denoted by σ, is used to represent a standard deviation of a set of data.
When using the uppercase Sigma, we can include an expression or formula that will be evaluated for each value in the sequence we're summing up. The expression can be anything that returns a numerical result, such as a simple arithmetic operation or a more complex function. For example, if we want to add up the square of each number in a list, we can write the expression "x**2" and use the Σ symbol to sum up the results.
The lowercase sigma, on the other hand, is used to calculate the standard deviation of a set of data. This is useful when working with statistics and data analysis. The formula for standard deviation involves taking the square root of the variance, which is calculated by finding the average of the squared differences between each value and the mean. The lowercase sigma symbol is used to denote the standard deviation of the data, and the formula can be written in Python using this symbol.
In conclusion, the two in Python are widely used in different contexts. The uppercase Sigma is used to calculate sums of values, while the lowercase sigma is used to calculate the standard deviation of data. By understanding these symbols, programmers can write more efficient and readable code for their projects.
Tips for Using Sigma Symbols
When working with sigma symbols in Python, there are a few key tips to keep in mind that can help you unlock their hidden power. First, make sure to define your sigma functions properly using the correct syntax: sigma (variable, start, end, expression). This will ensure that your sigma functions are correctly interpreted by Python and that your code operates as expected.
Another key tip is to use sigma symbols in conjunction with other Python functions, such as list comprehensions or lambda expressions. These functions can help to simplify your code and make it more efficient, allowing you to perform complex mathematical calculations with ease.
Another useful tip is to use conditional statements, such as if statements, to control the behavior of your sigma functions based on specific conditions. For example, you could use an if statement with the "name" keyword to check if a certain variable exists before executing your sigma function. This can help to prevent errors and ensure that your code runs smoothly.
Finally, don't be afraid to experiment with different code examples and techniques when working with sigma symbols in Python. This can help you to uncover new ways of using sigma functions and unlock their full potential for your own projects and applications. With these tips in mind, you'll be well on your way to mastering the hidden power of sigma symbols in Python.
Tricks for Maximizing the Use of Sigma Symbols
Sigma symbols are a powerful tool in Python programming that can be used to simplify complex mathematical formulas. Here are some in your code:
Understand the syntax: Sigma symbols are written as uppercase Greek letters, such as Σ. The syntax for a sigma symbol is as follows:
sum([expression], [index]). The
expressionis what you want to sum up, and the
indexis the variable that changes as you move through the summation.
Use shorthand notation: Python's syntax for summation can be quite verbose. To cut down on the amount of code you need to write, you can use the shorthand notation of a sigma symbol. For example,
sum(range(1,101))is equivalent to
Σ(i, i=1, 100).
Combine with other functions: Sigma symbols can be combined with other Python functions to create more complex formulas. For example, if you want to calculate the sum of the squares of all even numbers between 1 and 100, you can use the following code:
sum([i**2 for i in range(1,101) if i%2==0]).
Use multiple indices: If you need to sum over multiple indices, you can use nested sigma symbols. For example, to calculate the sum of all elements in a two-dimensional list, you can use the following code:
sum([Σ(j, j=0, n-1) for i in range(0,m)]).
Be aware of efficiency: Sigma symbols can be an elegant way of writing code, but they can also be slow to execute. If you need to sum over a large number of elements, it may be faster to use a for loop instead. You should always test the performance of your code to ensure it is running as efficiently as possible.
By understanding the syntax of sigma symbols, using shorthand notation, combining with other functions, using multiple indices, and being aware of efficiency, you can unlock the full power of sigma symbols in your Python code.
Code Examples for Sigma Symbols
Sigma symbols are an essential part of Python programming for working with summations of mathematical sequences. Here are a few code examples to help you get started with using sigma symbols in your Python programs.
Code Example 1
The following code uses sigma notation to calculate the sum of a sequence of numbers from 1 to 10:
total = 0
for i in range(1, 11):
total += i
The above code uses a for loop to iterate through the sequence of numbers from 1 to 10. The total variable is initialized to zero, and the loop adds each number in the sequence to the total. Finally, the total is printed to the console.
Code Example 2
The following code uses sigma notation to calculate the sum of a sequence of even numbers from 2 to 10:
total = 0
for i in range(2, 11, 2):
total += i
This code uses the range function to generate a sequence of even numbers from 2 to 10. The third argument to range specifies the step size, which in this case is 2 to generate only even numbers. The for loop then adds each number in the sequence to the total variable, and the total is printed to the console.
Code Example 3
The following code uses sigma notation to calculate the sum of a sequence of numbers from 1 to n, where n is a variable input by the user:
n = int(input("Enter a value for n: "))
total = 0
for i in range(1, n + 1):
total += i
Enter a value for n: 5
This code prompts the user to enter a value for n, which is then stored in the n variable. The for loop then iterates through the sequence of numbers from 1 to n, adding each number in the sequence to the total variable. Finally, the total is printed to the console.
These code examples should provide a good starting point for using sigma notation in your Python programs. With a solid understanding of how sigma notation works and how to use it in Python, you can unlock the hidden power of sigma symbols in your own programming work.
Common Mistakes When Using Sigma Symbols
When working with sigma symbols, there are a few common mistakes that beginner programmers often make. These mistakes can cause errors in your code, which can be frustrating to troubleshoot. Here are some of the most common mistakes to watch out for:
Incorrect use of variables: When using a sigma symbol, it's important to make sure that you're using the correct variables in your equation. Make sure that the variable you're using matches the variable in your sigma symbol, or you'll end up with incorrect results.
Incorrect range: The range in your sigma symbol specifies the values that your equation will be applied to. Make sure that you set the range correctly, or you may get unexpected results. Common errors include using the wrong start or end value, or using the wrong variable name.
Parenthesis: When using if statements with sigma symbols, it's important to remember to include parenthesis around your equation. This is because the if statement needs to evaluate the entire equation as a single unit, rather than evaluating each individual term in the equation separately.
Incorrect syntax: Finally, it's important to remember the correct syntax when using sigma symbols in Python. Make sure that you're using the correct operators (+, -, *, /) and that you're using the correct syntax for raising a number to a power (use ** rather than ^).
By avoiding these common mistakes, you can ensure that your sigma symbols are working correctly and producing accurate results. If you're having trouble with your code, double-check to make sure that you're not making any of these common errors. With practice, you'll be able to use sigma symbols with confidence and ease!
Conclusion: Mastering Sigma Symbols
In conclusion, mastering Sigma symbols can greatly enhance your Python programming skills. By understanding how the sum function works and how to use Sigma notation, you can write more efficient and elegant code. Additionally, utilizing the tips and tricks mentioned in this article, such as using list comprehensions and lambda functions, can further streamline your code and make it more concise.
Remember to always test your code and carefully consider the inputs you are using to ensure accuracy. It is also important to write code that is clear and understandable, both for yourself and for others who may be working with your code.
Overall, incorporating Sigma symbols into your programming toolkit can be a valuable asset, allowing you to solve complex mathematical problems with ease and efficiency. By practicing and experimenting with different techniques, you can unlock the hidden power of Sigma symbols and take your Python programming skills to the next level.
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The BITXOR function in Excel is a powerful tool that allows users to perform bitwise exclusive OR operations on numbers. This function is part of Excel's suite of Bitwise functions, which operate on the binary representations of numbers. Understanding the BITXOR function can help you manipulate data in new and interesting ways.
Understanding the BITXOR Function
The BITXOR function performs a bitwise exclusive OR operation on two numbers. In simple terms, this means that it compares the binary representations of two numbers, bit by bit, and returns a new number whose binary representation is determined by the rule of exclusive OR (XOR).
The XOR rule states that if the two bits being compared are the same, the result is 0. If the two bits are different, the result is 1. This rule is applied to each pair of corresponding bits in the binary representations of the two numbers. The result is a new number whose binary representation is the result of the XOR operation.
Using the BITXOR Function
The syntax for the BITXOR function is as follows: BITXOR(number1, number2). Here, number1 and number2 are the two numbers on which the XOR operation is to be performed. Both of these arguments must be non-negative integers.
For example, if you wanted to perform a bitwise XOR operation on the numbers 5 and 3, you would enter the following formula: BITXOR(5, 3). The result would be 6. This is because the binary representation of 5 is 101, and the binary representation of 3 is 011. When you perform the XOR operation on these two binary numbers, you get 110, which is the binary representation of 6.
Practical Applications of the BITXOR Function
The BITXOR function can be used in data analysis to compare two sets of binary data. By performing a bitwise XOR operation on the data sets, you can quickly identify differences between them. This can be particularly useful in fields such as computer science and information technology, where binary data is common.
For example, imagine you have two sets of binary data representing the states of a system at two different times. By using the BITXOR function, you can identify which bits have changed between the two time points. This can help you understand how the system has evolved over time.
The BITXOR function can also be used in error detection. In computer systems, data is often transmitted in binary form. To ensure that the data is transmitted correctly, an error detection code can be added to the data. One common type of error detection code is the parity bit, which is calculated using the XOR operation.
For example, imagine you are transmitting a binary number, such as 1011. You could calculate the parity bit by performing a bitwise XOR operation on the bits of the number. In this case, the parity bit would be 1, because there are an odd number of 1s in the number. If the number is received with a different parity bit, this indicates that an error has occurred during transmission.
Limitations of the BITXOR Function
While the BITXOR function is a powerful tool, it does have some limitations. One limitation is that it can only operate on integers. If you try to use the BITXOR function on a decimal number, Excel will truncate the number to an integer before performing the operation.
Another limitation is that the BITXOR function can only operate on non-negative numbers. If you try to use the BITXOR function on a negative number, Excel will return an error.
Finally, the BITXOR function can only operate on numbers that have a binary representation of up to 48 bits. If you try to use the BITXOR function on a number that has a binary representation of more than 48 bits, Excel will return an error.
The BITXOR function in Excel is a versatile tool that can be used in a variety of applications, from data analysis to error detection. While it does have some limitations, its ability to perform bitwise XOR operations on numbers makes it a valuable addition to any Excel user's toolkit.
By understanding how the BITXOR function works and how to use it, you can unlock new possibilities for data manipulation and analysis in Excel. Whether you're a seasoned Excel user or a beginner, the BITXOR function is a tool worth exploring.
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This tutorial introduces how to use Excel formulas and functions. Formulas are expressions entered by a user for the purpose of calculating cell values. Functions, on the other hand, are predefined formulas built into Excel that assist in data analysis.
Microsoft Excel has over 450 inbuilt functions, many introduced at different versions of Excel. For example, the VAR.S function was introduced in Excel 2010 while the CONCAT function was introduced in Office 2019. Between 2010 and 2019, over 100 new functions were introduced for effective data analysis.
For example, the worksheet below shows the difference between using a formula and a function in Excel.
When a formula is used, cell references are combined with mathematical operators to calculate the value of a cell. But some functions make it easy to perform calculations without introducing mathematical operators.
In this tutorial, we shall look at how to use Excel formulas and functions in the following headings:
- What are Excel formulas and functions?
- Inserting formulas and functions in a worksheet
- How to use Excel formulas and functions
What are Excel Formulas and Functions?
Formulas in Excel
In Microsoft Excel, formulas are expressions used to manipulate data within a cell or range. Formulas combine mathematical operators with cell references or addresses to determine the value of the selected cell.
In Microsoft Excel, users can insert different kinds of formulas depending on the calculation required. For example, the following formulas perform different calculations in Excel.
Formulas are tedious to enter and require carefulness to avoid violating operator precedence.
Functions in Excel
Functions are predefined formulas in Excel used to perform operations on data. Using them requires you to reference a cell or range of cells to compute cell values. There are different kinds of built-in functions in Excel. These functions fall into different categories. Some examples are:
|Used to calculate the internal rate of return
|Used to check if all arguments are true
|Used to convert text to capital case
|Date & Time
|Used to return the exact day of the month
|Lookup & Reference
|Used to return the number of rows in an array
|Math & Trig
|Used to return the absolute value of a number
|Used to calculate and return the mean of an array
|Used to convert a number from a unit to another
|Used to return a member from connection
|Used to check if a given value is a text or not
|Used to round a number down to the nearest level of significance
|Used to return data from a web service
These categories provide users with relevant functions to manipulate and analyze data with ease. However, users need to understand how these functions work in order to use them effectively. One good advantage of a function is that it increases productivity more than using a formula.
Advance and Basic Excel functions list
There are a lot of functions in Microsoft Excel that can be used in computation and data analysis. No user can use all these predefined formulas at once.
There are basic and advanced formulas that are mostly used by users that you can find useful too. Some of the functions appear under the AutoSum category and make calculations easy. Some of these functions include:
|Adds up values in the selected range
|Computes the average value of the selected range
|Counts and returns the value of the number of cells that has values within a selected range
|Counts and returns the number of cells that are not empty within a selected range
|Computes and returns the maximum value in the selected range
|Computes and returns the minimum value in the selected range
|Used to join text in different cells together
|Used to remove spaces before or after a text string in selected cells
|Used to round a number to a specified number of digits. The example rounds a number to 2 decimal places.
|Used to check whether a logical condition is true or false and returns the necessary value.
|Used to count the number of selected cells that meets a specified condition
|Used to calculate the sum of values in selected cells that meets a specified condition
|Looks for a value in the leftmost column and return a value in the same row from the column you specified
There are many more functions you will meet as you continue to explore Microsoft Excel. All you need to do is to understand how the function works and use it for your analysis.
Inserting Formulas and Functions in a Worksheet
If you must use a formula or function in Excel, you would have to insert the formula in your worksheet.
To add formula or function in Excel manually, you will need to insert the equal (=) sign first. Typing the (=) equal sign prepares the selected cell to receive formula or function.
There are five (5) ways you can insert a formula or function in a worksheet, namely:
- By manually typing the formula or function. To insert a formula in this manner requires you to type in the (=) sign first. Then the cell reference, followed by necessary operators. When working with a function, after typing the equal sign, follow it by typing the function you want.
- You can also use the fx icon on the formula bar. When you click on the icon, a dialog box will appear.
- On the dialog box that appears enter the function name and click on Go. The list of functionswill appear in the select a function window.
- Select your desired function and click OK.
- Another dialog box will appear. The features of the dialog box depend on the selected function.
- Click on the collapse button to enter the cell references.
- On the worksheet workspace, select the range of cells and click the collapse button to return to the dialog box.
- Click OK on the dialog box to calculate the result.
- Another way to insert a function is to use the AutoSum dropdown menu on the Home tab, Editing group.
The dropdown menu has a list of mostly used functions that calculates automatically when selected. However, to display the dialog box as shown in (2) above, select More Functions…
- The fourth way to insert a function in Excel is by using the Insert Function command on the Formula bar. When you select this command, the dialog box as discussed in (2) above will appear.
- The final way of inserting a function in Excel is by choosing a function from its category on the Formula bar. Choosing any of the functions categories will display the list of functions. Select a function from the list and follow the steps discussed in (2c) above.
Among these methods of inserting a formula, the manual method will help you work faster and smarter.
Edit a formula
While working with a formula, there may be a need for you to edit the formula and make adjustments. When working with Excel formulas and encounter an error, you do not need to delete the formula. You can edit the formula and continue your work.
To edit a formula,
- Select the cell that contains the formula.
- Click the formula bar to activate editing (you can also double click the cell)
- Make the necessary adjustments and press Enter on the keyboard or click the Enter icon on the formula bar.
When using the manual method to enter formulas in Excel, you should be mindful of the operator precedence. Operator precedence is the sequence by which mathematical operators are executed in an excel formula.
Microsoft Excel follows the general mathematical rule to carry out its mathematical operations. There are about six (6) arithmetic operators used by excel in formulas. When these operators are used together, the sequence of calculations is what we call operator precedence.
The operator precedence in Excel is as follows Parenthesis, Exponential, Multiplication and Division, and Addition and Subtraction.
This order means that whatever is in the parenthesis will be calculated first. Followed by the exponent, then multiplication and division. Addition and subtraction will be performed last in any excel formula.
For example, in the Excel formula [=C4*D4/E4+F4^2], the multiplication and division will be calculated first. That is, C4*D4/E4. This will be followed by the exponent, that is F4^2, before the addition.
However, using a parenthesis will change the whole order. For example, if the formula is rewritten thus, [=C4*D4/(E4+F4^2)] the entire result will change. This is because the formula in the bracket will be executed first before other ones.
How to Use Excel formulas and Functions
We have already discussed how to insert a formula/ function in excel. However, we shall use the manual method to illustrate how to use Excel formulas.
Addition and subtraction formula in excel
The addition formula is used to add values in Excel while the subtraction formula is used to subtract values. Because subtraction represents negative addition, the subtraction function does not exist.
For example, to subtract the following values: 10 – 2 – 3 – 1 = 10 – (2+3+1). Therefore, to implement subtraction, you can add the negative sign on values. By so doing, adding them will give you the required result.
In Microsoft Excel, the addition function is called the SUM and it is the simplest function to use in Excel. You can invoke the AutoSum or use the SUM function manually. Let us illustrate below.
- Select the cell you want to calculate addition on
- On the Home tab, under Editing, select AutoSum
- A range corresponding to the active cell will be automatically selected
- If the selected range is correct, press Enter
- If the selected range is wrong, use the mouse and select the correct range, and press Enter.
How to multiply using Excel formula
The asterisk (*) is used to perform multiplication in Excel. When a formula is entered manually, the asterisk is used to indicate necessary multiplications. For example, B3*C3.
But, if you are using the excel predefined formula, the multiplication function is called the PRODUCT. To use the PRODUCT function, do the following:
- Select the cell you want to calculate multiplication on
- On the active cell, enter the (=) sign and start typing product (=PRODUCT)
- A list of functions that begin with ‘pro’ will appear.
- Select PRODUCT from the list, a parenthesis opens with suggestions for number1, etc.
- Enter the cell references to be multiplied separated by a comma.
- Close the parenthesis when you are done, and press Enter.
When working with Excel functions manually, excel provides formula tips that will help you know which value to enter next.
When the first value is entered, separate it with a comma and continue until all the variables are entered.
Apart from these basic operations, you can use the QUOTIENT function for division and the POWER function for exponent.
Excel functions and formulas help us to perform basic and advance calculations in Excel. Formulas can be entered manually by using the predefined functions in Excel.
Using Excel formulas is easy when you understand how it functions. Whenever you do, pay attention to the formula tips provided by Excel. In our later tutorials, we shall discuss some of the advanced functions in excel, such as the IF, VLOOKUP, etc.
Our next edition will look at SUMIF and SUMIFS in Excel.
If you have any questions, please, kindly ask. In case you have not seen our previous tutorials, view them below:
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In the realm of mathematics, the term “sum” holds great significance. It refers to the result obtained by combining or adding two or more numbers or items. Whether you are a student exploring the basics of arithmetic or delving into complex mathematical concepts, understanding the meaning of sum is crucial. In this article, we will explore the definition of sum, various formulas associated with it, and examples to enhance your comprehension.
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What is Sum?
Sum, in its simplest form, represents the outcome of adding numbers or terms together. It is a fundamental operation that allows us to combine quantities and determine their total value. Whenever we encounter a situation that involves adding or combining numbers, the result obtained is referred to as the sum. Additionally, if you’re learning about mathematics, you might wonder, what does total mean in Math, since many people get it confused with the term ‘sum’. Total is another term used synonymously with sum, representing the combined value of a set of numbers or quantities.
Importance of Sum in Mathematics
Addition: The concept of sum forms the foundation of addition, which is one of the fundamental operations in arithmetic. It allows us to find the total value when two or more numbers are combined.
Subtraction: Subtraction, the inverse operation of addition, involves finding the difference between two quantities. It relies on the concept of sum to determine the value being subtracted.
Advanced Mathematical Concepts
Algebra: In algebraic equations, the sum represents the outcome of combining terms or expressions.
Calculus: The concept of sum is extended in calculus through the integration process, where sums are used to find the total value of a continuous function over a specific interval.
Formulas for Finding Sum
- Summation Notation (∑): In the realm of mathematics, the symbol ∑, known as sigma, is not just a way to represent a sum; it holds particular significance in fields like statistics, where sigma in statistics often refers to the standard deviation of a population. It is commonly employed when a long list or sequence of numbers needs to be added together.
- Sum of Two Numbers: To find the sum of two numbers, we simply add the numbers together. For example, the sum of 5 and 7 is 12.
- Sum of Digits: a. One-digit Numbers: The sum of two one-digit numbers, such as 3 and 4, can be obtained by adding the digits diagonally across the plus symbol. For instance, 3 + 4 = 7. b. Two-digit Numbers: When adding two-digit numbers, we add the ones place digits first, followed by the tens place digits, taking any carryovers into account.
- Sum of a Sequence: a. Sum of First n Natural Numbers: The sum of the first n natural numbers can be calculated using the formula S = n(n + 1)/2. b. Sum of Odd Numbers: The sum of the first n odd numbers can be determined using the formula S = n^2. c. Sum of Even Numbers: The sum of the first n even numbers can be found using the formula S = n(n + 1).
Sum of Two-Digit Numbers
Consider the numbers 89 and 22.
To find their sum: Step 1: Add the ones place digits: 9 + 2 = 11. Step 2: Add the tens place digits: 8 + 2 = 10. Step 3: The final sum is obtained by combining the results from Step 1 and Step 2, resulting in 111.
Sum of First n Natural Numbers
Let’s find the sum of the first 15 natural numbers using the formula S = n(n + 1)/2: S = (15 * 16)/2 = 120.
Understanding the concept of sum is integral to mastering various mathematical operations. From basic arithmetic to advanced concepts in algebra and calculus, the sum provides us with a valuable tool for combining and calculating quantities. By grasping the meaning of sum and familiarizing yourself with the associated formulas and examples, you will develop a solid foundation in mathematics and be better equipped to tackle more complex problems. So, embrace the power of the sum and let it unlock the wonders of mathematics for you.
What is the difference between sum and product in Math?
In mathematics, the sum refers to the result obtained by adding numbers or terms together, whereas the product represents the outcome of multiplying numbers or terms. The sum focuses on combining quantities to determine their total value, while the product emphasizes the result of multiplication.
How is the sum symbol represented in mathematical notation?
The sum symbol is represented by the capital Greek letter sigma (∑). It is commonly used in mathematical notation to indicate that a series of numbers or terms should be added together.
Are there any special rules for summing fractions in Math?
Yes, there are rules for summing fractions. To add fractions, you need to ensure they have a common denominator. If the fractions have different denominators, you must find the least common multiple (LCM) of the denominators and convert each fraction to an equivalent fraction with the common denominator. Then, you can add the numerators and keep the common denominator unchanged.
Can negative numbers be added in Math? How?
Yes, negative numbers can be added in mathematics. Adding negative numbers is equivalent to subtracting their absolute values. For example, adding -5 to -3 would result in -8. To add negative numbers, simply add their magnitudes and assign the negative sign to the sum.
How do you find the sum of an arithmetic series?
To find the sum of an arithmetic series, you can use the formula: S = (n/2) × [2a + (n-1)d], where S represents the sum, n is the number of terms in the series, a is the first term, and d is the common difference. Alternatively, you can also use the formula: S = (n/2) × (a + l), where l represents the last term of the series.
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- Cosmic distance ladder
The cosmic distance ladder (also known as the Extragalactic Distance Scale) is the succession of methods by which astronomers determine the distances to celestial objects. A real direct distance measurement of an astronomical object is possible only for those objects that are "close enough" (within about a thousand parsecs) to Earth. The techniques for determining distances to more distant objects are all based on various measured correlations between methods that work at close distances with methods that work at larger distances. Several methods rely on a standard candle, which is an astronomical object that has a known luminosity.
The ladder analogy arises because no one technique can measure distances at all ranges encountered in astronomy. Instead, one method can be used to measure nearby distances, a second can be used to measure nearby to intermediate distances, and so on. Each rung of the ladder provides information that can be used to determine the distances at the next higher rung.
At the base of the ladder are fundamental distance measurements, in which distances are determined directly, with no physical assumptions about the nature of the object in question. The precise measurement of stellar positions is part of the discipline of astrometry.
Direct distance measurements are based upon precise determination of the distance between the Earth and the Sun, which is called the Astronomical Unit (AU). Historically, observations of transits of Venus were crucial in determining the AU; in the first half of the 20th Century, observations of asteroids were also important. Presently the AU is determined with high precision using radar measurements of Venus and other nearby planets and asteroids, and by tracking interplanetary spacecraft in their orbits around the Sun through the Solar System. Kepler's Laws provide precise ratios of the sizes of the orbits of objects revolving around the Sun, but not a real measure of the orbits themselves. Radar provides a value in kilometers for the difference in two orbits' sizes, and from that and the ratio of the two orbit sizes, the size of Earth's orbit comes directly.
The most important fundamental distance measurements come from trigonometric parallax. As the Earth orbits around the Sun, the position of nearby stars will appear to shift slightly against the more distant background. These shifts are angles in a right triangle, with 2 AU making the short leg of the triangle and the distance to the star being the long leg. The amount of shift is quite small, measuring 1 arcseconds for an object at a distance of 1 parsec (3.26 light-years), thereafter decreasing in angular amount as the reciprocal of the distance. Astronomers usually express distances in units of parsecs; light-years are used in popular media, but almost invariably values in light-years have been converted from numbers tabulated in parsecs in the original source.
Because parallax becomes smaller for a greater stellar distance, useful distances can be measured only for stars whose parallax is larger than the precision of the measurement. Parallax measurements typically have an accuracy measured in milliarcseconds. In the 1990s, for example, the Hipparcos mission obtained parallaxes for over a hundred thousand stars with a precision of about a milliarcsecond, providing useful distances for stars out to a few hundred parsecs.
Stars can have a velocity relative to the Sun that causes proper motion and radial velocity. The former is determined by plotting the changing position of the stars over many years, while the latter comes from measuring the Doppler shift in their spectrum caused by motion along the line of sight. For a group of stars with the same spectral class and a similar magnitude range, a mean parallax can be derived from statistical analysis of the proper motions relative to their radial velocities. This statistical parallax method is useful for measuring the distances of bright stars beyond 50 parsecs and giant variable stars, including Cepheids and the RR Lyrae variables.
The motion of the Sun through space provides a longer baseline that will increase the accuracy of parallax measurements, known as secular parallax. For stars in the Milky Way disk, this corresponds to a mean baseline of 4 A.U. per year, while for halo stars the baseline is 40 A.U. per year. After several decades, the baseline can be orders of magnitude greater than the Earth-Sun baseline used for traditional parallax. However, secular parallax introduces a higher level of uncertainty because the relative velocity of other stars is an additional unknown. When applied to samples of multiple stars, the uncertainty can be reduced; the precision is inversely proportion to the square root of the sample size.
Moving cluster parallax is a technique where the motions of individual stars in a nearby star cluster can be used to find the distance to the cluster. Only open clusters are near enough for this technique to be useful. In particular the distance obtained for the Hyades has been an important step in the distance ladder.
Other individual objects can have fundamental distance estimates made for them under special circumstances. If the expansion of a gas cloud, like a supernova remnant or planetary nebula, can be observed over time, then an expansion parallax distance to that cloud can be estimated. Binary stars which are both visual and spectroscopic binaries also can have their distance estimated by similar means. The common characteristic to these is that a measurement of angular motion is combined with a measurement of the absolute velocity (usually obtained via the Doppler effect). The distance estimate comes from computing how far away the object must be to make its observed absolute velocity appear with the observed angular motion.
Expansion parallaxes in particular can give fundamental distance estimates for objects that are very far away, because supernova ejecta have large expansion velocities and large sizes (compared to stars). Further, they can be observed with radio interferometers which can measure very small angular motions. These combine to mean that some supernovae in other galaxies have fundamental distance estimates. Though valuable, such cases are quite rare, so they serve as important consistency checks on the distance ladder rather than workhorse steps by themselves.
Almost all of the physical distance indicators are standard candles. These are objects that belong to some class that have a known brightness. By comparing the known luminosity of the latter to its observed brightness, the distance to the object can be computed using the inverse square law. These objects of known brightness are termed standard candles.
In astronomy, the brightness of an object is given in terms of its absolute magnitude. This quantity is derived from the logarithm of its luminosity as seen from a distance of 10 parsecs. The apparent magnitude, or the magnitude as seen by the observer, can be used to determine the distance D to the object in kiloparsecs (where 1 kpc equals 103 parsecs) as follows:
where m the apparent magnitude and M the absolute magnitude. For this to be accurate, both magnitudes must be in the same frequency band and there can be no relative motion in the radial direction.
Some means of accounting for interstellar extinction, which also makes objects appear fainter and more red, is also needed. The difference between absolute and apparent magnitudes is called the distance modulus, and astronomical distances, especially intergalactic ones, are sometimes tabulated in this way.
Two problems exist for any class of standard candle. The principal one is calibration, determining exactly what the absolute magnitude of the candle is. This includes defining the class well enough that members can be recognized, and finding enough members with well-known distances that their true absolute magnitude can be determined with enough accuracy. The second lies in recognizing members of the class, and not mistakenly using the standard candle calibration upon an object which does not belong to the class. At extreme distances, which is where one most wishes to use a distance indicator, this recognition problem can be quite serious.
A significant issue with standard candles is the recurring question of how standard they are. For example, all observations seem to indicate that type Ia supernovae that are of known distance have the same brightness (corrected by the shape of the light curve). The basis for this closeness in brightness is discussed below; however, the possibility that the distant type Ia supernovae have different properties than nearby type Ia supernovae exists. The use of Supernovae type Ia is crucial in determining the correct cosmological model. If indeed the properties of Supernovae type Ia are different at large distances, i.e. if the extrapolation of their calibration to arbitrary distances is not valid, ignoring this variation can dangerously bias the reconstruction of the cosmological parameters, in particular the reconstruction of the matter density parameter.
That this is not merely a philosophical issue can be seen from the history of distance measurements using Cepheid variables. In the 1950s, Walter Baade discovered that the nearby Cepheid variables used to calibrate the standard candle were of a different type than the ones used to measure distances to nearby galaxies. The nearby Cepheid variables were population I stars with much higher metal content than the distant population II stars. As a result, the population II stars were actually much brighter than believed, and this had the effect of doubling the distances to the globular clusters, the nearby galaxies, and the diameter of the Milky Way.
Galactic distance indicators
With few exceptions, distances based on direct measurements are available only out to about a thousand parsecs, which is a modest portion of our own Galaxy. For distances beyond that, measures depend upon physical assumptions, that is, the assertion that one recognizes the object in question, and the class of objects is homogeneous enough that its members can be used for meaningful estimation of distance.
Physical distance indicators, used on progressively larger distance scales, include:
- Dynamical parallax, using orbital parameters of visual binaries to measure the mass of the system and the mass-luminosity relation to determine the luminosity
- Eclipsing binaries — In the last decade, measurement of eclipsing binaries' fundamental parameters has become possible with 8 meter class telescopes. This makes it feasible to use them as indicators of distance. Recently, they have been used to give direct distance estimates to the LMC, SMC, Andromeda Galaxy and Triangulum Galaxy. Eclipsing binaries offer a direct method to gauge the distance to galaxies to a new improved 5% level of accuracy which is feasible with current technology up to a distance of around 3 Mpc.
- RR Lyrae variables — red giants typically used for measuring distances within the galaxy and in nearby globular clusters.
- The following four indicators all use stars in the old stellar populations (Population II):
- In galactic astronomy, X-ray bursts (thermonuclear flashes on the surface of a neutron star) are used as standard candles. Observations of X-ray burst sometimes show X-ray spectra indicating radius expansion. Therefore, the X-ray flux at the peak of the burst should correspond to Eddington luminosity, which can be calculated once the mass of the neutron star is known (1.5 solar masses is a commonly used assumption). This method allows distance determination of some low-mass X-ray binaries. Low-mass X-ray binaries are very faint in the optical, making measuring their distances extremely difficult.
- Cepheids and novae
- Individual galaxies in clusters of galaxies
- The Tully-Fisher relation
- The Faber-Jackson relation
- Type Ia supernovae that have a very well-determined maximum absolute magnitude as a function of the shape of their light curve and are useful in determining extragalactic distances up to a few hundred Mpc. A notable exception is SN 2003fg, the "Champagne Supernova," a type Ia supernova of unusual nature.
- Redshifts and Hubble's Law
Main sequence fitting
When the absolute magnitude for a group of stars is plotted against the spectral classification of the star, in a Hertzsprung-Russell diagram, evolutionary patterns are found that relate to the mass, age and composition of the star. In particular, during their hydrogen burning period, stars lie along a curve in the diagram called the main sequence. By measuring these properties from a star's spectrum, the position of a main sequence star on the H-R diagram can be determined, and thereby the star's absolute magnitude estimated. A comparison of this value with the apparent magnitude allows the approximate distance to be determined, after correcting for interstellar extinction of the luminosity because of gas and dust.
In a gravitationally-bound star cluster such as the Hyades, the stars formed at approximately the same age and lie at the same distance. This allows relatively accurate main sequence fitting, providing both age and distance determination.
Extragalactic distance scale
Extragalactic distance indicators Method Uncertainty for Single Galaxy (mag) Distance to Virgo Cluster (Mpc) Range (Mpc) Classical Cepheids 0.16 15 - 25 29 Novae 0.4 21.1 ± 3.9 20 Planetary Nebula Luminosity Function 0.3 15.4 ± 1.1 50 Globular Cluster Luminosity Function 0.4 18.8 ± 3.8 50 Surface Brightness Fluctuations 0.3 15.9 ± 0.9 50 D - σ relation 0.5 16.8 ± 2.4 > 100 Type Ia Supernovae 0.10 19.4 ± 5.0 > 1000
The extragalactic distance scale is a series of techniques used today by astronomers to determine the distance of cosmological bodies beyond our own galaxy, which are not easily obtained with traditional methods. Some procedures utilize properties of these objects, such as stars, globular clusters, nebulae, and galaxies as a whole. Other methods are based more on the statistics and probabilities of things such as entire galaxy clusters.
Discovered in 1956 by Olin Wilson and M.K. Vainu Bappu, The Wilson-Bappu Effect utilizes the effect known as spectroscopic parallax. Certain stars have features in their emission/absorption spectra allowing relatively easy absolute magnitude calculation. Certain spectral lines are directly related to an object's magnitude, such as the K absorption line of calcium. Distance to the star can be calculated from magnitude by the distance modulus:
Though in theory this method has the ability to provide reliable distance calculations to stars roughly 7 megaparsecs (Mpc) away, it is generally only used for stars hundreds of kiloparsecs (kpc) away.
This method is only valid for stars over 15 magnitudes.
Beyond the reach of the Wilson-Bappu effect, the next method relies on the period-luminosity relation of classical Cepheid variable stars, first discovered by Henrietta Leavitt. The following relation can be used to calculate the distance to Galactic and extragalactic classical Cepheids:
Several problems complicate the use of Cepheids as standard candles and are actively debated, chief among them are: the nature and linearity of the period-luminosity relation in various passbands and the impact of metallicity on both the zero-point and slope of those relations, and the effects of photometric contamination (blending) and a changing (typically unknown) extinction law on Cepheid distances.
These unresolved matters have resulted in cited values for the Hubble Constant ranging between 60 km/s/Mpc and 80 km/s/Mpc. Resolving this discrepancy is one of the foremost problems in astronomy since the cosmological parameters of the Universe may be constrained by supplying a precise value of the Hubble constant.
Cepheid variable stars were the key instrument in Edwin Hubble’s 1923 conclusion that M31 (Andromeda) was an external galaxy, as opposed to a smaller nebula within the Milky Way. He was able to calculate the distance of M31 to 285 Kpc, today’s value being 770 Kpc.
As detected thus far, NGC 3370, a spiral galaxy in the constellation Leo, contains the farthest Cepheids yet found at a distance of 29 Mpc. Cepheid variable stars are in no way perfect distance markers: at nearby galaxies they have an error of about 7% and up to a 15% error for the most distant.
There are several different methods for which supernovae can be used to measure extragalactic distances, here we cover the most used.
Measuring a supernova's photosphere
We can assume that a supernova expands spherically symmetric. If the supernova is close enough such that we can measure the angular extent, θ(t), of its photosphere, we can use the equation
Where ω is angular velocity, θ is angular extent. In order to get an accurate measurement, it is necessary to make two observations separated by time Δt. Subsequently, we can use
Where d is the distance to the SN, Vej is the supernova's ejecta's radial velocity (it can be assumed that Vej equals Vθ if spherically symmetric).
This method works only if the supernova is close enough to be able to measure accurately the photosphere. Similarly, the expanding shell of gas is in fact not perfectly spherical nor a perfect blackbody. Also interstellar extinction can hinder the accurate measurements of the photosphere. This problem is further exacerbated by core-collapse supernova. All of these factors contribute to the distance error of up to 25%.
Type Ia light curves
Type Ia SN are some of the best ways to determine extragalactic distances. Ia's occur when a binary white dwarf star begins to accrete matter from its companion Red Dwarf star. As the white dwarf gains matter, eventually it reaches its Chandrasekhar Limit of . Once reached, the star becomes unstable and undergoes a runaway nuclear fusion reaction. Because all Type Ia SN explode at about the same mass, their absolute magnitudes are all the same. This makes them very useful as standard candles. All type Ia SN have a standard blue and visual magnitude of
Therefore, when observing a type Ia SN, if it is possible to determine what its peak magnitude was, then its distance can be calculated. It is not intrinsically necessary to capture the SN directly at its peak magnitude; using the multicolor light curve shape method (MLCS), the shape of the light curve (taken at any reasonable time after the initial explosion) is compared to a family of parameterized curves that will determine the absolute magnitude at the maximum brightness. This method also takes into effect interstellar extinction/dimming from dust and gas.
Similarly, the stretch method fits the particular SN magnitude light curves to a template light curve. This template, as opposed to being several light curves at different wavelengths (MLCS) is just a single light curve that has been stretched (or compressed) in time. By using this Stretch Factor, the peak magnitude can be determined.
Using Type Ia SN is one of the most accurate methods, particularly since SN explosions can be visible at great distances (their luminosities rival that of the galaxy in which they are situated), much farther than Cepheid Variables (500 times farther). Much time has been devoted to the refining of this method. The current uncertainty approaches a mere 5%, corresponding to an uncertainty of just 0.1 magnitudes.
Novae in distance determinations
Novae can be used in much the same way as supernovae to derive extragalactic distances. There is a direct relation between a nova's max magnitude and the time for its visible light to decline by two magnitudes. This relation is shown to be:
Where is the time derivative of the nova's mag, describing the average rate of decline over the first 2 magnitudes.
After novae fade, they are about as bright as the most luminous Cepheid Variable stars, therefore both these techniques have about the same max distance: ~ 20 Mpc. The error in this method produces an uncertainty in magnitude of about ± 0.4
Globular cluster luminosity function
Based on the method of comparing the luminosities of globular clusters (located in galactic halos) from distant galaxies to that of the Virgo cluster, the globular cluster luminosity function carries an uncertainty of distance of about 20% (or .4 magnitudes).
US astronomer William Alvin Baum first attempted to use globular clusters to measure distant elliptical galaxies. He compared the brightest globular clusters in Virgo A galaxy with those in Andromeda, assuming the luminosities of the clusters were the same in both. Knowing the distance to Andromeda, has assumed a direct correlation and estimated Virgo A’s distance.
Baum used just a single globular cluster, but individual formations are often poor standard candles. Canadian astronomer Racine assumed the use of the globular cluster luminosity function (GCLF) would lead to a better approximation. The number of globular clusters as a function of magnitude given by:
Where m0 is the turnover magnitude, and M0 the magnitude of the Virgo cluster, sigma the dispersion ~ 1.4 mag.
It is important to remember that it is assumed that globular clusters all have roughly the same luminosities within the universe. There is no universal globular cluster luminosity function that applies to all galaxies.
Planetary nebula luminosity function
Like the GCLF method, a similar numerical analysis can be used for planetary nebulae (note the use of more than one!) within far off galaxies. The planetary nebula luminosity function (PNLF) was first proposed in the late 1970s by Holland Cole and David Jenner. They suggested that all planetary nebulae might all have similar maximum intrinsic brightness, now calculated to be M = -4.53. This would therefore make them potential standard candles for determining extragalactic distances.
Astronomer George Howard Jacoby and his fellow colleagues later proposed that the PNLF function equaled:
Where N(M) is number of planetary nebula, having absolute magnitude M. M* is equal to the nebula with the brightest magnitude.
Surface brightness fluctuation method
The following method deals with the overall inherent properties of galaxies. These methods, though with varying error percentages, have the ability to make distance estimates beyond 100 Mpc, though it is usually applied more locally.
The surface brightness fluctuation (SBF) method takes advantage of the use of CCD cameras on telescopes. Because of spatial fluctuations in a galaxy’s surface brightness, some pixels on these cameras will pick up more stars than others. However, as distance increases the picture will become increasingly smoother. Analysis of this describes a magnitude of the pixel-to-pixel variation, which is directly related to a galaxy’s distance.
The D- σ relation, used in elliptical galaxies, relates the angular diameter (D) of the galaxy to its velocity dispersion. It is important to describe exactly what D represents in order to have a more fitting understanding of this method. It is, more precisely, the galaxy’s angular diameter out to the surface brightness level of 20.75 B-mag arcsec − 2. This surface brightness is independent of the galaxy’s actual distance from us. Instead, D is inversely proportional to the galaxy’s distance, represented as d. So instead of this relation imploring standard candles, instead D provides a standard ruler. This relation between D and σ is
Where C is a constant which depends on the distance to the galaxy clusters.
This method has the possibility of become one of the strongest methods of galactic distance calculators, perhaps exceeding the range of even the Tully-Fisher method. As of today, however, elliptical galaxies aren’t bright enough to provide a calibration for this method through the use of techniques such as Cepheids. So instead calibration is done using more crude methods.
Overlap and scaling
A succession of distance indicators, which is the distance ladder, is needed for determining distances to other galaxies. The reason is that objects bright enough to be recognized and measured at such distances are so rare that few or none are present nearby, so there are too few examples close enough with reliable trigonometric parallax to calibrate the indicator. For example, Cepheid variables, one of the best indicators for nearby spiral galaxies, cannot be satisfactorily calibrated by parallax alone. The situation is further complicated by the fact that different stellar populations generally do not have all types of stars in them. Cepheids in particular are massive stars, with short lifetimes, so they will only be found in places where stars have very recently been formed. Consequently, because elliptical galaxies usually have long ceased to have large-scale star formation, they will not have Cepheids. Instead, distance indicators whose origins are in an older stellar population (like novae and RR Lyrae variables) must be used. However, RR Lyrae variables are less luminous than Cepheids (so they cannot be seen as far away as Cepheids can), and novae are unpredictable and an intensive monitoring program — and luck during that program — is needed to gather enough novae in the target galaxy for a good distance estimate.
Because the more distant steps of the cosmic distance ladder depend upon the nearer ones, the more distant steps include the effects of errors in the nearer steps, both systematic and statistical ones. The result of these propagating errors means that distances in astronomy are rarely known to the same level of precision as measurements in the other sciences, and that the precision necessarily is poorer for more distant types of object.
Another concern, especially for the very brightest standard candles, is their "standardness": how homogeneous the objects are in their true absolute magnitude. For some of these different standard candles, the homogeneity is based on theories about the formation and evolution of stars and galaxies, and is thus also subject to uncertainties in those aspects. For the most luminous of distance indicators, the Type Ia supernovae, this homogeneity is known to be poor; however, no other class of object is bright enough to be detected at such large distances, so the class is useful simply because there is no real alternative.
The observational result of Hubble's Law, the proportional relationship between distance and the speed with which a galaxy is moving away from us (usually referred to as redshift) is a product of the cosmic distance ladder. Hubble observed that fainter galaxies are more redshifted. Finding the value of the Hubble constant was the result of decades of work by many astronomers, both in amassing the measurements of galaxy redshifts and in calibrating the steps of the distance ladder. Hubble's Law is the primary means we have for estimating the distances of quasars and distant galaxies in which individual distance indicators cannot be seen.
- ^ Ash, M.E., Shapiro, I.I., & Smith, W.B., 1967 Astronomical Journal, 72, 338-350.
- ^ Staff. "Trigonometric Parallax". The SAO Encyclopedia of Astronomy. Swinburne Centre for Astrophysics and Supercomputing. http://astronomy.swin.edu.au/cosmos/T/Trigonometric+Parallax. Retrieved 2008-10-18.
- ^ Perryman, M. A. C.; et al. (1999). "The HIPPARCOS Catalogue". Astronomy and Astrophysics 323: L49–L52. Bibcode 1997A&A...323L..49P.
- ^ Basu, Baidyanath (2003). An Introduction to Astrophysics. PHI Learning Private Limited. ISBN 8120311213.
- ^ Popowski, Piotr; Gould, Andrew (1998-01-29). "Mathematics of Statistical Parallax and the Local Distance Scale". arXiv:astro-ph/9703140 [astro-ph].
- ^ Bartel, N., et al., 1994, "The shape, expansion rate and distance of supernova 1993J from VLBI measurements", Nature 368, 610-613
- ^ Linden, Sebastian; Virey, Jean-Marc; Tilquin, André (2009). "Cosmological parameter extraction and biases from type Ia supernova magnitude evolution". A&A 506 (3): 1095–1105. Bibcode 2009A&A...506.1095L. doi:10.1051/0004-6361/200912811., and references therein.
- ^ Marinoni, C.; Saintonge, A.; Giovanelli, R.; Haynes, M. P.; Masters, J.-M.; Le Fèvre, O.; Mazure, A.; Taxil, P. et al. (2008). "Geometrical tests of cosmological models. I. Probing dark energy using the kinematics of high redshift galaxies". A&A 478 (1): 43–55. Bibcode 2008A&A...478...43M. doi:10.1051/0004-6361:20077116.
- ^ Bonanos, Alceste Z. (2006). "Eclipsing Binaries: Tools for Calibrating the Extragalactic Distance Scale". Binary Stars as Critical Tools and Tests in Contemporary Astrophysics, International Astronomical Union. Symposium no. 240, held 22–25 August 2006 in Prague, Czech Republic, S240, #008 240: 79. arXiv:astro-ph/0610923. Bibcode 2007IAUS..240...79B. doi:10.1017/S1743921307003845.
- ^ Ferrarese, Laura; Ford, Holland C.; Huchra, John; Kennicutt, Robert C., Jr.; Mould, Jeremy R.; Sakai, Shoko; Freedman, Wendy L.; Stetson, Peter B.; Madore, Barry F.; Gibson, Brad K.; Graham, John A.; Hughes, Shaun M.; Illingworth, Garth D.; Kelson, Daniel D.; Macri, Lucas; Sebo, Kim; Silbermann, N. A. (2000). "A Database of Cepheid Distance Moduli and Tip of the Red Giant Branch, Globular Cluster Luminosity Function, Planetary Nebula Luminosity Function, and Surface Brightness Fluctuation Data Useful for Distance Determinations". The Astrophysical Journal Supplement Series 128 (2): 431–459. arXiv:astro-ph/9910501. Bibcode 2000ApJS..128..431F. doi:10.1086/313391.
- ^ S. A. Colgate (1979). "Supernovae as a standard candle for cosmology". Astrophysical Journal 232 (1): 404–408. Bibcode 1979ApJ...232..404C. doi:10.1086/157300.
- ^ Adapted from Jacoby et al., Publ. Astron. Soc. Pac., 104, 499, 1992
- ^ "Assessing potential cluster Cepheids from a new distance and reddening parameterization and 2MASS photometry". MNRAS. arXiv:0808.2937. Bibcode 2008MNRAS.390.1539M. doi:10.1111/j.1365-2966.2008.13834.x.
- ^ Stanek, K. Z.; Udalski, A. (1999). "The Optical Gravitational Lensing Experiment. Investigating the Influence of Blending on the Cepheid Distance Scale with Cepheids in the Large Magellanic Cloud". Eprint arXiv:astro-ph/9909346: 9346. arXiv:astro-ph/9909346. Bibcode 1999astro.ph..9346S.
- ^ Udalski, A.; Wyrzykowski, L.; Pietrzynski, G.; Szewczyk, O.; Szymanski, M.; Kubiak, M.; Soszynski, I.; Zebrun, K. (2001). "The Optical Gravitational Lensing Experiment. Cepheids in the Galaxy IC1613: No Dependence of the Period-Luminosity Relation on Metallicity". Acta Astronomica 51: 221. arXiv:astro-ph/0109446. Bibcode 2001AcA....51..221U.
- ^ Ngeow, C.; Kanbur, S. M. (2006). "The Hubble Constant from Type Ia Supernovae Calibrated with the Linear and Nonlinear Cepheid Period-Luminosity Relations". The Astrophysical Journal 642: L29. arXiv:astro-ph/0603643. Bibcode 2006ApJ...642L..29N. doi:10.1086/504478.
- ^ Macri, L. M.; Stanek, K. Z.; Bersier, D.; Greenhill, L. J.; Reid, M. J. (2006). "A New Cepheid Distance to the Maser-Host Galaxy NGC 4258 and Its Implications for the Hubble Constant". The Astrophysical Journal 652 (2): 1133. arXiv:astro-ph/0608211. Bibcode 2006ApJ...652.1133M. doi:10.1086/508530.
- ^ Bono, G.; Caputo, F.; Fiorentino, G.; Marconi, M.; Musella, I. (2008). "Cepheids in External Galaxies. I. The Maser-Host Galaxy NGC 4258 and the Metallicity Dependence of Period-Luminosity and Period-Wesenheit Relations". The Astrophysical Journal 684: 102. Bibcode 2008ApJ...684..102B. doi:10.1086/589965.
- ^ Majaess, D.; Turner, D.; Lane, D. (2009). "Type II Cepheids as Extragalactic Distance Candles". Acta Astronomica 59: 403. Bibcode 2009AcA....59..403M.
- ^ Madore, Barry F.; Freedman, Wendy L. (2009). "Concerning the Slope of the Cepheid Period-Luminosity Relation". The Astrophysical Journal 696 (2): 1498. Bibcode 2009ApJ...696.1498M. doi:10.1088/0004-637X/696/2/1498.
- ^ Scowcroft, V.; Bersier, D.; Mould, J. R.; Wood, P. R. (2009). "The effect of metallicity on Cepheid magnitudes and the distance to M33". Monthly Notices of the Royal Astronomical Society 396 (3): 1287. Bibcode 2009MNRAS.396.1287S. doi:10.1111/j.1365-2966.2009.14822.x.
- ^ Majaess, D. (2010). "The Cepheids of Centaurus A (NGC 5128) and Implications for H0". Acta Astronomica 60: 121. Bibcode 2010AcA....60..121M.
- ^ Annual Review of Astronomy and Astrophysics. Bibcode 2008A&ARv..15..289T. doi:10.1007/s00159-008-0012-y.
- ^ Annual Review of Astronomy and Astrophysics. Bibcode 2010ARA&A..48..673F. doi:10.1146/annurev-astro-082708-101829.
- An Introduction to Modern Astrophysics, Carroll and Ostlie, copyright 2007
- Measuring the Universe The Cosmological Distance Ladder, Stephen Webb, copyright 2001
- The Cosmos, Pasachoff and Filippenko, copyright 2007
- The Astrophysical Journal, The Globular Cluster Luminosity Function as a Distance Indicator: Dynamical Effects, Ostriker and Gnedin, May 5, 1997
- The ABC's of distances (UCLA)
- The Extragalactic Distance Scale by Bill Keel
- The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
- The Hubble Constant, a historical discussion
- NASA Cosmic Distance Scale
- PNLF information database
- The Astrophysical Journal
Units of length used in AstronomyAstronomical system of units
- Dynamical parallax, using orbital parameters of visual binaries to measure the mass of the system and the mass-luminosity relation to determine the luminosity
Wikimedia Foundation. 2010.
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C programmers can pass arrays to functions in three different ways: by reference, as formal parameters, or as return values.
When a function is called with an array, the compiler converts the array declaration into an array pointer.
This means that changes made to the array will affect the corresponding value in the function’s formal parameter list. However, this can be avoided by adding the keyword const.
Arrays are passed by reference
Arrays are a powerful data structure that can store lists of elements. They are commonly used to manage large amounts of data in computer programs.
Unlike other types of data structures, arrays can be accessed very quickly by using their indices. This is because they allocate memory in contiguous memory locations for each element of the array. This allows the element to be retrieved very efficiently (random access, O(1) = constant time).
When you create an array, the space it takes up in your computer’s memory is “reserved” until you assign the values that will be stored in it. Once you have assigned the values, they are immediately allocated into the space that was previously reserved. This allows for efficient access of the array’s elements, and it means that you can insert new values into an array as often as necessary without having to worry about filling it up.
An array can be defined in many ways, and its size will depend on the type of data it will store. Typically, a strongly typed, compiled programming language will require that the elements in an array be of the same data type. However, dynamic scripting languages can also allow the elements to be of any data type.
To create an array, you need to define the type of elements it will store and the maximum size it can contain. In addition, you need to assign a name to the array.
Once you have completed this task, you need to set the base index of the array and its elements. These indexes can be a number, such as 0, 1, or n, and are used to locate an element in the array.
This is the most important step when creating an array, as it sets the ground rules for the entire array’s use. The first element of the array is referred to as the base index, and it is the starting point for all subsequent elements.
The second element of the array is referred to as its offset, and it is the number that is added to each value in the array to determine the location of the value. This is similar to adding an offset to a book to locate the page where the chapter starts.
Arrays are passed as formal parameters
Arrays are data structures that store multiple pieces of related information together. These types of data structures are useful for storing large amounts of data in a compact and efficient manner.
The C language allows arrays to be passed as formal parameters in a function without any restrictions. This means that the C compiler will convert the array to a pointer and pass it to the function as an argument. This makes it possible to use arrays in a variety of ways, such as recursively accumulating values for loops or for performing calculations on array elements.
In C, an array is a contiguously allocated nonempty sequence of objects of the same type stored at consecutive memory locations. The number of those objects (the array size) never changes during the life of the array. The elements are accessed by referring to the array’s index numbers within square brackets, which represent the position of an element in the array.
Since an array is stored contiguously in memory, each element has a unique index starting from “0” to the array’s size plus one. This is known as zero-based indexing.
During the declaration of an array, its name must be declared in accordance with the naming rules defined by C and its element type must also be declared in accordance with the identifier rules. This means that the array must have an integer value of size n and must have elements that are of the same data type.
A pointer to an array is a special type of memory object that stores the address of its first element in a contiguous block of memory. It is used to access the first element of an array, as well as to refer to the other elements in the array using their index numbers.
When an array is passed to a function, the size of the array can be specified in the form of a number or a variable-length expression. The number can be a positive or negative value, and can even be a decimal number, which is a rare exception.
Arrays are passed as return values
An array in C is an n-dimensional data structure where all elements are stored in a single contiguous memory location. These data elements can be accessed by referring to their index number inside square brackets .
An array is declared by defining a data type, such as int or float, followed by the name of the array. The size of the array must be specified at the time of declaration, and can only be changed after it is created.
Once an array is declared, the value of each element is uninitialized. This is because the value of each element is not set until it is referenced by an index to a unique identifier.
If you wish to change the values of an array in a function, you must use the & operator or declare the array as a static variable. This will ensure that the elements of the array do not become distributed when you modify them in the function.
Arrays are an important aspect of C programming, and they can be used to store and manipulate data in different ways. In particular, they are useful in storing and printing text, data from databases or user-inputted strings.
They can also be used to store large amounts of data, and to make it easy for the programmer to access them. An array can be passed as an argument to a function, or it can be returned as a return value.
The simplest way to pass an array as an argument is by sending the base address of the array, and the easiest way to return an array as a return value is by creating a user-defined data type that has a pointer to the array. The pointer is then stored in the variable, and the program can use it to access elements of the array.
Multidimensional arrays can also be passed to functions, but these are more complicated. In this case, you must specify a dimension for the array, and you can choose to pass it as an argument or return it as a return value.
Arrays are passed as user-defined data types
Arrays are an efficient data structure that can be used to store multiple values of a single type. They are usually used in cases where programmers need to keep track of large amounts of similar data.
In C, arrays are a group of data elements that are stored together in contiguous memory locations. Each element in an array is accessed by using its index number. The index starts from 0 and goes up to n-1 (where n is the length of the array).
Once the size of an array has been defined, it can be inserted into the computer’s memory. This can be done by creating a pointer to the array and assigning that pointer to the value of the desired variable. This is very useful when storing data on the computer’s memory because it allows you to access any of its elements at any time.
The array is also very convenient when you need to store different data types together. For example, you can put five integers in an array without having to declare each one as a separate variable. This can be extremely helpful when you need to store data of different types and sizes.
Moreover, it can be very useful in cases where you need to keep track of a large amount of information in a single variable. For example, if you need to keep track of student marks, it would be tedious to store each individual grade in a separate variable.
However, with an array, you can save a lot of space in your computer’s memory. All of the space in your array will be reserved until you insert new values into it.
Therefore, when you pass an array as a user-defined data type, you need to ensure that the size of the array is the same as the desired size. This is necessary because the array will be passed to the function as a pointer to its first element.
In C, arrays are a very common data structure that is used by many programmers. They are very useful in storing large amounts of data and can be very helpful when sorting or identifying variables. Fortunately, they are very easy to learn and use.
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Square brackets, also known as brackets, are a set of punctuation marks that are used in writing to add or alter information in a quote.
They are not used interchangeably with parentheses, which are another type of bracket.
Square brackets come in pairs, with an opening bracket [ and a closing bracket ]. They are often used to enclose words or phrases that are added to a quote to clarify or provide more information.
Here are some examples of how square brackets can be used in writing:
- To add missing words: “She [said] she would be here at 5 pm.”
- To clarify a pronoun: “John [Smith] said he would be here at 5 pm.”
- To correct a mistake: “He said he was born in 1990 .”
- To provide context: “The company [Microsoft] announced a new product today.”
Square brackets are also commonly used in academic writing to indicate edits or changes made to a quote. This is done to ensure that the original quote is not misrepresented or taken out of context.
The Purpose of Using Square Brackets in Writing
Square brackets, also known as brackets, are a type of punctuation mark that is used to enclose additional information within a sentence. They are used to clarify, modify, or add information to quoted material.
One of the main purposes of using square brackets in writing is to indicate that the words inside the brackets are not part of the original text. Square brackets are used to add information that is not present in the original text, such as clarifications, corrections, or explanations. They can also be used to replace a word or phrase in a quote to make it grammatically correct or to make it more understandable.
Another purpose of using square brackets is to indicate that a word or phrase has been changed or modified from the original text. For example, if you are quoting someone and they use a pronoun like “he” or “she,” but the gender is not clear from the context, you can use square brackets to replace the pronoun with the person’s name.
Square brackets can also be used to indicate that a word or phrase has been omitted from a quote. This is commonly done when a quote is too long or when a specific part of the quote is not relevant to the point being made.
Proper Usage of Square Brackets in Writing
When writing, it is crucial to use square brackets correctly to ensure that your text is clear and easy to understand. Here are some rules to follow when using square brackets in your writing:
- Correct usage: Square brackets are used to add information to quoted material, clarify nouns and pronouns in a quote that are unclear, translate foreign words or phrases in a quote, or enclose text added to a quote by someone other than the original speaker or writer.
- Formal writing: In formal writing, such as academic papers or professional documents, it is important to use square brackets correctly and sparingly. Overuse of square brackets can make your writing appear cluttered and confusing.
- Informal writing: In informal writing, such as emails or personal blogs, square brackets can be used more liberally to add additional information or clarify a point.
- APA style: If you are using APA style, square brackets are used to indicate changes or omissions in a quote. Use three-spaced periods in square brackets to indicate omitted words, and use square brackets to add words or changes to a quote.
Square Brackets in Formal Writing and Published Content
In formal writing and published content, square brackets are commonly used to add or clarify information within a quote or citation. This is particularly useful when the original text is unclear or when the writer wants to provide additional context for the reader.
One common use of square brackets in formal writing is to indicate editorial changes made to a quote. For example, if a quote contains a grammatical error or a typo, the writer can use square brackets to insert the correct word or letter. This helps to maintain the integrity of the original quote while also ensuring that the reader can understand it correctly.
Square brackets are also used in formal writing to provide additional information or context for the reader. For example, if a quote refers to a person or place that is unfamiliar to the reader, the writer can use square brackets to provide a brief explanation. This helps to ensure that the reader can fully understand the quote and its significance within the larger context of the text.
In published content, square brackets are commonly used to indicate changes or additions to a quote that have been made by the writer or editor. This is particularly important in academic writing, where accurate citations and references are essential.
By using square brackets to indicate changes or additions to a quote, the writer can ensure that their work is accurate and reliable.
Square Brackets in Quotations and Translations
There may be instances where you need to alter the text for clarity or to fit the context of your writing and this is where square brackets come in handy.
Square brackets can be used to add or replace words in a quotation without changing the meaning of the original text. This is particularly useful when you need to clarify a pronoun or add a word that was omitted from the original text. For example, if the original text reads “John said he was going to the store,” but you want to quote only the fact that John is going to the store, you can use square brackets to clarify: “[He] said he was going to the store.”
Square brackets can also be used in translations to indicate words that were added for clarity or omitted from the original text. This is particularly useful when translating from a language like Latin, where the word order and grammar can be quite different from English. For example, if you are translating a Latin text and need to add a word for clarity, you can use square brackets to indicate that the word was not in the original text: “The Latin text reads ‘veni, vidi, vici’ [I came, I saw, I conquered].'”
Using Square Brackets for Clarification
Square Brackets are often used to add additional information or to clarify a statement.
One common use of square brackets is to clarify a quotation by adding information that is not present in the original text. This can include explanations, definitions, or other clarifications that help the reader to understand the meaning of the quote.
For example, if you were quoting a passage from a book that used a pronoun that was unclear, you could use square brackets to clarify the identity of the pronoun. Similarly, if you were quoting a passage that used a technical term that might not be familiar to all readers, you could use square brackets to provide a definition of the term.
Square brackets can also be used to clarify the meaning of a word or phrase that might be ambiguous. For example, if you were writing about a product that had multiple meanings, you could use square brackets to clarify which meaning you were referring to.
Square Brackets in Informal Writing and Social Media
When it comes to informal writing and social media, the use of square brackets can vary depending on the context and platform. Here are some common scenarios where you may encounter square brackets in these settings:
Social Media Posts
In social media posts, square brackets can be used to clarify or add context to a message. For example, if you’re sharing a quote or excerpt from an article, you may use square brackets to indicate any changes or omissions you’ve made to the original text. This can help ensure that your message is clear and accurate.
In informal writing, such as personal emails or text messages, square brackets can be used to add additional information or clarify a point. For example, if you’re discussing a topic and want to provide some background information, you can use square brackets to indicate that this information is not part of the original message.
In comment sections on websites or social media platforms, square brackets can be used to add additional context or information to a discussion. For example, if you’re responding to a comment and want to provide some additional information or clarification, you can use square brackets to indicate that this information is not part of the original comment.
The use of square brackets in informal writing and social media can help ensure that your message is clear and accurate. However, always use them appropriately and not overuse them, as this can make your message difficult to read and understand.
Square Brackets vs. Round Brackets
What is the difference between square brackets and round brackets? In contrast to square brackets, round brackets are used to set off explanatory or supplementary information from surrounding text. They are used to provide non-essential information that may be useful but is tangential to the main meaning of the passage.
Can square brackets be used in place of parentheses? Yes, square brackets can be used in place of parentheses in certain situations. Take note that square brackets and parentheses have different uses and should not be used interchangeably.
Parentheses and square brackets are not the only types of brackets used in writing. Curly brackets, also known as braces, are often used in programming languages to indicate a block of code. Angle brackets, also known as chevrons, are commonly used in HTML coding to enclose tags.
Understanding the proper usage of brackets can help ensure that your writing is clear and accurate.
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In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The figure on the right illustrates the geometric relationship. Expressed algebraically:
where the Greek letter phi () represents the golden ratio. Its value is:
The golden ratio is also called the golden section (Latin: sectio aurea) or golden mean. Other names include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, and golden number.
Many 20th century artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing (see Applications and observations below). Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which can be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.
Two quantities a and b are said to be in the golden ratio φ if:
One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ,
it is shown that
Multiplying by φ gives
which can be rearranged to
Using the quadratic formula, two solutions are obtained:
Because of the fact that φ is the ratio between length and width of a rectangle, which are non-zero, the positive solution must be chosen:
The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years. According to Mario Livio:
Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.
Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. The Greeks usually attributed discovery of this concept to Pythagoras or his followers. The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol.
Euclid's Elements (Greek: Στοιχεῖα) provides the first known written definition of what is now called the golden ratio: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less." Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e., the golden ratio. Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio. Some of these propositions show that the golden ratio is an irrational number.
The name "extreme and mean ratio" was the principal term used from the 3rd century BCE until about the 18th century.
The modern history of the golden ratio starts with Luca Pacioli's De divina proportione of 1509, which captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio.
The first known approximation of the (inverse) golden ratio by a decimal fraction, stated as "about 0.6180340", was written in 1597 by Michael Maestlin of the University of Tübingen in a letter to his former student Johannes Kepler.
Since the 20th century, the golden ratio has been represented by the Greek letter Φ or φ ( phi, after Phidias, a sculptor who is said to have employed it) or less commonly by τ ( tau, the first letter of the ancient Greek root τομή—meaning cut).
Timeline according to Priya Hemenway:
- Phidias (490–430 BC) made the Parthenon statues that seem to embody the golden ratio.
- Plato (427–347 BC), in his Timaeus, describes five possible regular solids (the Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron), some of which are related to the golden ratio.
- Euclid (c. 325–c. 265 BC), in his Elements, gave the first recorded definition of the golden ratio, which he called, as translated into English, "extreme and mean ratio" (Greek: ἄκρος καὶ μέσος λόγος).
- Fibonacci (1170–1250) mentioned the numerical series now named after him in his Liber Abaci; the ratio of sequential elements of the Fibonacci sequence approaches the golden ratio asymptotically.
- Luca Pacioli (1445–1517) defines the golden ratio as the "divine proportion" in his Divina Proportione.
- Michael Maestlin (1550–1631) publishes the first known approximation of the (inverse) golden ratio as a decimal fraction.
- Johannes Kepler (1571–1630) proves that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers, and describes the golden ratio as a "precious jewel": "Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel." These two treasures are combined in the Kepler triangle.
- Charles Bonnet (1720–1793) points out that in the spiral phyllotaxis of plants going clockwise and counter-clockwise were frequently two successive Fibonacci series.
- Martin Ohm (1792–1872) is believed to be the first to use the term goldener Schnitt (golden section) to describe this ratio, in 1835.
- Édouard Lucas (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.
- Mark Barr (20th century) suggests the Greek letter phi (φ), the initial letter of Greek sculptor Phidias's name, as a symbol for the golden ratio.
- Roger Penrose (b.1931) discovered a symmetrical pattern that uses the golden ratio in the field of aperiodic tilings, which led to new discoveries about quasicrystals.
Applications and observations
De Divina Proportione, a three-volume work by Luca Pacioli, was published in 1509. Pacioli, a Franciscan friar, was known mostly as a mathematician, but he was also trained and keenly interested in art. De Divina Proportione explored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. Containing illustrations of regular solids by Leonardo da Vinci, Pacioli's longtime friend and collaborator, De Divina Proportione was a major influence on generations of artists and architects alike.
The Parthenon's façade as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles. Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Midhat J. Gazalé says, "It was not until Euclid, however, that the golden ratio's mathematical properties were studied. In the Elements (308 BC) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme ratios. Its occurrence in regular pentagons and decagons was duly observed, as well as in the dodecahedron (a regular polyhedron whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties." And Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 BC, showed how to calculate its value." Near-contemporary sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.
A 2004 geometrical analysis of earlier research into the Great Mosque of Kairouan reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz. They found ratios close to the golden ratio in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret. The authors note, however, that the areas where ratios close to the golden ratio were found are not part of the original construction, and theorize that these elements were added in a reconstruction.
The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."
Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's " Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.
Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.
In a recent book, author Jason Elliot speculated that the golden ratio was used by the designers of the Naqsh-e Jahan Square and the adjacent Lotfollah mosque.
The 16th-century philosopher Heinrich Agrippa drew a man over a pentagram inside a circle, implying a relationship to the golden ratio.
Leonardo da Vinci's illustrations of polyhedra in De divina proportione (On the Divine Proportion) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by anything in Leonardo's own writings. Similarly, although the Vitruvian Man is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.
Salvador Dalí, influenced by the works of Matila Ghyka, explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.
Mondrian has been said to have used the golden section extensively in his geometrical paintings, though other experts (including critic Yve-Alain Bois) have disputed this claim.
A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini). On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and root-5 proportions, and others with proportions like root-2, 3, 4, and 6.
According to Jan Tschichold,
There was a time when deviations from the truly beautiful page proportions 2:3, 1:√3, and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.
Some sources claim that the golden ratio is commonly used in everyday design, for example in the shapes of postcards, playing cards, posters, wide-screen televisions, photographs, and light switch plates.
Ernő Lendvaï analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis. In Bartok's Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau (Reflections in Water), from Images (1st series, 1905), in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position."
The musicologist Roy Howat has observed that the formal boundaries of La Mer correspond exactly to the golden section. Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.
Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent on this innovation.
Though Heinz Bohlen proposed the non-octave-repeating 833 cents scale based on combination tones, the tuning features relations based on the golden ratio. As a musical interval the ratio 1.618... is 833.090... cents (Play).
Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law. In connection with his scheme for golden-ratio-based human body proportions, Zeising wrote in 1854 of a universal law "in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form."
In 2010, the journal Science reported that the golden ratio is present at the atomic scale in the magnetic resonance of spins in cobalt niobate crystals.
Several researchers have proposed connections between the golden ratio and human genome DNA.
However, some have argued that many of the apparent manifestations of the golden mean in nature, especially in regard to animal dimensions, are in fact fictitious.
The golden ratio is key to the golden section search.
Studies by psychologists, starting with Fechner, have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.
Golden ratio conjugate
The negative root of the quadratic equation for φ (the "conjugate root") is
The absolute value of this quantity (≈ 0.618) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, b/a), and is sometimes referred to as the golden ratio conjugate. It is denoted here by the capital Phi (Φ):
Alternatively, Φ can be expressed as
This illustrates the unique property of the golden ratio among positive numbers, that
or its inverse:
This means 0.61803...:1 = 1:1.61803....
Short proofs of irrationality
Contradiction from an expression in lowest terms
- the whole is the longer part plus the shorter part;
- the whole is to the longer part as the longer part is to the shorter part.
If we call the whole n and the longer part m, then the second statement above becomes
- n is to m as m is to n − m,
To say that φ is rational means that φ is a fraction n/m where n and m are integers. We may take n/m to be in lowest terms and n and m to be positive. But if n/m is in lowest terms, then the identity labeled (*) above says m/(n − m) is in still lower terms. That is a contradiction that follows from the assumption that φ is rational.
Derivation from irrationality of √5
Another short proof—perhaps more commonly known—of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If is rational, then is also rational, which is a contradiction if it is already known that the square root of a non- square natural number is irrational.
The formula φ = 1 + 1/φ can be expanded recursively to obtain a continued fraction for the golden ratio:
and its reciprocal:
The convergents of these continued fractions (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ..., or 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) are ratios of successive Fibonacci numbers.
The equation φ2 = 1 + φ likewise produces the continued square root, or infinite surd, form:
An infinite series can be derived to express phi:
These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side, and similar relations in a pentagram.
The number φ turns up frequently in geometry, particularly in figures with pentagonal symmetry. The length of a regular pentagon's diagonal is φ times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles.
There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem). However, a useful approximation results from dividing the sphere into parallel bands of equal area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/φ ≅ 222.5°. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.
Dividing a line segment
- Having a line segment AB, construct a perpendicular BC at point B, with BC half the length of AB. Draw the hypotenuse AC.
- Draw a circle with centre C and radius BC. This circle intersects the hypotenuse AC at point D.
- Draw a circle with centre A and radius AD. This circle intersects the original line segment AB at point S. Point S divides the original segment AB into line segments AS and SB with lengths in the golden ratio.
Golden triangle, pentagon and pentagram
If angle BCX = α, then XCA = α because of the bisection, and CAB = α because of the similar triangles; ABC = 2α from the original isosceles symmetry, and BXC = 2α by similarity. The angles in a triangle add up to 180°, so 5α = 180, giving α = 36°. So the angles of the golden triangle are thus 36°-72°-72°. The angles of the remaining obtuse isosceles triangle AXC (sometimes called the golden gnomon) are 36°-36°-108°.
Suppose XB has length 1, and we call BC length φ. Because of the isosceles triangles XC=XA and BC=XC, so these are also length φ. Length AC = AB, therefore equals φ + 1. But triangle ABC is similar to triangle CXB, so AC/BC = BC/BX, and so AC also equals φ2. Thus φ2 = φ + 1, confirming that φ is indeed the golden ratio.
Similarly, the ratio of the area of the larger triangle AXC to the smaller CXB is equal to φ, while the inverse ratio is φ − 1.
In a regular pentagon the ratio between a side and a diagonal is (i.e. 1/φ), while intersecting diagonals section each other in the golden ratio.
George Odom has given a remarkably simple construction for φ involving an equilateral triangle: if an equilateral triangle is inscribed in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of two points, then these three points are in golden proportion. This result is a straightforward consequence of the intersecting chords theorem and can be used to construct a regular pentagon, a construction that attracted the attention of the noted Canadian geometer H. S. M. Coxeter who published it in Odom's name as a diagram in the American Mathematical Monthly accompanied by the single word "Behold!"
The golden ratio plays an important role in the geometry of pentagrams. Each intersection of edges sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's center) is φ, as the four-colour illustration shows.
The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomons.
The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are b, and short edges are a, then Ptolemy's theorem gives b2 = a2 + ab which yields
Scalenity of triangles
Consider a triangle with sides of lengths a, b, and c in decreasing order. Define the "scalenity" of the triangle to be the smaller of the two ratios a/b and b/c. The scalenity is always less than φ and can be made as close as desired to φ.
Triangle whose sides form a geometric progression
If the side lengths of a triangle form a geometric progression and are in the ratio 1 : r : r2, where r is the common ratio, then r must lie in the range φ−1 < r < φ, which is a consequence of the triangle inequality (the sum of any two sides of a triangle must be strictly bigger than the length of the third side). If r = φ then the shorter two sides are 1 and φ but their sum is φ2, thus r < φ. A similar calculation shows that r > φ−1. A triangle whose sides are in the ratio 1 : √φ : φ is a right triangle (because 1 + φ = φ2) known as a Kepler triangle.
Golden triangle, rhombus, and rhombic triacontahedron
A golden rhombus is a rhombus whose diagonals are in the golden ratio. The rhombic triacontahedron is a convex polytope that has a very special property: all of its faces are golden rhombi. In the rhombic triacontahedron the dihedral angle between any two adjacent rhombi is 144°, which is twice the isosceles angle of a golden triangle and four times its most acute angle.
Relationship to Fibonacci sequence
The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is:
- 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ....
The closed-form expression (known as Binet's formula, even though it was already known by Abraham de Moivre) for the Fibonacci sequence involves the golden ratio:
The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as originally shown by Kepler:
Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase, and:
where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when .
Furthermore, the successive powers of φ obey the Fibonacci recurrence:
This identity allows any polynomial in φ to be reduced to a linear expression. For example:
However, this is no special property of φ, because polynomials in any solution x to a quadratic equation can be reduced in an analogous manner, by applying:
for given coefficients a, b such that x satisfies the equation. Even more generally, any rational function (with rational coefficients) of the root of an irreducible nth-degree polynomial over the rationals can be reduced to a polynomial of degree n ‒ 1. Phrased in terms of field theory, if α is a root of an irreducible nth-degree polynomial, then has degree n over , with basis .
The golden ratio and inverse golden ratio have a set of symmetries that preserve and interrelate them. They are both preserved by the fractional linear transformations – this fact corresponds to the identity and the definition quadratic equation. Further, they are interchanged by the three maps – they are reciprocals, symmetric about , and (projectively) symmetric about 2.
More deeply, these maps form a subgroup of the modular group isomorphic to the symmetric group on 3 letters, corresponding to the stabilizer of the set of 3 standard points on the projective line, and the symmetries correspond to the quotient map – the subgroup consisting of the 3-cycles and the identity fixes the two numbers, while the 2-cycles interchange these, thus realizing the map.
The golden ratio has the simplest expression (and slowest convergence) as a continued fraction expansion of any irrational number (see Alternate forms above). It is, for that reason, one of the worst cases of Lagrange's approximation theorem and it is an extremal case of the Hurwitz inequality for Diophantine approximations. This may be why angles close to the golden ratio often show up in phyllotaxis (the growth of plants).
The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part in common with φ:
The sequence of powers of φ contains these values 0.618..., 1.0, 1.618..., 2.618...; more generally, any power of φ is equal to the sum of the two immediately preceding powers:
As a result, one can easily decompose any power of φ into a multiple of φ and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of φ:
If , then:
When the golden ratio is used as the base of a numeral system (see Golden ratio base, sometimes dubbed phinary or φ-nary), every integer has a terminating representation, despite φ being irrational, but every fraction has a non-terminating representation.
The golden ratio is a fundamental unit of the algebraic number field and is a Pisot–Vijayaraghavan number. In the field we have , where is the -th Lucas number.
The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is 4 ln φ.
The golden ratio's decimal expansion can be calculated directly from the expression
with √5 ≈ 2.2360679774997896964. The square root of 5 can be calculated with the Babylonian method, starting with an initial estimate such as xφ = 2 and iterating
for n = 1, 2, 3, ..., until the difference between xn and xn−1 becomes zero, to the desired number of digits.
The Babylonian algorithm for √5 is equivalent to Newton's method for solving the equation x2 − 5 = 0. In its more general form, Newton's method can be applied directly to any algebraic equation, including the equation x2 − x − 1 = 0 that defines the golden ratio. This gives an iteration that converges to the golden ratio itself,
for an appropriate initial estimate xφ such as xφ = 1. A slightly faster method is to rewrite the equation as x − 1 − 1/x = 0, in which case the Newton iteration becomes
These iterations all converge quadratically; that is, each step roughly doubles the number of correct digits. The golden ratio is therefore relatively easy to compute with arbitrary precision. The time needed to compute n digits of the golden ratio is proportional to the time needed to divide two n-digit numbers. This is considerably faster than known algorithms for the transcendental numbers π and e.
An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers F 25001 and F 25000, each over 5000 digits, yields over 10,000 significant digits of the golden ratio.
The golden ratio φ has been calculated to an accuracy of several millions of decimal digits (sequence A001622 in OEIS). Alexis Irlande performed computations and verification of the first 17,000,000,000 digits.
Both Egyptian pyramids and those mathematical regular square pyramids that resemble them can be analyzed with respect to the golden ratio and other ratios.
Mathematical pyramids and triangles
A pyramid in which the apothem (slant height along the bisector of a face) is equal to φ times the semi-base (half the base width) is sometimes called a golden pyramid. The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semi-base by apothem), joining the medium-length edges to make the apothem. The height of this pyramid is times the semi-base (that is, the slope of the face is ); the square of the height is equal to the area of a face, φ times the square of the semi-base.
The medial right triangle of this "golden" pyramid (see diagram), with sides is interesting in its own right, demonstrating via the Pythagorean theorem the relationship or . This " Kepler triangle" is the only right triangle proportion with edge lengths in geometric progression, just as the 3–4–5 triangle is the only right triangle proportion with edge lengths in arithmetic progression. The angle with tangent corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827... degrees (51° 49' 38").
A nearly similar pyramid shape, but with rational proportions, is described in the Rhind Mathematical Papyrus (the source of a large part of modern knowledge of ancient Egyptian mathematics), based on the 3:4:5 triangle; the face slope corresponding to the angle with tangent 4/3 is 53.13 degrees (53 degrees and 8 minutes). The slant height or apothem is 5/3 or 1.666... times the semi-base. The Rhind papyrus has another pyramid problem as well, again with rational slope (expressed as run over rise). Egyptian mathematics did not include the notion of irrational numbers, and the rational inverse slope (run/rise, multiplied by a factor of 7 to convert to their conventional units of palms per cubit) was used in the building of pyramids.
Another mathematical pyramid with proportions almost identical to the "golden" one is the one with perimeter equal to 2π times the height, or h:b = 4:π. This triangle has a face angle of 51.854° (51°51'), very close to the 51.827° of the Kepler triangle. This pyramid relationship corresponds to the coincidental relationship .
Egyptian pyramids very close in proportion to these mathematical pyramids are known.
In the mid-nineteenth century, Röber studied various Egyptian pyramids including Khafre, Menkaure and some of the Giza, Sakkara, and Abusir groups, and was interpreted as saying that half the base of the side of the pyramid is the middle mean of the side, forming what other authors identified as the Kepler triangle; many other mathematical theories of the shape of the pyramids have also been explored.
One Egyptian pyramid is remarkably close to a "golden pyramid"—the Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu). Its slope of 51° 52' is extremely close to the "golden" pyramid inclination of 51° 50' and the π-based pyramid inclination of 51° 51'; other pyramids at Giza (Chephren, 52° 20', and Mycerinus, 50° 47') are also quite close. Whether the relationship to the golden ratio in these pyramids is by design or by accident remains open to speculation. Several other Egyptian pyramids are very close to the rational 3:4:5 shape.
Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Temple Bell, mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem, nor any way to reason about irrationals such as π or φ.
Michael Rice asserts that principal authorities on the history of Egyptian architecture have argued that the Egyptians were well acquainted with the golden ratio and that it is part of mathematics of the Pyramids, citing Giedon (1957). Historians of science have always debated whether the Egyptians had any such knowledge or not, contending rather that its appearance in an Egyptian building is the result of chance.
In 1859, the pyramidologist John Taylor claimed that, in the Great Pyramid of Giza, the golden ratio is represented by the ratio of the length of the face (the slope height), inclined at an angle θ to the ground, to half the length of the side of the square base, equivalent to the secant of the angle θ. The above two lengths were about 186.4 and 115.2 meters respectively. The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. Similarly, Howard Vyse, according to Matila Ghyka, reported the great pyramid height 148.2 m, and half-base 116.4 m, yielding 1.6189 for the ratio of slant height to half-base, again more accurate than the data variability.
Examples of disputed observations of the golden ratio include the following:
- Historian John Man states that the pages of the Gutenberg Bible were "based on the golden section shape". However, according to Man's own measurements, the ratio of height to width was 1.45.
- Some specific proportions in the bodies of many animals (including humans) and parts of the shells of mollusks and cephalopods are often claimed to be in the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio. The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio. The nautilus shell, the construction of which proceeds in a logarithmic spiral, is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is proportioned by the golden ratio relative to the previous one; however, measurements of nautilus shells do not support this claim.
- The proportions of different plant components (numbers of leaves to branches, diameters of geometrical figures inside flowers) are often claimed to show the golden ratio proportion in several species. In practice, there are significant variations between individuals, seasonal variations, and age variations in these species. While the golden ratio may be found in some proportions in some individuals at particular times in their life cycles, there is no consistent ratio in their proportions.
- In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio. The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. Elliott wave principle and Fibonacci retracement). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.
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the dots in a scatter plot not only report the values of individual data points, but also patterns when the data are taken as a whole. in order to create a scatter plot, we need to select two columns from a data table, one for each dimension of the plot. it can be difficult to tell how densely-packed data points are when many of them are in a small area. a common modification of the basic scatter plot is the addition of a third variable.
rather than using distinct colors for points like in the categorical case, we want to use a continuous sequence of colors, so that, for example, darker colors indicate higher value. as noted above, a heatmap can be a good alternative to the scatter plot when there are a lot of data points that need to be plotted and their density causes overplotting issues. the scatter plot is one of many different chart types that can be used for visualizing data. violin plots are used to compare the distribution of data between groups.
scatter plot format
a scatter plot sample is a type of document that creates a copy of itself when you open it. The doc or excel template has all of the design and format of the scatter plot sample, such as logos and tables, but you can modify content without altering the original style. When designing scatter plot form, you may add related information such as scatter plot examples,scatter plot interpretation,scatter plot in research,scatter plot in data mining,scatter plot python
when designing scatter plot example, it is important to consider related questions or ideas, what is the scatter plot used for? how do you interpret a scatter plot? what are the 3 types of scatter plots? what does a scatter plot reveal?, scatter plot maker,examples of when to use a scatter plot,how to describe a scatter plot,what is a scatter plot and how does it help us,interpreting scatter plots examples
when designing the scatter plot document, it is also essential to consider the different formats such as Word, pdf, Excel, ppt, doc etc, you may also add related information such as scatter plot graph excel,scatter plot correlation,scatter plot notes,how to make a scatter plot
scatter plot guide
the data are displayed as a collection of points, each having the value of one variable determining the position on the horizontal axis and the value of the other variable determining the position on the vertical axis. a scatter plot can be used either when one continuous variable is under the control of the experimenter and the other depends on it or when both continuous variables are independent. if no dependent variable exists, either type of variable can be plotted on either axis and a scatter plot will illustrate only the degree of correlation (not causation) between two variables. if the pattern of dots slopes from upper left to lower right, it indicates a negative correlation. a line of best fit (alternatively called ‘trendline’) can be drawn to study the relationship between the variables.
the ability to do this can be enhanced by adding a smooth line such as loess. the researcher would then plot the data in a scatter plot, assigning “lung capacity” to the horizontal axis, and “time holding breath” to the vertical axis. the scatter plot of all the people in the study would enable the researcher to obtain a visual comparison of the two variables in the data set and will help to determine what kind of relationship there might be between the two variables. a plot located on the intersection of row and jth column is a plot of variables xi versus xj. a generalized scatter plot matrix offers a range of displays of paired combinations of categorical and quantitative variables.
scatter plots are the graphs that present the relationship between two variables in a data-set. the independent variable or attribute is plotted on the x-axis, while the dependent variable is plotted on the y-axis. the scatter diagram graphs numerical data pairs, with one variable on each axis, show their relationship. the line drawn in a scatter plot, which is near to almost all the points in the plot is known as “line of best fit” or “trend line“. we know that the correlation is a statistical measure of the relationship between the two variables’ relative movements. if the variables are correlated, the points will fall along a line or curve.
the scatter plot explains the correlation between two attributes or variables. there can be three such situations to see the relation between the two variables – when the points in the graph are rising, moving from left to right, then the scatter plot shows a positive correlation. it means the values of one variable are increasing with respect to another. it means the values of one variable are decreasing with respect to another. for data variables such as x1, x2, x3, and xn, the scatter plot matrix presents all the pairwise scatter plots of the variables on a single illustration with various scatterplots in a matrix format. a plot of variables xi vs xj will be located at the ith row and jth column intersection.
could you attach a small sample workbook demonstrating the problem (without sensitive data), or if that is not possible, make it available through onedrive, google drive, dropbox or similar? thanks! there are empty cells between the data points of each series. by default, these are displayed as gaps in the chart. to change this: the data points for a chart series are usually in adjacent cells. if there is an empty cell in between, it often means that there are missing data. but here, there are many empty cells between the data points because of the layout of the source range. so that’s different. the data points for a chart series are usually in adjacent cells. if there is an empty cell in between, it often means that there are missing data. but here, there are many empty cells between the data points because of the layout of the source range. so that’s different.
start with a free account to explore 20+ always-free courses and hundreds of finance templates and cheat sheets. a scatter plot is a chart type that is normally used to observe and visually display the relationship between variables. the positioning of the dots on the vertical and horizontal axis will inform the value of the respective data point; hence, scatter plots make use of cartesian coordinates to display the values of the variables in a data set. the most common use of the scatter plot is to display the relationship between two variables and observe the nature of the relationship. another common use of scatter plots is that they enable the identification of correlational relationships. scatter plots tend to have independent variables on the horizontal axis and dependent variables on the vertical axis.
data points can be grouped together based on how close their values are, and this also makes it easy to identify any outlier points when there are data gaps. seeing as scatter plots aid in the identification of correlations between variables, the nature of the correlations can also be estimated based on a specific confidence level. linear regression is part of the best-fit framework and is used for linear correlations. two common issues have been identified with the use of scatter plots – overplotting and the interpretation of causation as correlation. concerning correlation, it is important to remember that correlation does not mean that the changes observed in one variable are responsible for the changes observed in another variable. causation implies that an event occurring will have an impact on an outcome. gain unlimited access to more than 250 productivity templates, cfi’s full course catalog and accredited certification programs, hundreds of resources, expert reviews and support, the chance to work with real-world finance and research tools, and more.
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One of the most critical parts of an airplane is, obviously, its wings. The wings primarily create the lift to overcome the aircraft’s weight. However, they also have movable surfaces, such as ailerons, flaps, spoilers, trim tabs, etc., to develop additional aerodynamic forces to control the aircraft during its flight. Wings are geometrically defined in terms of their span (distance from wing tip to wing tip), planform, twist (pitch angle) distribution, and cross-section (i.e., airfoil section shape or profile shape).
The shape of a wing must be engineered to give good aerodynamic efficiency in lift production for the minimum amount of drag, i.e., the maximization of the lift-to-drag ratio, which is one fundamental goal in aerodynamic design. However, there will always be other aerodynamic requirements that will factor into the wing shape design, including its low-speed flight and stalling characteristics.
In addition, the wing structure must be strong enough and stiff enough to carry all of the aerodynamic and other loads acting on the wing, so the wing must be tailored to meet all structural requirements. Examples include:
- Minimizing the loads and deformations of the wing under the action of the aerodynamic forces and moments.
- Avoiding the onset of adverse aeroelastic effects and flutter.
- Carrying undercarriage loads or other point loads such as engines and various inertial loads.
The wings carry the entire weight of the aircraft, so large shear loads and bending moments are produced near the root of the wing. To this end, the wing shape typically needs to be much larger in chord and thicker in cross-section than at the wing tips to obtain the required structural strength and stiffness.
- Know the critical geometric parameters used to define the shape of a wing.
- Calculate the wing area and aspect ratio of an arbitrary wing planform.
- Understand the significance and use of mean wing chords.
Geometric Definition of a Wing
Engineers use various geometric parameters to describe the shapes of wings. Terms such as span, chord, mean chord, aspect ratio, and sweep angle are used routinely in wing design, and it is essential to understand what these terms mean. Other terms used in wing design include the wing’s twist or washout, the taper ratio, the dihedral or anhedral, and the thickness-to-chord ratio.
Wing Span & Semi-Span
The span of the wing or wing span, given the symbol , is defined as the distance from one wing tip to the other (for now, the effect of a winglet will not be considered), as shown in the figure below. Sometimes, the semi-span is used to define the wing in engineering analysis, given the symbol and equal to . Notice that the symbol means the centerline of the wing or aircraft.
Wing Chord and Planform
The wing chord is the distance from its leading edge to its trailing edge in the streamwise direction, i.e., parallel to the airplane’s longitudinal axis. The chord is given the symbol . On many airplanes, the chord changes along the wing’s span, i.e., as in the above figure, mainly for aerodynamic reasons. A primary aerodynamic goal for the wing is minimizing drag for a given amount of lift, i.e., maximizing the lift-to-drag ratio.
The shape of the wing is defined in terms of the chord distribution along the span of the wing, and when the wing is viewed from above, the resulting shape is called the planform. If is measured from the longitudinal centerline of the aircraft (i.e., not the wing root), as shown in the figure below, then the local value of the wing chord can be expressed as
where at the centerline of the aircraft and at the wing tip. The wing chord can also be expressed in terms of the non-dimensional span, i.e.,
where at the wing centerline and at the wing tip.
Wings are often linearly tapered in planform, for good engineering reasons, with different values of the root chord and the tip chord . In this case, the linear taper ratio of the wing is defined as
For a linear form of wing taper, as shown in the figure above, the chord distribution along the wing will be
The taper ratio may also be used in cases where the wing is not precisely linearly tapered to quantify the average taper of the wing planform.
Wings may not only taper in planform but also in thickness or, indeed, as combinations of taper and thickness, as shown in the figure below. Using both taper and thickness together gives considerable engineering latitude in tailoring the shape of the wing to meet a given level of aerodynamic performance and minimizes structural loads and weight. A further aerodynamic advantage may be gained by using different airfoil sections along the span, e.g., using a relatively thin airfoil section at the wing tip for low drag where the structural loads are lower.
The use of inverse wing taper is unusual, i.e., the chord increases toward the wing tip. However, it has been used to address a problem of adverse stall characteristics and a tendency to spin, a common issue with the first generation of jet aircraft with swept wings. The sweepback of the wing encourages a spanwise flow, making the wing tips more likely to stall first, also reducing the effectiveness of the ailerons. However, this latter approach was unsuccessful, and other (and simpler) methods were more effective in mitigating the stall problem with swept wings.
Sweepback of a Wing
Today, many airplanes have wings with sweepback, as shown in the figure below, but many lower-performance airplanes will have no sweepback. The primary aerodynamic purpose of sweepback on a wing is to delay the onset of compressibility effects and the build-up of wave drag to a higher flight Mach number and/or to reduce drag at a given Mach number, decreasing the propulsive thrust and fuel required for flight.
Aerodynamically, the flight Mach number component perpendicular to the wing’s leading edge primarily affects the lift and drag, assuming that the Mach number parallel to the leading edge makes no contribution, often referred to as the independence principle. Thus, for aerodynamic analysis, the free-stream Mach number, that is, the flight Mach number of the aircraft, is resolved into components normal and parallel to the wing’s leading edge based on the local sweep angle. Wings can also be swept forward to obtain the same effect. However, a problem with a forward-swept wing is that it is aeroelastically unstable and tends to twist nose-up under the action of aerodynamic loads, so forward sweep is rarely used.
Aircraft designed for sustained supersonic flight inevitably have much higher sweepback angles than subsonic airplanes; the best planform shape for supersonic flight approaches a classic “Delta” shape. However, the use of sweepback can have other effects on the aerodynamics of the wing and the airplane, including adverse stall and low-speed handling characteristics, so usually, as little as possible sweepback is used. Sweepback also tends to increase the lateral stability of the airplane. Some aircraft may have variable sweepback to optimize the flight aerodynamics, such as the B-1 bomber. Still, there is a significant structural weight penalty with such “swing-wing” designs, which will be at the expense of some useful load, i.e., fuel and payload.
The sweepback angle is often defined by the angle made by the location of the 1/4-chord points along the span of the wing, i.e., or it may be alternatively defined by the leading edge and trailing edge angles, i.e., by the values and , respectively. Wings may also be designed in parts with two different sweepback angles: one sweepback angle for the inboard wing panel (usually the smaller angle) and another (larger) angle for the outboard wing panel.
Wing Twist or Washout
Wings may also be slightly twisted along their span; the primary purpose of wing twist is to help give the desired distribution of aerodynamic forces over the span. In practice, most wings are twisted in some form, often subtly. It is known from aerodynamic theory and practice that the spanwise form of the lift distribution is critically important for minimizing induced drag. While the spanwise lift distribution is strongly affected by wing planform (i.e., the wing chord distribution), the additional use of wing twist can help tailor the wing lift distribution to obtain the desired aerodynamic effects.
Most wings are twisted nose-down from root to tip, i.e., the pitch angles change from wing section to section and become increasingly negative toward the wing tip. This effect is called “washing” the wing twist, and so is called washout. Typical washout values on a wing are between 0 and 7 degrees nose-down, anything more than this being somewhat unusual.
Although very uncommon for an airplane, if a wing is twisted nose-up along its span by increasing the wing twist from root to tip, it is called washin. Interestingly, some helicopter blades use both washout and washin, the washin component being used over the blade tip region to keep the tips from producing negative lift at higher forward airspeeds. Aerodynamic twist can also be incorporated into the wing design by changing the shape of the airfoil section, i.e., the angle of attack at which the section produces zero lift, which manifests similarly to what would be obtained by changing the pitch angle of the wing.
Wings can use various types and distributions of airfoil sections to suit the application of the aircraft. By cutting a slice out of an airplane wing and viewing it from the side, the wing cross-section is obtained, or what is usually called just the airfoil section or airfoil profile, with an example shown below. It is possible to describe the shape of airfoil sections by using camber, thickness, and nose radius as primary geometric parameters. Outside of the U.S., wing sections are usually called “aerofoils.”
In the evolution of wings, airfoil sections have progressed from simple curved plate-like shapes with little thickness inspired by birds’ wings to sophisticated shapes with camber and thickness to give high lift and low drag. Airfoils used on subsonic airplanes are usually relatively thicker and have camber, as shown in the figure below. Whereas for high-speed or supersonic aircraft, the airfoils are thinner with a small leading-edge radius and have slight camber. Notice that the term “thickness” usually means the section’s maximum thickness-to-chord ratio, so it measures how thick the wing section is relative to its chord, i.e., thickness/chord, which is usually quoted in percent. Therefore, a thickness-to-chord ratio of 0.12 would mean an airfoil that was 12% thick.
Most wings use different airfoils along their span, which can be progressively blended together to give an overall wing design that is better than could be obtained using a single airfoil. This latter approach is often necessary with large commercial aircraft. As shown in the figure below, the need for significant thickness to carry the bending and shear loads at the wing’s root makes the airfoil design for low drag at higher flight Mach numbers rather challenging, especially when the wing operates in transonic flow. The designer can use thinner airfoils better suited to high-speed flight and transonic flow conditions toward the wing tips, where the bending moments and structural stresses are lower. Even on low-performance airplanes, there can be significant aerodynamic and performance advantages in using different airfoils at the wing root compared to the wing tip sections.
Airfoil sections may also be used to introduce an aerodynamic twist along the wing span. Different cambered airfoils inevitably have different zero-lift angles of attack (although they may differ only by a degree or two at most), so other airfoils with different camber can be used to effectively twist the wing aerodynamically. The effects obtained are usually combined with a geometric twist to achieve the desired spanwise lift distribution to meet aerodynamic performance and other goals.
It is also found that using wing twist is particularly helpful in controlling the stall developments on the wing, especially in preventing the wing tips from stalling, allowing the designer to have some latitude in satisfying the low-speed handling qualities of the airplane. It is not unusual for newly designed airplanes to have the airfoils over the outer wing panels changed after their first flights to meet stalling and handling qualities requirements needed for certification and the award of a certificate of airworthiness.
Dihedral & Anhedral
The dihedral angle is the upward angle that the wing panels make relative to a reference axis for the aircraft, as shown in the figure below. The primary purpose of using dihedral is to improve the aircraft’s lateral (roll) stability; usually, only a few degrees are needed to enhance stability significantly. The horizontal tail may also have some dihedral, especially on larger aircraft, contributing somewhat to the lateral stability. Some amount of dihedral will come from wing-bending structural displacements.
An example of a Boeing 737 is shown in the photograph below, where it can be seen that both the main wing and horizontal tail have a notable amount of dihedral. Good stability about all three flight axes is desirable for most aircraft, especially an airliner, so the passengers experience a smooth and comfortable ride, especially through turbulence.
A downward wing angle is called anhedral and is somewhat less common to find on airplanes without wing sweepback or a high wing design because it decreases roll stability. However, airplanes with swept wings may use anhedral to offset the increase in roll stability from using sweepback. Airplanes with high-mounted wings also tend to have significant pendular lateral stability because the center of gravity lies below the center of the lift, as shown in the figure below. In this case, anhedral on the wing is needed so that the lateral stability is not too strong to make the aircraft difficult to turn and maneuver.
A good example of a wing with anhedral is found on the C-5 Galaxy military transport aircraft, as shown in the photograph below. The C-5 needed to have the loading deck as close to the ground as possible, so the only design choice for this aircraft was a high wing. While lateral stability (and stability, in general) is generally good, too much stability can make the aircraft less maneuverable and agile, and overall handling qualities can suffer. Sweepback, which is used to delay the onset of compressibility effects, can also introduce certain undesirable flight dynamic characteristics, which can be offset using anhedral.
Calculation of Wing Area
Several derived geometric characteristics of wings, including the wing area, are important in engineering analysis. The planform area of the wing, which is given the symbol , is obtained by integrating the distribution of wing chord along the span from one wing tip to the other, i.e.,
where , as shown in the figure below.
If both wings are of the same geometry, i.e., symmetrically disposed with respect to the longitudinal axis along the fuselage, and so are mirror images of each other (which is the case on nearly all airplanes), then
It is important not to confuse the wing area (capital ““) with the semi-span of the wing (lowercase ““). Of course, there can be a dilemma here, and a question to ask is whether the wing area is the actual area of the wing exposed to the airflow or something else. In the above equation, the former has been assumed, and the value of the area is called the planform or wing reference area. However, there can be circumstances when the actual area of the wing exposed to the flow needs to be known, called the wetted area, in which case the lower limit of integration would be adjusted accordingly to start from the side of the fuselage. Therefore, it is essential to verify the actual definition(s) of the wing area used in different types of engineering analysis.
Calculation of Wing Aspect Ratio
The aspect ratio of the wing, which is given the symbol , is defined as the ratio of the square of the wing span to the wing reference area, i.e.,
The aspect ratio of a wing is vital in aerodynamic analysis because a wing with a higher aspect ratio is generally more aerodynamically efficient and will have lower drag.
Every aircraft design will have a somewhat different wing aspect ratio. However, typical values range from 5 to 10 for a small general aviation aircraft to 9 to 15 for a commercial transport aircraft to 30 and higher for a glider (sailplane). As a point of reference, the Voyager aircraft that in 1986 flew around the World without refueling had a wing aspect ratio of 34, which is unusually high for an aircraft other than a sailplane.
It will be apparent that the aspect ratio of a wing is a physical measure of the geometric slenderness of the wing. The easiest way to understand this is to assume a wing of a constant chord, i.e., constant. In this case, then
which is just the ratio of the wing span to the chord. Therefore, a wing with a higher span and a narrower chord will have a higher aspect ratio. Hence, the numerical value of the aspect ratio becomes a measure of the slenderness of the wing.
However, the aspect ratio must be calculated using the exact wing planform; most wings will not have rectangular planforms and so the aspect ratio must be calculated by using the wing chord distribution and resulting area, i.e., using the formula
In some cases, the wetted aspect ratio may be specified, which will use the wetted wing area rather than the reference wing area, but this is rare.
The wing of a Supermarine Spitfire has an elliptical wing planform with a root chord of 100 inches and a span of 445 inches. Calculate the planform area and aspect ratio of this wing.
The chord for an elliptical wing planform shape can be expressed as
where is the root (centerline) chord. The planform area of the wing, , is
and substituting for the chord distribution gives
This is a standard integral, so
Therefore, the planform area of this wing is
and the aspect ratio, , is
The aspect ratio of a wing on an airplane can be increased by increasing the wing span and decreasing the wing chord, as shown in the figure below; this example also holds the wing area constant. The aerodynamic advantage in doing so is a significant reduction in induced drag because the wing tip vortices (the source of this type of drag) are further away from more of the wing. However, as the aspect ratio of a wing increases, it becomes more challenging to design the wing to have sufficient stiffness and strength without increasing its weight. As a result, a more extended wing is inevitably more flexible unless some additional structure is used to stiffen the wing.
In airplane design, the final selection of the wing aspect ratio is inevitably a compromise between aerodynamic, structural, weight, and aeroelastic considerations. Longer wings are also heavier and more susceptible to flutter problems because they are inevitably more flexible. Nevertheless, there are tremendous aerodynamic advantages in using a wing of the highest possible aspect ratio if all of the structural, weight, flutter, and other requirements for the airplane can be satisfied.
Sailplanes, which are high-performance gliders, typically have very high aspect ratio wings compared to powered aircraft, so they can achieve high lift-to-drag ratios and can glide long distances by design. The DG-800 in the photograph below exemplifies a modern sailplane with an aspect ratio of just over 27. These types of sailplanes may be able to glide more than 50 miles (in still air) from an altitude of only 5,000 feet.
Winglets can be used to increase the effective aspect ratio of the wing without substantially increasing the wingspan. Winglets were designed by Richard Whitcomb at NASA Langley to help move the wing tip vortices away from more of the wing and reduce the induced drag. Winglets do nothing to reduce the strength of the wing tip vortices. The effects are often apparent from the natural flow visualization, as shown in the photograph below, where it can be seen that the wing tip vortex is trailed from the very tip of the winglet. However, winglets also add some wetted area, increasing skin friction and overall profile drag, yet there is still a net reduction in total aircraft drag.
As shown in the images below, many different variations of winglets have been used on commercial airliners. Today, the trend is toward using more blended winglets with smooth chord variations in the wing-to-winglet transition area, which helps minimize the profile drag while maximizing induced drag reductions.
More recent variations of the winglet appear on aircraft such as the Boeing 737 MAX, which is designed further to reduce the induced drag from the lifting wing and improve the aircraft’s flight efficiency and range. Increasing the length (height) of a traditional winglet from the wing’s top surface helps to further reduce the induced drag, at least to a point, but it also increases the wing’s overall structural weight. By adding another winglet pointing downwards, the aerodynamic benefits are realized without as much weight penalty. It seems unlikely, however, that further reductions in induced drag can be realized by yet more permutations of the basic winglet design.
Other derived geometric parameters relevant to wings are the standard mean chord (SMC) and the aerodynamic mean chord or mean aerodynamic chord (MAC). The SMC, which is given the symbol , is defined as the ratio of the wing area to the wing span, i.e.,
It will be apparent then that the SMC is the average chord of an equivalent rectangular wing with the same area and span. For various reasons, including the fact that the SMC is purely a geometric quantity, the SMC is rarely used in practical aerodynamics.
The mean aerodynamic chord (MAC) is defined as the chord of an equivalent wing that would experience aerodynamic forces identical to those of the actual wing. This is essentially another type of representative or “average” aerodynamic chord of the wing, the basic principle being shown in the figure below.
The MAC is defined as
Finding the MAC of a wing is equivalent to finding the location of the aerodynamic center of pressure of the wing (i.e., where the resultant lift can be assumed to act) and the corresponding value of the chord at that location.
Notice that, in principle, the SMC and the MAC can be determined for any wing, including the horizontal and vertical stabilizers. However, in all cases, their evaluation will involve spanwise integration, either analytically or numerically. Mean wing chords are also used as reference lengths in other disciplines, such as flight dynamics, where the values of parameters such as neutral point, the center of gravity, etc., are often quoted as a fraction of the mean chord.
Linearly Tapered Wing
A linearly tapered wing is one of the most common types of wing planforms. Consider such a wing such that is the ratio of the chord at the tip of the wing to the chord at the root of the wing , i.e., . The planform area of the wing is
and the aspect ratio is
In general, a tapered wing has a chord distribution given by
For the SMC, then
and after integration and some algebra, gives
For the MAC, then
and after integration and some algebra, gives
A wing has an aspect ratio of 12. According to the equation below, the wing’s chord varies smoothly and continuously outward from the aircraft’s centerline. Calculate the span and planform area of this wing.
The planform area of the wing, , is
The chord is
and substituting for the chord distribution, , gives
noting that . The aspect ratio, , is
Therefore, solving for the span, , for an aspect ratio of 12 gives
The wing’s planform area is
Summary & Closure
The geometrical parameters that define the shape of a wing include its span, chord distribution, aspect ratio, washout or another type of twist, and airfoil section shape. Wings may also have winglets, which help reduce overall wing drag. The overarching design requirement is to engineer the wing for good aerodynamic efficiency in terms of its lift-to-drag ratio at normal flight conditions, as well as off-design conditions such as stall. The aerodynamics, however, also need to be balanced against the structural design of the wing in terms of its strength and stiffness, amongst other requirements. Using winglets in increasingly innovative forms has led to significant drag reductions on wings that can save substantial amounts of fuel, especially for an airliner.
- Do some research to determine what types of powered airplanes typically have the highest aspect ratio wings.
- Consider some non-engineering reasons for a large commercial airplane that may limit the wing span.
- Why do all sailplanes have high aspect ratio wings? Explain carefully.
- Why do many fighter jet airplanes use sweptback wings with anhedral? Are there any commercial airliners that use wings with anhedral?
- Why might an airplane use a forward-swept wing rather than an aft-swept wing? Have there been any aircraft built with a forward-swept wing?
- Have any airplanes successfully flown with a different left versus right wing?
- What might be the relative advantages of a conventional winglet versus a blended winglet?
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Hello friends, today we will know about the density of lead. Lead is denoted by Pb and it is a naturally occurring element found in small amounts in the earth’s crust. While it has some beneficial uses, it can be toxic to humans and animals, causing health effects. It can be found in all parts of the environment such as the air, the soil, the water, and even inside our homes. Lead is a cumulative toxicant that affects many body systems and is particularly harmful to young children. Lead in the body is distributed to the brain, liver, kidney, and bones. It is stored in the teeth and bones, where it accumulated over time. Let’s move toward the density of lead.
What Is The Density Of Lead?
The density of lead is 11.35 g/cc. Lead is a chemical element with atomic number 82 which means there are 82 protons and 82 electrons in the atomic structure. The chemical symbol for lead is Pd. Since the number of electrons is responsible for the chemical behaviour of atoms, the atomic number identifies the various chemical elements. Lead density is measured in several units, let’s see them.
The density of lead kg/m3 is 11342 kg/m, the density of lead g/cm3 is 11.35 g/cm3, the density of lead lb/in3 is 0.409 lb/in3, the density of lead lb/ft is 709 lb/ft3, and the density of lead g/ml is 11.4 g/ml.
Well, you have a great understanding about the density of lead g cm3 as well as the density of lead g ml. Let’s understand the definition of lead density in detail.
Definition Of Density Of Lead
The density of lead is defined as the measure of how much lead is in a given amount of space. For example, a block of the heavier element lead will be denser than the softer, lighter element gold. Density is also defined as the mass per unit volume of lead. Density is a physical property and it can be observed without changing the chemical makeup of the substance. And every substance, element and compound has a unique density associated with it. Means when lead element reacts with other elements then the new compound which is formed has different density than lead and other elements.
Density of Lead = mass of lead/volume of lead g/cm3
ρ = m v g/cm3
Where, ρ is the lead density
m is the mass of lead, and
v is the volume of lead
Now that you have understood the definition of lead density, let’s learn how to calculate the lead’s density lead with the help of an example.
Click here – What Is The Density Of Hydrogen?
How To Calculate Density Of Lead?
So, to calculate the density for lead, you can directly calculate it by using a density calculator. Also, you can calculate it with the help of a formula. Let’s calculate it.
- First of all, you need to know about the mass as well as the volume of lead.
- So, the easy way to measure the volume of lead is to immerse the block of lead in water and then measure the rise of water volume.
- For example, if you weigh the lead and it is 50 grams and you immerse it in water and the water rises by 5 ml, then you can use the density equation above to find the lead density, as 50/5 = 10 g/mL.
Let’s understand with the help of an example.
A block of lead of 100 grams immersed in the water and the water level rose by 10 ml. Then what will be the density of the lead block?
By using the density formula given,
Density of Lead = mass of lead/volume of lead g/mL
= 100/10 = 10 g/mL
So, the density of the lead block is 10 g/mL.
Let’s find out the densities by visiting Denseme
What Is The Density Of Lead Pb?
The density of Lead (Pb) is 11.34 g/cm3.
What Is The Density Of Lead English Units?
Different materials have different densities. For example, the mass density of gold is 19.3 g/cc, lead is 11.4 g/cc, copper is 9.0 g/cc, aluminum is 2.7 g/cc, water is 1.0 g/cc (1g/cc = 1 gram per cubic centimeter). If we want the density in English units we could use 11.1 oz/cubic inch as the density of gold.
Why Is Lead So Heavy?
Lead is a stable metal that’s often used as weights and sinkers. The reason it’s heavy in terms of mass per unit volume (or think about it as per teaspoon), is because the lead atoms are very close, making it a dense material.
Which Is Heavier Lead Or Steel?
Steel is less dense than lead. The pellets weigh one-third less than lead pellets of the same size. Steel retains less energy and may not kill birds cleanly at the same ranges.
Click here – What Is The Density Of Olive Oil?
Can Touching Lead Harm You?
Lead exposure occurs when a child comes in contact with lead by touching, swallowing, or breathing in lead or lead dust. Exposure to lead can seriously harm a child’s health and cause well-documented adverse effects such as: Damage to the brain and nervous system. Slowed growth and development.
What Is The Density Of Lead G Ml?
The density lead is 11.3 g/mL.
What Is The Densest Metal?
osmium (Os), chemical element, one of the platinum metals of Groups 8–10 (VIIIb), Periods 5 and 6, of the periodic table and the densest naturally occurring element. A gray-white metal, osmium is very hard, brittle, and difficult to work, even at high temperatures.
This article has given you all possible information about lead density and also the exact value of the lead density. If you know the formula of density then you can find a few variables. If you know the mass and density of a lead, you can calculate the volume. If you know that density and volume, you can calculate the mass of lead. And if you know the mass and volume, you can calculate the density of lead.
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What is the density of lead in g cm 3
What is the density of lead?
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Water has two closely linked dimensions: quantity and quality. Water quality is an important concept related to all aspects of ecosystems and human well-being such as the health of a community, food to be produced, economic activities, ecosystem health and biodiversity.
The quality of an aquatic environment can be defined as the set of concentrations, speciations, and physical partitions of inorganic or organic substances, the composition and state of aquatic biota in the water body and the description of temporal and spatial variations due to factors internal and external to the water body (http://www.who.int).
In other words; water quality refers to the condition of the water, including chemical, physical, and biological characteristics, usually relative to the requirements of one or more biotic species and/or to beneficiary use of water to any human need or purposes. Water quality helps ecological processes to sustain. Good water quality supports native fish populations, vegetation, wetlands and birdlife and poor water quality can pose health risk for people and for ecosystems. Besides, many human uses depend on water quality that is suitable for drinking, irrigation, recreation (swimming, boating), industrial processes, navigation and shipping, production of edible fish, shellfish and crustaceans, scientific study and education, etc.
Each freshwater body has an individual pattern of physical and chemical characteristics largely determined by the climatic, geomorphological and geochemical conditions of its drainage basin and the underlying aquifer. It should to be noted that water usually returns back to the hydrological system after its use and if discharged untreated, it can severely affect the environment. Thus, water quality is closely linked to the surrounding environment and land use. Water is affected by human uses such as agriculture, urban and industrial use, and recreation. Changes in water quality, including increases in levels of specific nutrients, can have serious adverse effects on aquatic life, thus in wildlife and eventually on human nature. Aquatic ecosystems play a crucial role in maintaining water quality. They are valuable indicators of water quality. If water quality is not maintained, the environment will suffer and the commercial and recreational value of our water resources will diminish, as well. Research studies indicate that worldwide water quality is declining mainly due to human activities. Increasing population growth, rapid urbanization, discharge of new pathogens and new chemicals from industries and invasive species are the main factors that contribute to the deterioration of water quality. In addition, climate change will further affect water quality.
From a management point of view, water quality is defined by its desired end use. Water for recreation, fishing, drinking, and habitat for aquatic organisms require higher levels of purity, whereas for hydropower, quality standards are much less important. It is important to know that different beneficial uses have different needs and thus, there is no single measure that constitutes good water quality. For example, while water that is suitable for drinking purpose can be used for irrigation, water used for irrigation may not meet drinking water standards. However, fish and wildlife have other requirements. Fish need water that contains enough oxygen and nutrients since they get all of their oxygen and food from water.
Therefore depending on the beneficiary use, we use guidelines, and standards must be met accordingly. For instance, the first edition of guidelines for drinking water quality was published by World Health Organisation (WHO) in 1984-1985 and was intended to supersede earlier European and international standards.
The standards are also set by the national agencies based on their political and technical/scientific decisions about how the water will be used and referred to their international commitments where exist. There are also international standards such as regulations of International Organization for Standardization (ISO) which is covered in the section of ICS 13.060. The European Union has established a framework for Community Action in the field of water policy in the EU Water Framework Directive (Directive 2000/60/EC of the European Parliament and of the Council of 23 October 2000). The primary objective of the directive is to prevent further water deterioration and to implement the necessary measures to achieve “good water status” in all EU waters by 2015.
Water quality guidelines and standards provide basic scientific information about water quality parameters and ecologically relevant toxicological threshold values to protect specific water uses. The most common standards used to assess water quality are related to health of ecosystems, safety of human contact and drinking water. Drinking Water Regulations are health-related standards that establish the Maximum Contaminant Levels. Drinking water should not present a risk of infection, or contain unacceptable concentrations of chemicals hazardous to health and should be aesthetically acceptable to consumer. The control of the feacal pollution depends on being able to access the risk from any water source and to apply suitable treatment to eliminate the identified risks.
In order to describe and access water quality of a river, stream, lake, groundwater or marine environment, we need to have parameters that can be measured. Measurements of these parameters can be used to determine and monitor changes in water quality, and determine whether it is suitable for the health of the natural environment and the required uses. Water quality is measured by several factors, such as the concentration of dissolved oxygen, bacteria levels, the amount of salt (or salinity), or the amount of material suspended in the water (turbidity). In some water bodies, the concentration of microscopic algae and quantities of pesticides, herbicides, heavy metals, and other contaminants may also be measured to determine water quality. These parameters are mainly categorized under physical, chemical, and biological properties of water.
Water quality is determined by measurements on site and by in situ examination of water samples or in the laboratory. Thus, on-site measurements, the collection and analysis of water samples, the study and evaluation of the analytical results, and the reporting of the findings are the main elements of water quality monitoring. The results of analyses conducted on a single water body are only valid for the particular location and time at which that sample was taken.
To gather sufficient data (by means of regular or intensive sampling and analysis) to assess spatial and/or temporal variations in water quality is therefore one of the purposes of a monitoring programme.
Physical measurements are those that include water temperature, depth, flow velocity and turbidity. These are all useful in analysing how pollutants are transported and mixed in the water environment, and can be related to habitat requirements for fish and other aquatic wildlife. For example, many fish have very specific temperature requirements, and cannot tolerate water that is either too cold or too hot.
With chemical measurements we measure concentrations of wide range of chemicals and chemical properties. Test results are defined as milligrams of chemical per liter of water (mg/l). Chemical water quality studies focus on the chemicals that are most important for the problem at stake, since even the purest water contains countless chemicals and it would be impossible to measure all of them. Therefore; while in agricultural areas, studies measure chemicals found in manure, fertilizers, and pesticides, in an industrial area studies focus on measuring chemicals used by the nearby industries.
With bacteriological analysis we measure the hygienic quality of water. The bacteriological quality of a water body is very important especially when we use the water body for drinking purposes.
To assess a water body in terms of quality, it is essential to gain a water quality data obtained in several intervals (monthly, seasonally, and annually) and monitor the changes of parameter values to point out the variations immediately after a specific intervention. The lack of water quality data and monitoring worldwide as well as lack of knowledge about the potential impact of natural and anthropogenic pollutants on the environment and on water quality is one of the major problems to define and solve water pollution problem. In many countries, the lack of prioritization of water quality has resulted in decreased allocation of resources, weak institutions and lack of coordination in addressing water quality challenges. Considering approximately 25% of the world’s population has no access to potable water, water quality is at most importance in human health.
Monitoring the water bodies is particularly important to point out the reliability of source accessed for public use especially for drinking purposes. Water quality measurements for drinking purposes generally focus on health of the community and aestethic aspects. It should be ensured that the water intended for human consumption can be consumed safely on a life-long basis, and this represents a high level of health protection. Monitoring and control technologies provide the surveillance of source water quality and the detection of biological and chemical threats. They lead us to define the boundary conditions for the subsequent treatment and provide early warning in case of unexpected contaminations. They are mandatory for the high quality of finished water in treatment processes. Moreover, detection of changes in water quality during distribution and monitoring drinking water quality at consumers’ tap is essential. Water quality deterioration in distribution systems, mainly caused by inappropriate planning, design and construction or inadequate operation and maintenance and water quality control, may be the cause of waterborne and water-related illness. Rapid urbanization, population growth and aging of infrastructure stress the distribution sysytems.
To ensure the water quality, the standards should be based on the latest scientific evidence and efficient and effective monitoring, assessment and enforcement of drinking water quality should be secured.
3.2. Physical and Chemical Quality of Water
Physical characteristics of water like temperature, colour, taste, odour and etc. can be determined by senses of touch, sight, smell and taste, together with adequate instruments. For example; by touching we can have an idea of its temperature, by smelling, its taste and odour, and we see the colour, floating debris, light penetration, turbidity and suspended solids. In general, the physical characteristics of water are not directly concerned for public health but they affect the aesthetic quality of the water. However, these affect the consumers’ perception and behaviour. Dimensions of the water body, flow velocity, hydrological balance, etc. are other physical characteristics of water.
Chemical characteristics of water provides information on whether or not it is safe to use, for human health, as well as plants and animals that live in and around streams. Chemical assessment of water quality includes measurements of many elements and molecules dissolved or suspended in the water. Chemical measurements can be used to detect pollutants and toxicity. A significant number of very serious problems may occur as a result of chemical contamination of water resources. Most chemicals arising in drinking-water are of health concern only after extended exposure of years, rather than months. (The principal exception is nitrate). Water from sources that are considered to have a significant risk of chemical or radiological contamination should be avoided. To provide information whether this type of a problem exists or not, a selected series of physicochemmical parameters have to be measured. Assessment of the acceptabality of the chemical quality of drinking-water relies on comparison of the results of water quality analysis with guideline values.
The source of chemical constituents are;
Naturally occuring (eg: rocks, soils and the effects of the geological setting and climate)
Industrial sources and human dwellings (eg: mining (extractive industries) and manufacturing and processing industries, sewage, solid wastes, urban runoff, fuel leakages)
Agricultural activities (eg: manures, fertilizers, intensive animal practices and pesticides)
Water treatment or materials in contact with drinking-water (eg: coagulants, DBPs, piping materials)
Pesticides used in water for public health (eg: larvicides used in the control of insect vectors of disease)
Cyanobacteria (eg: eutrophic lakes) Chemical parameters measured in natural waters mainly include pH, alkalinity, nitrates, nitrites and ammonia, ortho- and total phosphates, and dissolved oxygen and biochemical oxygen demand.
When the end use of a water body is for a community supply, additional measerements may include but may not be limited to, inorganics (metals, major ions, nutrients); and organics (total organic carbon, hydrocarbons and pesticides). Chlorination Disinfection By-Products (CDBPs), Trihalomethanes (THMs), Haloacetic Acids (HAAs) and Chlorine Residual Testing (Free and Total) should also be included to the parameters of monitoring of tap water. Treatment technologies play a significant role in safe water production from catchment to consumer. Each treatment step poses its own demands on the key-parameters to be monitored in order to guarantee an accurate and sustainable operation of the treatment process. Key-parameters for the monitoring of the quality of overall treatment effect/finished drinking water before entering distribution network, key-parameters for detection of quality changes during distribution, key-parameters for the monitoring of time related changes in water quality due to residence time in the distribution network and finally key-parameters for the monitoring of water quality at consumers’ tap should be identified according to the characteristics of water body used and national regulations to provide safe water.
Detailed information on some of the physical and chemical parameters is given below;
Low flow in surface waters may lead to bacteriological degredation and higher concentrations of pollutants. During treatment, changes in flow can adversely affect coagulation and sedimentation processes. Besides, filtration rate and contact time with disinfectant are significant in the production of safe drinking water. Changes in flow rates within distribution systems can result in suspension of sediments and deterioration of supplies.
Water temperature is an important physical property showing how hot or cold the water is. It is commonly measured on Celsius, Fahrenheit or Kelvin; but due to its universal use, water temperature is generally reported on the Celsius scale. The temperature of water can alter some of the important physical and chemical properties and characteristics of water: thermal capacity, density, specific weight, viscosity, surface tension, specific conductivity, salinity, solubility of oxygen and other dissolved gases, metabolic rates and photosynthesis production, compound toxicity, pH and etc. Besides, it can affect the metabolic rates and biological activity of aquatic organisms, i.e. the metabolic rates of aquatic organisms increase as the water temperature increases and, biological reaction rates increase with increasing temperature, as well. The temperature of water in streams and rivers throughout the world varies from 0 to 35°C. The existance of a fish species or an aquatic plant, besides other characteristics of water, depends on the temperature of that water body. In addition to this, high water temperatures can increase the solubility and thus toxicity of certain compounds including heavy metals and it can also affect the tolerance limit of organisms.
Dissolved oxygen concentrations in water bodies depend on temperature. Solubility of gases will decrease as temperature increases. The warmer the water, the less oxygen that it can hold. This is important for aquatic organisms to survive.
Water temperature plays a role in the shift between ammonium and ammonia in water. Ammonia is toxic at high pH levels, but temperature can also affect. For every 10°C increase in temperature, the ratio of unionized ammonia to ammonium doubles.
Moreover, water temperature can affect ionic activity and conductivity since it affects viscosity. An increase in temperature will decrease viscosity. Decrease in viscosity increases the mobility of ions in water, i.e. increases conductivity. Due to the increased mineral and salt ions, hot springs have high conductivity. Many salts are more soluble at higher temperatures. Thus in warm waters, the ionic concentration is often higher. These dissolved solutes are often called Total Dissolved Solids (TDS).
pH is also temperature dependent. pH is calculated by the number of hydrogen ions in solution. The hydrogen and hydroxyl ions have equal concentrations at a pH of 7 and it is neutral. However, the neutral concentration of 1 x 10-7 M only holds true at 25°C. As the temperature increases or decreases, the ion concentrations will also shift, thus shifting the pH without making it more acidic or basic.
Alteration in water temperature affects the density of water. Unlike most materials, the density of pure water decreases when it freezes. Pure water achieves its maximum density, 1.00 g/ml, at 4°C. This property ensures at least 4°C temperature at the bottom of a water body which sustains aquatic life.
Freezing point and maximum density is also affected by salinity. They decrease as salinity levels increase. Pressure shifts the freezing, boiling and maximum density points but does not affect the temperature of the water itself.
Water temperature’s impact on a variety of other parameters makes it an important factor in determining water quality.
Water temperature can be affected by many factors. Sunlight is the greatest source of heat transfer to water.
It is a form of thermal energy that is then transferred to water’s surface increasing the temperature of the water. This energy is absorbed until the sunlight is gone. The shallower water bodies tend to warm quicker than deeper water bodies. Like thermal energy from sunlight, atmospheric heat transfer occurs at the water’s surface, as well. Warm water will transfer energy to the air and cool off when the air is cold and if the air is hot, cold water will receive the energy and warm up. Water temperature fluctuates more gradually than air temperature. Turbidity can also increase water temperature. Turbid water has high amount of suspended solids. The suspended particles in the water absorb heat from sunlight more efficiently than water. The heat is then transferred to water increasing its temperature. Moreover; groundwater, streams and rivers can change the temperature of the water body into which they flow. There are also men-made influences. Man-made influences on water temperature include thermal pollution (commonly comes from municipal or industrial effluents), runoff (from parking lots and other impervious surfaces), deforestation (when trees are removed, a water body can become unusually warm) and impoundments (such as dams; the temperature will shift if the dam release unusually cool or unusually warm water into the stream). Shallow and surface waters are more easily influenced by the above mentioned factors than deep water.
For drinking waters; an aesthetic objective of 15°C has been established. However, it is not economical to change the temperature of the water in drinking water treatment plants. The temperature is hence largely determined by the selection of the raw water source and the depth at which the distribution system is buried.
The pH of natural water can give important information about many chemical and biological processes. It can be indicative of a number of different impairments. A high organic content will tend to decrease the pH because as microorganisms break down organic material, the by- product will be CO2 that will dissolve and equilibrate with the water forming carbonic acid (H2CO3). As a result of organic decomposition, some organic acids decrease pH. Besides, the acidity of natural waters can also be affected by mineral acids produced by the hydrolysis of salts of metals like aluminum and iron. Changes in pH indicate an industrial pollutant, photosynthesis or the respiration of algae that is feeding on a contaminant. Most ecosystems are sensitive to pH variations. pH is usually monitored to assess the health of aquatic ecosystem, recreational waters, irrigation sources and discharges, live stock, drinking water sources, industrial discharges, intakes, and storm water runoff.
In drinking water supply, disinfection with chlorine is highly dependent on pH. At pH above 8, disinfection is less effective. Generally, increasing pH values lead to decreased release of metals, due to decreased solubility at higher pHs. Therefore, raising the pH in water supplies has been used as a control measure to reduce lead concentrations.
Alkalinity of water may be due to the presence of one or more number of ions. These include hydroxides, carbonates and bicarbonates. However, borates, phosphates, silicates, and other bases also contribute to alkalinity if present. Hydroxide ions are always present in water, even if the concentration is extremely low. However, significant concentrations of hydroxides may be present after certain types of treatment. Carbonates may also be found in the water after lime soda has been used to soften the water. Bicarbonates are the most common sources of alkalinity. Moderate concentrations of alkalinity are desirable in most water supplies to balance the corrosive effects of acidity. However, strong alkaline water has an objectionable "soda" taste.
Alkalinity can have varied impacts on the release of hazardous chemicals from materials and fittings. Higher alkalinities decrease corrosion and the release of iron from pipes (Pisigan & Singley, 1987; Cantor, Park & Vaiyavatjamai, 2000; Sarin et al., 2003) and lime from cement pipes (Conroy et al., 1994). In contrast, water utility and laboratory results show that higher alkalinities increase copper release (Edwards, Jacobs & Dodrill, 1999; Cantor, Park & Vaiyavatjamai, 2000; Shi & Taylor, 2007).
Hardness is a measure of the concentration of divalent metallic cations (++ charged) dissolved in water and is generally expressed as the sum of calcium and magnesium concentrations expressed as equivalents of calcium carbonate. Other cations such as aluminium, barium, iron, manganese, strontium and zinc can contribute to hardness, but concentrations are usually much lower than calcium and magnesium. Hardness is most commonly expressed as milligrams of calcium carbonate equivalent per liter. Both calcium and magnesium are essential minerals and beneficial to human health in several respects. Excess calcium is excreted by the kidney in healthy people, however; the major cause of hypermagnesaemia is renal insufficiency associated with a significantly decreased ability to excrete magnesium. Drinking-water in which both magnesium and sulfate are present at high concentrations (above approximately 250 mg/l each) with continuous exposures can have a laxative effect. Hard waters can be problematic to low pressure and low flow watering systems due to the accumulation of insoluble calcium and magnesium carbonate deposits. Hard water can cause increased soap consumption as well. Producers located in karst regions should give additional consideration to hardness because elevated levels of calcium and magnesium are associated with limestone (karst) geology.
Sulfate is present in most water sources in the form of calcium, iron, sodium, and magnesium salts. High concentrations cause diarrhea and help development of polioencephalomalacia (a neurological disorder characterized by weakness, muscle tremors, lethargy, and even paralysis and death). The form of sulfur is important in determining toxicity. Sulphur bacteria may produce a dark slime or deposits of metal oxides that develop as a result of the corrosion of metal pipes. The slime or deposits can clog plumbing and stain clothing. Sulphates are discharged into the water body in wastes from industries that use sulphates and sulphuric acid, like mining and smelting operations, kraft pulp and paper mills, textile mills and tanneries. Salt water intrusion and acid rock drainage are also sources of sulphates in drinking water. Sulphate is one of the least toxic anions. The presence of sulphate in drinking water can also result in a noticeable taste.
3.2.7. Toxic Compounds
There hundreds of industrial and agricultural chemicals, including several known carcinogens, are posing risk in municipal water systems. The nation’s laws and enforcement programs have to keep pace with spreading contamination, posing significant health risks to millions. Those substances are; aluminium, antimony, arsenic, barium, benzo(a)pyrene, cadmium, chromium, copper, cyanide, disinfection by-products (including trihalomethanes, haloacetic acids and N-nitrosodimethylamine), fluoride, iron, lead, mercury, nickel, pesticides, petroleum hydrocarbons, selenium, silver, styrene, tin, uranium and vinyl chloride. Some risk factors are given below;
Lead - This poisonous metal can damage the blood, brain, and disrupt nervous system.
Mercury - Exposure to mercury can cause tremors, psychotic reactions, and suicidal tendencies.
Chlorine - Chlorine is a chemical element that is essential to human life. However, in anything other than trace amounts, it becomes a toxic gas that irritates the respiratory system.
PCB's - A class of organic compounds that cause skin, blood, and urine problems in humans.
Arsenic – It is an element that has been used for centuries as a deadly poison.
Fluoride - While this compound has many positive traits, such as the ability to clean our teeth, can also be quite toxic.
Cadmium - People who drink water containing cadmium well in excess of the maximum contaminant level (MCL) for many years could experience kidney damage.
Copper - Some people who drink water containing copper in excess of the action level may, with short term exposure, experience gastrointestinal distress, and with long-term exposure may experience liver or kidney damage.
Styrene - Some people who drink water containing styrene well in excess of the maximum contaminant level (MCL) for many years could have problems with their liver, kidney, or circulatory system problems.
MtBE - MtBE is a volatile, flammable, and colorless liquid that is used as an additive in gasoline.
DCPA - DCPA is an herbicide used on strawberries, melons, and cucumbers.
Hexachlorobenzene (HCB) – It is commonly used as a pesticide. HCB can cause cancer and disrupt the endocrine system and interfere with enzyme activity.
Dioxin – It is an organic compound which is known to increase the likelihood of cancer.
DDT – It is a deadly chemical used as an insecticide. It has been linked to diabetes and cancer.
3.2.8. Dissolved Oxygen
Oxygen is soluble in water.
The oxygen that is dissolved in water is called dissolved oxygen (DO) and is essential for all forms of aquatic life and necessary for a healthy aquatic ecosystem. The need for DO depends on the species and life stage.
Surface water, near the water-atmosphere interface and with sufficient light for photosynthesis, is generally saturated or even supersaturated with oxygen. Deeper water receives oxygen through mixing by wind, currents, and inflows. Mixing and aeration also occur at waterfalls and rapids. Dissolved oxygen can be reduced to very low levels during the winter months when water is trapped under ice.
In general, the concentration of dissolved oxygen in a water body is the result of biological activity. Photosynthesis of some aquatic plants increases DO level in a water body during day time. On the other hand, they consume DO at nights. Organic material is consumed by microorganisms that mostly their process depends on DO. DO concentrations of unpolluted fresh water is close to 10 mg/l. In waters polluted with antropogenic discharges like fertilizers, suspended material, or petroleum waste, microorganisms such as bacteria will break down the contaminants. During this process DO may be consumed to such levels that the water may become anaerobic. Typically fish cannot live in DO levels less than 2 mg/l. DO is depleted through chemical oxidation, as well.
DO can vary in daily and seasonal patterns and it is also correlated with temperature, salinity and elevation.
DO can affect the solubility and availability of nutrients, which can be released from sediments under conditions of low dissolved oxygen.
Usually membrane electrodes are used as in situ DO sensors. Laboratory tests for assessing the DO is the biological oxygen demand (BOD - the amount of oxygen required to biologically break down a contaminant) and the chemical oxygen demand (COD - the amount of oxygen that will be consumed directly by an oxidizing chemical contaminant).
A high DO level in a community water supply is preferred because it makes drinking water taste better. However, high DO levels accelerate corrosion in water pipes.
Colour in water is one of the physical parameters which concern mainly the aesthetic aspects. People may think coloured water unfits to drink but that water may be perfectly safe for drinking purposes. On the other hand, colour can indicate the presence of organic substances, such as algae or humic compounds. A blue coloured water body indicates a transparent water body with low dissolved solids; some algea cause red colour in water bodies; reddish-orange water bodies may indicate the presence of iron precipitation or silt; brown-yellow water bodies may contain dissolved organic materials, humic substances from soil, peat, or decaying plant material; and the water body can be green due to the rich presence of phytoplankton and other algea. Recently, colour is also used for quantitative assessment of the presence of potentially hazardous or toxic organic materials in water bodies. Drinking water should ideally be colourless.
3.2.10. Taste and Odour
Taste and odour are other physical parameters which are direcly related to human perceptions of water quality. While relatively simple compounds produce sour and salty tastes, sweet and bitter tastes are produced by more complex organic compounds. People can detect odour more effectively than tastes.
Biologically derived contaminants (actinomycetes and fungi, cyanobacteria and algae, invertebrate animal life, iron bacteria), chemically derived contaminants (aluminium, ammonia, chloramines, chloride, chlorine, chlorobenzenes, chlorophenols, copper, dissolved oxygen, ethylbenzene, colour, hardness, hydrogen sulfide, iron, manganese, petroleum oils, pH and corrosion, sodium, styrene, sulfate, synthetic detergents, toluene, total dissolved solids, turbidity, xylenes, zinc) and temperature affect the taste and odour of water.
Taste and odor are more significant parameters when we use the water for drinking purposes. Some substances of health concern have effects on the taste, odour or appearance of drinking-water, however; concentration limits of these substances determined for the rejection of the water are generally lower than those of concern for health since there is such a wide range in the ability of consumers to detect them by taste or odour.
Taste and odour can originate from biological sources or processes (e.g. aquatic microorganisms - organic materials discharged directly to water bodies, such as falling leaves, runoff, etc., go in biodegradation process in those water bodies in which tastes and odour-producing compounds released) and from natural inorganic and organic chemical contaminants. They can also originate in water treatment and transmission/storage/distribution facilities; such as contamination by synthetic chemicals, corrosion or they can be produced as a result of problems with water treatment (e.g. chlorination). The cause of taste and odour problem in municipal water supply facilities should be investigated, especially if there is a sudden or substantial change.
As a general definition; turbidity is a measure of the light-transmitting properties of water, in other words; an optical determination of water clarity.
Turbidity is significant for health and aesthetic considerations. Turbid water appears cloudy, murky or colored. Suspended solids and dissolved coloured material make the water body opaque, hazy or muddy and reduce water clarity.
Turbidity and total suspended solids are related. It is often used to indicate changes in the total suspended solids concentration in water body. The more solids present in the water, the less clear the water will be. Turbidity and water clarity are both visual properties of water based on light penetration. The clearer the water, the greater the potential for photosynthetic production is.
The clarity and transparency of natural water bodies are affected by human activities, organic matter such as algae, plankton and decaying material, suspended sediments such as silt or clay, and inorganic materials. Besides, turbidity can also include colored dissolved organic matter, fluorescent dissolved organic matter and other dyes. Chemical precipitates are also considered as a form of suspended solids. Salinity can be counted as one of the factors affecting water clarity. Salt ions collect and bind suspended particles together, thus their weights increase and settle to the bottom. Due to this mechanism, marine environment has a higher clarity (and lower turbidity) than fresh water.
Total suspended solids (TSS) are particles that are larger than 2 microns (both organic and inorganic) found in the water column. Particules smaller than 2 microns are considered as dissolved solid. Excessive total dissolved solids (TDS) can produce toxic effects on fish and fish eggs depending on their ionic properties. EPA, USPHS
and AWWA recommend an upper limit of 500 mg/L TDS. TDS can also affect water taste, and often indicates a high alkalinity or hardness.
Turbidity and total suspended solids often overlap. Turbidity measurement can be used to estimate the total dissolved solids concentration but there are a few factors that only contribute to one or the other.
Turbidity measurement does not include any settled solids or bedload. Moreover, they may be affected by colored dissolved organic matter which is not included in TSS measurements. Total suspended solids, on the other hand, is a specific measurement of all suspended solids, organic and inorganic, by mass. TSS is the direct measurement of the total solids, including settleable solids, and present in a water body. Therefore, sedimentation rates can be calculated by TSS, not by turbidity.
Presence of suspended solids in high concentrations can decrease water quality for aquatic and human life, hinder navigation and increase flooding risks. In addition to this, since they absorb additional heat from the sun, suspended solids can increase the temperature of water. This can also drop dissolved oxygen levels. Decrease in dissolved oxygen concentrations can be caused by blocked sunlight that inhibits photosynthesis. Without the necessary sunlight, plants below the water’s surface will not be able to continue photosynthesis and may die. When the plants die, as photosynthetic processes decrease more and less dissolved oxygen is produced, ending up in the further reduction of dissolved oxygen levels in a body of water. The decomposition of the dead plants can drop dissolved oxygen levels even lower. And, with the loss of the underwater vegetation, it can cause population declines up the food chain since the amount of vegetation available for other aquatic life to feed on is reduced.
Suspended sediment present in water bodies mostly comes from runoff and erosion.
An increase in turbidity can indicate erosion of stream banks. This may have a long-term effect on a water body.
Wastewater discharge increases turbidity. Pollutants such as dissolved metals and pathogens can attach to suspended particles and go into the water. Pollutants larger than 2 microns, they will also contribute to the total suspended solids concentration. This is why an increase in turbidity can also indicate potential pollution.
Turbidity and water flow are related. High flow velocities keep particles suspended instead of letting them settle to the bottom. So, rivers with high flow velocities are mostly turbid. Weather should be taken into consideration since it also affects water flow, which in turn affects turbidity.
Another important factor in increased turbidity and total suspended solids concentrations is inadequate land use. In construction, logging, mining areas and in other disturbed sites like agricultural areas exposed soil is increase and vegetation is decreased.
Turbidity is most often measured with a turbidity meter. Turbidity is reported in units called a Nephelometric Turbidity Unit (NTU), Jackson Turbidity Unit (JTU), Nephlometer Turbidity Units (NTU) or Formazin Nephelometric Units (FNU).
Total suspended solids can be measured by filtering and weighing a water sample and are reported in mg/L.
Water clarity, when not measured in terms of turbidity, is measured by Secchi depth. It measures how deep a person can see into the water. But this is used in shallow water.
Turbidity is significant in selection and efficiency of treatment processess. Turbidity can provide food and shelter for pathogens. If not removed, turbidity can promote regrowth of pathogens in the distribution system, leading to waterborne disease outbreaks.
As a basic definition, salinity is dissolved salts content in water. Salinity is a strong contributor to conductivity. Generally, salinity is derived from the conductivity measurement, it is not measured directly. This Practical Salinity Scale is acceptable in most situations, however; a new method of salinity measurement was adopted in 2010. This method, called TEOS-10, determines absolute salinity. Absolute salinity can be used to estimate salinity not only across the ocean, but at greater depths and temperature ranges. It gives more accurate values than other salinity methods when ionic composition is known.
The units used to measure salinity vary depending on application and reporting procedure. While parts per thousand or grams/kilogram (1 ppt = 1 g/kg) can be used, in some freshwater sources, salinity is reported in mg/L. Lately, salinity values are reported based on the unitless Practical Salinity Scale (sometimes denoted in practical salinity units as psu) and as of an Absolute Salinity calculation was developed, it is reported in g/kg and is denoted by the symbol S.
Salinity affects dissolved oxygen solubility in water body. In high salinity levels, dissolved oxygen concentration lowers. That’s why sea water has a lower dissolved oxygen concentration than freshwater sources.
Salinity influences organisms that can live in that area. In general, aquatic organisms can tolerate a specific salinity range.
Salinity also affects water density. One of the driving forces behind ocean circulation is the increase in density with salt levels.
Sodium levels in drinking water from most public water systems are unlikely to be a significant contribution to adverse health effects. There are aesthetic guideline levels for sodium.
Conductivity is a measure of water’s capability to pass electrical flow. This is related to the ion concentrations in the water (electrolytes). The more ions present in water, the higher the conductivity. Therefore, we can simply say that sea water has a very high conductivity. It is an early indicator of change in a water system. Conductivity change can indicate pollution.
The standardized method of reporting conductivity is specific conductance. It is a conductivity measurement made at or corrected to 25°C, since the temperature of water will affect conductivity readings. Conductivity is correlated with temperature and salinity/TDS. The higher the water temperature, the higher the conductivity level will be.
Conductivity is generally reported in micro- or millisiemens per centimeter (uS/cm or mS/cm). Less commonly, it can be measured in micromhos or millimhos/centimeter (umhos/cm or mmhos/cm).
In streams and rivers, conductivity depends on the surrounding geology.
Most of the salt in the seawater comes from runoff, sediment and tectonic activity.
The factors that affect water volume such as heavy rain or evaporation affect conductivity, as well.
Resistivity, which is defined as reciprocal of conductivity, is a measurement of water’s opposition to the flow of a current over distance.
Both total dissolved solids (TDS) and conductivity in drinking water indicate the total inorganic mineral content. Some water purification processes such as reverse osmosis can remove those inorganic contaminants from water.
3.2.14. Organic matter
Data of organic matter in treated water provide an indication of the potential for the regrowth of heterotrophic bacteria in reservoirs and distribution systems. Organic matter can be measured as Total Organic Carbon (TOC), BOD and COD. BOD is primarily used with wastewaters and polluted surface waters. TOC is the only parameter applicable to drinking water.
In aquatic ecosystems, nitrogen and phosphorus are the most important chemical elements that are essential to the growth and survival of living organisms.
126.96.36.199. Total Nitrogen (N2)
Nitrogen is essential to all life. Nitrogen can go through many complex chemical and biological changes in a continuing cycle called the nitrogen cycle. Nitrogen is present in natural waters in different forms as nitrate (NO3-), nitrite (NO2-), and ammonia (NH3). These three compounds are interrelated through the process of nitrification. It is the biological oxidation of ammonia to nitrate. Total nitrogen in natural waters refers to the sum of organic nitrogen containing compounds, and the inorganic nitrogen oxidation states present in solution. Total nitrogen can be calculated from the sum of the total kjeldahl nitrogen (organic and reduced nitrogen), nitrate and nitrite.
There are many sources of total nitrogen. Storm water runoff, livestock, fertilizers, waste water discharges, automobile exhausts finding way through precipitation into the water body contribute to the amount of total nitrogen. Natural break down of plant and animal material can be counted as natural sources of nitrogen.
Nitrogen as Ammonia (NH3)
Ammonia is highly toxic and ubiquitous in surface water systems. Therefore, it is one of the most important pollutants. Sources of ammonia are industrial, municipal and agricultural waste waters. Ammonia also occurs as a result of natural processes as breakdown of nitrogenous organic compounds in water and soil and the breakdown of biota. Ammonia has a toxic effect on healthy humans only if the intake becomes higher than the capacity to detoxify. It is not of direct importance for health in the concentrations to be expected in drinking-water.
Nitrogen as Nitrate (NO3-)
Nitrate is the essential nutrient for many photosynthetic species and it is the growth limit nutrient. Nitrate is less toxic to people than ammonia or nitrite but nitrate will become toxic especially to infants at high levels. Symptoms include shortness of breath and blue baby syndrome. The primary health hazard from drinking water with nitrate-nitrogen occurs when nitrate is transformed to nitrite in the digestive system.
Excessive nitrate concentrations with the presence of other essential nutrient factors, eutrophication and algal blooms can become a problem. Nitrate levels over 5 mg/l in natural waters generally indicate antropogenic pollution. Where agricultural use of land increase and urban areas expand, nitrate monitoring is an important tool in accessing and preventing man made sources of nitrate.
Faulty septics tanks, nearby animal feedlots, agricultural or residential fertilizer use may be the source of high levels of nitrate in wells. Nitrate contamination in well water from human or animal waste indicates that microbial contaminants are also present.
Nitrogen as Nitrite (NO2-)
Nitrite is extremely toxic to aquatic life. Fortunately, since it is rapidly oxidized to nitrate it is usually present only in trace amounts in most natural freshwater systems. The source of nitrite in a water body may be the discharge of wastewater treatment effluent with impeded process of nitrification. Nitrification process in a treatment plant can be affected by several factors, including pH, temperature and dissolved oxygen, number of nitrifying bacteria and presence of inhibiting compounds.
Infants below six months who drink water containing nitrite in excess of the maximum contaminant level (MCL) could become seriously ill and, if untreated, may die. Symptoms include shortness of breath and blue baby syndrome/methemoglobinemia.
Nitrogen as Total Kjeldahl (TKN)
Total Kjeldahl nitrogen is the sum of organic nitrogen, ammonia (NH3), and ammonium (NH4+) in the chemical analysis of soil, water and wastewater. To calculate Total Nitrogen (TN), the concentrations of nitrate-N and nitrite-N are determined and added to the total Kjeldahl Nitrogen. Nitrogen mainly occurs in wastewater in this form. It is a term that reflects the technique used in their determination.
Organic Nitrogen is the byproduct of living organisms. It may be in the form of a living organism, humus or in the intermediate products of organic matter decomposition. It usually occurs in only very small concentrations in most waters.
Phosphorus is essential for plant growth and metabolic reactions in animals and plants. It typically occurs in nature as phosphate (PO4-3). Both organic phosphate and inorganic phosphate forms are present in aquatic systems and may be either dissolved in water or suspended. With even small amounts it can cause significant plant growth that has adverse affects on aquatic life such as algal blooms causing DO depletion. The main environmental impact associated with phosphate pollution is eutrophication. High levels of phosphorus will be quickly consumed by plant and microorganisms, impairing the water by depleting the dissolved oxygen and increasing the turbidities. These impairments will kill or harm fish and other aquatic organisms. Inorganic phosphate is often referred to as orthophosphate or reactive phosphorous. It is the form most readily available to plants, and thus may be the most useful indicator of immediate potential problems with excessive plant and algal growth.
Sources of phosphate include animal wastes, sewage, detergent, fertilizer, disturbed land, and road salts used in the winter. In unpolluted waters, phosphorous can enter a water system from the weathering of phosphorous baring rocks and minerals.
Phosphates do not pose a human health risk except in very high concentrations. However, phosphate levels greater than 1.0 may interfere with coagulation in water treatment plants. As a result, organic particles that harbor microorganisms may not be completely removed before distribution. Sometimes public water systems add phosphates to the drinking water as a corrosion inhibitor to prevent the leaching of lead and copper from pipes and fixtures. It is measured in mg/L.
An automated drinking water system may contain residual disinfectants from the public drinking water supply. Chlorine is the most widely used disinfectant in water treatment. Proper measurement and control of disinfectant dose and contact time is obligatory. To maintain a minimal level of quality control of treated water disinfectant dose, residual obtained and time of contact should be measured. In addition to this, measurement of disinfectant residual concentration during and after disinfection is a required measurement in most water treatment plants. Residual concentration after contact should be continuously monitored.
3.3. Microbiological Quality of Water
Microbial quality is one of the primary indicators for the safety of a drinking water supply. Bacteriological examination of drinking water is made to determine whether the water consumed is contaminated, and microbial parameters can provide usefull information throughout the drinking water production process like catchment survey, source water characterisation, treatment efficiency and examination of distribution system.
Coliform organisms have been used to determine the biological characteristics of natural waters. Escherichia coli (E. coli) is generally used as an indicator organism. This organism is present in the intestine of warm-blooded animals, including humans. Therefore the presence of Escherichia coli in water samples indicates the presence of fecal matter and thus the possible presence of pathogenic organisms of human origin.
Faecal contamination is a common source of infectious microorganisms. These include bacteria, viruses and parasites that occur naturally in the gut of humans and other warm-blooded animals. The presence of waterborne disease causing microorganisms in drinking water may result in gastrointestinal illness or diarrhea and even lead to death. Microbiological contaminant parameters of EPA standards include Coliforms (total), Giardia lamblia, Heterotrophic Plate Count, Legionella, Pseudomonas sp., Pyrogens, Turbidity and Viruses).
Although treated, bacteria (Campylobacter jejuni/C. Coli, Esherichia coli, Vibrio cholerae, Salmonella typhi, Shigella, Legionella spp., Non-tuberculous Mycobacterium spp., Franciscella tularensis, etc.), viruses (Noroviruses, Rotaviruses, Enteroviruses, Adenoviruses, Hepatitis A, Hepatitis E, Sappoviruses, etc.), parasites (Cryptosporidium hominis/parvum, Entamoeba histolytica, Giardia intestinalis, Cyclosposa cayetanensis, Acanthamoeba, Naegleria fowleri, some invertebrates, including water mites, cladocerans and copepods, etc) and filamentous fungi and yeast (Aspergillus flavus, Stachybotrys chartarum, Psaudellescheria boydi, Mucor, Sporothrix, Cryptococcus, etc.) can be found in finished drinking-water, pipe biofilms and distribution systems. In a water distribution system faecal contamination may occur with an intrusion of faecal contaminant through broken mains and cross-connections or openings in storage tanks. In addition, construction, new pipe installation and repairs close to sewer lines can introduce contamination into the distribution system.
The presence of faecal pathogens is assessed by monitoring for indicator bacteria. The WHO Guidelines for Drinking-water Quality (WHO, 2011) recognize E. coli as the indicator of choice for faecal contamination, although thermotolerant coliforms (E. coli, Citrobacter, Klebsiella and Enterobacter) can be used as an alternative.
Although total coliforms are not a specific indicator group for contamination as for they can grow naturally in water and soils, they can be used to assess the cleanliness of distribution systems. Coliforms can arise from biofilm linings in pipes and fixtures or from contact with soil due to breaks or repair works. Testing for heterotrophic plate count bacteria is sometimes used for similar purposes. Total coliform numbers and heterotrophic plate count (also known as total or standart plate count) bacteria are used in operations as an indicator of system performance including a loss of disinfection efficiecy, intrusion of contaminants into drinking-water or the growth of biofilms that could support the presence of pathogens. The detection of any coliforms leads us a corrective action, such as increasing the chlorine dose at the water treatment plant, checking the operation of service reservoirs or pipe flushing and rechlorination of the affected area.
Standard plate count/HPC organisms are used for monitoring of efficiency of water treatment and disinfection processes or after growth in water distribution systems.
Total coliform bacteria (total coliforms) is used to evaluate quality of drinking water and related waters.
Fecal coliform bacteria (fecal coliforms) is used to evaluate the quality of wastewater effluents, river water, sea water at bathing beaches, raw water for drinking water supply and recreational waters.
Fecal streptococci (enterococci) are used in evaluation of treatment processes and recreational waters.
Clostridia (presumptive Clostridium perfringens) are used to indicate remote fecal pollution and to access efficacy of treatment and disinfection process.
Coliphages are used as indicators of the incidence and behaviour of human enteric viruses in the evaluation of drinking water. Also serves as an indicator of the presence of host bacteria.
Microbial pathogens including bacteria, viruses and protozoan parasites can be physically removed with other particles in treatment units such as coagulation / flocculation, clarification and filtration, or they can be chemically eliminated by disinfection. Since physical removal processes do not remove all microorganisms from the water, disinfection is important in maintaining the microbial quality of water. To control the microbial quality of water, the disinfectant residual that remains in the drinking water in the distribution system is important. It helps preventing bacterial re-growth after treatment and limiting the development of biofilms in the water pipes.
3.4. Characteristics of Pure Water
Webster defines pure water briefly as: "The liquid which descends from the clouds in rain, and which forms rivers, lakes, seas, etc. Pure ordinary water (H20) consists of hydrogen (11.1888 %) by weight and oxygen (88.812 %). It has a slightly blue color and is very slightly compressible. At its maximum density at 39.2 °F or 4 °C, it is the standard for the specific gravities of solids and liquids. Its specific heat is the basis for the calorie and the B.T.U. units of heat. It freezes at 32 °F or 0 °C".
However; in reality "pure water" (H20) occurs so rarely, that it can be called a non-existent liquid. Even the term "pure water" is vague having different implications to individuals in different fields. For example "pure water" is a sterile liquid for bacteriologists that do not contain any living bacteria in it. The chemist, on the other hand, might classify water as "pure" when it possesses no mineral, gaseous or organic impurities. Therefore, "pure water" as described is likely to be found only in laboratories and only under ideal conditions.
Environmental Protection Agencies (states/regional/territorial) and national regulatory bodies provide practical standards for water in terms of its suitability for drinking for aesthetic considerations in Water Regulations. In these regulations, the authorities commonly take into consideration adequate protection of water against the effects of contamination, both through natural processes and through artificial treatment. The standards in these regulations list requirements for bacterial count, physical and chemical characteristics.
Any source of water to meet basic requirements for a public water supply needs some form of treatment. In general, water to be used for public water supply;
Should contain no disease-producing organisms.
Should be colorless and clear.
Should be good-tasting, free from odors and preferably cool.
Should be non-corrosive.
Should be free from gases, such as hydrogen sulfide and staining minerals, such as iron and manganese.
Should be plentiful and low in cost.
A comparative table of both WHO and EU drinking water standards can be found below (Table 3.1);
Table 3.1. WHO and EU drinking water standards
5.0 mg/l O2
Cations (positive ions)
Nitrogen (total N)
Tin (Sn) inorganic
Anions (negative ions)
0 in 250 ml
0 in 250 ml
0 in 250 ml
0 in 100 ml
0 in 100 ml
Colony count 22oC
Colony count 37oC
Chlorine dioxide (ClO2)
Desirable: Less than 5 NTU
Desirable: 15 mg/l Pt-Co
Desirable: Less than 75% of the saturation concentration
Parameters of Water Quality, Interpretations and Standards, (Environmental Protection Agency (EPA)
Guidelines for Drinking-water Quality FIRST ADDENDUM TO THIRD EDITION Volume 1 Recommendations, World Health Organisation (WHO)
National Water Quality Management Strategy, Australian Drinking Water Guidelines 6, 2011, Version 3.1 Updated March 2015, Australian Government, Netional Health and Medical Research Council, Natural Resource Management Ministerial Council
Assessing Microbial Safety of Drinking Water, Improving Approaches and Methods, Published on behalf oft he World Health Organisation for Economic Co-operation and Development (OECD) by IWA Publishing.
Cantor AF, Park JK, Vaiyavatjamai P (2000). The effect of chlorine on corrosion in drinking water systems. Final report. Midwest Technology Assistance Center, University of Illinois and Illinois State Water Survey
(http://mtac.isws.illinois.edu/mtacdocs/CorrosionFinRpt/CorrosnFnlRpt00.pdf, accessed June 2013).
Chapra, S.C. (2008), Surface Quality Modelling, Waveland press, USA.
Conroy PJ, Kings K, Olliffe T, Kennedy G, Blois S (1994). Durability and environmental impact of cement mortar linings. Swindon, Wiltshire: Water Research Centre (Report No. FR 0473)
“Drinking Water Quality Management for Catchment to Consumer, A Practical Guide for Utilities Based on Water Safety Plans”, Edited By Bob BREACH, IWA Publication.
Edwards M, Jacobs S, Dodrill DM (1999). Desktop guidance for mitigating Pb and Cu corrosion byproducts. J Am Water Works Assoc. 91(5):66–77.
“Guidelines for Drinking-water Quality” Third Edition Volume I Recommendations, World Health Organization, Geneva, 2004.
Popek, E.P. (2003), Sampling and Analysis of Environmental Chemical Pollutants, Academic Press, USA.
Pisigan RA, Singley JE (1987). Influence of buffer capacity, chlorine residual, and flow rate on corrosion of mild steel and copper. J Am Water Works Assoc. 79(2):62–70.
Sarin P, Clement JA, Snoeyink VL, Kriven WM (2003). Iron release from corroded, unlined cast-iron pipe. J Am Water Works Assoc. 95(11):85–96.
Shi B, Taylor JS (2007). Iron and copper release in drinking-water distribution systems. J Environ Health. 70(2):29–36.
Taylor, J.S., Hong, S.K. (2000), Potable Water Quality and Membrane Technology, Laboratory Medicine, 31, 10, 563-568.
Tebbutt, T.H.Y. , Principles of Water Quality Control, Pergamon Press, UK.
UNESCO, WHO, UNEP
(2000) , Water Quality Assessment: A guide to the use of biota, sediments and water in environmental monitoring, Editor: Chapman, D.
Chapman and Hall Publ., ISBN: 0-412 448 40 8, USA.
“Water safety plans: Managing drinking-water quality from catchment to consumer” Prepared by Annette Davison, Guy Howard, Melita Stevens, Phil Callan, Lorna Fewtrell, Dan Deere and Jamie Bartram, World Health Organization Geneva 2005.
Disclaimer: The European Commission support for the production of this publication does not constitute endorsement of the contents which reflects the views only of the authors, and the Commission cannot be held responsible for any use
which may be made of the information contained therein.
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Where is chi square test used in real life?
Market researchers use the Chi-Square test when they find themselves in one of the following situations: They need to estimate how closely an observed distribution matches an expected distribution. This is referred to as a “goodness-of-fit” test. They need to estimate whether two random variables are independent.
What is chi square distribution with examples?
The Chi-Square Distribution The chi square distribution is the distribution of the sum of these random samples squared . The degrees of freedom (k) are equal to the number of samples being summed. For example, if you have taken 10 samples from the normal distribution, then df = 10.
Can chi-square be used for normal distribution?
The Chi Square Test for Normality can only be used if: Your expected value for the number of sample observations for each level is greater than 5. Your data is randomly sampled. The variable you are studying is categorical.
What are the different applications of chi-square test?
Applications of Chi-square Distribution: ii) To test the ‘goodness of fit’. iii) To test the independence of attributes. iv) To test the homogeneity of independent estimates of the population variance. v) To combine various probabilities obtained from independent experiments to give a single test of significance.
What is a Chi-square test used for?
A chi-square test is a statistical test used to compare observed results with expected results. The purpose of this test is to determine if a difference between observed data and expected data is due to chance, or if it is due to a relationship between the variables you are studying.
Why do we use chi-square distribution?
It is used to describe the distribution of a sum of squared random variables. It is also used to test the goodness of fit of a distribution of data, whether data series are independent, and for estimating confidences surrounding variance and standard deviation for a random variable from a normal distribution.
How does a chi-square test work?
The chi-square test of independence works by comparing the categorically coded data that you have collected (known as the observed frequencies) with the frequencies that you would expect to get in each cell of a table by chance alone (known as the expected frequencies).
How do you perform a chi-square test in a project?
Let us look at the step-by-step approach to calculate the chi-square value:
- Step 1: Subtract each expected frequency from the related observed frequency.
- Step 2: Square each value obtained in step 1, i.e. (O-E)2.
- Step 3: Divide all the values obtained in step 2 by the related expected frequencies i.e. (O-E)2/E.
What is the chi-square test for normally distributed data?
In all cases, a chi-square test with k= 32 bins was applied to test for normally distributed data. Because the normal distribution has two parameters, c= 2 + 1 = 3
How do you test for normality in chi square test?
Chi-square Test for Normality. The chi-square goodness of fit test can be used to test the hypothesis that data comes from a normal hypothesis. In particular, we can use Theorem 2 of Goodness of Fit, to test the null hypothesis:
What is the purpose of the chi square test?
The chi-square test (Snedecor and Cochran, 1989) is used to test if a sample of data came from a population with a specific distribution. An attractive feature of the chi-square goodness-of-fit test is that it can be applied to any univariate distribution for which you can calculate the cumulative distribution function.
How do you test if data comes from a normal distribution?
Chi-square Test for Normality The chi-square goodness of fit test can be used to test the hypothesis that data comes from a normal hypothesis. In particular, we can use Theorem 2 of Goodness of Fit, to test the null hypothesis: H0: data are sampled from a normal distribution.
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What is Vibration?
So, let’s get into it. What is vibration?
In its simplest form, vibration can be considered to be the oscillation or repetitive motion of an object around an equilibrium position. The equilibrium position is the position the object will attain when the force acting on it is zero. This type of vibration is called “whole body motion”, meaning that all parts of the body are moving together in the same direction at any point in time.
The vibratory motion of a whole body can be completely described as a combination of individual motions of six different types. These are translation in the three orthogonal directions x, y, and z, and rotation around the x, y, and z-axes. Any complex motion the body may have can be broken down into a combination of these six motions. Such a body is therefore said to possess six degrees of freedom. For instance, a ship can move in the fore and aft direction (surge), up and down direction (heave), and port and starboard direction (sway), and it can rotate lengthwise (roll), rotate around the vertical axis (yaw), and rotate about the port-starboard axis (pitch).
Suppose an object were restrained from motion in any direction except one. For instance, a clock pendulum is restricted from motion except in one plane. It is therefore called a single degree of freedom system. Another example of a single degree of freedom system is an elevator moving up and down in an elevator shaft.
The vibration of an object is always caused by an excitation force. This force may be externally applied to the object, or it may originate inside the object. It will be seen later that the rate (frequency) and magnitude of the vibration of a given object is completely determined by the excitation force, direction, and frequency. This is the reason that vibration analysis can determine the excitation forces at work in a machine. These forces are dependent upon the machine condition, and knowledge of their characteristics and interactions allows one to diagnose a machine problem.
Energy and Power Considerations
Energy is required to produce vibration and in the case of machine vibration, this energy comes from the source of power to the machine. This energy source can be the AC power line, an internal combustion engine, or steam driving a turbine, etc. Energy is defined as force multiplied by the distance over which the force acts, and the SI unit of energy is the Joule. One Joule of energy is equivalent to a force of one Newton acting over a distance of one meter. The physical concept of work is similar to that of energy, and the units used to measure work are the same as those for measuring energy.
The actual amount of energy present in the machine vibration itself is usually not very great compared to the energy required to operate the machine for its intended task.
Power is defined as the rate of doing work, or the rate of energy transfer, and according to the SI, it is measured in Joules per second, or Watts. One horsepower is equivalent to 746 watts. Power is proportional to the square of the vibration amplitude, just as electrical power is proportional to the voltage squared or the current squared.
According to the law of conservation of energy, energy cannot be created or destroyed, but it can be transformed into different forms. The vibratory energy in a mechanical system is ultimately dissipated in the form of heat.
Linear and Non-Linear Systems
To assist in understanding the transmission of vibration through a machine, it is instructive to investigate the concept of linearity and what is meant by linear and non-linear systems. Thus far, we have discussed linear and logarithmic amplitude and frequency scales, but the term “linear” also refers to the characteristics of a system which can have input and output signals. A “system” is any device or structure that can accept an input or stimulus in some form and produce a corresponding output or response. Examples of systems are tape recorders and amplifiers, which operate on electrical signals, and mechanical structures, whose inputs are vibration forces, and whose outputs are vibration displacements, velocities, or accelerations.
To get around the limitations in the analysis of the wave form itself, the common practice is to perform frequency analysis, also called spectrum analysis, on the vibration signal. The time domain graph is called the waveform, and the frequency domain graph is called the spectrum. Spectrum analysis is equivalent to transforming the information in the signal from the time domain into the frequency domain. The following relationships hold between time and frequency:
A train schedule shows the equivalence of information in the time and frequency domains:
The frequency representation in this case is much shorter than the time representation. This is a “data reduction”.
Note that the information is the same in both domains, but that it is much more compact in the frequency domain. A very long schedule in time has been compressed to two lines in the frequency domain. It is a general rule of the transformation characteristic that events that take place over a long time interval are compressed to specific locations in the frequency domain.
Logarithmic Frequency Scaling
So far, the only type of frequency analysis discussed has been on a linear frequency scale, i.e., the frequency axis is set out in a linear fashion. This is suitable for frequency analysis with a frequency resolution that is constant throughout the frequency range, commonly called “narrow band” analysis. The FFT analyzer performs this type of analysis.
There are several situations where frequency analysis is desired, but narrow band analysis does not present the data in its most useful form. An example of this is acoustic noise analysis where the annoyance value of the noise to a human observer is being studied. The human hearing mechanism is responsive to frequency ratios rather than actual frequencies. The frequency of a sound determines its pitch as perceived by a listener, and a frequency ratio of two is a perceived pitch change of one octave, no matter what the actual frequencies are. For instance if a sound of 100 Hz frequency is raised to 200 Hz, its pitch will rise one octave, and a sound of 1000 Hz, when raised to 2000 Hz, will also rise one octave in pitch. This fact is so precisely true over a wide frequency range that it is convenient to define the octave as a frequency ratio of two, even though the octave itself is really a subjective measure of a sound pitch change.
This phenomenon can be summarized by saying that the pitch perception of the ear is proportional to the logarithm of frequency rather than to frequency itself. Therefore, it makes sense to express the frequency axis of acoustic spectra on a log frequency axis, and this is almost universally done. For instance, the frequency response curves that sound equipment manufacturers publish are always plotted in log frequency. Likewise, when frequency analysis of sound is performed, it is very common to use log frequency plots.
The vertical axis of an octave band spectrum is usually scaled in dB.
The octave is such an important frequency interval to the ear that so-called octave band analysis has been defined as a standard for acoustic analysis. The figure below shows a typical octave band spectrum where the ISO standard center frequencies of the octave bands are used. Each octave band has a bandwidth equal to about 70% of it center frequency. This type of spectrum is called constant percentage band because each frequency band has a width that is a constant percentage of its center frequency. In other words, the analysis bands become wider in proportion to their center frequencies.
It can be argued that the frequency resolution in octave band analysis is too poor to be of much use, especially in analyzing machine vibration signatures, but it is possible to define constant percentage band analysis with frequency bands of narrower width. A common example of this is the one-third-octave spectrum, whose filter bandwidths are about 27 % of their center frequencies. Three one-third octave bands span one octave, so the resolution of such a spectrum is three times better than the octave band spectrum. One-third octave spectra are frequently used in acoustical measurements.
A major advantage of constant percentage band analysis is that a very wide frequency range can be displayed on a single graph and the frequency resolution at the lower frequencies can still be fairly narrow. Of course, the frequency resolution at the highest frequencies suffers, but this is not a problem for some applications such as fault detection in machines.
In the chapter on machine fault diagnosis, it will be seen the narrow band spectra are very useful in resolving higher-frequency harmonics and sidebands, but for the detection of a machine fault, no such high resolution is required. The vibration velocity spectra of most machines will be found to slope downwards at the highest frequencies, and a constant percentage band (CPB) spectrum of the same data will usually be more uniform in level over a broad frequency range. This means that a CPB spectrum takes better advantage of the dynamic range of the instrumentation. One-third octave spectra are sufficiently narrow at low frequencies to show the first few harmonics of run speed, and can be used effectively for the detection of faults if trended over time.
The use of constant CPB spectra for machine monitoring is not very well recognized in industry with a few notable exceptions such as the US Navy submarine fleet.
Logarithmic Amplitude Scaling
The spectrum above plots the logarithm of the vibration level rather than the level itself.
Since this spectrum is on a log amplitude scale, multiplication by any constant value simply translates the spectrum up on the screen without changing its shape or the relationship between the components.
Multiplication of the signal level translates into addition on a log scale. This means that if the amount of amplification of a vibration signal is changed, the shape of the spectrum is not affected. This fact greatly simplifies visual interpretation of log spectra taken at different amplification factors — the curves are simply translated up or down on the graph. With a linear scaling, the shape of the spectrum changes drastically with different degrees of amplification.
The next spectrum is presented in decibels, a special type of log scaling that is very important in vibration analysis
Linear and Logarithmic Amplitude Scales
It may seem to be best to look at vibration spectra with a linear amplitude scale because that is a true representation of the actual measured vibration amplitude. Linear amplitude scaling makes the largest components in a spectrum very easy to see and to evaluate, but very small components may be overlooked completely, or are at best difficult to assign a magnitude to. The eye is able to see small components about 1/50th as large as the largest ones in the same spectrum, but anything smaller than this is essentially lost. In other words, the dynamic range of the eye is about 50 to 1
Linear scaling may be adequate in cases where the components are all about the same size, but in the case of machine vibration, beginning faults in such parts as bearings produce very small signal amplitudes. If we are to do a good job of trending the levels of these spectral components, it is best to plot the logarithm of the amplitude rather than the amplitude itself. In this way, we can easily display and visually interpret a dynamic range of at least 5000 to 1, or more than 100 times better than the linear scaling allows.
To illustrate different types of amplitude presentations, the same vibration signature will be shown in linear and two different types of logarithmic amplitude scales.
It might be said that the dynamic range of the eye, when looking at linear spectra, is about 34 dB.
Linear Amplitude Scaling
Note that this linear spectrum shows the larger peaks very well, but lower level information is missing. In the case of machine vibration analysis, we are often interested in the smaller components of the spectrum, i.e., in the case of rolling element bearing diagnosis. This subject will be covered in detail in the chapter on Machine Vibration Monitoring.
The decibel (dB) is defined by the following expression:
where: LdB = The signal level in dB L1 = Vibration level in Acceleration, Velocity, or Displacement
Lref = Reference level, equivalent to 0 dB
The Bell Telephone Labs introduced the concept of the decibel before 1930. It was first used to measure relative power loss and signal to noise ratio in telephone lines. It was soon pressed into service as a measure of acoustic sound pressure level.
The vibration velocity level in dB is abbreviated VdB, and is defined as:
The Systeme Internationale, or SI, is the modern replacement for the metric system.
The reference, or “0 dB” level of 10-9 meter per sec is sufficiently small that all our measurements on machines will result in positive dB numbers. this standardized reference level uses the SI, or “metric,” system units, but it is not recognized as a standard in the US and other English-speaking countries. (The US. Navy and many American industries use a zero dB reference of 10-8 m/sec, making their readings higher than SI readings by 20 dB.)
The VdB is a logarithmic scaling of vibration magnitude, and it allows relative measurements to be easily made. Any increase in level of 6 dB represents a doubling of amplitude, regardless of the initial level. In like manner, any change of 20 dB represents a change in level by a factor of ten. Thus any constant ratio of levels is seen as a certain distance on the scale, regardless of the absolute levels of the measurements. This makes it very easy to evaluate trended vibration spectral data; 6 dB increases always indicate doubling of the magnitudes.
dB Values vs. Amplitude Level Ratios
The following table relates dB values to amplitude ratios:
|Linear Level Ratio
|Linear Level Ratio
It is strongly recommended that VdB be used as the vibration amplitude scaling because so much more information is available to the viewer compared to linear amplitude units. Also, compared to a conventional log scale, the dB scale is much easier to read.
Acceleration and Displacement can also be expressed on dB scales. The AdB scale is the most used one, and its zero reference is set 1 micro G, commonly abbreviated G.
It turns out that AdB = VdB at 159.2 Hz. VdB levels, AdB levels, and DdB levels are related by the following formulas:
Any vibration parameter — displacement, velocity, or acceleration can be displayed on a dB scale. The reference quantities for 0 dB on these scales were chosen such that the dB levels of all three quantities are the same at a frequency of 159.2 Hz, which is equal to 1000 radians per second.
Acceleration and Velocity in linear units are calculated from dB levels as follows:
It is convenient to remember the following rule of thumb: At 100 Hz, 1G = 120 AdB = 124 VdB = 2.8 mils p-p.
Note that the time domain wave form is always represented in linear amplitude units – it is not possible to use a log scale in the wave form plot because some of the values are negative, and the logarithm of a negative number is not defined.
VdB Levels vs. Vibration Levels in ips
Peak level is the de facto standard unit for vibration velocity measurements, even though RMS level would make more sense in most cases.
Following is a convenient conversion table for relating VdB levels to inches per second peak:
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Normal Force in Sliding Friction
by Ron Kurtus
The normal force in sliding friction is the perpendicular force pushing the object to the surface on which it is sliding. It is an essential part of the standard sliding friction equation.
That force can be due to the weight of an object or that caused by an external push.
When the weight is on an incline, the normal force is reduced by the cosine of the incline angle.
Questions you may have include:
- What is the standard friction equation?
- When it weight the normal force?
- What are examples of external normal force?
This lesson will answer those questions. Useful tool: Units Conversion
Standard sliding friction equation
The normal force is seen in the standard sliding friction equation:
Fs = μsN
N = Fs/μs
- N is the normal or perpendicular force pushing the two objects together
- Fs is the sliding force of friction
- μs is the sliding coefficient of friction for the two surfaces (Greek letter "mu")
(See Standard Friction Equation for details.)
Static and kinetic coefficients
The sliding coefficient of friction can be static when the object is stationary or kinetic when the object is sliding over the other surface.
The coefficient of sliding friction in the static mode of motion (μss) is greater that the coefficient in the kinetic or moving mode (μks).
μss > μks
(See Coefficient of Sliding Friction for more information.)
Weight as normal force
The normal force N can be the weight of an object as caused by gravity. This would apply in situations where you slide a heavy object across the floor or some horizontal surface.
Since weight is the force pushing the objects together, the friction equation becomes:
Fs = μsW
where W is the weight of the object.
Thus if a box weighs 100 pounds and the coefficient of friction between it and the ground is 0.7, then the force required to push the box along the floor is 70 pounds.
Likewise if a box weighs 500 newtons is placed on ice with a coefficient of friction of only 0.001, then it would only take 0.5 newtons to move the box.
Weight on incline
If the weight is on an incline, the normal force will be reduced by the cosine of the incline angle. The equation is
N = W*cos(β)
- N is the normal force on the incline
- W is the weight
- β is the incline angle (Greek letter beta)
- cos(β) is the cosine of the angle β
- W*cos(β) is W times cos(β)
Thus, the friction equation is:
Fs = μW*cos(β)
An illustration of the friction on a box on an incline is:
Normal force is weight times cosine of angle
(See Sliding Friction on an Inclined Surface for more information)
External normal force
Examples of external normal forces include pushing a sanding block on an object and a pair of pliers.
Pushing object sideways
If you push a sanding block against a wooden desk you were sanding, the normal force would be the amount of force you pushed on the block. You would move the sanding block in one direction and the force of friction would be in the opposite direction.
Applying normal force on sanding block and wooden desk
Two normal forces
Sometimes, two normal forces are used to cause the friction.
One example is a pair of pliers that applies a normal force on both sides of a piece of wood that the pair of pliers is holding. Another example are the calipers on automobile disc brakes that apply a force on both sides of the metal disc to slow down the car.
The normal force in the standard friction equation is the force pushing the two objects together, perpendicular to their surfaces. That force can be due to the weight of an object or that caused by an external push. When the weight is on an incline, the normal force is reduced by the cosine of the incline angle.
It all makes sense
Resources and references
Friction Resources - Extensive list
Friction Concepts - HyperPhysics
(Notice: The School for Champions may earn commissions from book purchases)
Friction Science and Technology (Mechanical Engineering Series) by Peter J. Blau; Marcel Dekker Pub. (1995) $89.95
Control of Machines with Friction (The International Series in Engineering and Computer Science) by Brian Armstrong-Hélouvry; Springer Pub. (1991) $179.00
Students and researchers
The Web address of this page is:
Please include it as a link on your website or as a reference in your report, document, or thesis.
Where are you now?
Normal Force in Sliding Friction
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I’m sure we’ve all been there before – that sinking feeling when you think you’ve just missed out on the next grade up by a fraction of a point. So, what’s the verdict? Does an 89.5 round up to a 90? It’s a question that’s caused quite a bit of controversy amongst students and teachers alike.
You might think that rounding up is a simple and fair way of determining grades, but it’s not quite that straightforward. Depending on where you’re studying, your professor, school or university might have different rules when it comes to rounding up. Some institutions might round up grades that fall within the half-point, while others might require you to have at least a full percentage point before bumping up your grade.
It’s important to remember that not all institutions have the same policies concerning rounding up grades. Additionally, some students might feel that the practice is unfair because it doesn’t account for the fact that an additional 0.5% could mean the difference between a pass and a fail. So why the discrepancy? Let’s delve deeper into the issue and find out if an 89.5 does indeed round up to a 90.
The Concept of Rounding
When dealing with numbers, rounding is a common practice that is used to simplify and approximate values. Rounding refers to the process of reducing a number to a certain degree of accuracy, typically by replacing it with another number that is a close approximation considering the scale of measurement being used. For example, rounding can be used to simplify a decimal number to a whole number or to a specific number of decimal places.
There are several reasons why rounding may be necessary or useful. One reason is to make calculations and comparisons easier. For instance, if you are working with a large set of decimal numbers, rounding can make them easier to compare and add. Rounding can also be helpful in reducing errors caused by minor variations in the accuracy of measurements or calculations. In finance, rounding is often used to standardize currency values and estimates based on demand and supply data.
- There are different types of rounding methods, and the choice of which one to use depends on the context and desired outcome.
- One common rounding method is to round to the nearest whole number. In this method, any number that is 0.5 or greater is rounded up to the next higher whole number, while any number that is less than 0.5 is rounded down to the current whole number. For example, 89.5 is rounded up to 90, while 89.4 is rounded down to 89.
- Another rounding method is to round to a certain number of decimal places. In this method, the number is rounded off to the specified number of digits after the decimal point, with the last digit being rounded up if it is 5 or greater. For example, 4.6752 rounded to two decimal places is 4.68.
Overall, rounding is a valuable tool that can simplify calculations and lead to more accurate results. By understanding the different methods of rounding and when to apply them, you can improve your ability to work with numbers and achieve the results you need with greater ease and precision.
Rules for Rounding in Mathematics
Mathematics is all about precision, and rounding plays an important role in ensuring the right level of accuracy. Whether you are balancing your checkbook or calculating your taxes, rounding can help simplify complex calculations. In this article, we will explore the rules for rounding in mathematics.
The Two Types of Rounding
- Round Half Up: This method rounds the number to the nearest integer. If the number is exactly halfway between two integers, it rounds up to the nearest even integer. For example, if you were rounding 2.5, it would round up to 3. However, if you were rounding 3.5, it would round down to 4.
- Round Half Down: This method rounds the number to the nearest integer. If the number is exactly halfway between two integers, it rounds down to the nearest even integer. For example, if you were rounding 2.5, it would round down to 2. However, if you were rounding 3.5, it would round up to 3.
Rules for Rounding Numbers
When rounding numbers, there are a few rules to follow:
- Identify the digit that you want to round. This is usually the last digit of the number.
- Look at the digit to the right of the one you want to round. If it is 5 or greater, round up. If it is less than 5, round down.
- If the digit you want to round is 5, use the round half up or round half down method to determine whether to round up or down.
- When rounding decimals, count the number of decimal places to determine the position of the digit you want to round.
Let’s look at a few examples of how to round numbers:
|Number to Round
|Round Half Up
|Round Half Up
|Round Half Down to the tenths place
Remember, rounding can be a useful tool in mathematics, but it is important to understand the rules and apply them correctly to ensure accurate calculations.
Rounding in different numbering systems
When it comes to rounding, it’s not just about dealing with whole numbers. In fact, rounding can also be done in different numbering systems such as binary, hexadecimal, and octal.
Let’s take a look at how rounding works in these different numbering systems.
Rounding in binary
- In binary, there are only two digits – 0 and 1.
- Rounding in binary is similar to rounding in decimal. The only difference is that we only have two digits to work with.
- If we want to round 1011.1111 to the nearest whole number, we would look at the digit to the right of the decimal point. In this case, it’s 1.
- If the digit to the right of the decimal point is 5 or greater, we would round up to the next whole number. If it’s less than 5, we would round down.
- So in this case, we would round up to 1100.
Rounding in hexadecimal
Hexadecimal is a numbering system that uses 16 digits – 0 to 9 and A to F.
- When rounding in hexadecimal, we would follow the same rules as rounding in decimal and binary.
- If we want to round 2F8.45 to the nearest whole number, we would look at the digit to the right of the decimal point. In this case, it’s 4.
- If the digit to the right of the decimal point is 8 or greater, we round up. If it’s less than 8, we round down.
- So in this case, we would round up to 2F8.
Rounding in octal
Octal is a numbering system that uses 8 digits – 0 to 7.
|Number to Round
When rounding in octal, we would also follow the same rules as rounding in decimal and binary.
To round 75.67 to the nearest whole number, we would again look at the digit to the right of the decimal point. In this case, it’s 6.
- If the digit to the right of the decimal point is 4 or less, we would round down to the nearest whole number.
- If the digit to the right of the decimal point is 5 or greater, we would round up to the next whole number.
- So in this case, we would round up to 76.
Significant figures and rounding
When it comes to numbers, there are rules that govern how we round them off and how many significant figures we should use. Without understanding these rules, we may end up with incorrect results that can have serious implications in fields such as science and engineering. In this article, we will shed light on rounding and significant figures.
- What are significant figures? Significant figures are the digits that are considered reliable in a number and carry meaning. Typically, significant figures do not include leading or trailing zeros.
- How do we determine the number of significant figures? To determine the number of significant figures in a number, start counting from the first non-zero digit and stop when you reach the last digit that carries meaning. For example, in the number 89.5, there are three significant figures.
- Why are significant figures important? Significant figures are vital because they dictate the level of precision in a number. If we use too many or too few significant figures, we can end up with incorrect results.
Now that we have a basic understanding of significant figures, let’s talk about rounding.
What is rounding? Rounding is the process of changing a number to make it more manageable or to fit a particular set of criteria. For instance, if we have the number 89.5, we may round it off to the nearest whole number, which is 90.
How do we round a number? The process of rounding depends on the set of criteria that we want to meet. For example, if we want to round off 89.5 to the nearest whole number, we check the first non-zero digit after the decimal point, which is 5. Since 5 is greater than 4 (the rounded threshold), we round up to the next whole number, which is 90.
Although it may seem simple, there are some instances where rounding can become complicated. In these cases, we may need to refer to tables to determine the appropriate behavior. For example, in statistical calculations, we use different tables to determine the correct rounding behavior depending on the type of calculation and the level of precision required.
|Five or less
|Ten or less
|Five or more
|Ten or more
As you can see, rounding is not always straightforward and requires a certain level of expertise to ensure accuracy.
In conclusion, rounding and significant figures are essential concepts in mathematics and science. Proper understanding of these concepts is vital in ensuring accurate calculations and results, hence avoiding costly errors.
How to Round Decimals
Many people struggle when it comes to rounding decimals. It can be confusing to know whether to round up or down. In this article, we will cover the basics of rounding decimals, including what to do when rounding up a 5.
The Number 5
- When rounding up, the general rule is to round up if the decimal is greater than or equal to 0.5.
- However, what do you do when the decimal is exactly 0.5? This is when the number 5 comes into play.
- When rounding up a 5, you should always round up. For example, 4.5 rounds up to 5.
- It’s important to note that this rule applies even when the number before the 5 is odd. For example, 3.5 rounds up to 4.
Remembering this rule can save you from unnecessary confusion or errors when rounding decimals.
General Rounding Rules
Aside from the number 5, there are general rules to follow when rounding decimals.
- When rounding to a whole number, look at the digit to the right of the decimal point. Round up if it is 5 or greater, and round down if it is 4 or less.
- When rounding to a specific decimal place, look at the digit in that place. If it is 5 or greater, round up; if it is 4 or less, round down.
It’s important to keep in mind that these rules apply to standard rounding. Certain fields may have different rules, such as always rounding up or always rounding down.
Here are some examples to help illustrate these rules:
|Number to Round
|Round to two decimal places
|Round to the nearest whole number
|Round to the nearest whole number
By following these rules and keeping the number 5 in mind, you can successfully round decimals in any situation.
Rounding to the nearest whole number
When it comes to rounding numbers to the nearest whole number, there are certain rules that must be followed. The most basic rule of rounding is if the number to be rounded is less than 0.5, it will be rounded down. If the number is equal to or greater than 0.5, it will be rounded up.
- For example, if we round 2.4 to the nearest whole number, the answer would be 2 because 0.4 is less than 0.5.
- However, if we round 3.6 to the nearest whole number, the answer would be 4 because 0.6 is equal to or greater than 0.5.
- If the number to be rounded ends in 0.5, it will be rounded up to the nearest even whole number. This method is also called “banker’s rounding”.
Another thing to keep in mind when rounding to the nearest whole number is the concept of significant digits. Significant digits are numbers that are important to the measurement and accuracy of a number and must be included in the final rounded value.
For example, if we have a measurement of 45.789 and we want to round it to the nearest whole number, the answer would be 46 because the number 5 is significant and we must round up.
It’s important to understand the rules of rounding and significant digits when dealing with decimal numbers and the rounding process to ensure accurate and precise calculations.
Rounding to the nearest tenth, hundredth, or thousandth
Rounding is a common practice that we use in our everyday life. Whether we are calculating our grocery bills or our income tax, rounding off numbers is an essential technique that helps us get a better idea of the values we are dealing with. When we round off numbers, we are simplifying them to a convenient level and making them easier to work with.
Rounding to the nearest tenth, hundredth, or thousandth
- Rounding to the nearest tenth: This involves rounding a number to the nearest multiple of ten. For example, 89.5 rounded to the nearest tenth becomes 89.6, as the digit in the hundredths place (5) is closer to 6 than it is to 5.
- Rounding to the nearest hundredth: This involves rounding a number to the nearest multiple of 0.01. For example, 89.525 rounded to the nearest hundredth becomes 89.53, as the digit in the thousandths place (5) is closer to 3 than it is to 2.
- Rounding to the nearest thousandth: This involves rounding a number to the nearest multiple of 0.001. For example, 89.5245 rounded to the nearest thousandth becomes 89.525, as the digit in the ten-thousandths place (5) is closer to 5 than it is to 4.
Rounding to the nearest tenth, hundredth, or thousandth
It is important to remember certain rules when rounding numbers. For instance, if the digit in the place being rounded is less than 5, we round down. If the digit is 5 or greater, we round up. However, if the digit in the place being rounded is exactly 5, then we need to look at the digit to the right of it. If that digit is odd, we round up, but if it is even, we round down.
The table below shows some examples of rounding numbers to the nearest tenth, hundredth, and thousandth:
|Rounded to nearest tenth
|Rounded to nearest hundredth
|Rounded to nearest thousandth
Rounding off numbers may seem like a small part of mathematical calculations, but it can make a significant difference in the final outcome. It allows us to simplify large numbers, work with easier-to-manage figures in our daily life, and make accurate calculations with greater ease.
Rounding up vs. rounding down
When it comes to rounding numbers, there are two options: rounding up or rounding down. Rounding up is when a number is increased to the nearest higher value, while rounding down is when a number is decreased to the nearest lower value.
- Round up: When a number ends in .5 or greater, it rounds up. For example, 6.5 rounds up to 7.
- Round down: When a number ends in .4 or less, it rounds down. For example, 6.4 rounds down to 6.
But what about the number 89.5? Does it round up to 90 or down to 89?
As you can see from the table, 89.5 rounds up to 90. This is because the number ends in .5 and therefore meets the criteria for rounding up. So, if you are ever faced with the question of whether 89.5 rounds up or down, the answer is up!
Rounding Errors and Their Impact
When dealing with decimals and fractions, rounding is a common practice. However, rounding can sometimes lead to errors and unexpected results.
The Number 9
- The number 9 plays a significant role in rounding. When a number ends in 9, the rounding process can be ambiguous.
- For example, if we round 89.5 to the nearest whole number, the result would be 90. However, if we round 88.5 to the nearest whole number, the result would be 89.
- This is because, in the first case, the tenths place is 5 or greater, and we round up to the next whole number. But in the second case, the tenths place is less than 5, so we round down.
- It’s important to consider the context and intended use of the rounded number when dealing with cases like these.
Overall, the number 9 can complicate rounding and lead to errors in calculations if not approached carefully.
Rounding in practical applications, such as accounting and measurement.
When it comes to practical applications, rounding plays an important role in areas such as accounting and measurement. Rounding helps in simplifying the data for better analysis and decision making. It eliminates the need for dealing with numbers that are too complex and avoids the risk of errors that can occur due to the presence of many decimal places.
- Rounding in Accounting: In accounting, rounding is used to round off financial statements such as income statements and balance sheets. It makes the data more readable to the stakeholders and helps them understand the financial performance of the company. For instance, if the income statement shows a net income of $89.5 million, it would be rounded off to $90 million to simplify the data and make it more comprehensible.
- Rounding in Measurement: In the field of measurement, rounding is used to limit the number of decimal places to a certain degree of accuracy. For instance, when weighing an object on a scale, if the scale measures the weight up to one-hundredth of a gram, it may be rounded off to the nearest tenth of a gram for practical purposes. This simplifies the data and makes it easier to work with.
- Rounding in Taxation: Rounding also plays a significant role in taxation. Governments use specific rules for rounding off tax calculations to ensure that the rounded figure doesn’t significantly affect the calculations of tax liability. This helps to avoid tax fraud and ensures that taxation calculations are consistent and fair for all taxpayers.
Understanding rounding is crucial in many fields as it helps us make more informed decisions based on the simplified and accurate data. However, it’s important to keep in mind that rounding can sometimes result in errors and should be used judiciously.
The table above shows an example of rounding. When rounding the number 89.5 to the nearest whole number, it would be rounded up to 90 using the original rounding method. However, using the half-up rounding method, it would also be rounded up to 90. This demonstrates that rounding methods may vary, and it’s important to know which method is appropriate to use in different situations.
Does an 89.5 Round Up to a 90? FAQs
1. Can an 89.5 grade be rounded up to 90?
Yes, an 89.5 grade can be rounded up to 90 since it is greater than or equal to 89.5 and less than 90.5.
2. What is the rounding rule for grades between 0.1 and 0.4?
For grades between 0.1 and 0.4, they should be rounded down to the nearest whole number. Therefore, an 89.4 grade would round down to 89.
3. What is the rounding rule for grades between 0.5 and 0.9?
For grades between 0.5 and 0.9, they should be rounded up to the nearest whole number. Therefore, an 89.5 grade would round up to 90.
4. Will rounding grades affect my GPA?
Yes, rounding grades can affect your GPA since it changes the numerical value of your grade. For example, an 89.5 grade would be a 3.5 GPA if not rounded, but it would be a 4.0 GPA if rounded up to 90.
5. How do I know if my teacher or professor rounds grades?
You can ask your teacher or professor if they round grades or if they follow a strict no-rounding policy. It’s always best to clarify their grading system beforehand to avoid any confusion.
6. Is rounding up grades fair?
Whether rounding up grades is fair or not is subjective and can vary depending on who you ask. Some argue that it rewards students for being close to the next highest grade, while others argue that it lowers the standard for what constitutes an excellent grade.
7. What are some other common rounding rules for grades?
Some other common rounding rules for grades include rounding to one decimal place, rounding to the nearest quarter, or rounding to the nearest half point.
Thanks for reading our FAQs on whether an 89.5 grade rounds up to a 90. We hope we were able to provide some helpful information for you. Remember to clarify with your teacher or professor whether they round grades or not and to keep striving for academic excellence. Come back soon for more educational content!
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A sphere has a radius of 5.5 cm. Determine its volume and surface area. A frustum of the sphere is formed by two parallel planes. One through the diameter of the curved surface of the frustum is to be of the surface area of the sphere. Find the height and volume of the frustum.
Did you find an error or inaccuracy? Feel free to write us. Thank you!
Tips for related online calculators
You need to know the following knowledge to solve this word math problem:
Related math problems and questions:
- Cross-sections of a cone
Cone with base radius 16 cm and height 11 cm divided by parallel planes to base into three bodies. The planes divide the height of the cone into three equal parts. Determine the volume ratio of the maximum and minimum of the resulting body.
- A lamp
A lamp shade like that of a frustum has a height of 12 cm and an upper and lower diameter of 10 cm and 20 cm. What area of materials is required to cover the curved surface of the frustum?
- A cone 4
A cone of radius 10 cm is divided into two parts by drawing a plane through the midpoint of its axis, parallel to its base. Compare the volumes of the two parts.
- A cone 2
A cone has a slant height of 10 cm and a square curved surface area of 50 pi cm. Find the base radius of the cone.
- A concrete pedestal
A concrete pedestal has a shape of a right circular cone having a height of 2.5 feet. The diameter of the upper and lower bases are 3 feet and 5 feet, respectively. Determine the lateral surface area, total surface area, and the volume of the pedestal.
- Quadrilateral 81385
A regular quadrilateral pyramid with base edge length a = 15cm and height v = 21cm is given. We draw two planes parallel to the base, dividing the height of the pyramid into three equal parts. Calculate the ratio of the volumes of the 3 bodies created.
- The block
The block, the edges formed by three consecutive GP members, has a surface area of 112 cm². The sum of the edges that pass through one vertex is 14 cm. Calculate the volume of this block.
- The diagram 2
The diagram shows a cone with a slant height of 10.5cm. If the curved surface area of the cone is 115.5 cm². Calculate to correct three significant figures: *Base Radius *Height *Volume of the cone
- Curved surface area CSA
A cylinder 5cm high has a base radius(7/2) cm. Calculate the curved surface area.
- Truncated cone
A truncated cone has bases with radiuses 40 cm and 10 cm and a height of 25 cm. Calculate its surface area and volume.
- Juice-soaked 17173
1. Find the dimensions of a 5-liter cylindrical container if the height of the container is equal to the radius of the base. 2. There is three dl of juice in a cylindrical glass with an inner diameter of 8 cm. Calculate the area of the juice-soaked port
- Calculate 73534
The sphere has a diameter of 70 cm. Calculate its surface area and volume.
- Cylindrical 16713
Twenty identical steel balls were dropped into a cylindrical container of water standing on a horizontal surface to submerge them below the surface. At the same time, the water level rose by 4 mm. Determine the radius of one sphere if the diameter of the
- Determine 81311
The surface of the rotating cone and its base area is in the ratio 18:5. Determine the volume of the cone if its body height is 12 cm.
- Frustrum - volume, area
Calculate the surface and volume of the truncated cone. The radius of the smaller figure is 4 cm, the height of the cone is 4 cm, and the side of the truncated cone is 5 cm.
- Spherical 81527
Sketch a spherical layer formed from a sphere with a radius of r= 8.5cm, given: v=1.5cm, r1=7.7cm, r2=6.8cm. What is its volume?
- Sphere slices
Calculate the volume and surface of a sphere if the radii of a parallel cut r1=32 cm, r2=47 cm, and its distance v=21 cm.
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STEP 4 Review the Knowledge You Need to Score High
12 Electric Circuits
IN THIS CHAPTER
Summary: Electric charge flowing through a wire is called current. An electrical circuit is built to control the current. In this chapter, you will learn how to predict the effects of current flow. Besides discussing circuits in general, this chapter presents a problem-solving technique: the V-I-R chart, an incredibly effective way to organize a problem that involves circuits.
The physics concepts of conservation of charge and conservation of energy, as expressed in Kirchhoff’s rules, explain the behavior of circuits.
Current is the flow of charge through the circuit.
A voltage source, such as batteries, generators, and solar cells, supply the energy required to create current.
Resistance is a restriction to the flow of current.
Resistance depends on the geometry of the resistor.
Not all resistors and circuit elements are ohmic.
Real batteries are not perfect and have internal resistance.
The current through series resistors is the same through each, whereas the voltage across series resistors adds to the total voltage.
The voltage across parallel resistors is the same across each, whereas the current through parallel resistors adds to the total current.
The brightness of a lightbulb depends on the power dissipated by the bulb.
A capacitor in a circuit blocks current and stores charge.
Changing the components in a circuit, like opening or closing a switch, has effects on the circuit that can be predicted.
The last chapter talked about situations where electric charges don’t move around very much. Isolated point charges, for example, just sit there creating an electric field. But what happens when you get a lot of charges all moving together? That, at its essence, is what goes on in a circuit.
A circuit is simply any path that will allow charge to flow.
Current: The flow of positive electric charge. In a circuit, the current is the amount of charge passing a given point per unit time.
A current is defined as the flow of positive charge. We don’t think this makes sense, because electrons—and not protons or positrons—are what flow in a circuit. But physicists have their rationale, and no matter how wacky, we won’t argue with it—the AP exam always uses this definition.
In more mathematical terms, current is defined as follows:
What this means is that the current, I, equals the amount of charge flowing past a certain point divided by the time interval during which you’re making your measurement. This definition tells us that current is measured in coulombs/second. 1 C/s = 1 ampere, abbreviated as 1 A.
Resistance and Ohm’s Law
You’ve probably noticed that just about every circuit drawn in your physics book contains a battery. The reason most circuits contain a battery is because batteries create a potential difference between one end of the circuit and the other. In other words, if you connect the terminals of a battery with a wire, the part of the wire attached to the “+” terminal will have a higher electric potential than the part of the wire attached to the “−” terminal. And positive charge flows from high potential to low potential. So, in order to create a current, you need a battery.
In general, the greater the potential difference between the terminals of the battery, the more current flows.
The amount of current that flows in a circuit is also determined by the resistance of the circuit.
Resistance: A property of a circuit that resists the flow of current.
Resistance is measured in ohms. One ohm is abbreviated as 1 Ω.
If we have some length of wire, then the resistance of that wire can be calculated. Three physical properties of the wire affect its resistance:
• The material the wire is made out of: the resistivity, ρ, of a material is an intrinsic property of that material. Good conducting materials, like gold, have low resistivities.1
• The length of the wire, L: the longer the wire, the more resistance it has.
• The cross-sectional area A of the wire: the wider the wire, the less resistance it has.
We put all of these properties together in the equation for resistance of a wire:
Now, this equation is useful only when you need to calculate the resistance of a wire from scratch. Usually, on the AP exam or in the laboratory, you will be using resistors that have a pre-measured resistance.
Resistor: Something you put in a circuit to change the circuit’s resistance.
Resistors are typically ceramic, a material that doesn’t allow current to flow through it very easily. Another common type of resistor is the filament in a light bulb. When current flows into a light bulb, it gets held up in the filament. While it’s hanging out in the filament, it makes the filament extremely hot, and the filament gives off light.
To understand resistance, an analogy is helpful. A circuit is like a network of pipes. The current is like the water that flows through the pipes, and the battery is like the pump that keeps the water flowing. If you wanted to impede the flow, you would add some narrow sections to your network of pipes. These narrow sections are your resistors.
The way that a resistor (or a bunch of resistors) affects the current in a circuit is described by Ohm’s law.
ΔV is the voltage across the part of the circuit you’re looking at, I is the current flowing through that part of the circuit, and R is the resistance in that part of the circuit. Ohm’s law is the most important equation when it comes to circuits, so make sure you know it well.
When current flows through a resistor, electrical energy is being converted into heat energy. The rate at which this conversion occurs is called the power dissipated by a resistor. This power can be found with the equation
This equation says that the power, P, dissipated in part of a circuit, equals the current flowing through that part of the circuit multiplied by the voltage across that part of the circuit.
Using Ohm’s law, it can be easily shown that . It’s only worth memorizing the first form of the equation, but any one of these could be useful.
Ohmic Versus Nonohmic
A circuit component that maintains the same resistance even when the voltage across it, or the current through it, are changed is said to be “ohmic.” Ohmic simply means that the resistance is constant. The exam writers like to ask questions about this. Look at this table of data showing the voltage and current through two resistors. Are the resistors ohmic?
An easy way to check is to rearrange Ohm’s law to solve for resistance and calculate the resistance .
First data point:
Second data point: . Close . . . let’s check another point.
Third data point: . Stop!
Resistor #1 is nonohmic, because the resistance does not stay constant.
When we repeat the same calculation for Resistor #2, we find that the resistance stays constant at 20 Ω. Resistor #2 is ohmic.
Another quick way to make this ohmic check is to graph the data. In a lab, you could vary the voltage applied to the resistor and measure the current. This would make ΔV the independent variable and I the dependent variable. Rearranging Ohm’s law we would get: , which means that plotting ΔV on the x-axis and I on the y-axis will result in a slope of 1/R. The slope will be constant (straight line) if the resistor is ohmic. In the above graph (a) is an ohmic material and graph (b) is a nonohmic material.
Resistors in Series and in Parallel
In a circuit, resistors can be arranged either in series with one another or parallel to one another. Before we take a look at each type of arrangement, though, we need first to familiarize ourselves with circuit symbols, shown next.
First, let’s examine resistors in series. In this case, all the resistors are connected in a line, one after the other after the other. In other words, there is only one pathway for the current to travel through.
To find the equivalent resistance of series resistors, we just add up all the individual resistors.
For the circuit in the above figure, Req = 3000 Ω. In other words, using three 1000-Ω resistors in series produces the same total resistance as using one 3000-Ω resistor.
Parallel resistors are connected in such a way that you create several paths through which current can flow. For the resistors to be truly in parallel, the current must split, go through only one resistor in each pathway, and then immediately come back together.
The equivalent resistance of parallel resistors is found by this formula:
For the circuit in the next figure, the equivalent resistance is 333 Ω. So hooking up three 1000-Ω resistors in parallel produces the same total resistance as using one 333-Ω resistor. (Note that the equivalent resistance of parallel resistors is less than any individual resistor in the parallel combination.)
A Couple of Important Rules
Rule #1—When two resistors are connected in SERIES, the amount of current that flows through one resistor equals the amount of current that flows through the other resistor and is equal to the total current through both resistors.
Rule #2—When two resistors are connected in PARALLEL, the voltage across one resistor is the same as the voltage across the other resistor and is equal to the total voltage across both resistors.
The reason why the current is the same for everything connected in a series is because of conservation of charge. Remember that charge is carried by real things, protons and electrons, and they won’t just disappear or be created in a circuit. Since there are no branches in a series pathway, all the charge that enters this single pathway must pass through every component in that pathway.
The reason why voltage is the same for resistors connected in parallel is due to conservation of energy. The voltage (energy per charge) used up in each parallel pathway must be the same. Think of this analogy: water in a river splits along parallel paths to flow around a large island in the middle of the stream. One path may go through rapids with rocks and drop off a waterfall, while the other path around the island may be calm and smooth. Even though the paths are different, the two paths begin and end together at the same height (the same gravitational potential). The same is true for current flowing through parallel paths in a circuit. No matter how different the paths may be, the current must begin and end at the same electric potential. Thus ΔV for parallel paths is always the same.
We will discuss conservation of charge and conservation of energy more a bit later in the section titled Kirchhoff’s Rules. For now, let’s discover a very useful tool: V-I-R charts.
The V-I-R Chart
Here it is—the trick that will make solving circuits a breeze. Use this method on your homework. Use this method on your quizzes and tests. But most of all, use this method on the AP exam. It works.
The easiest way to understand the V-I-R chart is to see it in action, so we’ll go through a problem together, filling in the chart at each step along the way.
Find the voltage across each resistor in the circuit shown below.
We start by drawing our V-I-R chart, and we fill in the known values. Right now, we know the resistance of each resistor, and we know the total voltage (it’s written next to the battery).
Next, we simplify the circuit. This means that we calculate the equivalent resistance and redraw the circuit accordingly. We’ll first find the equivalent resistance of the parallel part of the circuit:
Use your calculator to get .
Taking the reciprocal, we get
So we can redraw our circuit like this:
Next, we calculate the equivalent resistance of the entire circuit. Following our rule for resistors in series, we have
We can now fill this value into the V-I-R chart.
Notice that we now have two of the three values in the “Total” row. Using Ohm’s law, we can calculate the third. That’s the beauty of the V-I-R chart: Ohm’s law is valid whenever two of the three entries in a row are known.
Then we need to put on our thinking caps. We know that all the current that flows through our circuit will also flow through R1. (You may want to take a look back at the original drawing of our circuit to make sure you understand why this is so.) Therefore, the I value in the “R1” row will be the same as the I in the “Total” row. We now have two of the three values in the “R1” row, so we can solve for the third using Ohm’s law.
Finally, we know that the voltage across R2 equals the voltage across R3, because these resistors are connected in parallel. The total voltage across the circuit is 12 V, and the voltage across R1 is 6.5 V. So the voltage that occurs between R1 and the end of the circuit is
Therefore, the voltage across R2, which is the same as the voltage across R3, is 5.5 V. We can fill this value into our table. Finally, we can use Ohm’s law to calculate I for both R2 and R3. The finished V-I-R chart looks like this:
To answer the original question, which asked for the voltage across each resistor, we just read the values straight from the chart.
Now, you might be saying to yourself, “This seems like an awful lot of work to solve a relatively simple problem.” You’re right—it is.
However, there are several advantages to the V-I-R chart. The major advantage is that, by using it, you force yourself to approach every circuit problem exactly the same way.
So when you’re under pressure—as you will be during the AP exam—you’ll have a tried-and-true method to turn to.
Also, if there are a whole bunch of resistors, you’ll find that the V-I-R chart is a great way to organize all your calculations. That way, if you want to check your work, it’ll be very easy to do.
Tips for Solving Circuit Problems Using the V-I-R Chart
• First, enter all the given information into your chart. If resistors haven’t already been given names (like “R1”), you should name them for easy reference.
• Next, simplify the circuit to calculate Req, if possible.
• Once you have two values in a row, you can calculate the third using Ohm’s law. You CANNOT use Ohm’s law unless you have two of the three values in a row.
• Remember that if two resistors are in series, the current through one of them equals the current through the other. And if two resistors are in parallel, the voltage across one equals the voltage across the other.
Kirchhoff’s rules are expressions of conservation of charge and conservation of energy. They are useful in any circuit but especially in complicated circuits such as circuits with multiple resistors and batteries. Kirchhoff’s rules say:
1. Positive (+) currents entering a junction plus the negative (−) currents leaving a junction equal 0 (ΣIjunction = 0). Or, at any junction the current entering must equal the current leaving.
2. The sum of the voltages around a closed loop is zero (ΣΔVloop = 0).
The first law is called the “junction rule,” and the second is called the “loop rule.” To illustrate the junction rule, we’ll revisit the circuit from our first problem.
According to the junction rule, whatever current enters junction A must also leave junction A. So let’s say that a current of 1.3 A enters junction A from the left, and then that current gets split between the two branches. If we measured the current in the top branch and the current in the bottom branch, we would find that the two currents still add up to a total of 1.3 A. And, in fact, when the two branches came back together at junction B, we would find that exactly 1.3 A was flowing out through junction B and through the rest of the circuit.
Kirchhoff’s junction rule says that charge is conserved: you don’t lose any current when the wire bends or branches. This seems remarkably obvious, but it’s also remarkably essential to solving circuit problems.
Kirchhoff’s loop rule is a bit less self-evident, but it’s quite useful in sorting out difficult circuits.
As an example, I’ll show you how to use Kirchhoff’s loop rule to find the current through all the resistors in the circuit below.
We will follow the steps for using Kirchhoff’s loop rule:
• Arbitrarily choose a direction of current. Draw arrows on your circuit to indicate this direction.
• Follow the loop in the direction you chose. When you cross a resistor, the voltage is −IR, where R is the resistance, and I is the current flowing through the resistor. This is just an application of Ohm’s law. (If you have to follow a loop against the current, though, the voltage across a resistor is written +IR.)
• When you cross a battery, if you trace from the − to the +, add the voltage of the battery; subtract the battery’s voltage if you trace from + to −.
• Set the sum of your voltages equal to 0. Solve. If the current you calculate is negative, then the direction you chose was wrong—the current actually flows in the direction opposite to your arrows.
In the case of the following figure, we’ll start by collapsing the two parallel resistors into a single equivalent resistor of 170 Ω. You don’t have to do this, but it makes the mathematics much simpler.
Next, we’ll choose a direction of current flow. But which way? In this particular case, you can probably guess that the 9 V battery will dominate the 1.5 V battery, and thus the current will be clockwise. But even if you aren’t sure, just choose a direction and stick with it—if you get a negative current, you chose the wrong direction.
Here is the circuit redrawn with the parallel resistors collapsed and the assumed direction of current shown. Because there’s now only one path for current to flow through, we have labeled that current I.
Now let’s trace the circuit, starting at the top-left corner and working clockwise:
• The 170-Ω resistor contributes a term of −(170 Ω) I.
• The 1.5-V battery contributes the term of −1.5 volts.
• The 100-Ω resistor contributes a term of −(100 Ω) I.
• The 200-Ω resistor contributes a term of −(200 Ω) I.
• The 9-V battery contributes the term of +9 volts.
Combine all the individual terms, and set the result equal to zero. The units of each term are volts, but units are left off below for algebraic clarity:
By solving for I, the current in the circuit is found to be 0.016 A; that is, 16 milliamps, a typical laboratory current.
The problem is not yet completely solved, though—16 milliamps go through the 100-Ω and 200-Ω resistors, but what about the 300-Ω and 400-Ω resistors? We can find that the voltage across the 170-Ω equivalent resistance is (0.016 A)(170 Ω) = 2.7 V. Because the voltage across parallel resistors is the same for each, the current through each is just 2.7 V divided by the resistance of the actual resistor: 2.7 V/300 Ω = 9 mA, and 2.7 V/400 Ω = 7 mA. Problem solved!
Oh, and you might notice that the 9 mA and 7 mA through each of the parallel branches adds to the total of 16 mA—as required by Kirchhoff’s junction rule.
Since the AP exam has a lot of problems without numbers, let’s look at Kirchhoff’s rules symbolically. Look at the circuit in the next diagram. The battery has a voltage of ε; the resistors R1, R2, R3, and R4 are all the same size; and the currents are marked as I1, I2, and I3.
Let’s apply the junction rule to junction A:
That makes sense. The current into the junction equals the current exiting the junction. We get the same equation for junction B.
Now let’s work with the harder loop rule. We have several loops to choose from, but I’ve already marked loop #1 on the left and loop #2 that goes around the outside. Remember the procedures we have already learned!
If we rewrite the equation, we can see that the voltage supplied by the battery is completely consumed by the three resistors: ε = I1R1 + I2R2 + I2R3. This is exactly what conservation of energy says we should get. The voltage supplied to the loop is consumed by the components in the loop.
We can also make our math teacher proud and factor out I2: ε = I1R1 + I2(R2 + R3). That tells us that the resistors R2 + R3 add in series, but you already saw that didn’t you?
Now take a look at:
Solving for ε:
Compare our two equations:
They both equal ε, which means we can set them equal to each other:
Simplifying by canceling out the I1R1 term, we get:
Remember that ΔV = IR. This equation tells us that the voltage drop across R2 and R3 must equal the voltage drop across R4. But we already knew that because the two combined resistors R2 and R3 are in parallel with R4. That is the parallel rule!
Now for a good AP exam question:
Using this information and the previous figure:
(A) Rank the currents I1, I2, and I3 from greatest to least.
(B) Rank the voltage drops across the resistors R1, R2, R3, and R4 from greatest to least.
(A) Think about it. I1 has to be the greatest. I2 and I3 split off from I1. But, there is more resistance in the I2 path, so it must be the least.
Answer: I1 > I3 > I2
(B) Remember that all the resistors are the same size. Since ΔV = IR, the voltage drop will depend on the currents. More current = larger ΔV. (Notice that the current through resistors 2 and 3 must be the same because they are in the same pathway.)
Answer: ΔVR1 > ΔVR4 > ΔVR2 = ΔVR3
Get comfortable using Kirchhoff’s rules because it is common for the AP exam to ask you to write them and use them to justify a statement about the circuit.
Circuits from an Experimental Point of View
When a real circuit is set up in the laboratory, it usually consists of more than just resistors—lightbulbs and motors are common devices to hook to a battery, for example. For the purposes of computation, though, we can consider pretty much any electronic device to act like a resistor.
But what if your purpose is not computation? Often on the AP exam, as in the laboratory, you are asked about observational and measurable effects. A common question involves the brightness of lightbulbs and the measurement (not just computation) of current and voltage.
Brightness of a Bulb
The brightness of a bulb depends solely on the power dissipated by the bulb. (Remember, power is given by any of the equations I2R, IΔV, or (ΔV)2/R.) You can remember that from your own experience—when you go to the store to buy a lightbulb, you don’t ask for a “400-ohm” bulb, but for a “100-watt” bulb. And a 100-watt bulb is brighter than a 25-watt bulb. But be careful—a bulb’s power can change depending on the current and voltage it’s hooked up to. Consider this problem.
A lightbulb is rated at 100 W in the United States, where the standard wall outlet voltage is 120 V. If this bulb were plugged in in Europe, where the standard wall outlet voltage is 240 V, which of the following would be true?
(A) The bulb would be one-quarter as bright.
(B) The bulb would be one-half as bright.
(C) The bulb’s brightness would be the same.
(D) The bulb would be twice as bright.
(E) The bulb would be four times as bright.
Your first instinct might be to say that because brightness depends on power, the bulb is exactly as bright. But that’s not right! The power of a bulb can change.
The resistance of a lightbulb is a property of the bulb itself, and so will not change no matter what the bulb is hooked up to.
Since the resistance of the bulb stays the same while the voltage changes, by V2/R, the power goes up, and the bulb will be brighter. How much brighter? Since the voltage in Europe is doubled, and because voltage is squared in the equation, the power is multiplied by 4—choice E.
Ammeters and Voltmeters
Ammeters measure current, and voltmeters measure voltage. This is pretty obvious, because current is measured in amps, voltage in volts. It is not necessarily obvious, though, how to connect these meters into a circuit.
Remind yourself of the properties of series and parallel resistors—voltage is the same for any resistors in parallel with each other. So if you’re going to measure the voltage across a resistor, you must put the voltmeter in parallel with the resistor. In the next figure, the meter labeled V2 measures the voltage across the 100-Ω resistor, while the meter labeled V1 measures the potential difference between points A and B (which is also the voltage across R1).
Current is the same for any resistors in series with one another. So, if you’re going to measure the current through a resistor, the ammeter must be in series with that resistor. In the following figure, ammeter A1 measures the current through resistor R1, while ammeter A2 measures the current through resistor R2.
As an exercise, ask yourself, is there a way to figure out the current in the other three resistors based only on the readings in these two ammeters? Answer is in the footnote.2
Real Batteries and Internal Resistance
The batteries you use in your calculator or in the lab are not perfect. They have some internal resistance. This means that the voltage posted on the battery, the emf “ε,” is not what actually comes out of the battery, because some of the voltage is lost before it ever gets out! The real voltage supplied by the battery is called the terminal voltage, ΔVterminal. Using the ideas we learned from Kirchhoff’s rules, we can see that:
where ε is the emf or internal voltage of the battery, I is the current through the battery, and r is the internal resistance of the battery. Notice that more current through the battery means less terminal voltage supplied by the battery to the external circuit.
Here is a handy lab that you should know—how to find the internal resistance of a battery. Hook a battery up to a resistor. Using an ammeter and voltmeter, measure the current through the battery and the terminal voltage of the battery. Repeat for several different resistors to produce multiple data points. Plot ΔVterminal as a function of current I and you get a graph that looks like the diagram below.
Notice how the graph is linear. When we match up the terminal voltage equations with the equation of a line, we see that the slope equals the negative of the internal resistance r, the y-intercept equals the emf ε of the battery, and the x-intercept will be the maximum current that the battery can output.
Hint: This is one of those labs you might need to know about for the AP exam.
Changes in a Circuit—Switches
A common question on the AP exam is: How does closing a switch affect a circuit? This is an easy question when you know what to do.
• When there is an open switch, the part of the circuit that the switch is in goes dead. Pretend that line and anything in series with this line does not exist. Redraw the circuit eliminating the dead circuit lines so that you don’t get confused.
Let’s take a look at a circuit we used earlier, this time with a switch and let all the resistors be the same: R1 = R2 = R3 = R4 = R.
The circuit has an open switch. Therefore, I3 = 0 and resistor R4 receives no current. It is as if it isn’t even there. So, what does that tell us about the circuit? With the switch open, all we have is a simple series circuit with three resistors where I1 = I2:
Now close the switch. What happens to I1 and I2? With the switch closed, we now have a combination circuit with R4 in parallel with R2 and R3. Remember that when resistors are added in parallel, the total resistance of the circuit is lowered. The resistance of the parallel set between points A and B is:
The total resistance of the circuit is:
To find the total current (I1):
So the current I1 has gone up from .
What about I2? Well, the current I1 will split between I2 and I3, with I2 only getting one-third of the current because it has twice the resistance in its pathway. One-third of , which is less than the original current of before the switch was closed.
If the resistors in this circuit are lightbulbs, this is what happens when the switch is closed:
• Bulb 1 gets brighter.
• Bulbs 2 and 3 get dimmer.
• Bulb 4, which was originally off, comes on and will be brighter than bulbs 2 and 3 but dimmer than bulb 1.
Bottom line: When a switch is open, neglect that portion of the circuit and analyze the circuit as if it isn’t even there.
Capacitors were introduced in Chapter 11. Now let’s look at them in more detail. Capacitors are really very simple devices. They have two metal plates, separated from each other by air or a material called a dielectric. A charge builds on the plates, one side positive and the other side negative, and energy is stored in the electric field between the plates. Capacitance is a way to describe how much charge a capacitor can hold for each volt of potential difference you hook it up to. The equation to find the capacitance of a capacitor is:
• C is the capacitance in farads.
• κ is the dielectric constant of the material between the plates. It has no units.
• ε0 is the vacuum permittivity, the constant we have talked about already.
• A is the area of just one of the plates.
• d is the distance between the plates.
For any capacitor there is a direct relationship between the voltage across the plates and the charged stored on the plates:
• DV is the potential difference across the plates.
• Q is the charge that is stored on one plate. Be careful, if one plate has positive 2 C of charge on it and the other plate has −2 C of charge on it, the total charge is not 4 C or zero; it is 2 C.
• C is the capacitance of the capacitor in farads.
A capacitor is made of two plates 10 cm by 10 cm separated by 0.1 mm with a 100-V battery across it and a switch. The switch is closed and the capacitor is allowed to charge.
1. Find the charge stored across the capacitor.
2. Find the energy stored in the capacitor.
3. Find the electric field strength between the plates.
4. The switch is now opened and the distance between the plates is increased to 0.2 mm. What happens to:
(a) the capacitance
(b) the voltage across the plates
(c) the charge on the plates
(d) the energy stored in the capacitor
(e) the electric field strength between the plates
Part 1: First thing we need to do is find the area of the plates. Remember, everything must be in meters, so 0.1 m × 0.1 m = 0.01 m2.
Now let’s find the capacitance. Since again we must be in meters, distance between the plates is 1 × 10—4 m. Air is between the plates, so κ = 1:
To find the charge is now a snap!
Part 2: To find the energy stored, we could use either of these equations from the equation sheet. Both will give the same answer:
Part 3: To find electric field strength:
Now if we had been paying attention, we could have found electric field using That would have been a lot easier . . . we should pay better attention.
Part 4: Double the distance between the plates and what happens to everything else? This is a classic AP exam question. The first thing to do is decide what is changing and what is staying the same.
Changing: The distance d is doubled. New distance = 2d.
Staying the same: Area of plates A and the charge on the plates Q because the switch was opened and it has nowhere to go. The charge is stuck on the plates. (Note that if the battery has stayed connected to the capacitor, the voltage would stay the same and the charge could change.)
(a) OK, now we have a place to start! Look at the capacitance equation:
When we double the distance, everything else stays the same, so capacitance is cut in half.
(b) Next, look at the voltage equation for a capacitor:
With the capacitance cut in half, the potential difference doubles.
(c) We have already decided that the charge stays the same because the switch was open and the charge could not move anywhere.
(d) Energy stored on the capacitor? Take your pick of equations:
Both equations tell us that the stored energy will double.
(e) OK, this time we are paying attention! Let’s use the easy equation:
Hey! The electric field stays the same.
This is another one of those skills that the AP exam prizes—semi-quantitative reasoning. Make sure you understand how we worked part 4 of the problem, because it will be a skill that you will use a lot on the exam.
Parallel and Series Capacitors
Lucky for you, you already have a background in parallel and series, which will make your life much easier. Some hints to make solving problems easier:
1. Always draw a circuit diagram and label all the components clearly.
2. Make a chart to keep all your information clear and organized.
3. Keep in mind what is the same in the circuit—I can’t stress enough the importance of this. I’ll explain more as we go through each type of circuit.
Here is an example of a parallel capacitor circuit.
Remember that some things are the same as resistor circuits and some are different. For a parallel circuit:
• Electric potential difference is the same across each branch of a parallel circuit. This is the same for both capacitor circuits and resistor circuits:
• The total charge stored across the circuit equals the sum of the charges across each branch of the circuit.
• The total or equivalent capacitance of a parallel capacitor circuit equals the sum of the capacitance of each branch of the circuit. This is very different from a parallel resistance circuit, in which you add up reciprocals to find the total resistance:
Three capacitors, 3 μF, 6 μF, and 9 μF, are connected in parallel to a 300-V battery. What is the stored charge in the circuit and across each capacitor?
The first thing you want to do is create a C-ΔV-Q chart. Give each capacitor a name like C1, C2, C3, and so on. Give the total or equivalent values a T as shown.
Always ask yourself, “What is the same?” In the case of a parallel circuit, you know there are 300 volts across each leg of the circuit, so fill that into our chart.
There are a couple of ways you can solve from this point. Let’s find the total capacitance of the circuit:
Remember to fill that into your chart under the total capacitance. Now you can find the charge across each capacitor and the total charge using Q = CΔV.
Notice that we could have simply added up the charges across C1, C2, and C3 to find the total charge.
Just like in the parallel circuit, some things are the same as resistor circuits and some are different. For a series circuit:
• Electric potential difference adds up in a series circuit. This is the same for both capacitor circuits and resistor circuits:
• The charge in each series capacitor is the same:
• The total or equivalent capacitance of a series capacitor circuit equals the sum of the reciprocal of each capacitor in the circuit. This is very different from a series resistance circuit, in which you simply add the resistors to find the total resistance:
Three capacitors—6 μF, 10 μF, and 15 μF—are connected in series to a 300-V battery. What is the stored charge in the circuit and the electric potential across each capacitor?
Let’s make the chart C-ΔV-Q.
Always ask yourself, “What is the same?” In the case of a series circuit, the charge is constant. In this problem the change is unknown, so the only thing we have enough information to determine is the total capacitance:
Be careful about your math at this point! Take each reciprocal separately, add them up, and don’t forget to take a reciprocal at the end! You are trying to find CS, not !
Now you can use Q = CΔV to solve for the total charge, 3.0 × 10—6 F × 300 V = 9.0 × 10—4 C. Remember that charge is the same throughout a series circuit, so you can fill in the entire charge column.
Now use to find the electric potential across C1, C2, and C3. You know that in a series circuit, the potentials add up to the total, so it is easy to check your answers.
Combination Parallel Series Circuits
What if the capacitors are in both parallel and series? Don’t fret; just find the equivalent capacitance using the rules you have already learned.
What is the equivalent capacitance of this combination? First, we can see that the top two capacitors are in series. Their equivalent resistance is 2 μF. This 2-μF capacitor is in parallel with the 4-μF capacitor for a grand total of 6 μF. Easy peasy lemon squeezy.
The AP exam probably won’t ask you anything more complicated than that. But what the exam will ask you is how capacitors affect a circuit with resistors in it. So let’s move on to that.
Sometimes you will be asked to solve a problem with both resistors and capacitors in the circuits. These are called RC circuits. You will only be asked about RC circuits in two different states:
1. When you first connect a capacitor to a circuit, no charge has built up across the plates, so the capacitor freely allows charge to flow. Treat the capacitor like wire with no electrical potential across it.
2. After a long time, steady state will be achieved. The capacitor has built up its maximum charge, and no more current will flow onto the device. The capacitor is “full.” In steady state, the potential difference across the capacitor plates equals the voltage of whatever devices are connected in parallel with it. In steady state, treat the capacitor as an open switch. Every circuit component connected in a series to a capacitor in steady state will receive no current.
In the circuit in the diagram above, the capacitor is initially uncharged and the switch is opened.
1. Find the currents in the circuit when the switch is first closed.
2. Find the currents in the circuit and the charge on the capacitor after a long time.
When the switch is first closed, the uncharged capacitor acts like a wire. It is like the capacitor isn’t there, and there is just a parallel circuit as pictured in the diagram above. We can fill in the shaded part of our V-I-R chart just like we did earlier in the chapter.
When the switch has been closed for a long time (time approaches infinity), the capacitor becomes charged and it no longer allows current to flow. It becomes like an open switch, as seen in the diagram above. This means the 10-Ω resistor will no longer carry any current and the potential difference across it must be zero. It’s as if the top part of the circuit has been disconnected from the circuit by a switch! Filling in the shaded area of our V-I-R chart, we get:
Since the 15-Ω resistor has no current passing through it, the effective circuit is simply a 10-Ω resistor in series with the battery. Therefore, the total resistance of the circuit is just 10 Ω.
What is the voltage across the capacitor? It is connected in parallel to R1 and the battery. So, 12 volts. The charge on the capacitor would be:
In the circuit in the diagram above, we have moved the initially uncharged capacitor to the top in parallel with the resistor R2 and added another resistor R3, and ammeters A1, A2, and A3 to measure the currents.
Here is a great AP exam question: Rank the currents I1, I2, and I3:
(A) immediately after the switch is closed
(B) after the switch has been closed a long time
(A) When the switch is first closed, the uncharged capacitor acts like a wire, and we get the circuit pictured in the diagram above. Notice that the capacitor is acting like a short-circuit wire with no resistance at all. This means all current from the battery will bypass resistors R1 and R2. I1 and I2 will read zero.
Answer: I3 > I1 = I2
(B) After time goes by and the capacitor is full of charge, it acts like an open switch. The circuit is a simple combination circuit. Using what we have learned with Kirchhoff’s rules, we can find the answer.
Answer: I3 > I2 > I1
Note that the capacitor is in parallel with R1 and R2. To find the charge stored on the capacitor, we will need to find the potential difference across R1 and R2.
❯ Practice Problems
1. 100-Ω, 120-Ω, and 150-Ω resistors are connected to a 9-V battery in the circuit shown above. Which of the three resistors dissipates the most power?
(A) the 100-Ω resistor
(B) the 120-Ω resistor
(C) the 150-Ω resistor
(D) both the 120-Ω and 150-Ω resistors
2. A 1.0-F capacitor is connected to a 12-V power supply for a long time until it is fully charged. The capacitor is then disconnected from the power supply, and used to power a toy car. The average drag force on this car is 2 N. About how far will the car go?
(A) 36 m
(B) 72 m
(C) 144 m
(D) 24 m
3. The circuit shown in the figure above has two resistors, an uncharged capacitor, a battery, an ammeter, and a switch initially in the open position. After the switch is closed, what will happen to the current measured in the ammeter A?
(A) It will increase to a constant value.
(B) It will remain constant.
(C) It will decrease to a constant value.
(D) It will decrease to zero.
Questions 4 and 5 refer to the following material.
A student is investigating what effect wire diameter D has on a simple circuit. The student has five wires of various diameters made of the same material and length. She connects each wire to a power supply and measures the current I passing through the wire with an ammeter. The data from the investigation is given in the chart.
Output voltage of the power supply: 1.5 V
Length of wires: 1.0 m
4. What can be concluded from this data?
(A) I ∝ D2
(B) I ∝ D
(D) There does not seem to be a relationship between the diameter of the wire and the current.
5. Using the information already gathered, what additional steps would need to be taken to extend this investigation to see if there is a relationship between wire diameter and the resistance of the wire?
(A) Determine what material the wires are made of and look up the resistivity of the wire.
(B) Multiply the current through the wire by the voltage of the power supply.
(C) Divide the current through the wire by the voltage of the power supply.
(D) Divide the voltage of the power supply by the current through the wire.
6. Identical lightbulbs are connected to batteries in different arrangements, as shown in the figure above. Which of the following correctly ranks the brightness of the bulbs?
(A) C > B > A
(B) C > B = A
(C) C = B > A
(D) B > C > A
7. Three cylindrical resistors made of the same material but different dimensions are connected, as shown in the figure above. A battery is connected across the resistors to produce current. Which is the correct ranking of the currents for the resistors?
(A) IA = IB = IC
(B) IA > IB > IC
(C) IC > IA = IB
(D) IC > IB > IA
Questions 8 and 9
Two batteries and two resistors are connected in a circuit as shown in the figure below. The currents through R1, R2, and ε2 are shown.
8. Which of the following is a proper application of conservation laws to this circuit? (Select two answers.)
(A) ε1 − I1R1 − I2R2 = 0
(B) ε1 − ε2 − I1R1 = 0
(C) I1 + I2 − I3 = 0
(D) I2 + I3 − I1 = 0
9. The resistors R1 and R2 have the same resistance. If the potential differences of the batteries are ε1 = 9 V and ε2 = 6 V, which resistor will have the most current passing through it?
(C) R1 and R2 have the same current
(D) It is not possible to determine the currents through the resistors without more information
10. A single resistor (R) is connected to a variable voltage source that is increased from 1.5 V to 9 V. The power dissipated by the resistor for various voltages is shown in the two graphs. Which of the following can be deduced from the graphs? (Select two answers.)
(A) The slope of the lower graph can be used to find the resistance (R) of the resistor.
(B) The curve of the upper graph indicates that the voltage source must have internal resistance.
(C) The curve of the upper graph indicates that the resistor is non-ohmic.
(D) The power dissipated by the resistor is proportional to the voltage squared.
(A) Simplify the above circuit so that it consists of one equivalent resistor and the battery.
(B) What is the total current through this circuit?
(C) Find the voltage across each resistor. Record your answers in the spaces below.
(D) Find the current through each resistor. Record your answers in the spaces below.
(E) The 500-Ω resistor is now removed from the circuit. State whether the current through the 200-Ω resistor would increase, decrease, or remain the same. Justify your answer.
12. A circuit with four identical resistors (R1, R2, R3, and R4), a battery of potential difference (ε), and four ammeters (A1, A2, A3, and A4) is shown in the figure above.
(A) Use Kirchhoff’s junction rule to prove that currents I1 and I4 are the same.
(B) Write Kirchhoff’s loop rule for the loop that contains the battery and resistor R3.
(C) Rank the currents (I1, I2, I3, and I4) from greatest to least. Justify your answer.
(D) Write an expression for the equivalent resistance of the circuit in terms of known quantities.
13. Two lightbulbs have power ratings of 40 W and 100 W when connected to a potential difference of 120 V.
(A) Calculate the resistance of both bulbs. Show your work.
(B) Which bulb glows brightest when connected in parallel? Justify your answer.
(C) Which bulb glows brightest when connected in series? Justify your answer.
14. Four identical bulbs are attached in a circuit, as shown above. Rank the brightness of the bulbs. Justify your prediction in terms of power.
15. Using standard electrical schematic figures, sketch a circuit with bulbs, a battery, a capacitor, and switches that will accomplish the following.
(A) When the switch is closed, the bulb will immediately light but over time will go out.
(B) When the switch is closed, the bulbs will not immediately light, but over time they will glow brighter.
16. The circuit shown in the figure consists of three identical resistors, two ammeters, a battery, a capacitor, and a switch. The capacitor is initially uncharged, and the switch is open. Explain what happens to the readings of the two ammeters from the instant the switch is closed until a long time has passed.
17. The figure shows a circuit with two resistors, a battery, a capacitor, and a switch. Originally, the switch is open, and the capacitor is uncharged.
(A) Complete the voltage-current-resistance-power chart for the circuit immediately after the switch is closed.
(B) Complete the voltage-current-resistance-power chart for the circuit after the switch is closed for a long time.
(C) What is the energy stored in the capacitor after the switch has been closed a long time?
❯ Solutions to Practice Problems
1. (A) On one hand, you could use a V-I-R chart to calculate the voltage or current for each resistor, then use P = IV, I2R, or to find power. On the other hand, there’s a quick way to reason through this one. Voltage changes across the 100-Ω resistor, then again across the parallel combination. Because the 100-Ω resistor has a bigger resistance than the parallel combination, the voltage across it is larger as well. Now consider each resistor individually.
By power , the 100-Ω resistor has both the biggest voltage and the smallest resistance, giving it the most power.
2. (A) The energy stored by a capacitor is . By powering a car, this electrical energy is converted into mechanical work, equal to the force times the displacement. Solve for displacement, you get 36 m.
3. (C) When the switch is initially closed, the uncharged capacitor acts as a “wire” or “closed switch” with no resistance. Thus the initial circuit “looks” like a parallel circuit. As the capacitor charges, the current through the bottom resistor in series with the capacitor drops to zero, as the capacitor acts as a “broken wire” or “open switch” with infinite resistance. Thus, after a long time the circuit becomes a series circuit with current passing only through a single top resistor. As the circuit transitions from this “parallel to series,” the equivalent resistance of the circuit increases. This produces a current through the ammeter that drops from its maximum starting value to a steady-state lower value.
4. (A) As the diameter increases, the current also increases, indicating a direct relationship of some kind. Comparing trials 1 and 2, the diameter doubles and the current quadruples. This suggests a quadratic relationship. To confirm this, compare trials 1 and 3—the diameter is tripled and the current increases by a factor of 9.
5. (D) To investigate the relationship of diameter and resistance, the resistance of the wire needs to be calculated. We already know the output voltage of the power supply and the current through the wire. We just need to calculate the resistance for each entry in the table: .
6. (B) Using Kirchhoff’s loop rule, we can see that the batteries in arrangement C add up to double the voltage. Connecting batteries as in arrangement B does not add any extra voltage to the bulb, and it will not be any brighter than arrangement A.
7. (A) There is only one pathway. The current is the same.
8. (A) and (D) Applying Kirchhoff’s loop rule to the top loop, we get answer choice A. Applying Kirchhoff’s junction rule to the right junction, we get answer choice D.
9. (A) The emfs of both batteries point in the same direction for the outer loop, for a combined potential difference of 15 V. The emfs point in the opposite direction for the line that contains R2, for a combined potential difference of 3 V.
10. (A) and (D) Power (P) equals the Potential Difference (V) squared divided by the Resistance (R). Therefore, the slope of the lower graph in the figure would be equal to 1/R. The Power vs Voltage Squared graph shows power proportional to the voltage squared because the graph is linear. In addition, we can see in the upper graph of the figure that Power as a function of Voltage is quadratic. This also indicates that the Power is proportional to the Voltage squared.
11. (A) Combine each of the sets of parallel resistors first. You get 120 Ω for the first set, 222 Ω for the second set, as shown in the diagram below. These two equivalent resistances add as series resistors to get a total resistance of 342 Ω.
(B) Now that we’ve found the total resistance and we were given the total voltage, just use Ohm’s law to find the total current to be 0.026 A (also known as 26 mA).
(C) and (D) should be solved together using a V-I-R chart. Start by going back one step to when we began to simplify the circuit: 9-V battery, a 120-Ω combination, and a 222-Ω combination, shown above. The 26-mA current flows through each of these . . . so use V = IR to get the voltage of each: 3.1 V and 5.8 V, respectively.
Now go back to the original circuit. We know that voltage is the same across parallel resistors. So both the 200-Ω and 300-Ω resistors have a 3.1-V voltage across them. Use Ohm’s law to find that 16 mA goes through the 200-Ω resistor, and 10 mA through the 300 Ω. Similarly, both the 400-Ω and 500-Ω resistors must have 5.8 V across them. We get 15 mA and 12 mA, respectively.
Checking these answers for reasonability: the total voltage adds to 8.9 V, or close enough to 9.0 V with rounding. The current through each set of parallel resistors adds to just about 26 mA, as we expect.
(E) Start by looking at the circuit as a whole. When we remove the 500-Ω resistor, we actually increase the overall resistance of the circuit because we have made it more difficult for current to flow by removing a parallel path. The total voltage of the circuit is provided by the battery, which provides 9.0 V no matter what it’s hooked up to. So by Ohm’s law, if total voltage stays the same while total resistance increases, total current must decrease from 26 mA.
Okay, now look at the first set of parallel resistors. Their equivalent resistance doesn’t change, yet the total current running through them decreases, as discussed above. Therefore, the voltage across each resistor decreases, and the current through each decreases as well.
12. (A) For the junction on the left: I1 − I2 − I3 = 0. For the junction on the right: I3 + I2 − I4 = 0, therefore, I1 = I4.
(B) ε1 − I1R1 − I2R2 − I2R3 = 0
(C) I1 = I4 > I3 > I2. Currents 1 and 4 are the same and equal to the combination of currents 2 and 3. The currents 2 and 3 have the same potential but different resistances. The resistance is less for current 3; therefore, it is larger than current 2.
(B) . When connected in parallel, the bulbs receive the same electric potential, as shown by Kirchhoff’s loop rule. Therefore, the bulb with the smallest resistance (100 W) is brighter.
(C) P = I2R. When connected in series there is only one pathway for the current. The bulbs receive the same current. Therefore, the bulb with the largest resistance (40 W) is brightest.
14. D > A > C = B, P = I2R. All the bulbs have the same resistance. Therefore, the bulb that receives the greatest current will have the greatest power consumption and be brightest. Bulb D is in the main current pathway and receives the greatest current. The main current splits between the two pathways. Bulb A sits in the pathway with the least resistance, so it will receive more current than bulbs C and B. Bulbs C and B are in the same pathway with identical currents and receive the smallest current.
15. There is more than one way to draw each of these, but here is an example.
16. When the switch is first closed, the capacitor behaves like a wire. This creates a short circuit around ammeter 1, and it will read zero. All the current is flowing through only two resistors in series and ammeter 2. When the capacitor becomes fully charged, it behaves like an open switch in the circuit. The current is now passing through all three resistors in series. This increases the current in ammeter 1 but decreases the current in ammeter 2 because the total resistance of the circuit has increased.
(B) All numbers are rounded to two significant digits.
❯ Rapid Review
• Current is the flow of positive charge. It is measured in amperes.
• Resistance is a property that impedes the flow of charge. Resistance in a circuit comes from the internal resistance of the wires and from special elements inserted into circuits known as “resistors.”
• Resistance is related to current and voltage by Ohm’s law: ΔV = IR.
• Resistors that have a constant resistance no matter what the current through them or voltage across them are said to be ohmic. If the resistance changes, then these resistors are nonohmic.
• When resistors are connected in series, the total resistance equals the sum of the individual resistances. And the current through one resistor equals the current through any other resistor in series with it.
• When resistors are connected in parallel, the inverse of the total resistance equals the sum of the inverses of the individual resistances. The voltage across one resistor equals the voltage across any other resistor connected parallel to it.
• The V-I-R chart is a convenient way to organize any circuit problem.
• Kirchhoff’s junction rule says that the exact same amount of current coming into a junction will leave the junction. This is a statement of conservation of charge. Kirchhoff’s loop rule says that the sum of the voltages across a closed loop equals zero. This rule is helpful especially when solving problems with circuits that contain more than one battery.
• Ammeters measure current, and are connected in series; voltmeters measure voltage, and are connected in parallel.
• Real batteries have internal resistance that cuts the amount of voltage the battery supplies to the circuit.
• Bulbs are brighter when they are operating at a higher power.
• When a switch is open, the part of the circuit that is in series with the switch does not receive any current and is “dead.”
• When capacitors are connected in series, the inverse of the total capacitance equals the sum of the inverses of the individual capacitances. When capacitors are connected in parallel, the total capacitance just equals the sum of the individual capacitances.
• A capacitor’s purpose in a circuit is to store charge and energy. After it has been connected to a circuit for a long time, the capacitor becomes fully charged and prevents the flow of current.
• An uncharged capacitor behaves like a wire in a circuit, but once it is charged, it behaves like an open switch.
1Resistivity would be given on the AP exam if you need a value. Nothing here to memorize.
2The current through R5 must be the same as through R1, because both resistors carry whatever current came directly from the battery. The current through R3 and R4 can be determined from Kirchhoff’s junction rule: subtract the current in R2 from the current in R1 and that’s what’s left over for the right-hand branch of the circuit.
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3.1: Dilating, Again (10 minutes)
Students dilate a triangle from a center of dilation. Then, they think of things they notice and wonder about their completed drawings. The purpose of the activity is to elicit the idea that the triangle and its dilation can be viewed as representations of the cross sections of a three-dimensional pyramid. This will help prepare students for the next activity in which they create a representation of a pyramid by suspending several dilations of a rectangle. While students may notice and wonder many things about the figures they draw, the three-dimensional visualization is the most important discussion point.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Tell students that they will draw a dilated figure, then think of at least one thing they notice and at least one thing they wonder about the resulting drawing.
Dilate triangle \(BCD\) using center \(P\) and a scale factor of 2.
Look at your drawing. What do you notice? What do you wonder?
Choose a student’s drawing and display it for all to see. Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the displayed drawing. After all responses have been recorded without commentary or editing, ask students, “Is there anything on the list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the idea that the drawing resembles a pyramid does not come up during the conversation, ask students to discuss this idea. Remind students that pyramids, like prisms, are named for their bases, so this pyramid should be called a “triangular” pyramid.
3.2: Pyramid Mobile (25 minutes)
In this activity, students dilate rectangles and suspend them to resemble cross sections of a pyramid. This work connects to later lessons in which students use areas of cross sections in the development of the pyramid volume formula.
This task reinforces the idea that scaling by a factor of \(k\) adjusts all distances in a figure by a factor of \(k\), and foreshadows the upcoming concept that if a figure is scaled by a factor of \(k\), its area changes by a factor of \(k^2\).
Monitor for groups who choose different lengths of string, therefore creating pyramids of different heights.
Arrange students in groups of 4.
Draw a triangle and a square on the board. For each shape, hold a ball or another small object representing point \(A\) off the surface of the board, projected directly out from the center of the shape. Invite students to imagine the three-dimensional objects created by connecting \(A\) to each vertex of the shape: a triangular pyramid and a square pyramid.
Display this image for all to see. Tell students that it is a top-down view of the three-dimensional shapes they just imagined.
Ask students how they could dilate each two-dimensional figure by a scale factor of \(\frac12\) using \(A\) as the center of dilation. With students’ guidance, complete the dilations. Be sure students see that the dilation is created by measuring the distance from \(A\) to each vertex of the “base” and multiplying that distance by \(\frac12\) to find the halfway point. Ask students what the dilations could represent: cross sections of each solid.
Finally, explain to students that a mobile is a kind of sculpture in which materials are suspended in the air. Tell them they’ll be using the concepts just discussed to make a pyramid mobile.
Design Principle(s): Support sense-making
Supports accessibility for: Organization; Attention
Your teacher will give you sheets of paper. Each student in the group should take one sheet of paper and complete these steps:
- Locate and mark the center of your sheet of paper by drawing diagonals or another method.
- Each student should choose one scale factor from the table. On your paper, draw a dilation of the entire sheet of paper, using the center you marked as the center of dilation.
- Measure the length and width of your dilated rectangle and calculate its area. Record the data in the table.
- Cut out your dilated rectangle and make a small hole in the center.
|scale factor, \(k\)
|length of scaled rectangle
|width of scaled rectangle
|area of scaled rectangle
Now the group as a whole should complete the remaining steps:
- Cut 1 long piece of string (more than 30 centimeters) and 4 shorter pieces of string. Make 4 marks on the long piece of string an equal distance apart.
- Thread the long piece of string through the hole in the largest rectangle. Tie a shorter piece of string beneath it where you made the first mark on the string. This will hold up the rectangle.
- Thread the remaining pieces of paper onto the string from largest to smallest, tying a short piece of string beneath each one at the marks you made.
- Hold up the end of the string to make your cross sections resemble a pyramid. As a group, you may have to steady the cross sections for the pyramid to clearly appear.
Are you ready for more?
Is dilating a square using a factor of 0.9, then dilating the image using scale factor 0.9 the same as dilating the original square using a factor of 0.8? Explain or show your reasoning.
If students struggle to create the dilations, remind them of the demonstration from the activity launch. Suggest that they draw lines connecting the center of the paper to each vertex, and measure the lines. Then invite them to think about how the scale factor they’re using will apply to these distances.
The goal of the discussion is to make observations about the pyramid structure and about the relationships in the table students created. Here are some questions for discussion:
- “For the student who chose the scale factor \(k=1\), what did you have to do to the rectangle?” (Nothing! A scale factor of 1 yields the exact same figure.)
- “Where is the rectangle with scale factor \(k=1\) located in the pyramid?” (It is the pyramid’s base.)
- “When we did the dilations, we used the center of the rectangle as the center of dilation. If we imagine doing the dilations in 3 dimensions instead of 2, like in the activity launch, what is the center of dilation?” (It is the vertex at the top of the pyramid.)
- “Staying in 3 dimensions, what would a dilation by the scale factor \(k=0\) look like? Where would it be located in the pyramid?” (It would be at the center of dilation, the vertex at the top of the pyramid.)
- “How do the length and width of your cross sections relate to the scale factor you chose?” (The length and width of the dilated rectangles were changed by a scale factor of \(k\), to the limit of precision of the tools we’re using.)
- “Was the area of the dilated rectangle also changed by a factor of \(k\)?” (No. At this point, students do not need to draw any further conclusions about the area; this will be analyzed in an upcoming lesson.)
If time permits, display the applet for all to see.
Demonstrate the “hide pyramid” and “collapse layers” tools. Ask students to describe how the cross sections are related to the pyramid.
The goal of the discussion is to extend students’ understanding of cross sections and dilation. Here are some questions for discussion:
- “For 2 groups with pyramids of different heights, how did the placement of the scale factor \(k=0.5\) triangle differ in each pyramid?” (It was exactly halfway up the pyramid in each case.)
- “What is the difference between using the center of the rectangle as the center of dilation, and using the top vertex of the pyramid as the center?” (When we use the center of the rectangle, the dilation stays in the same plane. That’s what we did, but then we assembled those into a pyramid. The top vertex of the pyramid can be viewed as the center of dilation in 3 dimensions.)
- “What would happen if we dilated the rectangle by a scale factor of \(k=0.1\) or \(k=0.9\), using the top vertex of the pyramid as the center of dilation?” (The first would be a cross section near the base and the second would be a cross section near the top vertex of the pyramid.)
- “What is the range of scale factors that create cross sections of the pyramid?” (The scale factors between 0 and 1 create cross sections. Larger scale factors extend outside of the pyramid.)
3.3: Cool-down - Circle Dilation (5 minutes)
Student Lesson Summary
Imagine a triangle lying flat on your desk, and a point \(P\) directly above the triangle. If we dilate the triangle using center \(P\) and scale factor \(k=\frac12\) or 0.5, together the triangles resemble cross sections of a pyramid.
We can add in more cross sections. This image includes two more cross sections, one with scale factor \(k=0.25\) and one with scale factor \(k=0.75\). The triangle with scale factor \(k=1\) is the base of the pyramid, and if we dilate with scale factor \(k=0\) we get a single point at the very top of the pyramid.
Each triangle’s side lengths are a factor of \(k\) times the corresponding side length in the base. For example, for the cross section with \(k=\frac12\), each side length is half the length of the base’s side lengths.
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Decision trees are a powerful data visualization tool used in machine learning and data analysis. They are used to make predictions based on input variables, by creating a tree-like model of decisions and their possible consequences. The branches of the tree represent different decisions, and the leaves represent the outcome of those decisions. Decision trees are widely used in various fields such as finance, marketing, and healthcare to name a few. In this guide, we will explore the concept of decision trees, how they work, and when they are used.
Understanding Decision Trees
Decision trees are a type of machine learning algorithm used for both classification and regression tasks. They are graphical representations of decisions and their possible consequences. In other words, they are a series of if-then statements that help to determine the best course of action based on the available data.
Definition of Decision Trees
A decision tree is a flowchart-like tree structure where each internal node represents a "test" on an attribute, each branch represents the outcome of the test, and each leaf node represents a class label. In simpler terms, a decision tree is a series of questions that help to determine the outcome of a decision.
How Decision Trees Work
Decision trees work by partitioning the input space into regions, based on the attribute being tested, and assigning a class label to each region. The tree continues to split the data until it reaches a point where it can make an accurate prediction with a high degree of confidence.
The Structure of a Decision Tree
A decision tree consists of three main parts: the root node, the branches, and the leaf nodes. The root node represents the starting point of the tree, and the branches represent the possible outcomes of the test. The leaf nodes represent the final outcome of the decision tree.
Node Types in a Decision Tree
There are two main types of nodes in a decision tree: internal nodes and leaf nodes. Internal nodes represent the tests that are used to determine the outcome of the decision, while leaf nodes represent the final outcome of the decision tree.
Advantages and Disadvantages of Decision Trees
Decision trees have several advantages, including their ability to handle both categorical and continuous data, their ease of interpretation, and their ability to handle missing data. However, they also have some disadvantages, including their tendency to overfit the data, their lack of transparency, and their sensitivity to outliers.
Overall, decision trees are a powerful tool for making decisions based on data. They provide a simple and intuitive way to represent complex decisions and can be used in a wide range of applications, from medical diagnosis to financial analysis.
Building Decision Trees
Decision trees are a popular machine learning algorithm used for both classification and regression tasks. Before building a decision tree model, it is essential to prepare the data appropriately. This section will discuss the key steps involved in data preparation for decision tree algorithms.
The first step in data preparation is data preprocessing. This involves cleaning and transforming the raw data into a format that can be used by the decision tree algorithm. Some common preprocessing steps include:
- Handling missing values: Decision tree algorithms can be sensitive to missing values, so it is important to handle them appropriately. One approach is to impute the missing values with the mean or median of the respective feature.
- Feature scaling: Decision tree algorithms are sensitive to the scale of the input features. Feature scaling normalizes the input features to a common scale, such as between 0 and 1. This can improve the performance of the decision tree algorithm.
- Feature selection: Decision tree algorithms can handle a large number of input features. However, not all features may be relevant for the task at hand. Feature selection involves selecting a subset of the most relevant features to improve the performance of the decision tree algorithm.
Handling Categorical Variables
Decision tree algorithms can handle both numerical and categorical variables. Categorical variables need to be encoded before they can be used by the decision tree algorithm. One common encoding technique is one-hot encoding, which creates a new binary feature for each category. For example, if there are three categories, "A", "B", and "C", one-hot encoding would create three binary features, "A", "B", and "C".
Splitting the Dataset into Training and Testing Sets
Once the data has been preprocessed, the next step is to split the dataset into training and testing sets. The training set is used to build the decision tree model, while the testing set is used to evaluate the performance of the model. It is important to use a random split to ensure that the training and testing sets are representative of the entire dataset.
In summary, data preparation is a critical step in building decision tree models. It involves data preprocessing, handling categorical variables, and splitting the dataset into training and testing sets. By following these steps, you can ensure that your decision tree model is accurate and reliable.
Decision Tree Algorithms
Decision tree algorithms are a popular and powerful tool for creating decision trees. They work by recursively splitting the data into subsets based on the feature that provides the most information gain until a stopping criterion is reached. The following are some of the most popular decision tree algorithms:
- ID3 (Iterative Dichotomiser 3): ID3 is a simple, fast, and effective algorithm for constructing decision trees. It works by recursively selecting the best feature at each node, the one that provides the most information gain. ID3 has a built-in stopping criterion based on the information gain of the split, the minimum number of samples required to split the dataset, and the maximum depth of the tree.
- C4.5: C4.5 is an extension of ID3 that handles both continuous and categorical attributes. It uses information gain ratio to select the best feature, which takes into account the impurity of the data and the number of samples required to split the dataset. C4.5 also introduces the concept of a "threshold" to handle continuous attributes, which is the value above which the attribute is considered positive.
- CART (Classification and Regression Trees): CART is a widely used algorithm for creating decision trees. It works by recursively splitting the data based on the best feature, as determined by a measure of impurity. CART can handle both continuous and categorical attributes and is capable of handling both classification and regression tasks.
- Random Forests: Random forests are an ensemble method that consists of multiple decision trees. They work by randomly selecting subsets of the data and features at each node, which helps to reduce overfitting and improve the robustness of the model. Random forests are particularly effective for handling high-dimensional data and can be used for both classification and regression tasks.
Training and Evaluating Decision Trees
Training a decision tree model
Training a decision tree model involves providing the algorithm with a dataset that it can use to learn and make predictions. This dataset should be representative of the problem the decision tree will be used to solve.
The algorithm starts by selecting a feature to split the data based on. This feature is typically chosen based on the information gain or Gini index, which measures the impurity of the data. The algorithm then recursively splits the data until a stopping criterion is reached, such as a maximum depth or a minimum number of samples per leaf node.
Evaluating the performance of a decision tree model
Once a decision tree model has been trained, it is important to evaluate its performance to ensure that it is making accurate predictions. There are several metrics that can be used to evaluate the performance of a decision tree model, including accuracy, precision, recall, and F1 score.
Accuracy measures the proportion of correct predictions made by the model. Precision measures the proportion of positive predictions that are correct. Recall measures the proportion of true positive predictions that are made. The F1 score is a weighted average of precision and recall.
It is also important to check for overfitting and underfitting in the decision tree model. Overfitting occurs when the model is too complex and fits the noise in the training data, resulting in poor performance on new data. Underfitting occurs when the model is too simple and cannot capture the underlying patterns in the data, resulting in poor performance on both the training and new data.
Techniques to prevent overfitting
There are several techniques that can be used to prevent overfitting in decision tree models, including:
- Pruning: Removing branches of the tree that do not improve the performance of the model.
- Limiting the depth of the tree: Setting a maximum depth for the tree to prevent it from becoming too complex.
- Regularization: Adding a penalty term to the objective function to discourage overly complex models.
- Cross-validation: Using a technique to split the data into training and validation sets, and evaluating the performance of the model on the validation set to ensure that it is not overfitting.
Practical Applications of Decision Trees
Using decision trees for classification tasks
Decision trees are a popular machine learning algorithm used for classification tasks. Classification is the process of categorizing data into predefined classes. For example, an email can be classified as spam or not spam, or a patient can be classified as having a certain disease or not.
Decision trees are particularly useful for classification tasks because they can handle both continuous and categorical variables. The tree is constructed by recursively splitting the data into subsets based on the feature that provides the most information gain. This process continues until a stopping criterion is met, such as a maximum depth or minimum number of samples in a leaf node.
Examples of classification problems solved using decision trees
There are many real-world examples of classification problems that have been solved using decision trees. Some of these include:
- Spam email detection: Decision trees can be used to classify emails as spam or not spam based on features such as the sender's email address, the subject line, and the content of the email.
- Credit risk assessment: Decision trees can be used to predict the likelihood of a loan applicant defaulting on their loan based on features such as credit score, income, and employment history.
- Disease diagnosis: Decision trees can be used to diagnose a patient with a certain disease based on symptoms and medical history. For example, a decision tree could be used to diagnose a patient with pneumonia based on their temperature, respiratory rate, and blood oxygen saturation.
Decision trees are commonly used in regression problems, which involve predicting a continuous output variable based on one or more input variables. In regression tasks, decision trees are used to model the relationship between the input variables and the output variable.
Using decision trees for regression tasks
Decision trees are particularly useful for regression tasks because they can handle both numerical and categorical input variables. The tree structure also allows for the identification of important features that contribute to the prediction of the output variable.
Examples of regression problems solved using decision trees
There are many real-world applications of decision trees in regression tasks. For example, decision trees have been used to predict housing prices, stock market prices, and even the lifespan of electrical equipment.
In housing price prediction, decision trees are used to model the relationship between various features of a house, such as the number of bedrooms, square footage, and location, and the price of the house.
In stock market forecasting, decision trees are used to predict the future price of a stock based on various economic indicators, such as interest rates, inflation rates, and company earnings.
Housing price prediction
One of the most common applications of decision trees in regression tasks is housing price prediction. In this application, decision trees are used to model the relationship between various features of a house, such as the number of bedrooms, square footage, and location, and the price of the house. The decision tree model is trained on a dataset of houses and their prices, and then used to predict the price of new houses based on their features.
Stock market forecasting
Another common application of decision trees in regression tasks is stock market forecasting. In this application, decision trees are used to predict the future price of a stock based on various economic indicators, such as interest rates, inflation rates, and company earnings. The decision tree model is trained on a dataset of stock prices and economic indicators, and then used to predict the future price of a stock based on its current economic indicators.
Decision trees are particularly useful for stock market forecasting because they can handle non-linear relationships between the input variables and the output variable. Additionally, decision trees can identify important features that contribute to the prediction of the stock price, such as the relationship between interest rates and stock prices.
Feature Selection and Interpretability
Decision trees are widely used in machine learning for their ability to select relevant features and make predictions based on them. In this section, we will explore how decision trees can be used for feature selection and how they can be interpreted for better understanding.
Feature Selection using Decision Trees
Feature selection is the process of selecting a subset of relevant features from a larger set of available features. Decision trees can be used for feature selection by constructing a tree where each node represents a feature and each branch represents a decision based on the feature's value. The features that are most important for making predictions are those that are frequently used in the tree's branches.
For example, consider a dataset with two features, age and income, and a target variable, disease status. A decision tree constructed from this dataset might look like this:
|__ Under 40
|__ No disease
|__ 40 or older
In this tree, the age feature is used to split the data into two groups: under 40 and 40 or older. The income feature is not used in the tree, indicating that it is not as important for making predictions as age.
Importance of Features in Decision Trees
Decision trees assign a value to each feature to indicate its importance in making predictions. This value is called the Gini impurity or information gain and measures the probability of a randomly chosen instance being incorrectly classified if it were classified according to the class distribution in the node.
For example, in the above tree, the Gini impurity of the node containing the age feature is lower than the Gini impurity of the node containing the income feature, indicating that age is a more important feature for making predictions.
Visualizing and Interpreting Decision Trees
Decision trees can be visualized to better understand how they make predictions. The tree structure shows how the data is split into smaller and smaller subsets based on the most important features. This visualization can help identify which features are most important for making predictions and how the predictions are made.
For example, in the above tree, we can see that the data is split into two groups based on age and then further split into two groups based on disease status. This visualization helps us understand how the predictions are made and which features are most important for making them.
In conclusion, decision trees can be used for feature selection and interpretation, allowing machine learning models to select relevant features and make predictions based on them. By using decision trees for feature selection and visualization, we can better understand how the predictions are made and which features are most important for making them.
1. What are decision trees?
Decision trees are a popular machine learning algorithm used for both classification and regression tasks. They are graphical representations of decisions and their possible consequences. The tree consists of nodes that represent decisions, and leaves that represent the outcome of those decisions.
2. How do decision trees work?
Decision trees work by recursively splitting the data into subsets based on the feature that provides the most information gain. This process continues until a stopping criterion is met, such as reaching a maximum depth or minimum number of samples per leaf. The final tree is then used to make predictions by traversing down the tree based on the input features.
3. When are decision trees used?
Decision trees are used in a variety of applications, including finance, healthcare, marketing, and more. They are particularly useful in situations where the relationship between the input features and the output variable is complex and difficult to model. Decision trees can also be used for feature selection and to identify important variables in the data.
4. What are the advantages of using decision trees?
Decision trees have several advantages, including their ability to handle both numerical and categorical data, their simplicity and interpretability, and their effectiveness in handling missing data. They can also be used for both classification and regression tasks, and can be easily combined with other machine learning algorithms to improve performance.
5. What are the disadvantages of using decision trees?
Decision trees can be prone to overfitting, especially when the tree is deep and complex. They can also be sensitive to outliers and can be biased towards the feature distribution used to train the tree. Finally, they may not perform well when the relationship between the input features and the output variable is non-linear.
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Mathematics is an incredibly broad field that encompasses a wide range of concepts and topics, each with its own unique set of principles and characteristics. Among the many ideas within mathematics is the concept of fractionswhich is a crucial element in everyday arithmetic.
However, not all fractions are equal: Some are classified as proper fractions and others are complex fractions. Proper fractions are an essential component in elementary mathematics and continue to be relevant in more advanced fields. So what exactly is a proper fraction?
A proper fraction is a fraction where the numerator, or the number above, is smaller than the denominator, or the number below. In other words, the value of the fraction is always less than one. Proper fractions are integrals to represent a part or a portion of something, like a slice of pizza or a fraction of a budget.
Furthermore, they play an important role in many mathematical operations such as addition, subtraction, multiplication, and division.
Concept and definition of proper fractions
In mathematics, a proper fraction is a fraction in which the numerator (the top number) is smaller than the denominator (the number below). Simply put, proper fractions are those that represent a value less than one.
For example, 2/5 is a proper fraction because the numerator 2 is smaller than the denominator 5 and represents the value 0.4. By contrast, an improper fraction has a numerator that is equal to or greater than the denominator and represents a value greater than one.
For example, 7/5 is an improper fraction because the numerator 7 is greater than the denominator 5 and represents the value 1.4. In general, the concept of proper fractions is fundamental to studying fractions and understanding their characteristics in mathematics.
It is crucial to differentiate proper fractions from other types of fractions when performing operations such as addition, subtraction, multiplication, and division. Proper fractions have specific properties that clearly distinguish them from improper fractions and mixed numbers in mathematics.
What are the characteristics of proper fractions?
One of the main features of a proper fraction is that it always produces a decimal number that is less than one when divided. Another feature is that if the fraction is added to any whole number, the result will always be greater than one.
Furthermore, proper fractions often are represented in their simplest form, or in its reduced form, where the numerator and denominator have no common factors more than one. Finally, proper fractions can be converted to percentages or decimals to make it easier to calculate and compare with other numbers.
Some characteristics of fractions own are the following:
- Decimal value less than 1: As we mentioned earlier, the decimal value of a proper fraction is less than 1. For example, 3/4 has a decimal value of 0.75, which is less than 1.
- fractional part of a unit: Proper fractions represent a fractional part of a unit. For example, 2/3 represents two parts of a unit divided into three equal parts.
- Numerator less than denominator: The numerator of a proper fraction is always less than the denominator. If the numerator were equal to or greater than the denominator, it would be an improper fraction.
- always positive: Proper fractions are always positive, which means that their value is greater than zero. This is because the numerator and denominator are always positive numbers.
- can be simplified: Proper fractions can be simplified to their simplest form, dividing the numerator and denominator by their greatest common divisor. For example, the fraction 6/8 can be simplified by dividing both numbers by 2, leaving 3/4.
Proper fractions can be used in various mathematical equations and are essential for developing a strong understanding of mathematical concepts.
How do you make a proper fraction?
To make a proper fraction, we simply need make sure the numerator is smaller than the denominator. For example, the fraction 2/3 is a proper fraction, since the numerator 2 is less than the denominator 3.
To make a proper fraction, the following steps must be followed:
- choose a numerator: The numerator is the number that goes above the horizontal line of the fraction. It must be an integer and can be positive or negative.
- choose a denominator: The denominator is the number that goes under the horizontal line of the fraction. Must be an integer and cannot be zero.
- Verify that the numerator is less than the denominator: For a fraction to be proper, the numerator must always be less than the denominator. If the numerator is equal to or greater than the denominator, the fraction will be improper.
- Simplify, if necessary: If the numerator and denominator have a common factor, they can be simplified by dividing by the same number. For example, if we have the fraction 10/20, it can be simplified by dividing both numbers by 10, remaining as 1/2.
We can also represent a proper fraction as a decimal dividing the numerator by the denominator. In the case of 2/3, the decimal equivalent is recurring 0.666. In short, to make a proper fraction, we need to make sure that the numerator is smaller than the denominator.
What is the difference between proper and improper fractions?
In mathematics, fractions are represented by a numerator and a denominator, separated by a horizontal line. A proper fraction is a type of fraction where the numerator is smaller than the denominator.
In other words, the value of a proper fraction is less than one. For example, 1/3 and 4/5 are proper fractions, since the numerator is smaller than the denominator. Conversely, an improper fraction has a numerator that is equal to or greater than the denominator.
This type of fraction represents a value greater than one. For example, 5/3 and 8/7 are improper fractions, since the numerator is greater than the denominator. Understanding the difference between proper and improper fractions is important in various mathematical calculations, such as converting between fractions and mixed numbers, simplifying fractions, and comparing the value of fractions.
What are proper fractions? – Examples
Proper fractions are important in mathematics, as they are one of the fundamental concepts entering primary school. In terms of features, a proper fraction can be defined as a fraction where the numerator is always less than the denominator.
For example, 1/2, 2/3 and 3/4 are examples of proper fractions. Proper fractions can also be represented in decimal form by dividing the numerator by the denominator, the resulting decimal being a value between 0 and 1.
Proper fractions are used in many different applications in mathematics, including fractions, decimals, percentages, ratios and dimensions.
Proper fractions play an essential role in mathematics and are fundamental to understanding more complex mathematical concepts, their value is less than 1 and they can be easily convert to decimal numbers.
Also, they are used in many real-world situations, such as calculating probabilities, measuring proportions, and making comparisons.
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Edge Computing Tutorial
Last updated on 25th Sep 2020, Blog, Tutorials
Edge computing is a distributed information technology (IT) architecture in which client data is processed at the periphery of the network, as close to the originating source as possible. The move toward edge computing is driven by mobile computing, the decreasing cost of computer components and the sheer number of networked devices in the internet of things (IoT).
Depending on the implementation, time-sensitive data in an edge computing architecture may be processed at the point of origin by an intelligent device or sent to an intermediary server located in close geographical proximity to the client. Data that is less time-sensitive is sent to the cloud for historical analysis, big data analytics and long-term storage.
How does edge computing work?
One simple way to understand the basic concept of edge computing is by comparing it to cloud computing. In cloud computing, data from a variety of disparate sources is sent to a large centralized data center that is often geographically far away from the source of the data. This is why the name “cloud” is used — because data gets uploaded to a monolithic entity that is far away from the source, like evaporation rising to a cloud in the sky.
Unlike cloud computing, edge computing allows data to exist closer to the data sources through a network of edge devices.
By contrast, edge computing is sometimes called fog computing. The word “fog” is meant to convey the idea that the advantages of cloud computing should be brought closer to the data source. (In meteorology, fog is simply a cloud that is close to the ground.) The benefits of the cloud are brought closer to the ground (data source) and are spread out instead of centralized.
The name “edge” in edge computing is derived from network diagrams; typically, the edge in a network diagram signifies the point at which traffic enters or exits the network. The edge is also the point at which the underlying protocol for transporting data may change. For example, a smart sensor might use a low-latency protocol like MQTT to transmit data to a message broker located on the network edge, and the broker would use the hypertext transfer protocol (HTTP) to transmit valuable data from the sensor to a remote server over the Internet.
Edge infrastructure places computing power closer to the source of data. The data has less distance to travel, and more places to travel to than in a typical cloud infrastructure. Edge technology — sometimes called edge nodes — is established at the periphery of a network and may take the form of edge servers, edge data centers or other networked devices with compute power. Instead of sending massive amounts of data from disparate locations to a centralized data center, smaller amounts are sent to the edge nodes to be processed and returned, only being sent along to a larger remote data center if necessary.
Why does edge computing matter?
Edge computing is important because the amount of data being generated is growing as time goes on, and the array of devices that generate it are as well. The rise of IoT and mobile computing contribute to both factors.
Transmitting massive amounts of raw data over a network puts a tremendous load on network resources. It can also be difficult to process and maintain that massive amount when it is gathered from disparate sources. The data quality and types of data may vary significantly from source to source, and many resources are used to funnel it all to one centralized location. An edge network architecture can ease this strain on resources by decentralizing and processing the generated data closer to the source.
In some cases, it is much more efficient to process data near its source and send only the data that has value over the network to a remote data center. Instead of continually broadcasting data about the oil level in a car’s engine, for example, an automotive sensor might simply send summary data to a remote server on a periodic basis. A smart thermostat might only transmit data if the temperature rises or falls outside acceptable limits. Or an intelligent Wi-Fi security camera aimed at an elevator door might use edge analytics and only transmit data when a certain percentage of pixels significantly change between two consecutive images, indicating motion
Benefits of edge computing
A major benefit of edge computing is that it improves time to action and reduces response time down to milliseconds, while also conserving network resources. Some specific benefits of edge computing are:
- Increases capacity for low latency applications and reduced bottlenecks.
- Enables more efficient use of IoT and mobile computing.
- Enables 5G connectivity.
- Enables real-time analytics and improved business intelligence (BI) insights by using machine learning and smart devices within the edge compute
- Enables quick response times and more accurate processing of time-sensitive data.
- Centralizes management of devices by giving end-users more access to data processes, allowing for more specific network insight and control.
- Increases availability of devices and decreases strain on centralized network resources.
- Enables data caching closer to the source using content delivery networks (CDNs).
Challenges of edge computing
Despite its benefits, edge computing is not expected to completely replace cloud computing. Despite the ability to reduce latency and network bottlenecks, edge computing could pose significant security, licensing and configuration challenges.
- Security challenges: Edge computing’s distributed architecture increases the number of attack vectors. Meaning, the more intelligence an edge client has, the more vulnerable it becomes to malware infections and security exploits. Edge devices are also likely to be less protected in terms of physical security than a traditional data center.
- Licensing challenges: Smart clients can have hidden licensing costs. While the base version of an edge client might initially have a low ticket price, additional functionalities may be licensed separately and drive the price up.
- Configuration challenges: Unless device management is centralized and extensive, administrators may inadvertently create security holes by failing to change the default password on each edge device or neglecting to update firmware in a consistent manner, causing configuration drift.
Examples of edge computing
Edge computing offers a range of value propositions for smart IoT applications and use cases across a variety of industries. Some of the most popular use cases that will depend on edge computing to deliver improved performance, security and productivity for enterprises include:
For autonomous driving technologies to replace human drivers, cars must be capable of reacting to road incidents in real-time. On average, it may take 100 milliseconds for data transmission between vehicle sensors and backend cloud datacenters. In terms of driving decisions, this delay can have a significant impact on the reaction of self-driving vehicles.
Toyota predicts that the amount of data transmitted between vehicles and the cloud could reach 10 exabytes per month by the year 2025. If network capacity fails to accommodate the necessary network traffic, vendors of autonomous vehicle technologies may be forced to limit self-driving capabilities of the cars.
In addition to the data growth and existing network limitations, technologies such as 5G connectivity and Artificial Intelligence are paving the way for edge computing.5G will help deploy computing capabilities closer to the logical edge of the network in the form of distributed cellular towers. The technology will be capable of greater data aggregation and processing while maintaining high speed data transmission between vehicles and communication towers.
AI will further facilitate intelligent decision-making capabilities in real-time, allowing cars to react faster than humans in response to abrupt changes in traffic flows.
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Logistics service providers leverage IoT telematics data to realize effective fleet management operations. Drivers rely on vehicle-to-vehicle communication as well as information from backend control towers to make better decisions. Locations of low connectivity and signal strength are limited in terms of the speed and volume of data that can be transmitted between vehicles and backend cloud networks.
With the advent of autonomous vehicle technologies that rely on real-time computation and data analysis capabilities, fleet vendors will seek efficient means of network transmission to maximize the value potential of fleet telematics data for vehicles travelling to distant locations.
By drawing computation capabilities in close proximity to fleet vehicles, vendors can reduce the impact of communication dead zones as the data will not be required to send all the way back to centralized cloud data centers. Effective vehicle-to-vehicle communication will enable coordinated traffic flows between fleet platoons, as AI-enabled sensor systems deployed at the network edges will communicate insightful analytics information instead of raw data as needed.
The manufacturing industry heavily relies on the performance and uptime of automated machines. In 2006, the cost of manufacturing downtime in the automotive industry was estimated at $1.3 million per hour. A decade later, the rising financial investment toward vehicle technologies and the growing profitability in the market make unexpected service interruptions more expensive in multiple orders of magnitude.
With edge computing, IoT sensors can monitor machine health and identify signs of time-sensitive maintenance issues in real-time. The data is analyzed on the manufacturing premises and analytics results are uploaded to centralized cloud data centers for reporting or further analysis.
- Analyzing anomalies can allow the workforce to perform corrective measures or predictive maintenance earlier, before the issue escalates and impacts the production line.
- Analyzing the most impactful machine health metrics can allow organizations to prolong the useful life of manufacturing machines.
As a result, manufacturing organizations can lower the cost of maintenance, improve operational effectiveness of the machines, and realize higher return on assets.
Voice assistance technologies such as Amazon Echo, Google Home, and Apple Siri, among others, are pushing the boundaries of AI. An estimated 56.3 million smart voice assistant devices will be shipped globally in 2018. Gartner predicts that 30 percent of consumer interactions with the technology will take place via voice by the year 2020. The fast-growing consumer technology segment requires advanced AI processing and low-latency response time to deliver effective interactions with end-users.
Particularly for use cases that involve AI voice assistant capabilities, the technology needs go beyond computational power and data transmission speed. The long-term success of voice assistance depends on consumer privacy and data security capabilities of the technology. Sensitive personal information is a treasure trove for underground cybercrime rings and potential network vulnerabilities in voice assistance systems could pose unprecedented security and privacy risks to end-users.
To address this challenge, vendors such as Amazon are enhancing their AI capabilities and deploying the technology closer to the edge, so that voice data doesn’t need to move across the network. Amazon is reportedly working to develop its own AI chip for the Amazon Echo devices.
Prevalence of edge computing in the voice assistance segment will hold equal importance for enterprise users as employees working in the field or on the manufacturing line will be able to access and analyze useful information without interrupting manual work operations.
Edge computing, IoT and 5G possibilities
5G requires mobile edge computing, largely because 5G relies on a greater multitude of network nodes — more than 4G, which relies on larger centralized cell towers. 5G’s frequency band generally travels shorter distances and is weaker than 4G, so more nodes are required to pass the signal between them and mobile service users. More nodes means a higher likelihood that one may be compromised, so centralized management of these nodes is crucial. The nodes must also be able to process and collect real-time data that enables them to better serve users.
Edge computing is being driven by the proliferation and expansion of IoT. With so many networked devices, from smart speakers to autonomous vehicles to the industrial internet of things (IIoT), there needs to be processing power close to or even on those devices to handle the massive amounts of data they constantly generate.
IoT provides the devices and the data they collect, 5G provides a faster, more powerful network for that data to travel on, and edge computing provides the processing power to make use of and handle that data in real-time. The combination of these three makes for a tightly networked world that would make newer technologies like autonomous vehicles more feasible for public use. Together, the proliferation of 5G, IoT and edge computing could also make smart cities more feasible in places where they are not now.
Future of edge computing
According to the Gartner Hype Cycle 2017, edge computing is drawing closer to the Peak of Inflated Expectations and will likely reach the Plateau of Productivity in 2-5 years. Considering the ongoing research and developments in AI and 5G connectivity technologies, and the rising demands of smart industrial IoT applications, Edge Computing may reach maturity faster than expected.
Forms of Edge Computing
In this model, Edge Computing is taken to the customers in the existing environments. For example, AWS Greengrass and Microsoft Azure IoT Edge.
This model of Edge Computing is basically an extension of the public cloud. Content Delivery Networks are classic examples of this topology in which the static content is cached and delivered through a geographically spread edge locations.
Vapor IO is an emerging player in this category. They are attempting to build infrastructure for cloud edge. Vapor IO has various products like Vapor Chamber. These are self-monitored. They have sensors embedded in them using which they are continuously monitored and evaluated by Vapor Software, VEC(Vapor Edge Controller). They also have built OpenDCRE, which we will see later in this blog.
The fundamental difference between device edge and cloud edge lies in the deployment and pricing models. The deployment of these models — device edge and cloud edge — are specific to different use cases. Sometimes, it may be an advantage to deploy both the models.
Edges around you
Edge Computing examples can be increasingly found around us:
- 1. Smart street lights
- 2. Automated Industrial Machines
- 3. Mobile devices
- 4. Smart Homes
- 5. Automated Vehicles (cars, drones etc)
Data Transmission is expensive. By bringing compute closer to the origin of data, latency is reduced as well as end users have better experience. Some of the evolving use cases of Edge Computing are Augmented Reality(AR) or Virtual Reality(VR) and the Internet of things. For example, the rush which people got while playing an Augmented Reality based pokemon game, wouldn’t have been possible if “real-timeliness” was not present in the game. It was made possible because the smartphone itself was doing AR not the central servers. Even Machine Learning(ML) can benefit greatly from Edge Computing. All the heavy-duty training of ML algorithms can be done on the cloud and the trained model can be deployed on the edge for near real-time or even real-time predictions. We can see that in today’s data-driven world edge computing is becoming a necessary component of it.
There is a lot of confusion between Edge Computing and IOT. If stated simply, Edge Computing is nothing but the intelligent Internet of things(IOT) in a way. Edge Computing actually complements traditional IOT. In the traditional model of IOT, all the devices, like sensors, mobiles, laptops etc are connected to a central server. Now let’s imagine a case where you give the command to your lamp to switch off, for such simple task, data needs to be transmitted to the cloud, analyzed there and then lamp will receive a command to switch off. Edge Computing brings computing closer to your home, that is either the fog layer present between lamp and cloud servers is smart enough to process the data or the lamp itself.
If we look at the below image, it is a standard IOT implementation where everything is centralized. While Edge Computing philosophy talks about decentralizing the architecture.
Sandwiched between the edge layer and cloud layer, there is the Fog Layer. It bridges the connection between the other two layers.
The difference between fog and edge computing is described in this tutorial –
- Fog Computing — Fog computing pushes intelligence down to the local area network level of network architecture, processing data in a fog node or IoT gateway.
- Edge computing pushes the intelligence, processing power and communication capabilities of an edge gateway or appliance directly into devices like programmable automation controllers (PACs).
How do we manage Edge Computing?
The Device Relationship Management or DRM refers to managing, monitoring the interconnected components over the internet. AWS IOT Core and AWS Greengrass, Nebbiolo Technologies have developed Fog Node and Fog OS, Vapor IO has OpenDCRE using which one can control and monitor the data centers.
Following image (source — AWS) shows how to manage ML on Edge Computing using AWS infrastructure.
AWS Greengrass makes it possible for users to use Lambda functions to build IoT devices and application logic. Specifically, AWS Greengrass provides cloud-based management of applications that can be deployed for local execution. Locally deployed Lambda functions are triggered by local events, messages from the cloud, or other sources.
This GitHub repo demonstrates a traffic light example using two Greengrass devices, a light controller, and a traffic light.
We believe that next-gen computing will be influenced a lot by Edge Computing and will continue to explore new use-cases that will be made possible by the Edge.
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Table of Contents
What is Bone?
- Bones are integral components of the skeletal system in the human body. There are a total of 206 bones in the adult human skeleton, which can be classified into five main types based on their shape, placement, and additional properties. These types include flat, long, short, irregular, and sesamoid bones.
- The skeletal system serves several crucial functions in the body. First and foremost, bones provide somatic rigidity, giving the body its structural outline and enabling an erect posture and movement, such as the bipedal gait unique to humans. Additionally, bones act as protective structures, safeguarding internal organs and other vital structures within the body.
- Bone is a rigid connective tissue that constitutes the skeletal framework. It is comprised of various types of cells and possesses an internal matrix with a honeycomb-like structure, which imparts rigidity to the bones. The primary role of bones is to provide structural support to the body and facilitate mobility. Furthermore, bones play a role in the production of red blood cells (RBCs) and white blood cells (WBCs), and they also serve as storage sites for minerals.
- Bone tissue is a form of calcified connective tissue. It consists of a matrix composed of ground substance and collagen fibers, within which osteocytes are embedded. Osteocytes are the most abundant cells found in mature bone and are responsible for maintaining bone growth and density. Calcium and phosphate are abundant within the bone matrix, contributing to the strength and density of the bone structure.
- Each bone in the body is connected to one or more other bones through joints, with the exception of the hyoid bone. Through the attachment of tendons and musculature, the skeletal system acts as a lever, generating the force required for movement. The inner core of bones, known as the medulla, contains either red bone marrow, which serves as the primary site of hematopoiesis (formation of blood cells), or yellow bone marrow, which is predominantly composed of adipose tissue.
- Bone development can occur through two main processes: endochondral and membranous ossification. These processes determine the development of various bones in the body, including those in the skull. The specific characteristics of bone development, combined with the overall shape of the bone, are utilized in the classification of the skeletal system.
Definition of Bone
Bone is a rigid connective tissue that forms the framework of the skeletal system in humans and other vertebrates. It provides structural support, protects internal organs, enables movement, produces blood cells, stores minerals, and maintains overall body rigidity.
Features of Bone
Bones possess several notable features that contribute to their structure and function within the body:
- Rigidity: Bones are highly rigid and provide structural support to the body. Their hardness and strength come from the mineralized matrix of calcium salts, particularly hydroxyapatite, which gives bones their characteristic hardness.
- Cellular Composition: Bones are composed of various types of cells. Osteoblasts are responsible for bone formation, while osteoclasts are involved in bone resorption. Osteocytes are mature bone cells that maintain bone health and regulate mineral metabolism.
- Matrix: The matrix of bone is a unique combination of organic and inorganic substances. It consists of collagen fibers that provide flexibility and resilience, as well as a ground substance composed of proteins, glycoproteins, and proteoglycans. The inorganic component, primarily calcium and phosphate, gives bones their hardness.
- Marrow Cavity: Many bones contain a central marrow cavity, which can house either red bone marrow or yellow bone marrow. Red bone marrow is responsible for the production of blood cells, including red blood cells, white blood cells, and platelets. Yellow bone marrow is primarily composed of adipose tissue and serves as a site for energy storage.
- Joints and Articulations: Bones are connected to each other at joints or articulations. These connections can be immovable (as in the skull sutures) or movable (such as the ball-and-socket joint of the hip). Joints allow for movement and flexibility in the skeletal system.
- Blood Supply: Bones have a rich blood supply through a network of blood vessels. Blood vessels penetrate the bone through channels called Haversian canals, supplying nutrients and oxygen to the bone cells and aiding in waste removal.
- Growth and Remodeling: Bones have the ability to grow and remodel throughout life. During growth, the growth plates at the ends of long bones allow for longitudinal bone growth. Remodeling involves the ongoing process of bone resorption by osteoclasts and bone formation by osteoblasts, maintaining bone health and adjusting bone structure according to mechanical demands.
These features collectively contribute to the strength, flexibility, and functional adaptability of bones in the human body.
Gross Anatomy of Bones
The gross anatomy of bones reveals important structural features that contribute to their function and protection within the body:
- Long Bones: Long bones, such as the femur or humerus, consist of two main regions: the diaphysis and the epiphysis. The diaphysis refers to the tubular shaft that extends between the proximal and distal ends of the bone. It contains a hollow space called the medullary cavity, which is filled with yellow bone marrow in adults. The outer walls of the diaphysis are composed of dense and hard compact bone.
- Epiphysis: The wider sections at each end of a long bone are known as the epiphyses. Internally, the epiphyses contain spongy bone, which is another type of osseous tissue. In some long bones, such as the femur, red bone marrow fills the spaces within the spongy bone. The epiphyses meet the diaphysis at the metaphysis. During growth, the metaphysis contains the epiphyseal plate, a site of longitudinal bone elongation. In early adulthood, when bone growth ceases, the epiphyseal plate is replaced by an epiphyseal line.
- Endosteum and Periosteum: The inner lining of the bone adjacent to the medullary cavity is called the endosteum. It is a layer of bone cells that contribute to bone growth, repair, and remodeling throughout life. On the outside of the bone, there is the periosteum, a double-layered structure. The cellular layer of the periosteum is adjacent to the cortical bone and is covered by an outer fibrous layer of dense irregular connective tissue. The periosteum contains blood vessels, nerves, and lymphatic vessels that nourish compact bone. Tendons and ligaments attach to bones at the periosteum. The periosteum covers the entire outer surface of the bone, except at the regions where the epiphyses meet other bones to form joints. In these joint regions, the epiphyses are covered with articular cartilage, which acts as a friction-reducing and shock-absorbing layer.
- Flat Bones: Flat bones, such as those in the cranium, have a layered structure. They consist of an inner layer of spongy bone called diploë, sandwiched between two layers of compact bone. This layered arrangement provides protection to internal organs. If the outer layer of a cranial bone fractures, the intact inner layer still offers protection to the brain.
Bone markings refer to the surface features of bones, which vary depending on their function and location in the body. There are three general classes of bone markings: articulations, projections, and holes.
- Articulations: Articulations are where two bone surfaces come together to form a joint. These surfaces are designed to fit each other, such as one being rounded and the other cupped, to facilitate joint movement. An example of an articulation is the knee joint.
- Projections: Projections are raised markings on the surface of a bone. They serve as attachment points for tendons and ligaments, indicating the forces exerted through those attachments. Examples of projections include the spinous process of the vertebrae or the chin.
- Holes: Holes in bones allow the passage of blood vessels and nerves. The size and shape of these holes correspond to the size of the vessels and nerves that enter the bone. Examples of holes include the foramen (holes through which blood vessels pass) and the external auditory meatus.
The following table provides examples of different types of bone markings:
|Where two bones meet
|Spinous process of the vertebrae
|Holes and depressions
|Foramen magnum in the occipital bone
Understanding bone markings is essential for studying bone structure, identifying attachment points for muscles and ligaments, and comprehending the functional interactions between bones and other structures in the body.
Bone Matrix and Cells
- The bone matrix is a crucial component of osseous tissue, providing the structure and strength necessary for the function of bones. It is primarily composed of collagen fibers and calcium phosphate salt, with collagen accounting for about one-third of the matrix and the calcium phosphate salt comprising the remaining two-thirds.
- Collagen, a protein, forms a scaffold within the bone matrix, providing a surface for the attachment of inorganic salt crystals. These salt crystals, mainly hydroxyapatite, result from the combination of calcium phosphate and calcium carbonate. As the hydroxyapatite crystallizes, it also incorporates other inorganic salts like magnesium hydroxide, fluoride, and sulfate. These crystallized minerals give bones their hardness and strength.
- The collagen fibers within the bone matrix play a vital role in calcification. They provide a framework for the mineralization process, allowing the hydroxyapatite crystals to adhere and grow. Additionally, collagen fibers contribute to the flexibility of bones, preventing them from becoming brittle. Without the collagen matrix, bones would lack the necessary support and crumble easily.
- Conversely, if the inorganic matrix, or minerals, were removed from the bone while leaving the collagen intact, the bone would become overly flexible and lose its ability to bear weight. The balance between the organic collagen matrix and the inorganic mineral matrix is crucial for the proper functioning of bones, providing both strength and flexibility.
- Overall, the bone matrix, with its combination of collagen fibers and mineral salts, provides the necessary framework, strength, and flexibility that allow bones to withstand mechanical stress, support body structures, and perform their essential functions in the skeletal system.
Bone cells play a crucial role in the functioning and maintenance of bone tissue. There are four main types of bone cells: osteoblasts, osteocytes, osteogenic cells, and osteoclasts.
- Osteoblasts: Osteoblasts are responsible for the formation of new bone tissue. They are found in the growing portions of bone, such as the endosteum (inner surface) and the cellular layer of the periosteum (outer surface). Osteoblasts synthesize and secrete the collagen matrix and other proteins that make up the bone matrix.
- Osteocytes: Osteocytes are the most abundant type of bone cell and are considered the primary cells of mature bone. They develop from osteoblasts that become trapped within the calcified bone matrix. Each osteocyte resides in a small cavity called a lacuna. Osteocytes play a crucial role in maintaining the mineral concentration of the bone matrix through the secretion of enzymes. They communicate with each other and receive nutrients through long cytoplasmic processes that extend through canaliculi, which are channels within the bone matrix.
- Osteogenic cells: Osteogenic cells, also known as osteoprogenitor cells, are undifferentiated cells with a high mitotic activity. They are capable of dividing and differentiating into osteoblasts. Immature osteogenic cells are found in the cellular layer of the periosteum and the endosteum.
- Osteoclasts: Osteoclasts are responsible for bone resorption, which is the breakdown of old or damaged bone tissue. These multinucleated cells originate from monocytes and macrophages, types of white blood cells, rather than from osteogenic cells. Osteoclasts continuously break down old bone, while osteoblasts simultaneously form new bone. The balance between osteoblasts and osteoclasts is essential for the constant remodeling and reshaping of bone tissue.
The interaction and coordinated activity of these different bone cells ensure the dynamic nature of bone tissue, with new bone constantly being formed and old bone being broken down and replaced. This ongoing process helps maintain the integrity and strength of the skeletal system.
|Bone Cells (Table 6.3)
|Develop into osteoblasts
|Endosteum, cellular layer of the periosteum
|Endosteum, cellular layer of the periosteum, growing portions of bone
|Maintain mineral concentration of matrix
|Entrapped in matrix
|Endosteum, cellular layer of the periosteum, at sites of old, injured, or unneeded bone
Compact and Spongy Bone
Most bones have compact and spongy osseous tissue, but the distribution and concentration of this tissue varies depending on the bone’s overall function. Although compact and spongy bone are formed of the same matrix components and cells, they are structured differently. Compact bone is dense enough to sustain compressive stresses, whereas spongy bone (also known as cancellous bone) has open spaces and is supporting, but it is also lightweight and easily reshaped to fit changing body needs.
- Compact bone is the denser and stronger type of osseous tissue that forms the outer cortex of all bones and is in direct contact with the periosteum. It is characterized by a highly organized arrangement of concentric circles known as osteons or Haversian systems.
- When observed under a microscope, compact bone appears as a series of concentric circles resembling tree trunks. Each concentric circle is called a lamella, and it is composed of collagen fibers and calcified matrix. The collagen fibers in adjacent lamellae run in perpendicular directions to provide resistance against twisting forces.
- At the center of each osteon, there is a central canal, also known as the Haversian canal, which contains blood vessels, nerves, and lymphatic vessels. These vessels and nerves extend through perforating canals, called Volkmann’s canals, at right angles to reach the periosteum and endosteum. The central canal is lined by the endosteum, which allows for the removal, remodeling, and rebuilding of osteons over time.
- Osteocytes, the primary cells of mature bone, are located within small spaces called lacunae. These lacunae are situated at the borders of adjacent lamellae. Canaliculi, tiny channels, connect the lacunae and allow for communication and nutrient exchange between osteocytes. Despite the calcified nature of the matrix, this system of canaliculi enables the transport of nutrients to the osteocytes and the removal of waste materials from them.
- Compact bone provides strength and support to the skeletal system and plays a vital role in protecting internal organs and providing a framework for movement and locomotion. Its highly organized structure and the network of blood vessels, nerves, and osteocytes contribute to its resilience and ability to adapt to mechanical stress.
Spongy (Cancellous) Bone
- Spongy bone, also known as cancellous bone, differs from compact bone in terms of its structure and arrangement of osteocytes. While both types of bone contain osteocytes housed in lacunae, spongy bone does not have the concentric circles seen in compact bone. Instead, the osteocytes and lacunae are distributed within a lattice-like network of matrix spikes called trabeculae.
- The trabeculae in spongy bone are covered by the endosteum, a cellular layer that facilitates the remodeling of the bone. Although the trabeculae may appear random in their arrangement, each trabecula forms along lines of stress to efficiently direct forces towards the more solid compact bone, providing strength and support to the bone structure.
- One of the important functions of spongy bone is to balance the dense and heavy nature of compact bone. It achieves this by making bones lighter, allowing muscles to move them more easily. The lattice-like structure of spongy bone also serves to distribute forces and stresses throughout the bone, enhancing its overall resilience.
- Moreover, the spaces within some spongy bones contain red bone marrow, which is protected by the trabeculae. Red bone marrow is the site of hematopoiesis, the process of blood cell formation. This crucial function of spongy bone highlights its role in producing and maintaining the body’s blood cells.
- Overall, spongy bone complements the structural properties of compact bone, contributing to the strength, flexibility, and lightweight nature of the skeletal system, while also supporting essential physiological processes such as hematopoiesis.
Blood and Nerve Supply
- Blood and nerve supply play crucial roles in maintaining the health and functionality of bone tissue.
- The blood supply to bones involves arteries that pass through the compact bone to reach the spongy bone and medullary cavity. These arteries enter the bone through small openings called nutrient foramina, which are located in the diaphysis (the tubular shaft of a long bone). This allows for the delivery of oxygen, nutrients, and other essential substances to the bone tissue. The spongy bone receives nourishment from blood vessels of the periosteum, which penetrate the spongy bone and supply nutrients to the osteocytes. Additionally, blood circulates within the marrow cavities, providing nourishment to the bone marrow. As the blood passes through the marrow cavities, it is collected by veins, which then exit the bone through the nutrient foramina.
- Nerve supply to the bone follows a similar pathway as the blood vessels. Nerves enter the bone through the same channels and tend to concentrate in the metabolically active regions of the bone. These nerve fibers play important roles in the regulation of blood supply to the bone, as well as in bone growth. They are also responsible for sensing pain, which is vital for the body’s protective mechanisms. The concentration of nerves in metabolically active sites of the bone reflects their involvement in regulating the bone’s metabolic processes and responding to any potential damage or injury.
- The coordinated supply of blood and nerves ensures the proper functioning and maintenance of bone tissue. It enables the delivery of essential nutrients, oxygen, and regulatory signals, while also facilitating the removal of waste products. This intricate network of blood vessels and nerve fibers supports bone growth, repair, and overall bone health.
Classification of Bone
1. Long Bones
- Long bones are a specific type of bone characterized by their cylindrical shape, being longer than they are wide. It’s important to note that the term “long” refers to the shape of the bone, not its size. Long bones are primarily found in the limbs, such as the arms, legs, fingers, and toes. Examples of long bones include the humerus, ulna, radius, femur, tibia, fibula, metacarpals, phalanges, metatarsals, and phalanges.
- The main function of long bones is to act as levers, facilitating movement when muscles contract. They provide support, stability, and mobility for the body. The structure of long bones consists of a shaft called the diaphysis, which contains a marrow cavity filled with bone marrow. The diaphysis is surrounded by a dense layer of compact bone that provides strength and protection. At the ends of the long bones are the epiphyses, which have a rounded shape. The epiphyses are covered with a layer of smooth and slippery articular cartilage, which allows for smooth joint movement. Within the epiphyses, red bone marrow is present, responsible for the production of blood cells through a process called hematopoiesis.
- While most limb bones are classified as long bones, there are a few exceptions. The patella, or kneecap, is a sesamoid bone that develops within the tendon of the quadriceps femoris muscle and is located in the knee joint. Additionally, the bones of the wrist (carpals) and ankle (tarsals) are categorized as short bones due to their cube-like shape.
- Overall, long bones play a vital role in supporting the body, enabling movement, and serving as sites for blood cell production in the bone marrow.
2. Short Bones
- Short bones are a specific type of bone characterized by their cube-like shape, with approximately equal length, width, and thickness. Unlike long bones, short bones do not have a shaft or distinct epiphyses. They are primarily found in the wrists and ankles, specifically in the carpals of the wrists and the tarsals of the ankles.
- The main function of short bones is to provide stability and support to the body, as well as contribute to some limited motion. Due to their compact and sturdy structure, they play a crucial role in maintaining the overall integrity of the skeletal system. The cube-like shape of short bones allows for a greater degree of stability and resistance to compressive forces.
- In the wrists, the short bones known as carpals are arranged in two rows of four bones each, forming the structure of the wrist joint. These carpals work together to provide stability and flexibility to the wrist, allowing for movements such as flexion, extension, and lateral deviation.
- Similarly, in the ankles, the short bones called tarsals are arranged to form the structure of the ankle joint. The tarsals provide stability and support to the foot, allowing for movements such as dorsiflexion, plantarflexion, inversion, and eversion.
- While short bones primarily contribute to stability and support, they also participate in limited motion, facilitating smooth movements and weight distribution. Their compact and solid nature makes them resistant to bending and compression, enhancing their role in providing structural support to the body.
- Overall, short bones, such as the carpals and tarsals, are essential components of the skeletal system, providing stability, support, and some degree of motion in the wrists and ankles. Their cube-like shape and compact structure contribute to the overall strength and integrity of the skeleton.
3. Flat Bones
- Flat bones are a specific type of bone characterized by their thin and broad structure. These bones play a crucial role in providing extensive protection to vital organs and offering broad surfaces for muscle attachment. They are primarily found in areas of the body where protection and muscle attachment are of utmost importance.
- Examples of flat bones include the sternum, ribs, scapulae (shoulder blades), and the roof of the skull. The sternum, commonly known as the breastbone, is a flat bone located in the center of the chest. It serves as a protective shield for the underlying organs, such as the heart and major blood vessels.
- The ribs are another example of flat bones. They form a curved structure that wraps around the thoracic cavity, providing protection to vital organs like the lungs and heart. The ribs also serve as attachment points for muscles involved in respiration and movements of the upper body.
- The scapulae, or shoulder blades, are large flat bones located on the upper back. They provide attachment sites for several muscles involved in arm and shoulder movements, contributing to the stability and flexibility of the shoulder joint.
- The roof of the skull, known as the cranial vault, is composed of flat bones. These bones, including the parietal and frontal bones, encase and protect the brain. The flat nature of these bones helps distribute forces and protect the delicate neural tissue from external trauma.
- Due to their thin and broad structure, flat bones offer a larger surface area for muscle attachment. Muscles, tendons, and ligaments can attach to these broad surfaces, allowing for efficient movement and stability. Additionally, the flat shape of these bones helps to distribute forces and absorb impacts, reducing the risk of injury to underlying organs.
- In summary, flat bones are thin and broad bones that serve important functions in the body. They provide extensive protection to vital organs and offer broad surfaces for muscle attachment. Examples of flat bones include the sternum, ribs, scapulae, and the roof of the skull. Their flat shape and structural characteristics contribute to the overall stability, protection, and functionality of the skeletal system.
4. Irregular Bones
- Irregular bones are a category of bones that do not conform to a specific easily recognizable shape or fit into any other classification of bone shapes. They exhibit complex and unique forms, often serving specialized functions in the body. These bones are typically found in areas where protection and structural support are crucial, but their shapes do not fit the standard descriptions of long, short, or flat bones.
- One prominent example of an irregular bone is the vertebra, which forms the spinal column and protects the delicate spinal cord from compressive forces. The vertebrae are stacked upon one another to create a flexible and protective structure that allows for movement and provides support to the body. Their irregular shape, with distinct features such as the vertebral arch and processes, enables them to interlock and form the spinal canal, safeguarding the spinal cord.
- Many bones of the face are also classified as irregular bones. These include the bones that house the paranasal sinuses, such as the frontal, ethmoid, sphenoid, and maxillary bones. The irregular shape of these facial bones accommodates the sinuses, air-filled cavities that help lighten the skull and contribute to the resonance of the voice. Additionally, the irregular bones of the face, such as the mandible (lower jaw) and the bones of the orbit (eye socket), provide protection for delicate structures like the teeth, eyes, and surrounding soft tissues.
- Irregular bones demonstrate a wide range of shapes and structures that are specifically adapted to their functions. Their unique characteristics make them well-suited for their roles in the body, including protection, support, and housing of specialized features. While irregular bones do not fit into the traditional categories of bone shapes, their complex forms serve important purposes in maintaining the overall structure and function of the skeletal system.
5. Sesamoid Bones
- Sesamoid bones are unique small bones that resemble the shape and size of a sesame seed. They are typically found embedded within tendons, which are fibrous tissues connecting bones to muscles. These specialized bones develop in locations where tendons are subjected to significant pressure within a joint. By forming within tendons, sesamoid bones serve to protect the tendons and assist in overcoming compressive forces during movement.
- The number and placement of sesamoid bones can vary among individuals, but they are commonly found in association with the feet, hands, and knees. In fact, the patellae, commonly known as the kneecaps, are the only sesamoid bones present in every person. The patellae are situated within the quadriceps tendon, which connects the thigh muscles to the tibia bone of the lower leg. These sesamoid bones enhance the mechanical advantage of the quadriceps muscle by improving its leverage during movements such as walking, running, and jumping.
- The presence of sesamoid bones in other areas of the body, particularly in the hands and feet, aids in reducing friction and stress on the tendons as they pass over bony prominences. For example, sesamoid bones can be found near the base of the thumb or big toe, where the corresponding tendons experience high pressure and repetitive motion. These sesamoid bones act as pulleys, altering the direction and reducing the strain on the tendons, thereby improving their efficiency and protecting them from wear and tear.
- While the number and location of sesamoid bones can vary, their general function remains consistent. They play a crucial role in maintaining the integrity and proper functioning of tendons, especially in areas where pressure and stress are prominent. The sesamoid bones help distribute forces evenly, reduce friction, and protect tendons from excessive wear, ultimately contributing to the smooth and efficient movement of the joints.
|Cylinder-like shape, longer than it is wide
|Femur, tibia, fibula, metatarsals, humerus, ulna, radius, metacarpals, phalanges
|Cube-like shape, approximately equal in length, width, and thickness
|Provide stability, support, while allowing for some motion
|Thin and curved
|Points of attachment for muscles; protectors of internal organs
|Sternum, ribs, scapulae, cranial bones
|Protect internal organs
|Vertebrae, facial bones
|Small and round; embedded in tendons
|Protect tendons from compressive forces
Bone Formation or Ossification
The embryo’s skeleton is made up of fibrous membranes and hyaline cartilage in the early stages of development. The true process of bone growth, ossification (osteogenesis), begins by the sixth or seventh week of embryonic life. There are two osteogenic pathways: intramembranous ossification and endochondral ossification, although bone is the same no matter which pathway creates it.
- Cartilage serves as a vital template for the formation of bone during skeletal development. In the early stages of fetal development, a flexible and semi-solid matrix is created by specialized cells called chondroblasts. This matrix, produced by chondroblasts, consists of components such as hyaluronic acid, chondroitin sulfate, collagen fibers, and water. As the matrix surrounds and encapsulates the chondroblasts, they become mature cartilage cells known as chondrocytes.
- Unlike many other connective tissues, cartilage is avascular, meaning it lacks a direct blood supply. Consequently, cartilage relies on diffusion through its matrix to obtain nutrients and eliminate metabolic waste products. This avascular nature of cartilage makes it less capable of self-repair compared to other tissues in the body.
- During fetal development and throughout childhood, bone gradually replaces the cartilaginous template. The process of ossification involves the deposition of a mineral matrix onto the existing cartilage scaffold. By the time a fetus is born, a significant portion of the initial cartilage has been transformed into bone. However, bone growth and development continue during childhood, leading to the replacement of additional cartilage with bone. While most cartilage is replaced by bone, some remnants of cartilage persist in the adult skeleton, primarily in areas requiring flexibility and shock absorption, such as the joints and the external ear.
- The cartilage template plays a critical role in establishing the framework for bone formation and subsequent skeletal development. It serves as a foundation upon which bone can be deposited, providing the necessary structure and shape. Over time, the cartilage is gradually replaced by the stronger and more rigid mineralized matrix of bone, contributing to the growth and maturation of the skeleton.
- Intramembranous ossification is a process in which bone develops directly from mesenchymal connective tissue without the presence of a cartilage precursor. This type of ossification is responsible for the formation of flat bones in the face, most cranial bones, and the clavicles.
- The process begins with mesenchymal cells gathering together in the embryonic skeleton and undergoing differentiation into specialized cells. Some of these cells differentiate into capillaries, while others become osteogenic cells and then osteoblasts. The osteoblasts, initially clustered together in an area called an ossification center, secrete osteoid, which is an uncalcified bone matrix.
- Over time, the osteoid calcifies as mineral salts are deposited onto it, resulting in the hardening of the matrix. This calcification process entraps the osteoblasts within the bone matrix, causing them to transform into osteocytes. Simultaneously, the surrounding osteogenic cells differentiate into new osteoblasts, ensuring the ongoing formation of bone tissue.
- As the osteoblasts become osteocytes, the osteoid secreted around the capillaries forms a trabecular matrix. Additionally, the osteoblasts on the surface of the newly formed spongy bone give rise to the periosteum, which is a protective layer surrounding the bone. The periosteum eventually generates a superficial layer of compact bone.
- During intramembranous ossification, the trabecular bone begins to crowd nearby blood vessels, leading to the condensation of these vessels into red marrow. This red marrow contributes to the blood cell production within the bone.
- Intramembranous ossification initiates during fetal development and continues throughout adolescence. At birth, certain bones, such as the skull and clavicles, are not fully ossified, and the sutures of the skull remain open. This flexibility allows for deformations during passage through the birth canal. The final bones to undergo intramembranous ossification are the flat bones of the face, which attain their adult size by the end of the adolescent growth spurt.
- Endochondral ossification is the process by which bone develops by replacing hyaline cartilage. Unlike in intramembranous ossification, where bone forms directly from mesenchymal connective tissue, in endochondral ossification, cartilage serves as a template that is gradually replaced by bone. This process takes longer compared to intramembranous ossification. Bones at the base of the skull and long bones, such as the femur and humerus, are formed through endochondral ossification.
- During endochondral ossification, at around 6 to 8 weeks after conception, some mesenchymal cells differentiate into chondrocytes, which are cartilage cells that form the cartilaginous skeleton precursor of the bones. The cartilage is covered by a membrane called the perichondrium. As the cartilage matrix is produced, chondrocytes in the center of the cartilage model increase in size. The matrix eventually calcifies, leading to the death of chondrocytes and the disintegration of the surrounding cartilage.
- Blood vessels invade the resulting spaces left by the disintegrating cartilage, enlarging the cavities and carrying osteogenic cells with them. These osteogenic cells differentiate into osteoblasts, which are responsible for bone formation. The enlarging spaces eventually merge to form the medullary cavity within the bone.
- Simultaneously, as the cartilage continues to grow, capillaries penetrate it, initiating the transformation of the perichondrium into the periosteum, which is essential for bone production. Osteoblasts in the periosteum form a collar of compact bone around the cartilage in the diaphysis (shaft) of the bone. This marks the formation of the primary ossification center.
- As the bone develops, chondrocytes and cartilage continue to grow at the ends of the bone, known as the epiphyses. This contributes to the bone’s lengthening while the cartilage in the diaphysis is replaced by bone. By the time the fetal skeleton is fully formed, cartilage remains only at the joint surfaces as articular cartilage and between the diaphysis and epiphysis as the epiphyseal plate, which allows for longitudinal bone growth. After birth, a similar sequence of events occurs in the epiphyseal regions, leading to the formation of secondary ossification centers.
- In summary, endochondral ossification involves the replacement of hyaline cartilage by bone. It begins with the formation of a cartilaginous model, followed by the invasion of blood vessels and the differentiation of osteogenic cells into osteoblasts. The process continues throughout fetal development and into postnatal growth, ultimately resulting in the formation of mature bone.
Bones Growth in Length
- The growth of bones in length occurs at the epiphyseal plate, which is a layer of hyaline cartilage located at the ends of long bones. The epiphyseal plate is responsible for ossification and bone growth in immature bones. The process of bone growth in length involves different zones within the epiphyseal plate.
- Starting from the epiphyseal side of the plate, cartilage is formed in the reserve zone, which is the region closest to the epiphysis. The chondrocytes in this zone secure the epiphyseal plate to the osseous tissue of the epiphysis but do not actively participate in bone growth.
- Moving towards the diaphysis, the proliferative zone is the next layer of the epiphyseal plate. It consists of slightly larger chondrocytes that undergo mitosis to generate new chondrocytes. These new cells replace the ones that die at the diaphyseal end of the plate.
- Adjacent to the proliferative zone is the zone of maturation and hypertrophy. Chondrocytes in this layer are older and larger than those in the proliferative zone. The cells become more mature as they approach the diaphyseal end of the plate. Cellular division in the proliferative zone and the maturation of cells in the zone of maturation and hypertrophy contribute to the longitudinal growth of the bone.
- The zone closest to the diaphysis is the zone of calcified matrix. In this zone, most chondrocytes are dead because the surrounding matrix has calcified. Capillaries and osteoblasts from the diaphysis invade this zone. Osteoblasts deposit bone tissue onto the remaining calcified cartilage. This process connects the epiphyseal plate to the diaphysis and adds osseous tissue to the diaphysis, resulting in the growth of the bone in length.
- Bone growth in length continues until early adulthood and is regulated by hormones. Once the chondrocytes in the epiphyseal plate stop their proliferation and bone replaces the cartilage, longitudinal growth ceases. The epiphyseal plate then becomes the epiphyseal line, indicating the closure of the growth plate.
- In summary, bones grow in length at the epiphyseal plate through a process involving the proliferation and maturation of chondrocytes. As new bone tissue is added to the diaphysis, the bone grows in length. This growth is controlled by hormones and ceases when the epiphyseal plate closes and becomes the epiphyseal line.
Bones Growth in Diameter
- Bones have the ability to grow in diameter through a process called appositional growth. Unlike longitudinal growth, which determines bone length, appositional growth allows bones to increase in thickness even after longitudinal growth has ceased.
- Appositional growth involves two key activities: bone resorption and bone deposition. Osteoclasts are responsible for resorbing or breaking down old bone that lines the medullary cavity. This resorption of old bone helps create space within the medullary cavity. On the other hand, osteoblasts, through intramembranous ossification, produce new bone tissue beneath the periosteum, which is the dense connective tissue covering the outer surface of bones.
- As osteoclasts resorb old bone along the medullary cavity, osteoblasts deposit new bone beneath the periosteum. This simultaneous process of resorption and deposition increases both the diameter of the diaphysis (the shaft of a long bone) and the diameter of the medullary cavity. The deposition of new bone beneath the periosteum adds layers of compact bone, contributing to the increase in diameter.
- This process of modeling, where old bone is eroded and new bone is formed, allows bones to grow in diameter and become stronger and more structurally sound. It helps maintain the balance between bone resorption and bone deposition, ensuring that bone remains healthy and adapts to mechanical stresses.
- Appositional growth continues throughout life and is influenced by various factors, including mechanical stress, hormonal regulation, and nutritional factors. It plays a crucial role in maintaining bone integrity and adapting bone structure to the changing demands placed on the skeleton.
Resorption and Remodeling of Bone
- Bone remodeling is a continuous process that occurs throughout life to maintain the strength and integrity of the skeleton. It involves the coordinated activity of bone-resorbing cells called osteoclasts and bone-forming cells called osteoblasts. The process of remodeling allows for the removal of old or damaged bone tissue and the replacement with new bone.
- During bone remodeling, matrix resorption and deposition occur on the same bone surface. Osteoclasts are responsible for resorbing or breaking down the existing bone matrix, while osteoblasts are involved in the synthesis and deposition of new bone matrix. This dynamic process helps repair microdamage, adapt bone to mechanical stresses, and regulate calcium and phosphate levels in the body.
- While bone modeling primarily occurs during periods of growth, such as childhood and adolescence, bone remodeling continues in adulthood. Various factors can influence bone remodeling, including injury, exercise, hormonal regulation, and nutritional factors. For example, in response to mechanical stress from weight-bearing activities or exercise, bone remodeling is stimulated to enhance bone strength and density in specific regions.
- Interestingly, even without external influences like injury or exercise, a certain percentage of the skeleton is remodeled annually. Approximately 5 to 10 percent of the bone tissue is remodeled each year, involving the destruction of old bone by osteoclasts and the subsequent renewal of fresh bone by osteoblasts. This ongoing remodeling process helps maintain bone health, repair minor damage, and ensure the replacement of aging bone tissue.
- Bone remodeling is a tightly regulated process that relies on the intricate coordination between osteoclasts and osteoblasts. Imbalances in bone remodeling can lead to conditions such as osteoporosis, where bone resorption exceeds bone formation, resulting in weakened and fragile bones.
- Overall, bone remodeling plays a crucial role in the maintenance, repair, and adaptation of the skeleton, allowing bones to withstand mechanical forces and maintain their structural integrity throughout life.
Fractures: Bone Repair
A fracture is defined as a broken bone. It will heal whether or not it is returned to its anatomical location by a physician. If the bone is not properly reset, the healing process will keep it in the distorted position.
A closed reduction occurs when a broken bone is manipulated and placed into its normal position without the use of surgery. Surgery is required for open reduction to expose the fracture and fix the bone. While some fractures are modest, some are severe and cause serious problems. A broken diaphysis of the femur, for example, has the ability to release fat globules into the bloodstream. These can become trapped in the lungs’ capillary beds, causing respiratory distress and, if not treated promptly, death.
Types of Fractures
Fractures, or broken bones, can be classified based on various characteristics such as their complexity, location, and specific features. Understanding the different types of fractures is important for proper diagnosis and treatment. Here are some common types of fractures:
- Transverse Fracture: This type of fracture occurs straight across the long axis of the bone. The break is perpendicular to the bone’s length.
- Oblique Fracture: An oblique fracture is characterized by a diagonal break that occurs at an angle other than 90 degrees. The fracture line runs obliquely across the bone.
- Spiral Fracture: Spiral fractures are the result of a twisting or rotational force applied to the bone. This causes the bone segments to be pulled apart, resulting in a spiral-shaped fracture pattern.
- Comminuted Fracture: In comminuted fractures, the bone breaks into several pieces, resulting in many small fragments between two main segments. This type of fracture often occurs due to high-energy trauma.
- Impacted Fracture: Impacted fractures involve one fragment of the bone being driven or wedged into another fragment. This commonly occurs due to compression forces, causing the bone ends to impact each other.
- Greenstick Fracture: Greenstick fractures are partial fractures seen primarily in children. In this type of fracture, one side of the bone is broken, while the other side bends or remains intact. The fracture resembles the way a green stick breaks.
- Open (or Compound) Fracture: An open fracture occurs when at least one end of the broken bone pierces and tears through the overlying skin. This type of fracture carries a higher risk of infection due to the exposure of the bone to the external environment.
- Closed (or Simple) Fracture: In contrast to an open fracture, a closed fracture does not break through the skin. The skin remains intact despite the underlying bone being fractured.
It’s worth noting that some fractures may exhibit features of more than one type. For example, an open fracture can also be transverse or oblique, depending on the orientation of the fracture line.
Understanding the type of fracture is essential for appropriate treatment planning, including immobilization, realignment (reduction), and potential surgical intervention. Different fractures may require different approaches to promote proper healing and restore normal bone function.
Bone repair is a complex process that occurs when a bone is fractured. Here’s a breakdown of the stages involved in bone repair:
- Formation of Fracture Hematoma: When a bone breaks, blood vessels in the periosteum, osteons, and medullary cavity are torn, leading to bleeding. Within a few hours after the fracture, the blood begins to clot, forming a fracture hematoma. The lack of blood flow to the bone results in the death of bone cells around the fracture.
- Callus Formation: Within approximately 48 hours, chondrocytes from the endosteum (internal membrane lining the medullary cavity) start creating an internal callus. These chondrocytes secrete a fibrocartilaginous matrix between the broken bone ends. At the same time, periosteal chondrocytes and osteoblasts (bone-forming cells) generate an external callus. The external callus consists of hyaline cartilage and bone, stabilizing the fracture.
- Bone Resorption and Osteoblast Activity: Over the next few weeks, specialized cells called osteoclasts resorb the dead bone tissue. Osteogenic cells, which are precursor cells, become active, divide, and differentiate into osteoblasts. These osteoblasts are responsible for producing new bone tissue.
- Cartilage Replacement by Trabecular Bone: The cartilage present in the calli is gradually replaced by trabecular bone through a process called endochondral ossification. This process involves the transformation of cartilage into bone tissue, resulting in the formation of a bony callus.
- Remodeling and Healing Completion: As the healing process continues, the internal and external calli unite, and the fractured bone becomes more stable. Compact bone replaces spongy bone at the outer margins of the fracture, restoring the bone’s strength. Over time, the bone undergoes remodeling, which involves resorption of unnecessary bone tissue and deposition of new bone to achieve the optimal structure and strength. This remodeling phase helps to restore the bone to its original shape and function.
In most cases, after complete healing, there may be no visible evidence of the fracture externally. However, slight swelling or remodeling may occur on the outer surface of the bone, but it usually resolves, leaving no noticeable signs of the previous fracture.
It’s important to note that the timeline and progression of bone repair can vary depending on the individual, the location and severity of the fracture, and other factors such as age and overall health. Proper medical management and support, including immobilization and rehabilitation, are crucial for successful bone repair and functional recovery.
Functions of Bone
- Support: One of the fundamental functions of bones is to provide structural support for the body. They form the framework that gives shape and stability to the body, allowing us to maintain an upright posture and perform various movements.
- Protection: Bones act as protective shields for vital organs and tissues. For example, the skull protects the brain, the ribcage safeguards the heart and lungs, and the vertebrae shield the spinal cord. The hard and durable nature of bone helps prevent damage from external forces.
- Movement: Bones work together with muscles, tendons, and ligaments to enable movement. The attachment points of muscles on bones, through tendons, allow muscles to exert force and generate movement around joints. The skeletal system provides a lever system that facilitates the body’s ability to perform various types of motion.
- Blood Cell Production: Within the bone marrow, a soft and spongy tissue located in the center of many bones, the production of blood cells occurs. Red blood cells, white blood cells, and platelets are produced in the bone marrow through a process called hematopoiesis. These blood cells are essential for carrying oxygen, fighting infections, and promoting clotting.
- Mineral Storage and Homeostasis: Bones act as reservoirs for minerals, especially calcium and phosphorus. These minerals are stored in the bone matrix and can be released into the bloodstream as needed to maintain mineral balance and support vital functions such as muscle contraction, nerve conduction, and blood clotting. Bones also play a role in regulating the levels of calcium in the body through interactions with hormones like parathyroid hormone and calcitonin.
- Endocrine Regulation: Bones produce and release hormones that are involved in various physiological processes. One example is osteocalcin, a hormone produced by bone cells, which plays a role in regulating glucose metabolism and insulin sensitivity.
- Acid-Base Balance: Bones participate in maintaining the acid-base balance of the body by acting as a buffer system. They can release or absorb ions, such as bicarbonate and phosphate, to help regulate the pH level of body fluids.
Collectively, these functions highlight the crucial role that bones play in supporting the body’s structure, protecting vital organs, enabling movement, and contributing to various physiological processes necessary for overall health and well-being.
What is the skeletal system?
The skeletal system is the framework of bones, cartilage, and connective tissues that provide support, protect organs, and enable movement in the human body.
How many bones are in the human body?
An adult human body typically has 206 bones. However, the number can vary slightly depending on factors such as age and individual variations.
How are bones classified?
Bones can be classified into four main types: long bones (e.g., femur), short bones (e.g., wrist bones), flat bones (e.g., skull bones), and irregular bones (e.g., vertebrae).
How do bones grow?
Bones grow through a process called ossification. During growth, bones increase in length through the growth plates (epiphyseal plates) and in diameter through appositional growth.
What is a fracture?
A fracture is a broken bone. It can occur due to trauma, such as a fall or impact, or as a result of certain medical conditions weakening the bone.
How do bones heal after a fracture?
After a fracture, bones heal through a process called bone remodeling. The broken ends of the bone are first stabilized, and then new bone tissue is formed to bridge the gap. Over time, the bone remodels and strengthens.
What is osteoporosis?
Osteoporosis is a condition characterized by low bone density and increased bone fragility, leading to an increased risk of fractures. It is more common in older individuals, particularly women after menopause.
Can bones repair themselves?
Yes, bones have the ability to repair themselves. When a bone is fractured, the body initiates a healing process in which new bone tissue is formed to reconnect the broken ends.
What is the role of calcium in bone health?
Calcium is a vital mineral for bone health. It is a major component of the bone matrix and is necessary for the strength and integrity of bones. Adequate calcium intake, along with other nutrients, is essential for maintaining healthy bones.
How can I keep my bones healthy?
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25 Division Word Problems for Year 2 to Year 6 With Tips On Supporting Pupils’ Progress
Division word problems are important in building proficiency in division. Division is one of the bedrocks of mathematics alongside addition, subtraction and multiplication. Therefore, it is vital that pupils have a deep understanding of division, its function within arithmetic and word problems, and how to apply both short division and long division with success.
Division itself is the mathematical process of breaking a number up into equal parts and then finding out how many equal parts you can have. It may be that you have a remainder following this division or you may have no remainder and so a whole number as your answer.
All Kinds of Word Problems Division
Download this free pack of division word problems to develop your class' word problem solving skills
- What are division word problems?
- Division word problems in the national curriculum.
- Why are word problems important for childrens’ understanding of division?
- How to teach division word problem solving in primary school.
- Example of a division word problem
- Examples of division word problems in the primary setting
- Year 2 division word problems
- Year 3 division word problems
- Year 4 division word problems
- Year 5 division word problems
- Year 6 division word problems
- More word problems resource
What are division word problems?
Division word problems are an extension of the arithmetic method whereby they are word problems with division at the heart.Pupils will be expected to use the process of division to find a solution to the word problem.
Typically, word problems use a story as a scenario and are based on a real life situation where pupils are expected to interpret what the word problem is asking and then apply their division knowledge to find the answer. Division can also be introduced early through the idea of grouping before advancing to the formal method of short division and long division.
To help you with the division journey, we have put together a collection of division word problems which can be used for children between Year 2 to Year 6 – also aimed at both 3rd grade and 4th grade pupils in America.
Division word problems in the national curriculum.
Division word problems in KS1
The national curriculum states that division and word problems should be encountered from Key Stage 1 and throughout our pupils’ primary school journey. Practical resources such as counters, dienes cubes and base ten can be used to supplement the teaching of division.
In Key Stage 1 the focus is on simple addition and subtraction word problems, however, it is standard practice to introduce the concept of grouping and sharing small quantities and to calculate the answer using concrete objects and pictorial representations. It is at this point that the idea of finding fractions of objects can also be introduced and that whilst multiplication of two numbers can be done in any order (the commutative theory), the division of one number by another cannot.
Division word problems in KS2
As children enter Lower Key Stage 2 they begin to develop their mental and written strategies for division. Pupils will begin to use their multiplication knowledge and times tables to assist in their solving of division problems and how they can use the corresponding division facts and multiplication facts to answer questions.
By the end of Year 4, pupils are expected to recall their multiplication and division facts for multiplication times tables up to 12 x 12. They should also use their knowledge of place value, and known and derived facts to assist with simple division such as dividing by 1 and halving.
Introducing short division
Short division is the next step in Lower Key Stage 2. Pupils practise their fluency of short division, also known as ‘the bus stop method’, in order to answer division word problems that have a whole number answer, and those with a remainder.
Before entering Upper Key Stage 2, pupils encounter division word problems and multi step problems with increasingly harder numbers going from a simple short division problem, such as, ‘If we have 30 pupils in our class and we are divided into groups of 5, how many pupils will be in each group?’ to ‘If there are 56 books in our library and they are shared amongst 7 children, how many books would each child get?’
Introducing long division
As our pupils enter Upper Key Stage 2, long division is introduced. By the end of KS2 pupils should be fluent in both multiplication and division and the written strategies, and be able to apply knowledge in fraction word problems and percentage word problems.
Year 5 pupils work towards being able to divide up to 4 digit numbers by a one digit number using short division and being able to interpret remainders in the correct context – even presenting the remainder as a decimal or fraction. Pupils should also be able to divide mentally and know how to divide by 10, 100 and 1000 and how place value works alongside dividing a number so it is 10, 100 or 1000 times smaller.
Year 6 pupils are expected to consolidate on the above formal methods of short division before being able to divide a four digit number by a two digit number using the formal method of long division and to again be able to understand remainders within this and present them in the correct context.
This also flows into division word problems as children should be able to read a multi step problem and know how to correctly interpret it, apply their divisional knowledge and solve the problem successfully. The concept of multi step problems is built upon at each stage of the national curriculum.
Why are word problems important for childrens’ understanding of division?
Word problems, alongside the use of concrete objects and pictorial representations, are important in helping children understand the complexities and possible abstract nature of division.
Whilst children may understand that when we divide our answer will be smaller, before providing a child with word problem worksheets, just like with exploring arrays to support multiplication word problems, it is important to visually explore how division looks – from grouping and beyond.
Applying maths to real life situations
Word problems are important because they provide a real life context for children to understand division and how we encounter it in real life. By allowing children to see how division is used in everyday situations, children will find it more meaningful and relevant which in turn develops a deeper understanding of the four operations as a whole.
Building problem solving skills
Word problems are also vital to developing problem solving skills. Firstly, they must read and understand the problem before being able to identify the relevant information within the contextual problem and apply their knowledge to find a solution. This naturally builds critical thinking and a child’s ability to reason, which is an important skill for any mathematician.
Developing mathematical language skills
Finally, the importance of moving from simple division word problems to more challenging ones enhances pupils’ vocabulary and language skills. For children to develop an understanding of vocabulary such as divisors, quotient and remainders means they must first understand these key words and apply it to the process of division and be able to communicate clearly what they are aiming to do.
Deepening understanding of the inverse relationship between division and multiplication
Division word problems also solidify the connection between multiplication and division. Understanding these inverse operations and being able to interchange the skills of multiplication and division will help make connections between different mathematical concepts and deepen pupils’ learning.
How to teach division word problem solving in primary school.
Having taught the concept of division to pupils using concrete examples, for example how to group or share counters and cubes, the next step is to advance to division word problems.
As with all word problems, it is important that pupils are able to read the question carefully and interpret it so they know what they are being asked. Do they need to add, subtract, multiply or divide? Do they need to solve a multi step problem and so need to do more than one step? They may decide what operation to do, in this blogs’ case – division, and then choose to represent it pictorially.
Example of a division word problem
There are 40 sweets ready to go in the party bags for Laura’s birthday. They are to be shared between 8 friends. How many sweets will each child get?
How to solve this:
Firstly we need to interpret the question. Laura has invited 8 friends to her party and she has 40 sweets to share equally between her friends. So we know:
- There are 40 sweets in total
- They are to be divided amongst 8 friends in total
- We therefore need to divide the total number of sweets (40) by the number of friends (8). So to solve this problem we could put the total number of sweets (the dividend – 40 ) in the ‘bus stop’ for short division and divide by the total number of friends (the divisor – 8). If we do this, we would get the answer of 5 – the quotient. Each friend would get 5 sweets each as 40 divided by 8 is 5.
- Alternatively, we could use the inverse – multiplication – to solve this problem. We may not know the division fact that 40 divided by 8 = 5 but if we look to the inverse we may know what numeral multiplied by 8 equals 40. If we did our 8 times table we would get the answer of 5 – the correct answer.
How can we show this pictorially?
We could show 8 circles – each circle to represent a child – and place a sweet in each circle until we have placed all 40 sweets. This would mean we have shared the sweets equally between the friends and would result in each child having 5 sweets.
We could represent the division word problem as a bar model. We could split the bar model into 8 sections. There are 40 sweets and so we share them between the 8 sections. We will again see each section gets 5 sweets.
The below visuals show how this would look:
Word problems are an important aspect of our learning at Third Space Learning’s one-to-one tuition programme. Tutors will work with our tutees to break down the word problems and identify the correct operation needed to solve the word problem.
Examples of division word problems in the primary setting
Below are examples of what can be expected at each year group from Years 2 to 6. Through our tutoring programme at Third Space Learning, our tutees will become familiar with word problems throughout their learning. They will encounter word problems on a regular basis with each lesson personalised to develop the learning our tutees need. The word problems will increase their confidence, familiarity with vocabulary and mathematical understanding.
Division word problems are essential to developing problem solving skills and mathematical reasoning.
Year 2 division word problems
In Year 2 pupils use division facts for the 2, 5 and 120 times table and solve problems using concrete materials, arrays and see word problems in context.
Rosie picks 12 apples on a summer walk and wants to share them equally into 4 baskets. How many apples will be in each basket?
12 divided by 4 = 3
Pictorially this would look like:
Amy loves baking and has baked 20 cup cakes. She wants to divide them between ten friends. How many cupcakes does each friend get?
20 divided by 10 = 2
If I have a pizza and it is cut into 16 slices, and I share it amongst 4 people. How many slices will each person get?
16 divided by 4 = 4
A multiplication fact that would help with this question would be to know 4 x 4 = 16
Rhys says, ‘I have 36 football trading cards and I am going to share them between my 2 friends. Each of my friends will get 14 cards.’ Is Rhys correct?
Answer: he is incorrect
Rhys is incorrect. If he correctly shares 36 cards between his 2 friends, each friend will get 18 cards because 36 divided by 2 = 18.
If a picnic bench can fit 8 children on it, and there are 24 children in our class, how many picnic benches will we need for our class to all have a seat?
Answer: We will need 3 benches.
24 divided by 3 = 8
Year 3 division word problems
With word problems for Year 3, pupils should begin using their recall of the 3, 4 and 8 times table to help with division word problems and be able to divide two digit numbers by one digit numbers using mental and short division.
if a school has 90 pupils in Year 3 and there are 3 classes in Year 3, how many pupils are in each class?
90 shared equally into 3 classes = 30 children per class
Every day a school gets a delivery of milk in a crate. There are 96 cartons of milk in the crate. If there are 8 milk cartons in a pack, how many packs will be in the crate?
96 divided by 12 = 8.
There are 8 cartons of milk in a pack.
A delivery of 124 footballs arrives at school for sports day. They are to be shared equally between 4 classes. How many footballs does each class get?
124 divided by 4 = 31 footballs per class
Year 3 is going to the beach on a school trip. If there are 150 children in Year 3 and only 10 children can go on one mini bus, how many mini buses does Mr. Pearson need to book?
150 children divided 10 = 15 mini buses.
Everly has a bag of 66 marbles. She says ‘If I share these marbles equally between 8 people, I will have 2 left over.’ Is Everly correct?
Answer: Yes, Everly is correct.
If we divide 66 into 8 groups we will have 2 marbles left.
This is because 66 divided 8 is 8 equal groups with 2 left over, because 8 x 8 is 64 and so there will be 2 marbles left.
Year 4 division word problems
With word problems for year 4, pupils should be using their full knowledge and recall of times tables to 12 x 12 to help with short division of 2 digit and 3 digit numbers by a 1 digit number. Word problems may also involve multi step problems. Remainders may also be within the answer.
If you have 61 flowers and divide them into four flower pots, how many flowers are in each pot? Are there any left over?
Answer: 15 flowers in each pot with 1 flower left over
If we divide 61 into 4 equal groups then we can use the short division method.
We put 61 into the ‘bus stop’ and divide it by 4.
We ask ourselves, how many 4’s go into the first number ‘6’ – and we know 4 x 1 = 4, so only one 4 goes into 6 but we have 2 remaining. We put that 2 next to the 1 in 61 and we now have the number ‘21’. How many 4’s go into 21? 4 x 5 is 20. Therefore, 5 lots of 4 go into 20. We now have the answer 15 flowers….but we would have 1 left over as we had to divide 21 by 4.
So the answer to 61 divided by 4 = 15 remainder 1.
This would look like:
A plate can hold 9 cereal bars. There are 180 cereal bars to put out. How many plates do we need?
Answer: 20 with no remainder.
180 divided by 9 = 20 with no remainder.
We may also know that 9 x 2 = 18 and so can use our place value knowledge to know that 9 x 20 = 180 as the answer would be ten times bigger than 9 x 2.
Amy is calculating 188 divided by 11 and thinks that as the number 188 ends in an 88, that there will be no remainder. Is she correct?
Answer: Amy is incorrect.
If we do 188 divided by 11 we will get 17 remainder 1.
There are 216 animals in a zoo and they are spread out across 8 different zones. How many animals are in each zone?
216 divided by 8 = 27
At a sports tournament there are 6 players in each team. There are 132 players altogether.
How many teams are there?
132 divided by 6 = 22
Year 5 division word problems
Word problems for Year 5 centre around dividing a 4 digit number by a 1 digit number using the formal method of short division. They will also interpret remainders correctly depending on the context. Division and remainders is often demonstrated through money word problems.
Ronan has a ball of string that is 819cm long. He cuts it into 7 equal pieces. How long is 1 piece of string?
819 divided by 7 = 117
In Key Stage 2 there are 1,248 coloured pencils. If there are 6 classes in Key Stage 2, how many pencils would each class receive?
We use the short division method to divide 1,248 by 6 and we get 208 as the result.
Mia buys three computer games for £84.99. How much is one computer game?
Whilst this involves decimal division due to it being monetary with pounds and pence, the process of short division is the same.
We divide £84.99 by 3 and we get £28.33.
The area of the school hall is 1,704m and needs to be split into four quadrants. What would be the area of each quadrant?
We take the total area of the school and divide it by 4 to represent each quadrant. In doing so, we would have 426m for each quadrant.
To check this is the correct answer, we could do the inverse and multiply 426 by 4 and we would get 1,704m.
Packets of sweets are put into multi packs of 8. The multi packs are then placed into boxes of 6. Today, 7800 packets of sweets were packed. How many boxes of sweets were packed?
Answer: 163 boxes
This is a two step problem. First we need to multiply the number of packets in a multi pack – 8 – by the number of boxes of multi packs – 6. We would get the answer 48.
We then have to take the total packets of sweets – 7800 – and divide this by 48. This would be our introduction to long division. If we do this we will get the answer 162.5.
Now we cannot have 162 and a half boxes and so we would round this up to 163 boxes – but the 163rd box would only be half full.
Year 6 division word problems
Word problems for year 6 will be preparing for their SATs exams in May. They would be familiar with the concept of long division and needing to divide a 4 digit number by a 2 digit number using the formal methods of both short and long division.
A school is selling tickets at £6 each to attend the Big Christmas Fair. Over 15 weeks it has earned an amazing £9,720! On average, how many tickets were sold each week?
Answer: 108 tickets per week
First, we need to use the formal method of long division to divide the grand total – £9720 by 15. If we do this correctly we will have the answer 648.
Then, we need to take this answer of 648, which is how much is earned each week, and then divide this by £6, the amount each ticket is.
This will result in the number of tickets sold each week – 108 tickets.
A square sports field has a perimeter of 2.696km. How long is each side of the field?
To answer this we need to be able to convert the 2.696km into metres. There are 1000 metres in a kilometre so that would be 2,696m. Then we divide this by 4 and get 674m for one side.
Keira is given a toy blocks kit containing 2,208 individual blocks. She wants to split the toy blocks evenly between 15 friends and herself to work on making a toy block city together. How many blocks should she give each of her friends?
Answer: 138 blocks
We need to use the formal method of long division to solve this. We also need to ensure we include Keira and her 15 friends so we have the number 16 as the divisor.
When we divide 2,208 by 16 using long division we get the answer 138.
Wesleigh was running in the cross country race. He ran for a distance of 3,569m and it took him 11 minutes to complete the race. How many metres did he run per minute? Give your answer to the nearest whole metre.
Answer: 324 metres
We need to use long division to divide 3,569 by 11. That will give us an answer of 324.45. As the decimal can be rounded down, the answer is 324 metres.
Sophia is preparing her sweet stall for the fair. She can fit 18 tins of sweets into one crate. How many crates will be needed to153 tins of sweets?
Answer: 9 cratesWe divide 153 by 18 using long division and we have an answer of 8 remainder 5. Therefore, having 8 crates would not be enough as we would have 85 tins left over and so we need a further tin to house the 5 tins left over. So, 9 crates are needed.
More word problems resource
Are you looking for more word problems resources? Take a look at our library of word problems practice questions including: time word problems, ratio word problems, addition word problems and subtraction word problems.
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PLC stands for Programmable Logic Controller which is a specialized computer designed for industrial automation and control. Programmable logic controller’s are used extensively in industrial processes around the world today. So what exactly is PLC Programming and Ladder Logic?
PLC programming uses one or more programming languages to develop software to control industrial processes. This software is then compiled and downloaded into a programmable logic controller. One way to accomplish this is to use a programming language known as Ladder Logic.
It’s important to note that Ladder Logic Programming is just one way to program a PLC. There are other programming languages commonly used to accomplish this task as well. However, Ladder Logic Programming is without question the most common.
In this article we are going to perform an in-depth analysis of the following:
- What is a PLC?
- What is PLC Programming?
- What is Ladder Logic Programming?
- What are basic Ladder Logic Instructions?
- How to program a PLC using Ladder Logic?
What is a PLC?
As mentioned at the beginning, PLC is an abbreviation for Programmable Logic Controller and in a nutshell, is a small industrial-grade computer with two core functionalities: monitor various inputs and outputs in an industrial system, and to make logic-based decisions to automatically controls processes and components in machines or automated systems.
PLCs have been around for quite some time and were introduced back in the late 1960’s by the inventor Richard Morley. They were originally designed to replace relay logic systems since relay systems back then tend to create delays and were rigged with various issues.
The PLC was then introduced to eliminate these issues, such as high power consumption of the relay systems (with more relays, more power is required). Since PLCs feature a modular design (that is, they can be plugged into various setups), they don’t use relay switching and so will use less power.
Also, relay systems can cause undesired arcing between contacts, which can generate high temperatures that can lead to mechanical failure. PLCs help prevents these overheating issues.
PLC’s are robust and durable and can perform well in harsh conditions like severe heat/cold, cold, and extreme moisture, making them ideal for industrial applications.
The PLC is arguably the single-most important invention used in industrial manufacturing even until today, six decades after its invention. You can learn further about the definition and concept of the PLC, the differences to relay systems, the hardware, and how a PLC operates in programmable logic controllers article here.
What is PLC Programming?
Simply put, PLC programming is the effort of developing the internal programming logic for a PLC. Without any program, the PLC is just a computer with a processor, an empty shell. So, we program the logic we want the PLC to understand and base its operation in a programming language that is known by the PLC.
If, for example, we want to tell the PLC to “if condition 1 is true, then turn the output of device A”, we use a language proprietary to the PLC, something like “If I1.0=1 then OA.0=1”. PLC programming essentially makes the decision based on input from the real world, and tell the PLC how it should interact with the real world through the outputs.
We’ve assembled a detailed step-by-step guide to PLC programming that we recommend you check out here.
Different Languages Used in PLC Programming
However, there are 5 PLC programming languages defined by the IEC 61131-3 standard, and below we will discuss them briefly. For a more indepth look a the different programming languages available, check out our article here.
Ladder Diagram (LD)
Ladder Diagram is the most common language used to program PLC and is also commonly known as Relay Ladder Logic (RLL). The Ladder Diagram is a graphical language (rather than text-based language) that visualizes the relationships between the inputs and outputs of the system.
Every Ladder Diagram is arranged similarly to an electrical diagram, and so we can say that the resulting ladder diagrams are virtual circuits.
They are called “ladder” because the diagram literally resembles a ladder with two vertical rails (the supply power/contact) and horizontal lines (the rungs) to represent the control circuits/relay coils.
None of these supply power and control circuits seen in a Ladder Diagram program are real, but they are virtual bits in the PLC memory.
Function Block Diagram (FBD)
Function Block Diagram (FBD) is based on—as the name suggests—, the function blocks. It is also a graphical, visual language, and for those who are familiar with Boolean systems, FBD would feel more intuitive than the Ladder Diagram language.
The FBD, like the LD, also resembles electrical wiring diagrams. However, if in LD the diagram is more akin to relay logic systems, in FBD it’s more like we are wiring the function blocks together.
In FBD, inputs are placed on the left of the function blocks and outputs on the right.
We can say that FBD is a simpler, more intuitive version of the LD language. However, to start programming in FBD language, we need to understand what the function blocks do, and so might have a steeper learning curve than LD.
Due to its similarities with LD, both the FBD and LD have a similar weakness: when the program becomes long or complex, it can be very difficult to trace every step and troubleshooting.
Structured Text (ST)
Unlike LD and FBD, Structured Text is—as the name suggests— is a text-based language.
While it’s more intuitive to use graphical programming languages for PLC programming, like the LD or FBD, they have their weakness, as we have briefly discussed above.
A text-based PLC program will take up a much smaller file size, and when the program gets very long/complex, it will be easier to read and understand than a graphical language.
Another advantage of the ST language is that we can easily combine it with different programming languages. For example, we can have FBD function blocks containing functions written in ST.
Instruction List (IL)
Similar to ST, the Instruction List (IL) is not a graphical programming language, but it’s a low-level programming language more similar to assembly language.
As the name suggests, the IL is a set of instructions.
There are common mathematical operators listed, like addition, subtraction, multiplication, and divisive functions. There are also functions to call and return from different functions.
The main benefit of the IL compared to other PLC programming languages is speed. Like any other low-level programming languages, IL is low in overhead and so will execute much faster than a graphical language. It also takes up less memory, which can be a huge benefit in a PLC with limited memory space.
However, there is also a major drawback in using IL: assembly language is simply not that common, and so learning to use it can be a challenge since the available resources are also relatively limited.
Sequential Function Charts (SFC)
The Sequential Function Charts language is also a graphical PLC programming language (not text-based) and is ideal for breaking down a large and complex process into smaller bits.
There are two core elements of a sequence in the SFCs: steps and transitions. A ‘step’ is any function within the industrial system and a transition is a condition between one step and another.
An SFC program can also include various standard logical programming like parallel/alternative branching and feedback loops.
If the PLC manufacturer offers SFC programming, typically they will provide additional documentation to allow users to implement and customize the SFC programs.
More About Ladder Diagram/Ladder Logic
As mentioned, Ladder Diagram (LD) or Relay Ladder Logic (RLL) is the most popular programming language in PLC programming, and so here we will delve further into it.
Ladder Logic is essentially a rule-based language that approximates an electrical wiring/circuit diagram as a graphical programming language.
Ladder Logic is most commonly applied for designing logical switching units (i.e. on/off, start/stop button).
However, it is versatile enough and can be implemented to create networks like the FBD (Function Block Diagram), so you can use Ladder Logic to control other program blocks.
The Ladder Logic program consists of networks: on the left side, these networks are bound by a vertical line (the bus bar), and a network contains a circuit diagram of POUs, contacts, coils, and connecting lines.
On the left side, there is a series of contacts that relays the ‘1’ (on) or ‘0’ (off) state, which corresponds to the true or false boolean values.
A boolean variable is associated with each of these contacts, and if the variable is true, the status is then relayed from the left to right via the line, and the coil in the right side of the network will receive either the ‘0’ or ‘1’ value coming from the left.
The value true or false is then written accordingly. We will further discuss this by discussing the basic ladder logic instructions below.
What are basic Ladder Logic Instructions?
Below, we will discuss five basic instructions that are common in Lader Logic (Ladder Diagram):
1. XIC (Examine if Closed)
Examine if Closed, or XIC, examines a bit status of data to see if it is ‘true’. It is active/true when the bit status is in a ‘1’ state, and it is not active/false if the bit status is in a ‘0’ state. XIC instruction is applied in your ladder logic to determine if a bit is ‘true’ or ON. This instruction is always found on the left side of a ladder rung.
- If the binary bit is true, rung condition is set to true
- If the binary bit is false, rung condition is set to false
Common applications of XIC instruction are:
- Start/stop button
- Limit switch
- Proximity switch
- Light on/off
- Internal bit
2. XIO (Examine if Open)
Examine of Open or XIO is essentially an instruction that performs the opposite to the XIC (Examine if Closed).
So, it evaluates a bit status for an ‘off’ or 0 state, and only then it will evaluate to true. When that’s the case, the XIO instruction will execute the rest of the rung.
The XIO instruction is commonly used to evaluate the status of inputs on the PLC, but can be used on any boolean to examine whether the program’s boolean is energized.
This instruction is always found on the left side of a ladder rung.
- If the binary bit is false, rung condition out is set to true
- If the binary bit is true, rung condition out is set to false
Common applications of XIO instruction are mostly similar to XIC:
- Start/stop button
- Limit switch
- Proximity switch
- Light on/off
- Internal bit
3. OTE (Output Energize)
The OTE (Output Energize) instruction will either set (energize) or clear a data bit depending whether the input leading to it is true or false. This instruction is always found on the right side of a ladder rung.
It will turn a bit into a high state (energized) if the preceding input evaluates to be true. On the other hand, it will turn a bit into a low state (cleared) if the preceding input evaluates to be false.
- If the OTE instruction is true, the controller sets the data bit
- If the OTE instruction is false, the controller clears the data bit
Common applications of OTE instruction are:
- Motor run signal
- Internal bit
4. OTL (Output Latch)
The OTL or Output Latch will force a bit into a ‘1’ state if all the preceding inputs leading to it are true.
This instruction is always found on the right side of a ladder rung, and will switch a bit status to a high state. OTL will maintain this condition and unlike OTE instruction, will never turn the bit low.
OTL is commonly paired with OTU (which we will discuss below).
OTL and OTU can be used to energize any boolean within the program, so they are quite versatile. At first glance, it might seem like a good idea to introduce OTU and OTL for all bits, but there might be scenarios where the logic will depend on the rung’s position.
So, avoid using OTL and OTU unless they are absolutely needed.
- OTL sets the bit to “1” when the rung becomes true, and retains this state when a power cycle occurs or when the rung loses continuity
- OTL is a retentive output, and can only turn on a bit to a ‘1’ state
- Ladder logic can examine a bit controlled by OTL as often as required.
- The output device wired to the screw terminal is energized when a bit is set. When rung conditions become false, the bit remains energized, and the corresponding output also remains energized.
- The latch input causes the function to change state. The function then stays on even when the latch input is turned off. If we want to turn off the function, we must use OTU instruction to unlatch.
- OTU (Output Unlatch)
OTU, or Output Unlatch will set the bit status to ‘0’ when all the preceding conditions leading to it evaluate to true. As with OTL, this instruction is always found on the right side of a ladder rung and will turn the bit to a ‘0’ (low) state when all preceding conditions are true.
So, it’s the exact opposite of the OTL instruction, and it will retain the ‘0’ state and never turn a bit to ‘1’.
Again, it might seem like a good idea to introduce OTU and OTL for all bits, but there might be scenarios where the logic will depend on the rung’s position.
So, avoid using OTL and OTU unless they are absolutely required.
- OTU resets the bit to “0” when the rung becomes true and will maintain this state.
- OTU is a retentive output instruction and can only turn off a bit to ‘0’ state
- Ladder logic can examine a bit status controlled by OTU as often as required
- The output device wired to the screw terminal is de-energized when a bit is cleared. When rung conditions become true, the bit remains de-energized, and the corresponding output also remains de-energized.
How to program a PLC using Ladder Logic?
Before we get going with PLC programming and using ladder logic. It is highly recommended that you familiarize yourself with PLC hardware and the basics of PLC programming and boolean logic. These articles will teach you a lot.
As mentioned ladder logic programming is graphical be design. This makes it very easy to program boolean expressions and logic. We’ve discussed basic ladder logic instructions above. You’d be surprised how many processes you can program with just these instructions alone.
Let’s take a look at an example. In this example we will make use of all the instructions we discussed, namely, the XIC (eXamine If Closed), the XIO (eXamine If Open), and the OTE (OuTput Energize) to control a simple motor start/stop circuit (with seal-in) using ladder logic programming.
Below is an image of this circuit built using ladder logic on Rockwell’s ControlLogix platform of programmable automation controllers. The instructions that are illuminated green indicate that the instruction is in its TRUE state.
Let’s analyze this circuit starting with the Starting with the Start_Motor_PB.
- The Start_Motor_PB is using an XIC instruction to monitor the ON state of Input Point 0 of the input module residing in Slot #1 of the PLC chassis as indicated by the text between the carrots <Local:1:I.Data.0>. This type of construct is known as Aliasing — see the YouTube video at the end of this section for more information on this. This push button would be normally open (N.O) and momentary – meaning – we push the button to start the motor and release it.
- The Stop_Motor_PB is also using an XIC instruction, however, this would typically be wired to a normally closed (N.C.) push button. This means the Input Point 1 would have voltage at all times until the operator presses the push button to break the circuit. Which would subsequently turn off the motor.
- The next input instruction is an XIC and it is monitoring the overload relay of the motor. You can see in this case we are monitoring the OFF condition to be TRUE. In other words, if the motor overload is NOT TRIPPED the instruction is TRUE.
- The next instruction is the output instruction or OTE. This output module is residing in Slot #2 of the PLC chassis and when this output turns ON, voltage and current is sent to the Output Point 0 which is wired to the coil of the contactor controlling the motor.
- The last instruction we have not mentioned is the XIC that is located around or Branched (this indicates a logical OR condition) around the Start_Motor_PB. We are using the Motor_Contactor_Enable tag itself to seal-in the momentary start push button.
- What this does is allows the operator to remove her finger from the start push button and if the motor turns on (Stop_Motor_PB AND Motor_Overload_Tripped are both in their TRUE state), the motor output will stay on even after the push button is released and the instruction goes FALSE.
- Then the motor output would remain TRUE until either the Stop_Motor_PB goes FALSE (meaning the operator presses the stop push button), or the Motor_Overload_Tripped goes FALSE (meaning the overload of the motor has tripped – remember where are using an XIO here to monitor the OFF condition).
This is just a brief example of how to perform PLC programming using Ladder Logic. I encourage you to work through our playlist of videos on our YouTube channel. Here is a few videos from that playlist. You can also view them on our Learn PLC Programming page.
How To Become a PLC Programmer?
Being a PLC programmer requires a specialized skill set that combines electrical, mechanical and software engineering. A PLC programmer must possess strong analytical thinking and problem-solving skills.
In general, here are the required skills of a PLC programmer:
- Software engineering skills: PlC programmers are expected to understand common programming languages for PLC programming, algorithms, and logical flow of program implementations
- Mechanical and electrical knowledge: PLC programmers are expected to be involved in physically implementing the PLC programs, and in some cases, mechanical troubleshooting might be necessary
- Logical reasoning: Programmers are expected to use rational steps and mathematical concepts to design the flow of a program.
- Design skills: Strong program design and architecture principles are required to design graphical programs and to draft 3D blueprints and drawings.
- Problem-solving: PLC programmers must be able to identify the source of a problem and create a solution.
The current changes in software development and increasing complexity have allowed most employers to prefer candidates with at least a bachelor’s degree. PLC programmers typically hold at least a college degree in the fields of computer science, electrical engineering, or mechanical engineering. There is currently no degree or direct vocational training that trains you as a PLC programmer specialist.
PLC programmers are expected to:
- Analyze the specific situation or problem to be solved
- Decide on a possible solution
- Design the solution into logic
- Write that logic with a proper programming language
There are many paths to becoming a PLC programmer. We did a comprehensive article on the different paths to PLC programming here.
How Much Does a PLC Programmer Make?
According to the U.S. Bureau of Labor Statistics (BLS), the median annual wage of a PLC programmer is $56,749. The top 10 percentile earns more than $87.970, and the lowest 10 percentile earns less than $36,550.
For more information on how much a PLC programmer makes, I encourage you to check out our article here.
Some final words from PLCGurus.NET
If you haven’t already done so yet, I do encourage you to become a member of the site. PLCGurus.NET is quickly becoming one of the largest and fastest growing communities of professional engineers, technicians and technologists who all share a passion for industrial automation and control systems.
Registration is and will always remain completely free. Register Here! Also, be sure to check out the PLCGurus.NET YouTube Channel to see some great videos…and don’t forget to like and subscribe to our channel.
Lastly, if you run into any problems in your day-to-day engineering activities please be sure to check out our Live and Interactive PLC Forum! And if you so desire, assist other community members by replying or offering helpful information to the questions or challenges they may be facing right now! Thanks for reading.
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Chapter 3: The Frequency Domain
Section 3.2: Phasors
In Chapter 1 we talked about the basic atoms of soundthe sine wave, and about the function that describes the sound generated by a vibrating tuning fork. In this chapter were talking a lot about the frequency domain. If you remember, our basic units, sine waves, only had two parameters: amplitude and frequency. It turns out that these dull little sine waves are going to give us the fundamental tool for the analysis and description of sound, and especially for the digital manipulation of sound. Thats the frequency domain: a place where lots of little sine waves are our best friends.
But before we go too far, its important to fully understand what a sine wave is, and its also wonderful to know that we can make these simple little curves ridiculously complicated, too. And its useful to have another model for generating these functions. That model is called a phasor.
Description of a Phasor
Think of a bicycle wheel suspended at its hub. Were going to paint one of the spokes bright red, and at the end of the spoke well put a red arrow. We now put some axes around the wheelthe x-axis going horizontally through the hub, and the y-axis going vertically. Were interested in the height of the arrowhead relative to the x-axis as the wheelour phasorspins around counterclockwise.
As time goes on, the phasor goes round and round. At each instant, we measure the height of the dot over the x-axis. Lets consider a small example first. Suppose the wheel is spinning at a rate of one revolution per second. This is its frequency (and remember, this means that the period is 1 second/revolution). This is the same as saying that the phasor spins at a rate of 360 degrees per second, or better yet, 2\2 radians per second (if were going to be mathematicians, then we have to measure angles in terms of radians). So 2 radians per second is the angular velocity of the phasor.
This means that after 0.25 second the phasor has gone /2 radians (90 degrees), and after 0.5 second its gone radians or 180 degrees, and so on. So, we can describe the amount of angle that the phasor has gone around at time t as a function, which we call (t).
Now, lets look at the function given by the height of the arrow as time goes on. The first thing that we need to remember is a little trigonometry.
The sine and cosine of an angle are measured using a right triangle. For our right triangle, the sine of , written sin() is given by the equation:
This means that:
Well make use of this in a minute, because in this example a is the height of our triangle.
Similarly, the cosine, written cos(), is:
This means that:
This will come in handy later, too.
Now back to our phasor. Were interested in measuring the height at time t, which well denote as h(t). At time t, the phasors arrow is making an angle of (t) with the x-axis. Our basic phasor has a radius of 1, so we get the following relationship:
We also get this nice graph of a function, which is our favorite old sine curve.
Then we get this nice curve, which is another kind of sinusoid (bigger!).
Now lets start messing with the frequency, which is the rate of revolution of the phasor. Lets ramp it up a notch and instead start spinning at a rate of five revolutions per second. Now:
This is easy to see since after 1 second we will have gone five revolutions, which is a total of 10 radians. Lets suppose that the radius of the phasor is 3. Again, at each moment we measure the height of our arrow (which we call h(t)), and we get:
Now we get this sinusoid:
In general, if our phasor is moving at a frequency of revolutions per second and has radius A, then plotting the height of the phasor is the same as graphing this sinusoid:
Now were almost done, but there is one last thing we could vary: we could change the place where we start our phasor spinning. For example, we could start the phasor moving at a rate of five revolutions per second with a radius of 3, but start the phasor at an angle of /4 radians, instead.
Now, what kind of function would this be? Well, at time t = 0 we want to be taking the measurement when the phasor is at an angle of /4, but other than that, all is as before. So the function we are graphing is the same as the one above, but with a phase shift of /4. The corresponding sinusoid is:
Our most general sinusoid of amplitude A, frequency , and phase shift has the form:
A particularly interesting example is what happens when we take the phase shift equal to 90 degrees, or /2 radians. Lets make it nice and simple, with equal to one revolution per second and amplitude equal to 1 as well. Then we get our basic sinusoid, but shifted ahead /2. Does this look familiar? This is the graph of the cosine function!
You can do some checking on your own and see that this is also the graph that you would get if you plotted the displacement of the arrow from the y-axis. So now we know that a cosine is a phase-shifted sine!
Fouriers theorem tells us that any periodic function can be expressed as a sum (possibly with an infinite number of terms!) of sinusoids. (Well discuss Fouriers theorem more in depth later.) Remember, a periodic function is any function that looks like the infinite repetition of some fixed pattern. The length of that basic pattern is called the period of the function. Weve seen a lot of examples of these in Chapter 1.
In particular, if the function has period T, then this sum looks like:
If T is the period of our periodic function, then we now know that its frequency is 1/Tthis is also called the fundamental (frequency) of the periodic function, and we see that all other frequencies that occur (called the partials) are simply integer multiples of the fundamental.
If you read other books on acoustics and DSP, you will find that partials are sometimes called overtones (from an old German word, "übertonen") and harmonics. Theres often confusion about whether the first overtone is the second partial, and so on. So, to be specific, and also to be more in keeping with modern terminology, were always going to call the first partial the one with the frequency of the fundamental.
Example: Suppose we have a triangle wave that repeats once every 1/100 second. Then the corresponding fundamental frequency is 100 Hz (it repeats 100 times per second). Triangle waves only contain partials at odd multiples of the fundamental. (The even multiples have no energyin fact, this is generally true of wave shapes that have the "odd" symmetry, like the triangle wave.) Click on Applet 3.2 and see a triangle wave built by adding one partial after another.
How should we define an arithmetic of arrows? It sounds funny, but in fact its a pretty natural generalization of what we already know about adding regular old numbers. When we add a negative number, we go backward, and when we add a positive number, we go forward.
Our regular old numbers can be thought of as arrows on a number line. Adding any two numbers, then, simply means taking the two corresponding arrows and placing them one after the other, tip to tail. The sum is then the arrow from the origin pointing to the place where "adding" the two arrows landed you.
Really, what we are doing here is thinking of numbers as vectors. They have a magnitude (length) and a direction (in this case, positive or negative, or better yet 0 radians or radians).
Now, to add phasors, we need to enlarge our worldview and allow our arrows to get not just 2 directions, but instead a whole 2 radians worth of directions! In other words, we allow our arrows to point anywhere in the plane. We add, then, just as before: place the arrows tip to tail, and draw an arrow from the origin to the final destination.
So, to recap: to add phasors, at each instant as our phasors are spinning around, we add the two arrows. In this way, we get a new arrow spinning around (the sum) at some frequencya new phasor. Now its easy to see that the sum of two phasors of the same frequency yields a new phasor of the same frequency. We can also see that the sum of a cosine and sine of the same frequency is simply a phase-shifted sine of the same frequency with a new amplitude given by the square root of the sum of squares of the two original phasors. Thats the Pythagorean theorem!
Sampling and Fourier Expansion
The decomposition of a complex waveform into its component phasors (which is pretty much the same as saying the decomposition of an acoustic waveform into its component partials) is called Fourier expansion.
In practice, the main thing that happens is that analog waveforms are sampled, creating a time-domain representation inside the computer. These samples are then converted (using what is called a fast Fourier transform, or FFT) into what are called Fourier coefficients.
Figure 3.17 shows a common way to show timbral information, especially the way that harmonics add up to produce a waveform. However, it can be slightly confusing. By running an FFT on a small time-slice of the sound, the FFT algorithm gives us the energy in various frequency bins. (A bin is a discrete slice, or band, of the frequency spectrum. Bins are explained more fully in Section 3.4.) The x-axis (bottom axis) shows the bin numbers, and the y-axis shows the strength (energy) of each partial.
The slightly strange thing to keep in mind about these bins is that they are not based on the frequency of the sound itself, but on the sampling rate. In other words, the bins evenly divide the sampling frequency (linearly, not exponentially, which can be a problem, as we’ll explain later). Also, this plot shows just a short fraction of time of the sound: to make it time-variant, we need a waterfall 3D plot, which shows frequency and amplitude information over a span of time. Although theoretically we could use the FFT data shown in Figure 3.17 in its raw form to make a lovely, synthetic gamelan sound, the complexity and idiosyncracies of the FFT itself make this a bit difficult (unless we simply use the data from the original, but that’s cheating).
Figure 3.18 shows a better graphical representation of sound in the frequency domain. Time is running from front to back, height is energy, and the x-axis is frequency. This picture also takes the essentially linear FFT and shows us an exponential image of it, so that most of the "action" happens in the lower 2k, which is correct. (Remember that the FFT divides the frequency spectrum into linear, equal divisions, which is not really how we perceive soundits often better to graph this exponentially so that theres not as much wasted space "up top.")
The waterfall plot in Figure 3.18 is stereo, and each channel of sound has its own slightly different timbre.
Heres a fact that will help a great deal: if the highest frequency is B times the fundamental, then you only need 2B + 1 samples to determine the Fourier coefficients. (Its easy to see that you should need at least 2B, since you are trying to get 2B pieces of information (B amplitudes and B2 phase shifts).)
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Data Types in Computer Programming Languages: A Comprehensive Guide with a Focus on Java
Data types are fundamental building blocks in computer programming languages, providing a means of classifying and organizing data. Understanding the different data types available in a programming language is essential for writing efficient and error-free code. This comprehensive guide aims to provide an in-depth exploration of data types in computer programming languages with a specific focus on Java.
Consider the following scenario: imagine that you are developing software for a financial institution to analyze customer transaction data. In order to accurately process this information, it becomes crucial to properly define and manipulate various types of data such as numerical values, text strings, dates, and boolean values. Without a clear understanding of how these different data types work within Java or any other programming language, errors can occur during execution leading to incorrect results or even system failures.
This article will delve into the world of data types by first defining what they are and why they are important in computer programming. It will then proceed to explore the most commonly used data types in Java, including primitive (e.g., integers, floating-point numbers) and non-primitive (e.g., classes, arrays) types. Additionally, it will discuss type casting, which allows for converting one type of data into another. By gaining a thorough understanding of data types and their nuances in Java, programmers can write code that is more efficient, robust, and accurate.
Data types in Java are used to specify the type of data that a variable can hold. They define the range of values that a variable can take and the operations that can be performed on it. By explicitly specifying the data type of a variable, you can ensure that only valid values are assigned to it, reducing the chance of errors and making your code more reliable.
In Java, there are two main categories of data types: primitive and non-primitive.
Primitive Data Types: These are the basic building blocks provided by Java for storing simple values. There are eight primitive data types in Java:
- byte: 8-bit signed integer
- short: 16-bit signed integer
- int: 32-bit signed integer
- long: 64-bit signed integer
- float: single-precision floating-point number
- double: double-precision floating-point number
- char: 16-bit Unicode character
- boolean: true or false
Non-Primitive Data Types (Reference Types): These data types do not store actual values but rather references to objects in memory. They include classes, interfaces, arrays, and enumerations. Non-primitive data types are created using class definitions provided by Java or custom-defined classes.
Understanding how to work with these different data types is crucial in programming tasks such as arithmetic operations, string manipulation, conditional statements, loops, and array manipulation.
Type casting is another important aspect of working with data types in Java. It allows you to convert one data type into another. There are two types of casting:
- Implicit casting (widening): Automatically converting a smaller type to a larger type without any loss of information.
- Explicit casting (narrowing): Manually converting a larger type to a smaller type where there may be potential loss of information.
By mastering the concepts of data types, their limitations, and how to manipulate them effectively, you can write more efficient and error-free code in Java.
Primitive Data Types
In the world of computer programming, data types are essential elements that allow programmers to organize and manipulate information efficiently. One crucial category is primitive data types, which represent basic values directly supported by a programming language without requiring any additional processing or manipulation. Understanding these fundamental data types is pivotal for developers as they form the building blocks upon which more complex structures are constructed.
To illustrate the significance of primitive data types, let us consider a hypothetical e-commerce application developed in Java. In this scenario, imagine a user adding items to their cart before proceeding to checkout. The application needs to store various details about each item, such as its name, price, quantity, and availability. To accomplish this task effectively, appropriate selection and utilization of primitive data types become imperative.
Bullet Point List – Emotional Response:
- Efficiency: By utilizing primitive data types, programs can optimize memory usage and execution speed.
- Simplicity: These data types provide simple representations of basic values without unnecessary complexities.
- Reliability: Primitive data types offer reliable operations due to their direct support from the programming language itself.
- Portability: As standard features provided by languages like Java, primitive data types ensure code compatibility across different platforms.
Table – Emotional Response:
|Represents integer numbers
|Stores floating-point decimal numbers
|Represents true/false values
|Holds single characters
Transition into Composite Data Types Section:
As we delve deeper into the realm of computer programming and explore more sophisticated applications, it becomes evident that relying solely on primitive data types may not be sufficient. Therefore, our journey now leads us towards understanding composite data types – an essential concept where multiple primitive data types combine to form more complex structures.
Composite Data Types
Transition from the previous section
Having explored primitive data types, we now turn our attention to composite data types. These are more complex and versatile than primitive types, allowing programmers to create custom structures that can hold multiple values of different data types within a single entity. In this section, we will delve into the characteristics and applications of composite data types in computer programming languages, with a specific focus on Java.
Composite Data Types: An Overview
To illustrate the concept of composite data types, let’s consider an example scenario where we need to store information about students in a class. Rather than using separate variables for each student’s name, age, and grade point average (GPA), we can make use of composite data types to create a cohesive structure that holds all these attributes together. This allows us to efficiently manage and manipulate student records as a unified unit.
When working with composite data types, it is important to understand their key features:
- Encapsulation: Composite data types encapsulate related pieces of information into a single object or record. This promotes code organization and enhances readability by grouping relevant data together.
- Abstraction: By defining custom composite data types, developers can abstract away implementation details and work with higher-level concepts. This improves code maintainability and facilitates modular design.
- Flexibility: Composite data types offer flexibility in terms of representing complex relationships between objects or entities. They allow us to define hierarchical structures like trees or graphs, enabling us to model real-world scenarios effectively.
- Data Integrity: Using composite data types helps ensure consistent handling of related information. Changes made to one attribute within the type automatically propagate throughout the entire structure, reducing the risk of errors or inconsistencies.
To further grasp the significance of composite data types, let’s consider the following comparison table:
|Primitive Data Types
|Composite Data Types
As evident from this table, composite data types offer a broader range of capabilities compared to primitive ones. While primitive types are essential for basic operations, composite types expand our programming toolkit and enable us to handle more complex scenarios.
In the subsequent section, we will delve into numeric data types within the realm of computer programming languages. Understanding how numbers are represented and manipulated is crucial in various applications, making it an important topic to explore. So now, let’s dive deeper into the world of numeric data types.
[Next section H2: Numeric Data Types]
Numeric Data Types
Transitioning from the discussion of numeric data types, we now delve into composite data types. These are data structures that can hold multiple values or elements of different data types within a single variable. One example is the array in Java, which allows us to store and access a collection of elements using a single identifier.
Composite data types offer several advantages over individual variables for storing related information. Firstly, they enable efficient storage and retrieval of large amounts of data by grouping related items together. This enhances code readability and organization, making it easier to understand and maintain complex programs. Secondly, composite data types provide flexibility in handling varying lengths of collections since their size can be dynamically adjusted during runtime.
To further illustrate the significance of composite data types, consider a case where you need to track student grades for an entire semester. Rather than creating separate variables for each grade, effectively cluttering your codebase, you could use an array to store all the grades in one place. This not only simplifies accessing and manipulating the grades but also facilitates statistical analysis such as calculating averages or finding the highest score.
The benefits of utilizing composite data types can be summarized as follows:
- Enhanced code clarity and maintainability.
- Efficient storage and retrieval of large volumes of related data.
- Flexibility in handling varying lengths or sizes of collections.
|Enhanced Code Clarity
|Grouping related items together improves code readability and organization by reducing clutter and enhancing overall program structure.
|Efficient Storage & Retrieval
|Composite data types allow efficient storage and retrieval of large amounts of related information, optimizing memory usage while facilitating easy access to individual elements within the collection.
|Flexible Collection Handling
|The dynamic resizing capability provided by composite data types enables flexible handling of collections with varying lengths or sizes during runtime, eliminating the need to predefine fixed sizes and allowing for more adaptive data structures.
|Simplified Data Analysis
|Composite data types simplify complex data analysis tasks by providing a unified structure to store related information, enabling streamlined operations such as calculating averages or identifying extreme values without the need for convoluted code constructs.
Moving forward, we will now explore the concept of character data types, which are fundamental in representing textual information within computer programs. The utilization of these types is crucial in various domains, including text processing, natural language understanding, and user interaction scenarios.
Character Data Types
Character Data Types
Section H2: Character Data Types
Transitioning from the previous section on Numeric Data Types, let us now delve into another fundamental aspect of data types in computer programming languages – Character Data Types. Just as numeric data types are essential for representing numerical values, character data types play a vital role in storing and manipulating textual information within programs.
Consider an example where we have developed a Java program to analyze customer feedback. The program prompts users to enter their comments and then processes these inputs accordingly. In this scenario, character data types enable us to store individual characters or strings of characters that make up the customers’ feedback. By utilizing character data types effectively, programmers can manipulate and process textual information with ease.
- Flexibility: Character data types provide flexibility by allowing programmers to handle various tasks related to text manipulation.
- Compatibility: These data types ensure compatibility across different platforms and systems, enabling seamless exchange of textual information.
- Efficiency: Efficient memory allocation is achieved through character data types specifically designed for handling textual content.
- Internationalization: With Unicode support, character data types facilitate the representation of diverse writing systems and languages.
Additionally, it is important to note that there are several subtypes within character data types that cater to specific requirements such as single characters (char) or sequences of characters (String). Table 1 provides an overview of some commonly used character data type variants:
Table 1: Common Character Data Type Variants
|Data Type Variant
|A primitive type that represents a single Unicode character.
|A reference type capable of holding multiple characters forming a sequence.
|An array capable of storing integer representations of each character in a string.
|A built-in method used to determine the length of a string.
In summary, character data types are crucial for handling textual information within computer programming languages. By utilizing their flexibility, compatibility, efficiency, and internationalization capabilities, programmers can effectively manipulate and process text-based inputs. In the subsequent section on Boolean Data Types, we will explore yet another important category of data types used in programming languages.
Transitioning into the next section about “Boolean Data Type,” let us now examine how it functions within various programming contexts without further delay.
Boolean Data Type
After exploring the character data types used in computer programming languages, let us now delve into another essential category of data types – numeric data types. These data types are employed to represent numerical values such as integers and floating-point numbers. To better understand their significance, consider a hypothetical scenario where we are developing a program to calculate the average temperature for each month of the year based on historical weather data.
One crucial aspect when working with numeric data types is precision. Different numeric data types have varying levels of precision, which determines the range and accuracy of values they can hold. Here are some key considerations:
- Integer Data Type: An integer data type allows us to store whole numbers without any fractional component. For instance, we could use an integer data type to represent the number of days in a month or the total rainfall in millimeters.
- Floating-Point Data Type: Unlike integer data types, floating-point data types permit storing decimal numbers that may have a fractional part. This flexibility is useful when dealing with temperature readings or calculating averages.
To illustrate further using our case study, imagine having monthly temperatures recorded with two decimal places for enhanced accuracy. In this situation, employing a floating-point data type would be more appropriate than an integer one due to its ability to handle fractions.
Let’s explore these concepts further by examining a table summarizing different numeric data types commonly used in programming languages:
|Represents signed 8-bit integers
|-128 to 127
|Represents signed 16-bit integers
|-32,768 to 32,767
|Represents signed 32-bit integers
|Represents signed 64-bit integers
|Represents single-precision floating point
|IEEE 754 standard
|Represents double-precision floating point
|IEEE 754 standard
With this table in mind, we can choose the appropriate numeric data type based on our program’s requirements and constraints. In the case of our average temperature calculation program, using either a
double data type would be suitable depending on the desired level of precision.
Transitioning seamlessly into the next section about “Data Type Conversion,” understanding these numeric data types is crucial as it forms the foundation for conversions between different data types. By manipulating these values effectively, programmers can ensure accurate calculations and efficient storage utilization within their programs.
Data Type Conversion
After understanding the boolean data type, it is crucial to delve into the concept of data type conversion, which plays a significant role in computer programming languages like Java. Data type conversion refers to the process of converting one data type into another. This can be necessary when we need to perform operations or assign values between variables of different types.
To illustrate this concept, let’s consider an example scenario where we have two variables:
num1 is an integer with a value of 5, while variable
num2 is a floating-point number with a value of 3.14. Now, suppose we want to add these two numbers together and store the result in a new variable called
sum. In order to do so, we would need to convert the integer value of
num1 into a floating-point number before performing the addition operation.
When dealing with data type conversion, there are certain rules that programmers must keep in mind:
- Some conversions may result in loss of precision or truncation.
- Certain combinations of data types may not be compatible for direct conversion.
- Implicit conversion (also known as automatic or widening conversion) occurs automatically by the compiler when no information is lost during the conversion.
- Explicit conversion (also known as casting or narrowing conversion) requires manual intervention from the programmer using special syntax to indicate the desired target data type.
It is important to understand these rules thoroughly as improper usage of data type conversions could lead to unexpected results or program errors. The table below provides further insight into various types of conversions commonly encountered in programming languages:
|int -> long
|Converts smaller range datatype into larger range datatype
|double -> int
|Converts larger range datatype into smaller range datatype
|int -> float
|Automatically converts one type to another without loss of precision
|double -> int (casting)
|Manually converts one type to another with possible data loss
By understanding the concept of data type conversion and adhering to the rules associated with it, programmers can effectively manipulate and utilize different data types within their programs. This knowledge empowers them to perform complex operations and ensure accurate results in the world of computer programming.
In summary, data type conversion is a crucial aspect of programming languages like Java that allows for seamless manipulation and utilization of various data types. By following certain rules and utilizing both implicit and explicit conversions, programmers can harness the power of different data types while avoiding potential errors or unexpected outcomes.
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- This article is about tides in the Earth's oceans.
Tides are the cyclic rising and falling of the Earth's ocean surface caused by the tidal forces of the Moon and Sun acting on the oceans. Tides cause changes in the depth of the marine and estuarine water bodies and produce oscillating currents known as tidal streams, making prediction of tides important for coastal navigation (see Tides and navigation). The strip of seashore that is submerged at high tide and exposed at low tide, called the intertidal zone, is an important ecological product of ocean tides (see Intertidal ecology).
The changing tide produced at a given location is the result of several factors, including the changing positions of the Moon and Sun relative to the Earth, the effects of Earth's rotation, and local water depth. Sea level measured by coastal tide gauges may also be strongly affected by wind. More generally, tidal phenomena can occur in other systems besides the ocean, whenever a gravitational field that varies in time and space is present (see Other tides).
Introduction and tidal terminology
A tide is a repeated cycle of sea level changes in the following stages:
- Over several hours the water rises or advances up a beach in the flood tide.
- The water reaches its highest level and stops at high tide. Because tidal currents cease this is also called slack water or slack tide. The tide reverses direction and is said to be turning.
- The sea level recedes or falls over several hours during the ebb tide.
- The level stops falling at low tide. This point is also described as slack or turning.
Tides may be semidiurnal (two high tides and two low tides each day), or diurnal (one tidal cycle per day). In most locations, tides are semidiurnal. Because of the diurnal contribution, there is a difference in height (the daily inequality) between the two high tides on a given day; these are differentiated as the higher high water and the lower high water in tide tables. Similarly, the two low tides each day are referred to as the higher low water and the lower low water. The daily inequality changes with time and is generally small when the Moon is over the equator.
The various frequencies of astronomical forcing which contribute to tidal variations are called constituents. In most locations, the largest is the "principal lunar semidiurnal" constituent, also known as the M2 (or M2) tidal constituent. Its period is about 12 hours and 24 minutes, exactly half a tidal lunar day, the average time separating one lunar zenith from the next, and thus the time required for the Earth to rotate once relative to the Moon. This is the constituent tracked by simple tide clocks.
Tides vary on timescales ranging from hours to years, so to make accurate records tide gauges measure the water level over time at fixed stations which are screened from variations caused by waves shorter than minutes in period. This data is compared to the reference (or datum) level usually called mean sea level.
Constituents other than M2 arise from factors such as the gravitational influence of the Sun, the tilt of the Earth's rotation axis, the inclination of the lunar orbit and the ellipticity of the orbits of the Moon about the Earth and the Earth about the Sun. Variations with periods of less than half a day are called harmonic constituents. Long period constituents have periods of days, months, or years.
Tidal range variation: springs and neaps
The semidiurnal tidal range (the difference in height between high and low tides over about a half day) varies in a two-week or fortnightly cycle. Around new and full moon when the Sun, Moon and Earth form a line (a condition known as syzygy), the tidal forces due to the Sun reinforce those of the Moon. The tide's range is then maximum: this is called the spring tide, or just springs and is derived not from the season of spring but rather from the verb meaning "to jump" or "to leap up." When the Moon is at first quarter or third quarter, the Sun and Moon are separated by 90° when viewed from the earth, and the forces due to the Sun partially cancel those of the Moon. At these points in the lunar cycle, the tide's range is minimum: this is called the neap tide, or neaps. Spring tides result in high waters that are higher than average, low waters that are lower than average, slack water time that is shorter than average and stronger tidal currents than average. Neaps result in less extreme tidal conditions. There is about a seven day interval between springs and neaps.
The changing distance of the Moon from the Earth also affects tide heights. When the Moon is at perigee the range is increased and when it is at apogee the range is reduced. Every 7½ lunations, perigee and (alternately) either a new or full moon coincide causing perigean tides with the largest tidal range, and if a storm happens to be moving onshore at this time, the consequences (in the form of property damage, etc.) can be especially severe.
Tidal phase and amplitude
Because the M2 tidal constituent dominates in most locations, the stage or phase of a tide, denoted by the time in hours after high tide, is a useful concept. It is also measured in degrees, with 360° per tidal cycle. Lines of constant tidal phase are called cotidal lines. High tide is reached simultaneously along the cotidal lines extending from the coast out into the ocean, and cotidal lines (and hence tidal phases) advance along the coast.
If one thinks of the ocean as a circular basin enclosed by a coastline, the cotidal lines point radially inward and must eventually meet at a common point, the amphidromic point. An amphidromic point is at once cotidal with high and low tides, which is satisfied by zero tidal motion. (The rare exception occurs when the tide circles around an island, as it does around New Zealand.) Indeed tidal motion generally lessens moving away from the continental coasts, so that crossing the cotidal lines are contours of constant amplitude (half of the distance between high and low tide) which decrease to zero at the amphidromic point. For a 12 hour semidiurnal tide the amphidromic point behaves roughly like a clock face, with the hour hand pointing in the direction of the high tide cotidal line, which is directly opposite the low tide cotidal line. High tide rotates about once every 12 hours in the direction of rising cotidal lines, and away from ebbing cotidal lines. The difference of cotidal phase from the phase of a reference tide is the epoch.
The shape of the shoreline and the ocean floor change the way that tides propagate, so there is no simple, general rule for predicting the time of high tide from the position of the Moon in the sky. Coastal characteristics such as underwater topography and coastline shape mean that individual location characteristics need to be taken into consideration when forecasting tides; high water time may differ from that suggested by a model such as the one above due to the effects of coastal morphology on tidal flow.
Isaac Newton laid the foundations for the mathematical explanation of tides in the Philosophiae Naturalis Principia Mathematica (1687). In 1740, the Académie Royale des Sciences in Paris offered a prize for the best theoretical essay on tides. Daniel Bernoulli, Antoine Cavalleri, Leonhard Euler, and Colin Maclaurin shared the prize. Maclaurin used Newton’s theory to show that a smooth sphere covered by a sufficiently deep ocean under the tidal force of a single deforming body is a prolate spheroid with major axis directed toward the deforming body. Maclaurin was also the first to write about the Earth's rotational effects on motion. Euler realized that the horizontal component of the tidal force (more than the vertical) drives the tide. In 1744 D'Alembert studied tidal equations for the atmosphere which did not include rotation. The first major theoretical formulation for water tides was made by Pierre-Simon Laplace, who formulated a system of partial differential equations relating the horizontal flow to the surface height of the ocean. The Laplace tidal equations are still in use today. William Thomson rewrote Laplace's equations in terms of vorticity which allowed for solutions describing tidally driven coastally trapped waves, which are known as Kelvin waves.
The tidal force produced by a massive object (Moon, hereafter) on a small particle located on or in an extensive body (Earth, hereafter) is the vector difference between the gravitational force exerted by the Moon on the particle, and the gravitational force that would be exerted on the particle if it were located at the center of mass of the Earth. Thus, the tidal force depends not on the strength of the gravitational field of the Moon, but on its gradient.
The gravitational force exerted on the Earth by the Sun is on average 179 times stronger than that exerted on the Earth by the Moon, but because the Sun is on average 389 times farther from the Earth, the gradient of its field is weaker. The tidal force produced by the Sun is therefore only 46% as large as that produced by the Moon.
Tidal forces can also be analyzed from the point of view of a reference frame that translates with the center of mass of the Earth. Consider the tide due to the Moon (the Sun is similar). First observe that the Earth and Moon rotate around a common orbital center of mass, as determined by their relative masses. The orbital center of mass is 3/4 of the way from the Earth's center to its surface. The second observation is that the Earth's centripetal motion is the averaged response of the entire Earth to the Moon's gravity and is exactly the correct motion to balance the Moon's gravity only at the center of the Earth; but every part of the Earth moves along with the center of mass and all parts have the same centripetal motion, since the Earth is rigid.
On the other hand each point of the Earth experiences the Moon's radially decreasing gravity differently; the near parts of the Earth are more strongly attracted than is compensated by the centripetal motion and experience a net tidal force toward the Moon; the far parts have more centripetal motion than is necessary for the reduced attraction, and thus feel a net force away from the Moon.
Finally only the horizontal components of the tidal forces actually contribute tidal acceleration to the water particles since there is small resistance. The actual tidal force on a particle is only about a ten millionth of the force caused by the Earth's gravity.
The ocean's surface is closely approximated by an equipotential surface, (ignoring ocean currents) which is commonly referred to as the geoid. Since the gravitational force is equal to the gradient of the potential, there are no tangential forces on such a surface, and the ocean surface is thus in gravitational equilibrium.
Now consider the effect of external, massive bodies such as the Moon and Sun. These bodies have strong gravitational fields that diminish with distance in space and which act to alter the shape of an equipotential surface on the Earth. Gravitational forces follow an inverse-square law (force is inversely proportional to the square of the distance), but tidal forces are inversely proportional to the cube of the distance. The ocean surface moves to adjust to changing tidal equipotential, tending to rise when the tidal potential is high, the part of the Earth nearest the Moon, and the farthest part. When the tidal equipotential changes, the ocean surface is no longer aligned with it, so that the apparent direction of the vertical shifts. The surface then experiences a down slope, in the direction that the equipotential has risen.
Laplace tidal equation
The depth of the oceans is much smaller than their horizontal extent; thus, the response to tidal forcing can be modeled using the Laplace tidal equations which incorporate the following features: (1) the vertical (or radial) velocity is negligible, and there is no vertical shear—this is a sheet flow. (2) The forcing is only horizontal (tangential). (3) the Coriolis effect appears as a fictitious lateral forcing proportional to velocity. (4) the rate of change of the surface height is proportional to the negative divergence of velocity multiplied by the depth. The last means that as the horizontal velocity stretches or compresses the ocean as a sheet, the volume thins or thickens, respectively.
The boundary conditions dictate no flow across the coastline, and free slip at the bottom. The Coriolis effect steers waves to the right in the northern hemisphere and to the left in the southern allowing coastally trapped waves. Finally, a dissipation term can be added which is an analog to viscosity.
Tidal amplitude and cycle time
The theoretical amplitude of oceanic tides due to the Moon is about 54 cm at the highest point, which corresponds to the amplitude that would be reached if the ocean possessed a uniform depth, there were no landmasses, and the Earth were not rotating. The Sun similarly causes tides, of which the theoretical amplitude is about 25 cm (46% of that of the Moon) with a cycle time of 12 hours. At spring tide the two effects add to each other to a theoretical level of 79 cm, while at neap tide the theoretical level is reduced to 29 cm. Since the orbits of the Earth about the Sun, and the Moon about the Earth, are elliptical, the amplitudes of the tides change somewhat as a result of the varying Earth-Sun and Earth-Moon distances. This causes a variation in the tidal force and theoretical amplitude of about ±18 percent for the Moon and ±5 percent for the Sun. If both the Sun and Moon were at their closest positions and aligned at new moon, the theoretical amplitude would reach 93 cm.
Real amplitudes differ considerably, not only because of variations in ocean depth, and the obstacles to flow caused by the continents, but also because the natural period of wave propagation is of the same order of magnitude as the rotation period: about 30 hours. If there were no land masses, it would take about 30 hours for a long wavelength ocean surface wave to propagate along the equator halfway around the Earth (by comparison, the natural period of the Earth's lithosphere is about 57 minutes).
The tidal forcing is essentially driven by orbital energy of the Earth Moon system at a rate of about 3.75 Terawatts. The dissipation arises as the basin scale tidal flow drives smaller scale flows which experience turbulent dissipation. This tidal drag gives rise to a torque on the Moon that results in the gradual transfer of angular momentum to its orbit, and a gradual increase in the Earth-Moon separation. As a result of the principle of conservation of angular momentum, the rotational velocity of the Earth is correspondingly slowed. Thus, over geologic time, the Moon recedes from the Earth, at about 3.8 cm/year, and the length of the terrestrial day increases, meaning that there is about 1 less day per 100 million years.
Tidal observation and prediction
From ancient times, tides have been observed and discussed with increasing sophistication, first noting the daily recurrence, then its relationship to the Sun and Moon. Eventually the first tide table in China was recorded in 1056 C.E. primarily for the benefit of visitors to see the famous tidal bore in the Qiantang River. In Europe the first known tide-table is thought to be that of John, Abbott of Wallingford (d. 1213), based on high water occurring 48 minutes later each day, and three hours later at London than at the mouth of the Thames. William Thomson led the first systematic harmonic analysis to tidal records starting in 1867. The main result was the building of a tide-predicting machine (TPM) on using a system of pulleys to add together six harmonic functions of time. It was "programmed" by resetting gears and chains to adjust phasing and amplitudes. Similar machines were used until the 1960s.
The first known sea-level record of an entire spring–neap cycle was made in 1831 on the Navy Dock in the Thames Estuary, and many large ports had automatic tide gages stations by 1850.
William Whewell first mapped co-tidal lines ending with a nearly global chart in 1836. In order to make these maps consistent, he hypothesized the existence of amphidromes where co-tidal lines meet in the mid-ocean. These points of no tide were confirmed by measurement in 1840 by Captain Hewett, RN, from careful soundings in the North Sea.
In most places there is a delay between the phases of the Moon and the effect on the tide. Springs and neaps in the North Sea, for example, are two days behind the new/full Moon and first/third quarter. This is called the age of the tide.
The exact time and height of the tide at a particular coastal point is also greatly influenced by the local bathymetry. There are some extreme cases: the Bay of Fundy, on the east coast of Canada, features the largest well-documented tidal ranges in the world, 16 meters (53 ft), because of the shape of the bay . Southampton in the United Kingdom has a double high tide caused by the interaction between the different tidal harmonics within the region. This is contrary to the popular belief that the flow of water around the Isle of Wight creates two high waters. The Isle of Wight is important, however, as it is responsible for the 'Young Flood Stand', which describes the pause of the incoming tide about three hours after low water.
There are only very slight tides in the Mediterranean Sea and the Baltic Sea owing to their narrow connections with the Atlantic Ocean. Extremely small tides also occur for the same reason in the Gulf of Mexico and Sea of Japan. On the southern coast of Australia, because the coast is extremely straight (partly due to the tiny quantities of runoff flowing from rivers), tidal ranges are equally small.
Careful Fourier and data analysis over a 19 year period (the National Tidal Datum Epoch in the US) uses carefully selected frequencies called the tidal harmonic constituents. This analysis can be done using only the knowledge of the period of forcing, but without detailed understanding of the physical mathematics, which means that useful tidal tables have been constructed for centuries.
The resulting amplitudes and phases can then be used to predict the expected tides. These are usually dominated by the constituents near 12 hours (the semidiurnal constituents), but there are major constituents near 24 hours (diurnal) as well. Longer term constituents are 14 day or fortnightly, monthly, and semiannual. Most coastline is dominated by semidiurnal tides, but some areas such as the South China Sea and the Gulf of Mexico are primarily diurnal. In the semidiurnal areas, the primary constituents M2(lunar) and S2(solar) periods differ slightly so that the relative phases, and thus the amplitude of the combined tide, change fortnightly (14 day period).
In the M2 plot above each cotidal line differs by 1 hour from its neighbors, and the thicker lines show tides in phase with equilibrium at Greenwich. The lines rotate around the amphidromic points counterclockwise in the northern hemisphere so that from Baja California to Alaska and from France to Ireland the M2 tide propagates northward. In the southern hemisphere this direction is clockwise. On the other hand M2 tide propagates counterclockwise around New Zealand, but this because the islands act as dam and permit the tides to have different heights on opposite sides of the islands. But the tides do propagate northward on the eastside and southward on the west coast, as predicted by theory.
The exception is the Cook Strait where the tidal currents periodically link high to low tide. This is because cotidal lines 180° around the amphidromes are in opposite phase, for example high tide across from low tide. Each tidal constituent has a different pattern of amplitudes, phases, and amphidromic points, so the M2 patterns cannot be used for other tides.
Tidal flows are of profound importance in navigation and very significant errors in position will occur if they are not taken into account. Tidal heights are also very important; for example many rivers and harbors have a shallow "bar" at the entrance which will prevent boats with significant draft from entering at certain states of the tide.
The timings and velocities of tidal flow can be found by looking at a tidal chart or tidal stream atlas for the area of interest. Tidal charts come in sets, with each diagram of the set covering a single hour between one high tide and another (they ignore the extra 24 minutes) and give the average tidal flow for that one hour. An arrow on the tidal chart indicates the direction and the average flow speed (usually in knots) for spring and neap tides. If a tidal chart is not available, most nautical charts have "tidal diamonds" which relate specific points on the chart to a table of data giving direction and speed of tidal flow.
Standard procedure to counteract the effects of tides on navigation is to (1) calculate a "dead reckoning" position (or DR) from distance and direction of travel, (2) mark this on the chart (with a vertical cross like a plus sign) and (3) draw a line from the DR in the direction of the tide. The distance the tide will have moved the boat along this line is computed by the tidal speed, and this gives an "estimated position" or EP (traditionally marked with a dot in a triangle).
Nautical charts display the "charted depth" of the water at specific locations with "soundings" and the use of bathymetric contour lines to depict the shape of the submerged surface. These depths are relative to a "chart datum," which is typically the level of water at the lowest possible astronomical tide (tides may be lower or higher for meteorological reasons) and are therefore the minimum water depth possible during the tidal cycle. "Drying heights" may also be shown on the chart, which are the heights of the exposed seabed at the lowest astronomical tide.
Heights and times of low and high tide on each day are published in tide tables. The actual depth of water at the given points at high or low water can easily be calculated by adding the charted depth to the published height of the tide. The water depth for times other than high or low water can be derived from tidal curves published for major ports. If an accurate curve is not available, the rule of twelfths can be used. This approximation works on the basis that the increase in depth in the six hours between low and high tide will follow this simple rule: first hour - 1/12, second - 2/12, third - 3/12, fourth - 3/12, fifth - 2/12, sixth - 1/12.
Intertidal ecology is the study of intertidal ecosystems, where organisms live between the low and high tide lines. At low tide, the intertidal is exposed (or ‘emersed’) whereas at high tide, the intertidal is underwater (or ‘immersed’). Intertidal ecologists therefore study the interactions between intertidal organisms and their environment, as well as between different species of intertidal organisms within a particular intertidal community. The most important environmental and species interactions may vary based on the type of intertidal community being studied, the broadest of classifications being based on substrates - rocky shore and soft bottom communities.
Organisms living in this zone have a highly variable and often hostile environment, and have evolved various adaptations to cope with, and even exploit, these conditions. One easily visible feature of intertidal communities is vertical zonation, where the community is divided into distinct vertical bands of specific species going up the shore. Species ability to cope with desiccation determines their upper limits, while competition with other species sets their lower limits.
Intertidal regions are utilized by humans for food and recreation, but anthropogenic actions also have major impacts, with overexploitation, invasive species and climate change being among the problems faced by intertidal communities. In some places Marine Protected Areas have been established to protect these areas and aid in scientific research.
Biological rhythms and the tides
Intertidal organisms are greatly affected by the approximately fortnightly cycle of the tides, and hence their biological rhythms tend to occur in rough multiples of this period. This is seen not only in the intertidal organisms however, but also in many other terrestrial animals, such as the vertebrates. Examples include gestation and the hatching of eggs. In humans, for example, the menstrual cycle lasts roughly a month, an even multiple of the period of the tidal cycle. This may be evidence of the common descent of all animals from a marine ancestor.
In addition to oceanic tides, there are atmospheric tides as well as earth tides. All of these are continuum mechanical phenomena, the first two being fluids and the third solid (with various modifications).
Atmospheric tides are negligible from ground level and aviation altitudes, drowned by the much more important effects of weather. Atmospheric tides are both gravitational and thermal in origin, and are the dominant dynamics from about 80 km to 120 km where the molecular density becomes too small to behave as a fluid.
Earth tides or terrestrial tides that affect the entire rocky mass of the Earth. The Earth's crust shifts (up/down, east/west, north/south) in response to the Moon's and Sun's gravitation, ocean tides, and atmospheric loading.
While negligible for most human activities, the semidiurnal amplitude of terrestrial tides can reach about 55 cm at the equator (15 cm is due to the Sun) which is important in GPS calibration and VLBI measurements. Also to make precise astronomical angular measurements requires knowledge of the earth's rate of rotation and nutation, both of which are influenced by earth tides. The semi-diurnal M2 Earth tides are nearly in phase with the Moon with tidal lag of about two hours.
Terrestrial tides also need to be taken in account in the case of some particle physics experiments. For instance, at the CERN or SLAC, the very large particle accelerators were designed while taking terrestrial tides into account for proper operation. Among the effects that need to be taken into account are for circular accelerators and particle beam energy
Since tidal forces generate currents of conducting fluids within the interior of the Earth, they affect in turn the Earth's magnetic field itself.
The galactic tide is the tidal force exerted by galaxies on stars within them and satellite galaxies orbiting them. The effects of the galactic tide on the Solar System's Oort cloud are believed to be the cause of 90 percent of all observed long-period comets.
When oscillating tidal currents in the stratified ocean flow over uneven bottom topography, they generate internal waves with tidal frequencies. Such waves are called internal tides.
Tsunamis, the large waves that occur after earthquakes, are sometimes called tidal waves, but this name is due to their resemblance to the tide, rather than any actual link to the tide itself. Other phenomena unrelated to tides but using the word tide are rip tide, storm tide, hurricane tide, and black tide, referring to oil spills; or red tides, that refer to algae blooms.
- The orientation and geometry of the coast affects the phase, direction, and amplitude of coastal waves as well as resonant seiches (standing waves) in bays. In estuaries, seasonal river outflows influence tidal flow.
- Tide tables usually list mean lower low water (mllw, the 19 year average of mean lower low waters), mean higher low water (mhlw), mean lower high water (mlhw), mean higher high water (mhhw), as well as perigean tides. These are mean in the sense that they are predicted from mean data. Glossary, A Guide to National Shoreline Data and Terms, NOAA Shoreline Website. Retrieved March 31, 2020.
- The Moon orbits in the same direction the Earth spins. Compare this to the minute hand crossing the hour hand at 12:00 and then again at about 1:05 (not at 1:00).
- Ocean service education, Tidal lunar day, NOAA. Do not confuse with the astronomical lunar day on the Moon. A lunar zenith is the Moon's highest point in the sky. Retrieved March 31, 2020.
- Y. Accad, and C. L. Pekeris. 1978. Solution of the Tidal Equations for the M2 and S2 Tides in the World Oceans from a Knowledge of the Tidal Potential Alone. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 290 (1368): 235-266.
- Semidiurnal and long term constituents phase are measured from high tide, diurnal from maximum flood tide. This and the discussion that follows is only precisely true for a single tidal constituent.
- Generally clockwise in the southern hemisphere, and counterclockwise in the northern hemisphere
- The reference tide is the hypothetical constituent equilibrium tide on a landless earth that would be measured at 0° longitude, the Greenwich meridian.
- Zuosheng, Yang, K. O. Emery, Xui Yui. 1989. Historical Development and Use of Thousand-Year-Old Tide-Prediction Tables. Limnology and Oceanography 34(5): 953-957.
- David E. Cartwright, Tides: A Scientific History (Cambridge, UK: Cambridge University Press, 1999, ISBN 0521797462).
- Compare this to passengers on a turning bus. The bus's overall motion follows the center of mass, but passengers sitting in different parts of the bus experience different forces, and so may shift within the bus. The rigid body of the bus redistributes the road traction forces through its frame and seats to the passengers, who experience the sideways traction of the seat. There is relatively small difference between the way the entire bus responds to the turn compared to individual passengers, and their movement relative to the bus is much smaller than the turning motion of the bus. Like the bus, the Earth does deform some, but the oceans still are subject to a residual forcing.
- Hypothetically, if the ocean were a constant depth, there were no land, and the Earth did not rotate, high water would occur as two bulges in the height of the oceans, one facing the Moon and the other on the opposite side of the earth, facing away from the Moon. There would also be smaller, superimposed bulges on the sides facing toward and away from the Sun.
- Myrl Hendershott, Lecture 2: The Role of Tidal Dissipation and the Laplace Tidal Equations. GFD Proceedings, 2004.
- C.J.R. Garrett et al., The age of the tide and the “Q” of the oceans Deep Sea Research and Oceanographic Abstracts 8(5) (1971): 493-503. Retrieved May 6, 2020.
- Everything You Need to Know to Experience the Bay of Fundy Tides Bay Ferries Limited. Retrieved May 6, 2020.
- Center for Operational Oceanographic Products and Services, National Ocean Service, Tide and Current Glossary National Oceanic and Atmospheric Administration, 2000. Retrieved May 6, 2020.
- About Harmonic Constituents, NOAA. Retrieved May 6, 2020.
- Charles Darwin, The Descent of Man, and Selection in Relation to Sex (London, UK: John Murray. Facsimile reprint: Adamant Media Corporation, 2000, ISBN 1402184352).
- Accelerator on the move, but scientists compensate for tidal effects Stanford Report, March 29, 2000. Retrieved May 6, 2020.
- L. Arnaudon, et al., "Effects of Tidal Forces on the Beam Energy in LEP" IEEE, 1993. Retrieved May 56, 2020.
- Masaru Takao and Taihei Shimada, "Long term variation of the circumference of the spring-8 storage ring", Proceedings of EPAC 2000, Vienna, Austria. Retrieved May 6, 2020.
- P. Nurmi, M.J. Valtonen, and J.Q. Zheng, Periodic variation of Oort Cloud flux and cometary impacts on the Earth and Jupiter Monthly Notices of the Royal Astronomical Society 327(4) (2001): 1367-1376. Retrieved May 6, 2020.
ReferencesISBN links support NWE through referral fees
- Accad, Y., and C.L. Pekeris. Solution of the Tidal Equations for the M2 and S2 Tides in the World Oceans from a Knowledge of the Tidal Potential Alone. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 290(1368) (1978): 235-266.
- Anthoni, J. Floor. Oceanography: tides Seafriends.org, 2000. Retrieved March 15, 2020.
- Cartwright, David E. Tides: A Scientific History. Cambridge, UK: Cambridge University Press, 1999. ISBN 0521797462
- Darwin, Charles. The Descent of Man, and Selection in Relation to Sex. London, UK: John Murray. Facsimile Reprint: Adamant Media Corporation, 2000. ISBN 1402184352
- Mccully, James Greig. Beyond the Moon: A Conversational, Common Sense Guide to Understanding the Tides. Singapore: World Scientific Publishing Company, 2006. ISBN 9812566449
- Open University. Waves, Tides and Shallow-Water Processes. Burlington, MA: Butterworth-Heinemann, 2000. ISBN 0750642815
All links retrieved April 30, 2023.
- Our Restless Tides: NOAA's practical & short introduction to tides.
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Centripetal acceleration is the acceleration towards the center of the circle while centrifugal force is the apparent force that seems to pull objects outwards. Centrifugal force is not a real force, but rather the result of inertia.
An acceleration, according to Newton’s second law of motion, is impossible to achieve without an object being subjected to a force. A body experiences centripetal acceleration when it is subjected to a centripetal force. However, unlike centripetal force, a centrifugal force is not real. Even though we experience this pair of forces almost every day, physicists deem the latter — in the jargon of physics — apparent. How is centripetal acceleration different from linear acceleration? Why is the centripetal force that imparts this acceleration considered real, while centrifugal force isn’t? Here’s how, and why.
What Is Centripetal Acceleration?
A body experiences centripetal acceleration when it is forced to travel in a circle, or at least in an arc. The force that causes it to do so is called a centripetal force. As the name suggests, the force is directed towards the center of the circle that the body traces or the center of the circle the arc would draw if it were allowed to be completed. For this reason, the centripetal acceleration imparted is also referred to as radial acceleration.
A body traces a circle by changing its trajectory constantly, at every instant throughout its motion, and at each of those instants, it experiences centripetal acceleration. What is surprising is that a body experiences centripetal acceleration even when it is tracing an arc or a circle at a constant velocity. This is counterintuitive, as we typically define acceleration as a change in velocity. However, velocity is a vector, meaning that it exhibits magnitude as well as direction. Therefore, even though the body might travel at a constant velocity, it still accelerates because, at each point on the circle, it changes its direction.
At each point on the circle, the body is subjected to two accelerations, the directions of which are perpendicular to each other. One is the centripetal acceleration, directed towards the center or inward, and second is the linear acceleration, which is directed immediately forward of the body, or outward, but more precisely, tangential to the point of concern on the circle.
Consider the simplest example – a thread on one end of which you exert tension with your fingers that causes the stone attached to the other end to whirl around you. The stone’s centripetal acceleration is directed towards the center of the circle — that is, along the tension in the thread, towards your hand, the tension’s source — while the stone experiences linear acceleration when it is unleashed abruptly. You would observe that the stone is then hurled in a single direction immediately – the instant it is free of the centripetal force, its inertia forces it to move linearly – it then travels along the tangent drawn to that point in that very instant.
Centripetal acceleration is directed towards the center of the circle because the centripetal force that imparts it is directed towards the center. As demonstrated by Newton, a body accelerates in the same direction in which the force is applied. This is because it is in this direction in which the force causes the change in velocity. The diagram below illustrates this, and it is this diagram that we will use to derive the expression for centripetal acceleration.
The circle represents the circular trajectory of a body, which at point P has a linear velocity v, and later, at point P’, has a linear velocity v’. Observe how the velocities are tangential to the two points, and therefore perpendicular to the radii r and r’ by which they are tethered to the center. Logically, then, the change in velocity Δv is also perpendicular to the change in radius Δr. As a consequence, the centripetal acceleration a(c), which is directed towards the change in velocity Δv, is also perpendicular to Δr. One can infer from the diagram that a line bisecting Δr perpendicularly goes through the center of the circle. This can also be illustrated with a triangle of vectors. Observe the small triangle drawn adjacent to our circle: the resultant vector of v and v’ – Δv — points inward, or more precisely, towards the center of the circle. We have determined centripetal acceleration’s direction, now let’s determine its magnitude.
The two isosceles triangles – one formed by the radii and the chord (PCP), and the other formed by the velocities and the resultant (IGH) – are similar. This means that the ratios of their sides are equal. For an infinitesimally small angle α, the arc Δs becomes indistinguishable from the chord Δr. At an infinitesimally small angle, the arc becomes one of the infinitesimally small lines that together constitute the entire, huge circle. Considering that the two triangles are similar and that Δs = Δr, we find that:
Divide the two sides by Δt. The ratio Δv/Δt is equal to centripetal acceleration a(c) and the ratio Δs/Δt is equal to v. According to Newton’s second law, the force F that imparts the acceleration is a product of the body’s mass m and the acceleration a.
One can understand from this expression why taking sharp turns, particularly at a high velocity, is astonishingly difficult. In fact, the magnitude of acceleration is directly proportional not only to the magnitude of velocity, but to the square of the magnitude of velocity, which means that a car of a certain mass m, to turn when speeding at 100 km/hr, would require four times the centripetal force it would require to turn when traveling at 50 km/hr. Furthermore, the magnitude is inversely proportional to the radius of the circle the arc would have drawn; therefore, the car would require an even greater centripetal force to curve sharply. However, at least the force would be real; centripetal force really exists. Centrifugal force, on the other hand, does not.
What Is Centrifugal Force?
Centrifugal force is often referred to as the equal and opposite reaction to centripetal force. If the centripetal force issued by your hand on one end of the string makes the bucket attached to the other end whirl around you, then it is the centrifugal force that prevents the water in the bucket from spilling out. The smaller the centripetal force, or the slower you rotate the bucket, the smaller the centrifugal force, or the more liable the water is to spill due to gravity. Such an equal and opposite force is also experienced by passengers in a turning car. As the car turns left, the passengers drift to the right. However, while its effects are palpable, physicists assert that the force doesn’t really exist in nature.
Fundamentally, there exist in the Universe only four forces: strong, weak, electromagnetic and gravitational. Every other force we encounter is actually a manifestation of one or several of these forces. For instance, remember that centripetal force is just a force that acts towards the center or constrains the motion of a body to a circle. It is the electromagnetic interactions between my fingers and the rope that cause the stone attached to its other end to whirl around me. Electromagnetic forces manifest as the centripetal force that constantly pulls the stone towards the center.
Another example of such a manifestation is our orbit around the Sun. It is the Sun’s gravitational force that pulls us towards it, the center of our practically circular orbit, thereby manifesting as centripetal force. This is why centripetal force is deemed real — it is not a new force — but instead, it is any fundamental force acting towards the center of a circle. Centrifugal force is fictitious or apparent because there is no instance where any of the four fundamental forces manifest as centrifugal force. For this reason, it doesn’t exist in nature. It is merely the inevitable consequence of how we do physics.
Merriam-Webster defines a frame of reference as “an arbitrary set of axes with reference to which the position or motion of something is described or physical laws are formulated.” A frame of reference is a system that is subjected to several forces. The problem is that a frame of reference is individualistic — the passenger and an observer observing the turning car from outside each have their own frames of reference.
A frame of reference is inertial when the forces to which it is subjected negate each other, such that the net force experienced by the system is zero, and therefore, so is the acceleration. No acceleration doesn’t necessarily mean no motion – according to Newton’s first law of motion, a system experiences no acceleration when it is either at rest or moving at a constant speed in a straight line. However, remember that even though the car turns at a constant speed – because, to trace a circle, it must change its direction constantly — it is accelerating. Such a frame of reference is therefore non-inertial.
What the passenger inhabits is a non-inertial frame of reference, or more precisely, a rotating frame. And because a non-inertial frame accelerates, it cannot exist without the presence of an additional force. This is the force that causes the passenger to feel that she is being pushed in the opposite direction. Also, because she witnesses a change in direction or trajectory (of the other passengers as well), she’s certain that some force is present. We call this force the centrifugal force.
However, the system appears drastically different to an observer observing the car from outside, or to a person inhabiting an inertial frame of reference. To such an observer, no such push is apparent. What she observes is simply the passenger moving inwards, as directed by the centripetal force — the real force generated by the friction, and more fundamentally, the electromagnetic interactions, between the tires and the road.
A centrifugal force becomes necessary and must be added to non-inertial frames, such as a rotational frame, because it makes calculations much simpler, more convenient and intuitive. A centrifugal force wouldn’t arise if we didn’t study rotational motion with a rotational frame. However, we would find the calculations to be quite tedious and cumbersome, which is why the force is a consequence of how we do physics.
In a rotational frame, it is as equal and opposite in nature as is the pull you feel when the vehicle you’re traveling in suddenly accelerates. The centrifugal force that one feels is just a consequence of one’s inertia, one’s tendency to resist a change in motion. The greater the change, the greater is the “pull”. In fact, you have surely experienced that when a vehicle suddenly accelerates to a very high velocity, the pull is seemingly impossible to overcome. This is why the water doesn’t spill when the bucket is rotated at a high velocity.
Scientists have actually found an ingenious way to exploit the force’s inertial nature. In the same way that centrifugal force pins the water to the bucket’s surface, it can also pin the passengers of a rotating spaceship to its sides and surface. If you’re wondering why the Endurance was so furiously rotating in the film Interstellar, it was doing so to simulate gravity with the centrifugal force derived through the process. The majority of NASA’s gravity-simulating prototypes “roll” ahead or “spin” in place. The merely fictitious or apparent force then emulates a real, fundamental force.
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R= sine 1'= cos. 0: = tan. cot.
Sine acord 2 a
Sine (12+a) + cos. a
Cord a 2 sine a
Cos. (29+a) = + sine a
END OF PLANE TRIGONOMETRY.
THE following definitions and properties of spherical triangles belong to spherical geometry, and are premised here as principles on which the demonstrations of the propositions in spherical trigonometry depend.*
1. A sphere or globe is a solid contained under one uniform round surface, which is every where equally distant from a point within it called the centre.
A sphere may be conceived to be formed by the revolution of a semicircle about its diameter, which remains unmoved, and is called the axis of the sphere.
2. A diameter of a sphere is a straight line passing through the centre, and terminated both ways by the convex surface. 3. The circles of the sphere are of two denominations, great circles, and small circles.
A great circle of the sphere is that which divides its surface into two equal parts.
A small circle of the sphere is that which divides its surface into two unequal parts.
Thus, the equator of a common globe is a great circle, and any parallel of latitude is a small circle.
4. Hence the plane of any great circle passes through the
These definitions and elementary propositions, with their demonstrations, will be annexed to the second edition of Playfair's Geometry, under the title of Elements of Spherical Geometry. They constitute no part of spherical trigonometry, according to the genuine meaning of the terms which express the title of that science; and therefore cannot, with propriety, be prefixed to a regular and systematic treatise of spherical trigonometry.
centre of the sphere, and divides the sphere into two equal parts.
5. The poles of any circle are the two extremities of that diameter of the sphere which is perpendicular to the plane of the circle.
6. Hence either pole of any circle is equidistant from every part of its circumference; and each pole of a great circle is 90° from the circumference.
7. A spherical angle is an angle on the surface of a sphere, contained between the arcs of two great circles which intersect each other.
8. The measure of a spherical angle is the arc of a great circle intercepted between the two arcs which form the angle, and drawn at the distance of 90° from the angular point.
9. A spherical triangle is a portion of the surface of a sphere contained by the arcs of three great circles which intersect one another.
Note. The three arcs are called the sides of the triangle, and the three angles which every two arcs form by their intersection are called its angles.
Also, both the sides and the angles of spherical triangles are computed in degrees, minutes, and seconds, in the same manner as the angles of plane triangles.
10. A right-angled spherical triangle is that which has one right angle, or an angle of 90°.
11. A quadrantal spherical triangle is that which has one of its sides a quadrant, or 90°.
12. An oblique-angled spherical triangle is that which has each of its sides, or angles, greater or less than 90°.
13. Any two sides, or angles, of a spherical triangle are said to be like, or of the same kind, or of the same affection, when they are both greater or less than 90°.
14. If one side or angle of a spherical triangle be equal to, or greater than 90°, and the other side or angle less, they are said to be unlike, or of different kinds, or of different affections.
Properties of the Sphere.
15. Every section of a sphere by a plane passing through it is a circle.
16. The centre of a sphere is the centre of all its great circles, and its axis is the common section of all the great circles which pass through its two extremities.
17. A great circle can be drawn through any two points on
the surface of a sphere, and a small circle can be drawn through any three points on its surface.
18. All parallel circles of the sphere have the same pole; and no two great circles can have a common pole.
19. Any two great circles of the sphere cut each other twice at the distance of 180o, and make the angles at the intersections equal.
20. A great circle of the sphere is perpendicular to any other circle, when its plane is perpendicular to the plane of the other; and conversely.
21. A great circle passing through the poles of any other great circle cuts the other circle at right angles; and if a great circle cut any other circle at right angles it will pass through its poles.
Note. Most of these principles will be evident by inspecting the nature and position of the circles drawn on an artificial globe. The sixth article becomes obvious by observing that all the meridians pass through the north and south poles of the globe, and are perpendicular to the equator, and to all the parallels of latitude.
22. If two arcs of great circles intersect each other, the vertical or opposite angles will be equal.
23. An angle made by the intersection of any two great circles of the sphere is equal to the angle of inclination of the planes of those circles.
24. The distance of the poles of any two great circles of the sphere is equal to the angle of inclination of the planes of those circles.
General Properties of Spherical Triangles.
25. Any side, or any angle, of a spherical triangle is less than 180°, or two quadrants.
26. The greater side is opposite to the greater angle, and the less side to the less angle.
27. The sum of any two sides is greater than the third side, and their difference is less than the third side.
28. The difference of any two sides is less than 180°, or a semicircle; and the sum of the three sides is less than 360°, or two semicircles.
29. The sum of the three angles is greater than 180°, or two right angles; and less than 540°, or six right angles.
30. The sum of any two angles is greater than the supplement of the third angle.
31. A spherical triangle is equilateral, isosceles, or scalene, according as its angles are all equal, or only two of them equal, or all unequal.
32. If each of the three angles be acute, or right, or obtuse, then each of the three sides will be less than 90°, or equal to 90°, or greater than 90°; and conversely.
33. Half the sum of any two sides is of the same kind as half the sum of their opposite angles.
Or, the sum of any two sides is of the same kind, in respect of 180°, as the sum of their opposite angles.
34. If three arcs of great circles be described from the angular points of any spherical triangle, as poles, the sides and angles of the new triangle, so formed, will be the supplements of the opposite angles and sides of the former triangle; and conversely.
Again, AB 180°F, BC= 180°
– D, AC=180o — E, and A = 180o F -EF, B=180° FD, C=180°
Affections of Right-angled Spherical Triangles.
35. The sides are of the same kind as their opposite angles; and conversely.
36. The hypothenuse is less or greater than 90°, according as a side and its adjacent angle, or the two sides, or the two angles, are like or unlike.
37. A side is less or greater than 90°, according as its adjacent angle and the hypothenuse, or the other side and the hypothenuse, are like or unlike.
38. An angle is acute or obtuse according as its adjacent side and the hypothenuse, or the other angle and the hypothenuse, are like or unlike.
Other Properties of Right-angled Spherical Triangles.
39. If the hypothenuse be 90°, one of the sides and its opposite angle will be 90° each; and the other side and angle will be of the same number of degrees.
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Angles – Meaning, Types, Measurement and Examples
Table of Contents
Angles are fundamental elements in mathematics that play a crucial role in various geometric and trigonometric concepts. Understanding the concept of angles is essential for solving problems related to shapes, lines, and measurements. Let’s dive into the world of angles and explore their properties and applications.
Analogy of Definition
What is an Angle?
In geometry, an angle is formed by two rays or line segments that share a common endpoint, known as the vertex. The two rays are referred to as the sides of the angle. Angles are typically measured in degrees and are used to describe the amount of rotation between the two sides.
Parts of an Angle
An angle consists of several components, including the vertex, arms, and interior. The vertex is the common endpoint of the two rays, while the rays themselves are known as the arms of the angle. The space between the arms, known as the interior of the angle, determines the angle’s measurement.
Types of Angles
There are several types of angles based on their measurements and characteristics. These include:
1. Acute Angle: An angle that measures less than 90 degrees.
2. Obtuse Angle: An angle that measures more than 90 degrees but less than 180 degrees.
3. Right Angle: An angle that measures exactly 90 degrees.
4. Straight Angle: An angle that measures exactly 180 degrees, forming a straight line.
5. Reflex Angle: An angle that measures more than 180 degrees but less than 360 degrees.
6. Complete Angle: An angle that measures exactly 360 degrees, forming a complete circle.
Interior and Exterior Angles
In geometric figures such as polygons, the interior angles are the angles formed inside the shape, while the exterior angles are formed outside the shape. The sum of the interior angles of a polygon depends on the number of sides, while the sum of the exterior angles is always 360 degrees.
In the given figure, ∠OAB, ∠OBA and ∠AOB, are interior angles while ∠XOY is exterior angle.
Complementary and Supplementary Angles
Complementary angles are two angles that add up to 90 degrees, while supplementary angles are two angles that add up to 180 degrees. Understanding these relationships is essential for solving problems involving angle measurements and calculations.
Find more fun activites related to angles on our site, ChimpVine.
How to Draw and Measure Angles
Drawing and measuring angles can be done using various tools such as a protractor or a compass. To draw an angle, you can use a protractor to measure the desired angle and then use a ruler to draw the lines accordingly. Measuring angles involves aligning the protractor with the vertex and reading the measurement from the scale.
Steps to Measure Angles
Step 1: Place the center of the protractor on the vertex of the angle and align the bottom of the protractor with the ray.
Step 2: Read where the second arm of the angle intersects the protractor.
The angle is 30° .
Steps to Draw an Angle
Step 1: Draw a ray and place the center of the protractor on one point of the ray like so.
Step 2: To make an angle of 60°, find 60 on te protractor and mark a point named Z above it.
Step 3: Remove the protractor and draw a line beginning at X using a ruler. Hence, ∠XYZ = 60° .
Identifying Angle Types
Example 1: Identify the type of angle formed by the hands of a clock at 3:00.
Answer: Right Angle (90°)
Example 2: Identify the type of angle formed by the hands of a clock at 6:00.
Answer: Straight Angle (180°)
Tips and Tricks
1. Angle Identification Tip
Tip: Remember that a right angle measures 90 degrees, a straight angle measures 180 degrees, and an acute angle measures less than 90 degrees.
2. Interior and Exterior Angle Calculation
Tip: For a polygon with n sides, the sum of the interior angles can be calculated using the formula (n-2) * 180 degrees.
3. Complementary and Supplementary Angle Identification
Tip: Complementary angles add up to 90 degrees, while supplementary angles add up to 180 degrees. Look for angle pairs that satisfy these conditions.
4. Drawing and Measuring Angles Technique
Tip: Place the center of the protractor on the vertex of the angle, align one arm with the 0-degree mark, and read the measurement from the scale.
5. Angle Properties Understanding
Tip: The Triangle Sum Property states that the sum of the interior angles of a triangle is always 180 degrees, regardless of the triangle’s size or shape.
Real life application
Story: “The Angle Adventures of Maya and Ethan”
Maya and Ethan, two curious students, embarked on a series of adventures that required them to apply their knowledge of angles to solve real-life challenges.
Challenge 1: The Architect’s Blueprint
Maya and Ethan visited an architect’s office, where they were tasked with analyzing the blueprints of a new building. They had to identify the types of angles formed by the intersecting lines and determine the measurements of the interior angles of the rooms. By understanding the properties of angles, they were able to assist the architect in creating accurate designs.
Challenge 2: The Geocaching Quest
As part of a geocaching adventure, Maya and Ethan encountered a series of clues that required them to calculate the exterior angles of various geometric shapes hidden in the park. By applying their knowledge of angle properties, they were able to decipher the clues and locate the hidden treasures.
Challenge 3: The Surveyor’s Mission
In their final challenge, Maya and Ethan joined a team of surveyors to map out a new hiking trail. They had to measure the angles of the trail’s turns and calculate the complementary and supplementary angles to ensure the trail’s safety and accessibility. Their understanding of angle relationships proved to be invaluable in completing the surveying mission.
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Scales of Measurement
The first rule of a scientific investigation is to report honestly and as accurately as possible what a scientist saw and under what conditions. A scientist describes what has been seen and the conditions and procedures followed. The need to provide others an opportunity to corroborate those findings makes this a high-priority requirement. Towards actuality, defining the observational parameters is the initial step in measuring a certain occurrence.
Scales of Measurement
Measurement is making observations and writing down the data gathered as part of the research. The relationship between the values that are allocated to the characteristics, emotions, or views of a variable is referred to as the level of measurement. For instance, there are various qualities for the variable "if the taste of fast food is good," including very good, good, neither good nor bad, bad, and very awful. We can give the five qualities, in order, the values 1, 2, 3, 4, and 5, to analyze this variable's results. The level of measurement describes the relationship between these five variables. Here, the numbers serve as shorter substitutes for the longer text phrases.
Nominal or Categorical Scale
The simplest, most basic, and weakest type of measurement is when we can replace real objects with symbols or numbers (without understanding their numerical meanings). In other words, we only describe or categorize things, people, or even features using these symbols or numbers. At the most basic level, a scientist needs to develop a classification system that will allow all recorded occurrences to fit into it. For ease of identification, we assign each category of event or item a name, a number, or a symbol.
Then, a nominal or classificatory scale comprises these symbols or numbers. The scale's categories must be mutually exclusive (each observation may only be classified under one category), exhaustive (there must be enough categories to classify every observation), and unordered. Typically, the categories that make up a nominal scale are called characteristics. As a result, there are only two kinds of sex for mammals: male and female. In a nominal scale, the scaling operation entails dividing a given class into a number of mutually incompatible subclasses. Any subclass member must be equivalent in the scaled property or feature. Equivalence is the sole connection utilized in this scale.
Statistical Tests for Nominal Scale
The only descriptive statistics that can be used are those that would not be influenced or altered by such interchange since the symbols or labels assigned to each category are arbitrary and can be modified without changing the scale's fundamental information. Crude mode, proportion, and frequency are what they are. However, the data on a nominal scale can be used to test a hypothesis about how occurrences are distributed among the classes. For this, it is possible to employ the chi-square test, the contingency coefficient, and a few more tests based on the binomial expansion.
The ordinal scale allows the researcher to group people, things, or survey responses according to a certain trait they share. For instance, there are occasions when objects in one class on a nominal scale are not just distinct from those in another class on the same scale but also have some relationship with one another. The members of one class typically possess more of a certain quality or characteristic than members of other classes. Such a connection is frequently indicated with the symbol carat (>), which stands for "more than."
All relationships between classes, including "preferred to," "more than," "greater than," "higher than," etc., are expressed with the symbol >. The ordinal numbers express the relative position or magnitude of the characters about other characteristics. The rank of a category is determined by how many categories come before it in terms of the quantity of the feature being compared, not by how many classes come after it. The discrepancies in ordinal numbers do not indicate the precise variations in the percentage of a characteristic the objects possess.
Statistical Tests for Ordinal Scale
The best way to determine the central tendency of scores on an ordinal scale is to use the median. For such data, quartile deviation is undoubtedly the best way to gauge dispersion. Numerous non-parametric tests, such as the runs test, sign test, median test, Mann-Whitney U-test, etc., can be used to test a hypothesis with scores on an ordinal scale. The terms "order statistics" and "ranking statistics" frequently describe these tests. Rankings of two sets of observations on the same group of people can be used to calculate interrelations. For these circumstances, Spearman's Rank Difference or Kendall Rank Correlation coefficients are suitable.
When ranking qualities on an interval scale, numerically equal distances on the scale correspond to similar distances in the characteristic being measured, and an interval scale provides for comparison of the distance or difference between qualities while still containing all the data of an ordinal scale.
Although there is the lowest endpoint, the zero point, the ratio of any two periods denoted by real numbers is independent of the unit of measurement. This results in a ratio of 1:5, which has no unit, between two intervals of 32 cm and 40 cm and 100 cm and 140 cm. The ratio between the two intervals remains the same if a constant, such as 10 cm, is added to each interval point, resulting in new intervals of 42 cm - 50 cm and 110 cm - 150 cm, respectively.
When analyzing differences between two or more qualities, interval measurement should be done with appropriate caution. When the origin, zero, for both scales is the same, and the measurement units are the same, comparisons are meaningful. Interval scales are used when measuring temperature using a thermometer, time from a chosen beginning point, and altitude relative to mean sea level.
Having all the characteristics of a nominal scale (equivalence relation), an ordinal scale (greater than or transitivity relation), and an ordered metric scale, an interval scale is also a metric scale (transitively related concerning the distance between classes). The ratio of any two intervals can also be specified using this scale. Using units that measure equal distance intervals, interval scale can place things or occurrences into a continuum. The scale's zeroth position is picked at random.
Statistical Tests for Interval Scale
Even though the numbers linked with an object's position may change according to a regular system, the interval scale preserves both the ordering of objects and the relative differences between them. If the information allows for a linear transformation, a set of observations will be scalable by interval size.
In other words, a set of real numbers is said to be in an interval scale if the equation y = a + bx, where a and b are two positive constants, fulfills the set of real numbers. Data that follow an interval scale can be subjected to all typical parametric tests, including arithmetic mean, median, standard deviation, product-moment correlation, etc. For statistical significance, non-parametric tests like Z, t, and F can also be used on interval scale data.
The ratio scale offers the most accurate measurement since it satisfies all interval scale requirements and an additional, crucial one: it has an invariant or absolute zero. The mathematics operations take on a new dimension because of this invariant zero point. The numbers associated with scale points can also be written as ratios independent of the unit of measurement, much like the ratio of intervals between two classes.
Most frequently, the ratio scale is used in the physical sciences. Regardless of whether two objects are weighed in pounds or kilograms, the ratio remains the same. The same holds for how long two objects are or how long it takes two people to finish a particular task. Suppose the four relations of I equivalence (ii) larger than (iii) the known ratio of any two intervals and (iv) the known ratio of any two real values connected with any two locations on the scale can be operationally attained. In that case, a measurement is said to be in ratio scale.
Statistical Tests for Ratio Scale
The ratios between two numbers and intervals maintain all the information contained in the scale because the values in a ratio scale are real numbers with a true zero (no upper limit) and only the unit of measurement is arbitrary. This is true even if these true numbers are multiplied by a true positive constant. When a ratio scale is employed, any statistical test, parametric or non-parametric, can be applied. Statistical tools like the geometric mean and coefficient of variation, which need to know the true scores, can be applied to data using ratio scales.
Criteria for Judging the Measuring Instruments
A measurement must also meet several requirements. The following list of the most crucial factors to consider while assessing a measurement tool.
Unidimensionality − For a ruler, a scale should only measure one feature at a time, such as length rather than temperature.
Linearity − A scale must adhere to the straight-line concept to be considered linear. It is necessary to create a scoring system based on movable units. An inch is an inch, whether at one end of a ruler or the other. However, this interchangeability cannot be guaranteed for altitude scales. In these circumstances, ranking is preferred.
Validity − The capacity of a scale to measure what it is intended to measure is referred to here.
Reliability − Consistency has this quality. A scale should produce reliable results.
Accuracy and Precision − A tool should provide a precise and accurate measurement of the thing we are trying to gauge.
Simplicity − A scale should be as simple as feasible; otherwise, it may become unnecessarily complex, expensive, or even useless.
Practicability − This covers a wide range of issues, including affordability, practicality, and interpretability. Usually, a trade-off must be made between the "perfect" instrument and what the budget will allow. The benefit received must be equal to the expense incurred.
The four levels are nominal, ordinal, interval, and ratio measurements. These scales make up a hierarchy, with the nominal scale of measurement having significantly fewer statistical applications than scales higher. Nominal scales provide data on categories; ordinal scales provide sequences, magnitudes between points on the scale are revealed by interval scales, and ratio scales explain the order and absolute distance between any two points on the scale.
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In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than the absolute value of the divisor. A fundamental property is that the quotient and the remainder exist and are unique, under some conditions. Because of this uniqueness, Euclidean division is often considered without referring to any method of computation, and without explicitly computing the quotient and the remainder. The methods of computation are called integer division algorithms, the best known of which being long division.
Euclidean division, and algorithms to compute it, are fundamental for many questions concerning integers, such as the Euclidean algorithm for finding the greatest common divisor of two integers, and modular arithmetic, for which only remainders are considered. The operation consisting of computing only the remainder is called the modulo operation, and is used often in both mathematics and computer science.
Euclidean division is based on the following result, which is sometimes called Euclid's division lemma.
Given two integers and, with, there exist unique integers and such that
In the above theorem, each of the four integers has a name of its own: is called the, is called the, is called the and is called the .
The computation of the quotient and the remainder from the dividend and the divisor is called, or in case of ambiguity, . The theorem is frequently referred to as the (although it is a theorem and not an algorithm), because its proof as given below lends itself to a simple division algorithm for computing and (see the section Proof for more).
Division is not defined in the case where ; see division by zero.
For the remainder and the modulo operation, there are conventions other than, see .
See main article: Euclidean domain. Although originally restricted to integers, Euclidean division and the division theorem can be generalized to univariate polynomial over a field and to Euclidean domains.
In the case of polynomials, the main difference is that the inequalities
In the generalization to Euclidean domains, the inequality becomes
Although "Euclidean division" is named after Euclid, it seems that he did not know the existence and uniqueness theorem, and that the only computation method that he knew was the division by repeated subtraction.
Before the discovery of Hindu–Arabic numeral system, which was introduced in Europe during the 13th century by Fibonacci, division was extremely difficult, and only the best mathematicians were able to do it. Presently, most division algorithms, including long division, are based on this notation or its variants, such as binary numerals. A notable exception is Newton–Raphson division, which is independent from any numeral system.
The term "Euclidean division" was introduced during the 20th century as a shorthand for "division of Euclidean rings". It has been rapidly adopted by mathematicians for distinguishing this division from the other kinds of division of numbers.
Suppose that a pie has 9 slices and they are to be divided evenly among 4 people. Using Euclidean division, 9 divided by 4 is 2 with remainder 1. In other words, each person receives 2 slices of pie, and there is 1 slice left over.
This can be confirmed using multiplication, the inverse of division: if each of the 4 people received 2 slices, then 4 × 2 = 8 slices were given out in total. Adding the 1 slice remaining, the result is 9 slices. In summary: 9 = 4 × 2 + 1.
In general, if the number of slices is denoted
If 9 slices were divided among 3 people instead of 4, then each would receive 3 and no slice would be left over, which means that the remainder would be zero, leading to the conclusion that 3 evenly divides 9, or that 3 divides 9.
Euclidean division can also be extended to negative dividend (or negative divisor) using the same formula; for example −9 = 4 × (−3) + 3, which means that −9 divided by 4 is −3 with remainder 3.
The following proof of the division theorem relies on the fact that a decreasing sequence of non-negative integers stops eventually. It is separated into two parts: one for existence and another for uniqueness of
For proving the existence of Euclidean division, one can suppose
This proves the existence in all cases. This provides also an algorithm for computing the quotient and the remainder, by starting from
The pair of integers and such that is unique, in the sense that there can be no other pair of integers that satisfy the same condition in the Euclidean division theorem. In other words, if we have another division of by, say with, then we must have that
To prove this statement, we first start with the assumptions that
Subtracting the two equations yields
.So is a divisor of . As
by the above inequalities, one gets
Since, we get that and, which proves the uniqueness part of the Euclidean division theorem.
In general, an existence proof does not provide an algorithm for computing the existing quotient and remainder, but the above proof does immediately provide an algorithm (see Division algorithm#Division by repeated subtraction), even though it is not a very efficient one as it requires as many steps as the size of the quotient. This is related to the fact that it uses only additions, subtractions and comparisons of integers, without involving multiplication, nor any particular representation of the integers such as decimal notation.
In terms of decimal notation, long division provides a much more efficient algorithm for solving Euclidean divisions. Its generalization to binary and hexadecimal notation provides further flexibility and possibility for computer implementation. However, for large inputs, algorithms that reduce division to multiplication, such as Newton–Raphson, are usually preferred, because they only need a time which is proportional to the time of the multiplication needed to verify the result—independently of the multiplication algorithm which is used (for more, see Division algorithm#Fast division methods).
The Euclidean division admits a number of variants, some of which are listed below.
In Euclidean division with as divisor, the remainder is supposed to belong to the interval of length . Any other interval of the same length may be used. More precisely, given integers
In particular, if
See main article: Montgomery modular multiplication. Given integers
a=mq+R-1 ⋅ r
Given an element and a non-zero element in a Euclidean domain equipped with a Euclidean function (also known as a Euclidean valuation or degree function), there exist and in such that and either or .
Uniqueness of and is not required. It occurs only in exceptional cases, typically for univariate polynomials, and for integers, if the further condition is added.
Examples of Euclidean domains include fields, polynomial rings in one variable over a field, and the Gaussian integers. The Euclidean division of polynomials has been the object of specific developments.
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Modifications of sediment deliver by rivers, climate change and human impact are modifying the dynamics of beaches leading to departures from equilibrium and consequent beach evolution. When erosion occurs, concerns may arise for its impact on societal use of beaches and safety of human settlements and infrastructures. In these cases, beach preservation strategies allow us to mitigate the above impacts.
Groins are a widely used structure that is used to protect beaches from erosion. We define groins as shore-perpendicular structures aimed at either (1) maintaining the beach behind them, or (2) controlling the amount of sand moving alongshore. An example of beach protection with groins is shown in Figure 1.
Figure 1. Beach recovery strategy at Cogoleto (Liguria region, Italy; from APAT, Atlante delle Opere di Sistemazione Costiera, Manuali e Linee Guida, 2007)
Groins can be classified as either "long" or "short," depending on how far across the surf zone they extend. Groins that traverse the entire surf zone are considered "long," whereas those that extend only part way across the surf zone are considered "short." These terms are relative since the width of the surf zone varies with the prevailing wave height and beach slope. During periods of low waves, a groin might function as a "long" groin, whereas during storms it might be "short." Groins can also be classified as either "high" or "low," depending on how high their crest is relative to prevailing beach berm levels. "High" groins have crest elevations above the normal high-tide level and above the limit of wave runup on the beach (USACE, 1992).
Being directed perpendicularly to the shoreline, groins may not be effective if cross-shore sediment transport is dominant, such as is typical in shallow beaches. In fact, groins function best on beaches with a predominant alongshore transport direction. They may also be inefficient if there is a large tidal range, allowing sand to bypass the structures at low tide or to overpass at high tide. Bypass of groins by sediments is defined as the movement of sediments along shore beyond the groins in the seaward direction. Overpassing is the movement of sediments above the groins when these are submerged.
Groins may be effective where sand movement alongshore is to be managed, such as where there is a divergent nodal region in alongshore transport, for instance where the curvature of the coast changes, where intruding sand is to be managed, such as in correspondence of the banks of inlets, at the down-drift side of a large harbor breakwater, and other situations where alongshore sediment transport is a reason of concern.
Groins may or may not be permeable to sediments. Impermeable groins are constructed of various materials, typically by boulders, and are designed to completely interrupt any available littoral drift in the proximity of the undisturbed shoreline. Thus, the littoral drift is forced around the seaward end of the groin or, if sufficient beach material has been trapped by the groin, to pass over its top. Permeable groins are designed so that an appreciable quantity of the available littoral drift will pass directly through its structural components. They are frequently built with wood elements.
Groin functional design has been discussed by several authors, including Bruun (1952, 1972), Balsillie and Berg (1972), Balsillie and Bruno (1972), Nayak (1976), Fleming (1990), and the U.S. Army Corps of Engineers (USACE, 1992). They offer the basis for empirical design based on laboratory experiments and field observations. The former may suffer from distorsion in small scale physical models while the latter suffer from specificity of beach conditions, wave forcing and sediment dynamics in general. A comprehensive discussion has been provided by Kraus et al. (1995).
Wave behaviours are leading parameters as they determine alongshore sediment transport. Therefore, a rigorous statistical analysis of wave forcing, and in particular wave height and variability, is needed. Groin length is a leading parameter as well, and in particular the sea depth at the tip of the groin. Furthermore, another leading parameter for groin fields is the groin length to spacing ratio.
Practical experience reveals that modification of the shoreline accretion induced by groins rarely brings the shoreline itself to reaching the seaward end of groins. Typically, the updrift shoreline reaches only a modest distance to the groin tip. Such a behaviour indicates that sand bypassing and permeability of the groin to sand, as well as variability in wave forcing, play an important role in determining transport around the groin and resultant local and regional shoreline change. About the time of development of the new beach configuration, Nersesian et al. (1992) found that groin compartments were still slowly filling in the predominant direction of the shoreline sediment transport after almost three decades after displacement. Such functioning can be explained by the process of bypassing, whereby each compartment deprives sand to neighboring downdrift compartments during shoreline evolution.
Permeability is a desired feature of groins, in order to avoid accumulation of large volume of sand against them, which can be transported offshore during dominant cross-shore sediment transport. For the same reason, groins should not extend too far offshore beyond the average wave breaker line.
The evolution of the shoreline profile after the displacement of groins can be represented trough the progress in time of a suitable spatial coordinate x(t) which can be expressed through an analytical relationship. The variables that are important in groin design can be determined through dimensional analysis. In engineering and science, dimensional analysis allows one to inspect the relationships between different physical quantities by identifying their fundamental dimensions and units of measure and tracking these dimensions as calculations or comparisons are performed. We are not inspecting the details of dimensional analysis here. Dimensional analysis for groin functioning indicate that several parameters may be effective to determine the shoreline evolution. The engineer can design some of them, like groin spacing, length and permeability, which is in turn determined by groin elevation and porosity. Other parameters related to wave forcing can be controlled by displacing other protection structures like breakwaters.
Kraus et al. (1994) provide a comprehensive overview of parameters governing the effect of groins. Parameters related to groin structure are:
- Spacing between groins;
- Angle to the shoreline;
- Shape (as straight, angled, T-head, spurred etc.).
Parameters related to beach and sediment are:
- Depth at tip of groin;
- Beach morphology (slope, berm height, shape of the shoreline etc.);
- Depth at the average breaker line or beach closure;
- Sediment availability;
- Grains size and variability;
- Sediment density.
Parameters related to waves, wind and tide are:
- Wave height and variability;
- Wave period and variability;
- Wave direction and variability;
- Wind speed and variability;
- Wind direction and variability;
- Wind duration and variability;
- Tidal range.
Furthermore, functioning of groins may be affected by wave diffraction if it occurs. The impact of diffraction may be difficult to predict.
Design of groins can be carried out by applying empirical relationships derived from laboratory experiments and analysis of observations. The use of rules of thumb and empirical formulations is widely diffused as it is difficult to provide a physically-based interpretation of processes that are affected by large uncertainty. However, after a preliminary shape has been determined for groins it is suggested to carry out a numerical simulation of the beach evolution, which can provide support to the design and subsequent monitoring activities.
Preliminary design is often carried out by following the so-called “λ shoreline” rule of thumb, originally termed the “one-third rule” (Bodge 1998). The rule says that the post-project coastline during mean low water level is located between about λ=1/3 and λ=2/3 of the distance between adjacent groins measured behind the structures' seaward face (see Figure 2). Larger λ values, that may even exceed 2/3, may be appropriate for energetic environments, larger distances between groins, uncertainty in long term sediment supply and the whole design process. In general, larger λ values imply a more prudent design (Bodge, 2003). These values for λ are confirmed by several field observations that are summarized by Bodge (2003). The denomination “one-third rule” is explained by the above indicated minimum λ=1/3 value. The rule is considered applicable where the angle between the wave crest and the line connecting the tips of the structures (control line) is small, that is, lower than 25°-30° (that is, wave crest is nearly parallel to the control line).
Figure 2. λ rule for the preliminary design of groins. LWS and HWS indicate low and high water shoreline, respectively.
Actually, the λ shoreline rule would predict a rectilinear line shape for the equilibrium profile of the beach when straight (non T-shaped) groins are displaced orthogonally to the shoreline. However, settlement of sediments tends to originate a curved shape of the after-project shoreline along the sides of the groin, for the effect of diffraction and sediment interception by groins. Empirical rules have been proposed to determine the actual equilibrium profile of the beach, which are not discussed here.
Another example of application of the lambda rule is shown in Figure 3, with irregularly shaped groins.
Figure 3. λ rule for the preliminary design of groins with an irregular shape. Redrawn from Bodge (2003).
We adopt here the following steps for groin design as summarized by Bodge (2003).
- Step 1: estimation of the wave statistics, in particular the wave angle, and longshore sediment transport potential, at selected locations of interest. If an extended data set is not available, useful indications can be derived by looking at the beach morphology and/or displacement of sand in different seasons against existing structures or irregularities in the beach profile.
- Step 2: identification of the extension of the intervention, by taking into account the dynamics of longshore sediment transport. Structures should be placed in correspondence of relevant transport and should end where the sediment transport gradient is low.
- Step 3: identification of the design location of the post-project berm.
- Step 4: estimation of the probable post-project slope of the beach, by looking at adjacent beaches. We should take into account that the post-project beach is expected to be slightly gentler sloped than the non-stabilized beach, because of the decreased (diffracted) wave energy that reaches the beach.
- Step 5: prediction of the horizontal distance W between the mean low water and mean high water shorelines. We should also predict the horizontal distance S between the mean low water shoreline and the berm.
- Step 6: identification of the number of beach cells, n, and average gap width, G, and groin head width, H. For T shaped groins H > z, where z is the width of the groin main body. Therefore, the total shorefront length of the intervention, L, which is composed of n beach cells, is expected to L=n(G+H).
- Step 7: design of the length and head-width of groines, H (see Figure 2).
- Depending on the design position of the low water shoreline, length and shape of the groin, as well as head-width, are determined by applying the λ rule. A T shaped groin allows one to reduce the groin width, as the T implies that the low water shoreline is shifted seaward according to the λ rule itself. If one wants to bring the low water shoreline up to the groin head, then a T shape should be adopted, and then λ rule implies that H=2λG + z (see Figure 2). In some cases it may be required that the mean high water shoreline reaches the head. Namely, H= 2(λG+W) + z. In general, then, H=2(λG+X) + z
where 0≤X≤W. Therefore one obtains
L = n(G+2λG+2X+z),
L = nG+2nλG+2nX+nz,
If the gap width reduces too much, then a rectilinear groin may be adopted and its length increased in order to reach the desired position of the shoreline.
In practice, we select a number of beach cells, n, for a identified value of λ and desired value of X in order to develop a physically reasonable gap width, G. G values less than about 20 m are not generally recommended, at least for recreational beaches. In fact, small openings have limited aesthetic; moreover, strong rip currents may be originated with small openings. On the other hand, gap widths greater than about 100 m are not recommended as well, because distant structures may be not effective in determining an extended reshaping of the beach.
Additionally, for recreational beaches, it is advisable that the head widths are smaller than the gap widths (H≤G).
- Step 8: positioning of the gaps and heads along a line that is S+λG seaward of the target (design) berm location, where S is the distance from the new low water profile and the berm.
- Step 9: orientation of the gaps to make them more closely parallel with the average, or principal, crest orientation of the breaking waves at each cell. The aim is to minimize the wave crest angle with the orientation of gaps. If the terminal structure in a field employs a head or spur on its downdrift side, it is better to offset it seaward than landward. While intuition suggests that a landward offset would offer a more natural transition from the structure to the downdrift shoreline, the λ rule suggests that it inherently induces a crenulate bay that will erode into the native beach.
- Step 10: prediction of the impact of extreme waves. Once satisfied that the project lay-out's predicted shoreline and berm will satisfy the "target" shoreline for the principal wave direction, then the shorelines and berm locations are predicted for extreme wave directions. This aims to assess the degree to which the structures might be exposed to seasonal or storm events where the waves deviate from their average direction. Adjustments may be necessary to accommodate the wave extremes.
- Step 11: the landward head of the groins are extended to reach the berm and buried within the beach fill.
- Step 12: The elevation of the heads should be determined to minimize wave overtopping during the design event of interest, which typically includes typical seas at higher high tides. The elevation of the stems may be lower than the heads, but of sufficient elevation to prevent tidal overtopping at higher high tides. The profile elevations of the terminal structures should be higher than the design beach profile predicted adjacent to the structure. In the case of a rock structure, the crest elevation should be not less than 1⁄2 armor stone diameter above the predicted design profile (plus some contingency).
Groin fields should be built from the downdrift to updrift direction. A desired evolution of groin field is that they are filled.
Main parameters for groin design are given in Figure 4.
Figure 4. Structure of a groin.
Orders of magnitude for the principal parameters of groins are as follows.
- h - depth of the basement of the groin head with respect to mean sea level (msl)- From 2.5 to 4 m.
- Rc - height of the groin top with respect to mean sea level - From 0.5 to 1.5 m.
- Depth of the of the groin basement landward with respect to mean sea level - From 1.0 to 2.0 m.
- B - Width of the groin top - From 3.0 to 6.0 m.
- 1/np - Slope of the groin sides - From 1:1 to 1:2.
- Thickness of foundation - From 0.5 to 1.0 m.
Stability of groins should be verified against extreme sea conditions. If groins are made by boulders, the stability of the single stone should be checked against the expected wave energy.
The preliminary design of groins based on empirical rules should be verified through numerical modeling of beach evolution. Models are based on subdividing the shoreline in adjacent cells for which sediment transport rate is estimated by applying suitable relationship, like for instance the CERC formula. Limiting configurations of the beach, for instance due to emerging rock layers, are subsequently identified, as well as sources and sinks of sediments.
Once the model is set up, groins are displaced along the shoreline and modified sediment transport rates are computed. To this end, one need to estimate the volume of sediments that is intercepted by groins and the volume of sediments that bypass or overtop the structure.
Numerical models were proposed by the literature to estimate beach evolution after the displacement of groins. We summarise here below the theory of the GENESIS model (GENEralized model for SImulating Shoreline change; Hanson, 1988; Kraus et al., 1995). GENESIS simulates shoreline change as caused primarily by wave action. The model is based on the one-line concept, which assumes that the beach profile remains unchanged, thereby allowing beach change to be described uniquely in terms of the shoreline position. In fact, it is frequently observed that the beach profile maintains an average shape that is characteristic of that particular coast. Beach features like slope and singularities are preserved in the long term. Although seasonal changes in wave climate cause the position of the shoreline to move in a cyclical manner, the departure from the characteristic shape are relatively small. Therefore the beach profile responds to wave action by moving back and forth during erosion and accretion, under the assumption that the profile moves parallel to itself. If the profile shape does not change, any point on it is sufficient to specify the location of the entire profile with respect to a baseline. Groins may act as fixed points along the shoreline therefore inducing a seaward shift of the shoreline.
Important implications of the above assumption of steady profile are that only longshore sand transport can be taken into account and that the profile is always in equilibrium. However, cross-shore transport can be simulated within GENESIS in a schematic way, in terms of non-wave induced sources and/or sinks of sands along the coast.
The second major assumption of the model is that sand actively moves over the profile to a certain limiting depth, beyond which the bottom does not move. This depth is called the depth of closure Dc. GENESIS uses a simple relation for calculating the depth of closure according to the relationship
where Hmax is the maximum annual wave height. Evolution of beach occurs according the scheme presented in Figure 5.
Figure 5. Computation of beach profile under the effect of longshore sediment transport according to the one-line model. Source: tutorial of prof. Musumeci, University of Catania
The model is based on the application of mass conservation in each beach cell. For applications involving bypassing of sand at structures, knowledge of the depth to which sand is actively transported alongshore is required. This depth, assumed to be related to the incident wave conditions which vary with time, is called the depth of longshore transport. The latter parameter affect the rate of sand that is transported by waves around groins. A through analysis of groins bypassing should include the cross-shore distribution of the longshore sand transport rate, as well as the two-dimensional horizontal pattern of sand transport. In view of the uncertainties related to evaluating these processes, a simple assumption producing reasonable results was adopted.
The fraction F of sand that is transported over a groin and through it is represented by a permeability factor P, and the amount passing around the seaward end is represented by a bypassing factor B (Hanson & Kraus 1989), such that
where 0≤P≤1, and 0≤B≤1. P is estimated depending on the shape of groins and in particular the presence of apertures in their cross shore direction. P values are suggested by the literature for typical groin shapes. Groins made by boulders have a P value that is close to zero. The actual volumetric sediment transport rate at the groin Q'G is related to the calculated potential rate at the groin QG as
The permeability factor is assigned based on groin elevation, groin porosity, and tide range, and the bypassing factor B is calculated in the model at each time step through the relationship
where DG is the sea depth at the groin seaward end at a particular time step, and DLT is the depth of active longshore sediment transport, which can be assumed to be about 1.6 times the significant breaking wave height (Hanson & Kraus 1989). Therefore groin bypassing is dictated by sea depth at the groin seaward end and wave behaviors.
Empirical formulation are used within GENESIS to estimate bypassing for angled groins, shape of the sea bottom profile, impact of refraction and diffraction. We are not interested in the details herein. Once the after-project sediment transport rate is estimated, beach profile can be determined. The λ rule and other empirical rules of beach evolution can provide a benchmark to verify the reliability of the simulation. A user interface is available in GENESIS to allow implementation to a diverse variety involving arbitrary configuration and combinations of groins.
The stabilization of beach fill by structures may be warranted at sites where erosion stress is sufficiently severe to require otherwise impractical (frequent) renourishment intervals; or where the proximity of natural resources or marine structures preclude construction of a wide beach fill; or where the project shoreline is advanced far seaward of the adjacent shoreline. Adverse impacts to downdrift shorelines may be minimized by (i) advance-nourishment of the structures’ impoundment field with imported beach fill, (ii) use of T-head or other headland structures that do not promote rip currents and offshore losses, and (iii) termination of the structural field in non-accelerating longshore transport potential.
Particular care should be given to the preliminary interpretation of the processes leading to shoreline shape. Observation is the basis for a successful design.
Balsillie, J.H. and Berg, D.W. 1972. State of Groin Design and Effectiveness. Proc. 13th Coastal Eng. Conf., ASCE, 1367-1383.
Balsillie, J.H. and Bruno, R.O. 1972. Groins: An Annotated Bibliography. Miscell. Paper No. 1-72, Coastal Eng. Res. Center, Vicksburg, Miss.
Bodge, K. R., 2003. Design Aspects of Groins and Jetties. In: Advances in coastal structure design. Ed. R. Mohan, O. Magoon, M. Pirrello. American Society of Civil Engineers (ASCE). Reston, VA. ISBN 0-7844-0689-8. Pp. 181-199
Bruun, P. 1952. Measures Against Erosion at Groins And Jetties. Proc, 3rd Coastal Eng. Conf., ASCE, 137-164.
Bruun, P. 1972. The History and Philosophy of Coastal Protection. Proc. 13th Coastal Eng, Conf., ASCE, 33-74.
Fleming, C.A. 1990. Principles and Effectiveness of Groynes. Coastal Protection, Pilarczyk, K.W., (Ed.), Balkema Press, Rotterdam, 121-156.
Hanson H., 1988. GENESIS-A generalized shoreline change numerical model. Journal of Coastal Research, 5(1), 1-27, Charlottesville (Virginia). ISSN 0749-0208.
Hanson, H. and Kraus, N. 2001. Chronic Beach Erosion Adjacent to Inlets and Remediation by Composite (T-Head) Groins. U. S. Army Corps of Engineers, Waterways Experiment Station, Vicksburg, MS, ERDC/CHL Coastal Engineering Technical Note CHETN-IV-36, 15 pp, June 2001.
Kraus, N. C., Hanson, H., & Blomgren, S. H. (1995). Modern functional design of groin systems. In Coastal Engineering 1994 (pp. 1327-1342).
Nayak, U.B. 1976. On the Functional Design and Effectiveness of Groins in Coastal Protection. Ph.D. Dissertation, U. of Hawaii, 205 pp. USACE, (1992). Coastal Groins and Nearshore Breakwaters. In: Engineer Manual, (1992).
Last updated on March 29, 2023
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Pi is a significant mathematical constant that connects a circle's diameter and circumference. It is an irrational number, which means that its decimal digits are infinite and non-repeating (Beckmann, 2023). Pi is widely used in formulas and calculations that include circles and spheres. Furthermore, pi appears unexpectedly in mathematical contexts that appear to be unconnected to circles.
This ScienceShot presents an overview of pi, including its definition, origin, mathematical history, and applications. The experimental derivation of pi is shown. Key historical breakthroughs in finding the digits of pi are noted.
Definition and Derivation of Pi
Pi (π) is defined as the ratio of a circle's circumference (the perimeter or distance around the circle) to its diameter (the straight line passing through the center of the circle connecting one side to the other). This ratio holds true for circles of any size.
Pi can be derived geometrically by a simple experiment. Using a compass, construct a circle on a piece of paper. Take a string and align it so that it wraps exactly once along the circumference. Carefully remove the string and straighten it out, then measure its length using a ruler. This length represents the circumference (C) of the constructed circle.
Next, using the ruler, directly measure the diameter (d) of the circle from one edge, straight through the center point, to the opposite edge. Regardless of the original circle's size, dividing its circumference (C) by its diameter (d) will result in a ratio that is approximately equal to 3.14.
Mathematically, this relationship can be summarized by the equation for circumference, where π represents the constant ratio between the circumference and diameter:
Circumference = π x Diameter
C = πd
Rearranging this formula leads to the definition for pi:
π = Circumference / Diameter
π = C/d
Furthermore, pi links the diameter or radius of a circle with its area. Consider a circle of radius r. Its area can be divided into small wedge-shaped slices radiating out from the center point. The curved edge of each wedge section approximates a straight line segment of length πr. As the number of wedges approaches infinity, summing their areas gives the total area. This yields the standard formula for the area (A) of a circle:
A = πr2
Interestingly, pi relates diagonal lengths within squares and their circumscribed circles. Begin with a circle and draw a square outward so that the sides are tangent to the circle. If the circle has diameter d, the diagonal length of this square will also be d.
Now place a smaller circle inside the square so that it touches all four sides. The diameter of this smaller circle is equal to the side length (s) of the original square. Applying the circumference formula shows its circumference is πs.
Noting the inner circle's circumference must equal the outer square's perimeter of 4s, an interesting geometrical derivation of pi results:
πs = 4s
π = 4
This demonstrates pi naturally linking the geometry of circles to squares.
Thus, pi fundamentally connects a circle's diameter to its circumference, area, and to other geometric shapes like squares. This deep relationship to basic geometric properties gives the ratio now known as pi great mathematical and scientific importance.
History of Pi Usage
Ancient civilizations recognized the existence and utility of the constant ratio of circumference to diameter now known as pi. Babylonians and Egyptians used approximate numerical values for pi in mathematical calculations as early as 2000 B.C (Beckmann, 2023). For example, the Babylonians used 25/8 (approximately 3.125) as their value, while the Egyptians used 256/81 (approximately 3.16) (Jones, 2013).
These ancient civilizations applied these numerical approximations for pi in various formulas related to circles and spheres. Evidence indicates the Babylonians utilized their rough value to estimate areas, volumes, and perimeters of circular shapes in architectural and engineering design problems (Jones, 2013). Additionally, the Egyptians applied their pi approximation in semidirect formulas to estimate enclosed areas and volumes (Jones, 2013).
Later, Greek mathematicians like Archimedes produced more accurate estimations through innovative approaches using polygon perimeters bounded by circumscribed circles. Archimedes established upper and lower bounds for pi which tightened the approximations considerably (Beckmann, 2023). His estimate of 22/7 (approximately 3.14) became widely adopted.
Subsequently, Chinese mathematicians like Zu Chongzhi used iterative algorithms building on these techniques to calculate pi to seven decimal places by 480 A.D. (Roy, 1990). Arab mathematicians continued incremental improvements between the 9th to 11th centuries A.D., achieving levels equivalent to nine decimal places (Fowler, 1987).
Jones (1706) first represented this important geometric ratio with the Greek letter π, likely abbreviating periphery. This notation was popularized in mathematics around 1736 (Beckmann, 2023). Since then, π became the standard symbol used to represent this constant in formulas across mathematics and the sciences.
With computational advances, more precise values for π have been obtained. By 1900 A.D., over 500 digits were established. Modern computers have allowed determination of over 6 billion known digits of π (Beckmann, 2023). However, as an irrational number, the digits of π are never ending.
Applications and Unexpected Occurrences of Pi
The ubiquity of pi relates to its presence across numerous mathematical and scientific contexts beyond simple geometry. Pi arises in many formulas in physics, engineering, and data science.
In physics, pi is an integral component of equations describing wave motion, electrostatics, quantum mechanics, and Einstein's field equations for relativity (Arfken et al., 2005). For example, the formula to determine the wavelength λ of a wave is:
λ = 2πc/ω
Where c is the wave propagation speed and ω is the angular frequency.
In engineering, approximation of shapes as circular is convenient for calculations like stress distributions in cylindrical pipes. Here pi naturally appears in equations for hoop stress based on radii and pressures (Boresi et al., 1993).
Additionally, pi is essential in various data transformations used for time series analyses in data science applications (Brillinger, 2001). The Fourier transform utilized in signal analysis contains integrals over angular frequencies defined in units containing pi.
Furthermore the below infinite series shows an example of the unexpected emergence of pi.
1 + 1/4 + 1/9 + 1/16 + 1/25 + ... + 1/n2 + ... = π2/6
This illustrates how pi arises in areas of mathematics not overtly related to geometry. Namely, analytic number theory and analysis of converging infinite series (Weisstein, 2004).
In summary, the ubiquity of pi across the quantitative sciences relates to its presence in diverse mathematical contexts beyond simple circumference and area formulas in basic geometry.
In summary, pi arose historically as a constant ratio relating circular dimensions. Standardized mathematical notation as π developed shortly after 1700 A.D. Computation has revealed pi to over 6 billion digits, although as an irrational number, the digits are endless. While originally derived geometrically, pi emerges surprisingly across many mathematical contexts and has wide-ranging utility for quantitative sciences.
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The Latin breakdown: “Arthro” = Joint “Itis” = Inflammation Osteoarthritis (OA) is a common, degenerative, painful disease with chronic onset and acute flare-ups . This means it most commonly takes a long time to develop, but will also have periods of increased sensitivity and pain (acute flare-ups). This disease involves the whole synovial structure – which is one of the reasons it is difficult to prevent and to manage. The synovial structure includes the attached connective tissue that covers tendons, muscles, bones and enclosed joints… which means it can majorly affect movement. The smooth articular surfaces become injured and inflamed, which result in lameness and compensation. The primary and secondary problems associated with managing arthritis are numerous. Layers of a joint: A: shows what a normal bone/joint looks like, B: diagram representing cartilage and the outer most edge of bone, C: demonstrates normal cartilage and subchondral bone, and D represents damaged cartilage and subchondral bone which is indicative of arthritis. Features of joints and arthritis 1. The articular surface (the surface between two bones) is made of smooth and softer cartilage. This allows frictionless motion between bones at the joint. 2. Between these surfaces is a thick fluid made of hyaluronic acid and other chemicals which provide a frictionless and shock-absorptive liquid to cushion the joint ends. 3. The gradient of cells and tissues which develop from the core of the bone to the articular cartilage surface is highly interwoven . This means it’s not distinct and the relation from the bone to the end of the joint is extensive. This gradient also represents a change in the elasticity/“softness” of the levels. This is marginal but does provide for some of the shock-absorptive properties of joints. 4. The “cracking” and loss of smooth surface observed in damaged tissue, and the loss of an even distribution of cells and tissue across the layers of joint tissue indicates arthritis. This alters the frictionless, smooth surface effectively, and becomes a downward cycle which increases joint inflammation and damage in turn. Effects of/evidence of arthritis include : - Narrowing of joint space - Osteophyte formation (joint mice, also known as little bony spurs that grow in or around the joint cavity). - Subchondral sclerosis – This is the thickening of bone ends just under the cartilage - Subchondral cyst – a fluid-filled space which appears at the end of a bone as a result of arthritis. - Reduction in cartilage mineral density in early phases of arthritis Chondrocytes are cartilage-producing cells. Losing them reduces the ability of the joint to repair and produce cartilage. Chondrocytes decrease in activity as the animal ages; however increased strain, damage and pressure to the joints, which could cause these microscopic fragments of cartilage to break, will increase the “wear and tear” associated with aging animals. Âge -> cartilage degeneration Increased workload -> cartilage degeneration Cartilage degeneration -> increased joint damage, inflammation and pain Joint damage and pain -> biomechanical compensations and discomfort https://www.drwilliamhatten.com/blog/2017/6/2/lets-talk-more-about-oa Causes of arthritis are not easily defined; 1. As a chronic disease, the precise moment that it starts happening is nearly impossible to define. However, some metabolic and physiological changes have been linked to early stages of OA. 2. As an acute onset, direct damage to cartilage in a sudden injury can result in a string of events which cause arthritis too. A very brief overview of factors that can increase arthritis onset: 1. Joint conditioning when young - Studies shown that growing and moving young horses “primes” their joints for good cartilage layers at their joints, and that these joints are better prepared for increased pressure later in life. Injury - Joint injury, soft tissue, bone fractures, - Overcompensation (increased pressure on other joints) 2. Poor conformation - Straight hocks (check out the blog: ) 3. Discipline - Racing, showjumping, cross country, dressage… all have different predisposition to arthritis in particular joints 4. Location of the joint - Interestingly, different joints have been categorized in various studies to be far more prone to problems (sometimes labelled “radiographic lesions”). Bones of the distal limb, aka the Phalanges, are short, compact bones that appear to have evolved to take the brunt of force applied to limbs when moving . Because of this function, these joints are exposed to the greatest force, and most likely to suffer arthritic changes as a result. Common locations: (for those who paid attention, this would be the answer to the question posed on my social media this week!) - The coffin joint of the forelimbs are the most commonly affected with arthritis. This is associated with the high level of force these joints absorb as the first point of contact with the ground. - Hocks are the second most affected – this is so commonly related to conformation, back problems and as a result saddle and training problems. - Hindlimb fetlocks are also highly affected, likely as a result of compensation from other problems. Treating arthritis NO TREATMENT WILL REVERSE THE EFFECTS OF ARTHRITIS, NOR SAVE THE JOINT ENTIRELY. TREATMENT OF ARTHRITIS MEANS TREATMENT OF SYMPTOMS. THIS MEANS ADDRESSING PAIN AND COMPENSATION, WHICH INVOLVES A VARIETY OF APPROACHES. Treating arthritis at the level of the joint space and surface is reserved for the Veterinarian. Joint injections in the form of Hyaluronic acid and corticosteroids are commonly used to address inflammation (remember inflammation itself can and does cause pain and further problems) in the joint directly, and to aid in “plumping” the joint fluid already there: this is the Veterinarian’s domain. Pain management and lameness work-ups should all be used in the diagnosis and management of this chronic disease, particularly if you’re expecting your horse to work hard and compete. Treatment by the Veterinary Physiotherapist is in support of the comfort and functionality of the horse, and to reduce pain caused by the acute flare-ups. Target 1: The pain. This is a wide-ranging target for this disease. Pain is likely to be in the affected joint… this will however be followed by that dark shadow that follows pain: compensation. This is likely to be causing pain elsewhere. Increased bracing activity of muscles in other limbs, or bracing the limb itself mean muscles are prone to fatigue quickly: causing further injury risk, discomfort and pain. On top of that, the increased muscle activity elsewhere is likely to increase the pressure on their associated joints, tendons and muscles, which can cause pain and even increase the rate of arthritis onset. Furthermore, and I can’t stress this enough, pain will hinder muscle development and strengthening. If your horse’s pain and comfort levels are not being adequately managed, your training will be an uphill battle. Training is hard enough as it is. Monitor your horse’s comfort and improve your awareness of their indicators of fatigue. LASER: is a fantastic tool for addressing painful and inflamed sites. This means muscle knots resulting from fatigue can be addressed without too much manual force. This is an advantage as chronic diseases can heighten the pain felt in the body. Horses with increased pain sensation rarely enjoy manual pressure and handling, therefore using a LASER to very effectively treat pain on a biochemical level, which can then improve the sensation and allow manual handling afterwards makes it a powerful tool for these cases. Manual therapy: Massage, myofascial release and stretching offer release of the tight muscles developing from compensation, plus an element of strengthening and improved suppleness when the time is right. Increased blood flow, soft manipulation of tight muscles, movement of lymphatic fluid and gentle, physical treatment of muscle knots and tension all benefit the heightened state of discomfort the horse will be feeling. Target 2: The joint. Reducing further inflammation and maintaining the condition of the joint remaining is crucial. This is where the Vet’s involvement and direction is crucial. However, there are some powerful modalities a Veterinary Physiotherapist can offer to help your horse LASER has been shown to increase the rate of production by chondrocytes: meaning increased cartilage production. This makes this an excellent choice to use at joint articulation sites, as it may help to power the chondrocytes that are remaining at the damaged joints. CRYOTHERAPY: Ice applied to an inflamed joint after work reduces heat and the further damage that can cause. Ice further acts as a local mild painkiller after a certain period. It should not be applied for more than 20 minutes duration after work. Arthritic joints generally don’t like moving or being moved; so applying ice prior to work is not ideal, for pre-work preparation, HEAT therapy is more beneficial to these cases. Heat will increase the blood flow and lymphatic fluid movement to the joint, increasing its comfort and functionality in preparation for work. Target 3: Maintaining and improving function. Treating to maintain and improve function will increase the strength and condition of muscles, which should hopefully reduce the severity of acute flare-ups. Massage, stretching, myofascial release accompanied by a targeted remedial exercise program of controlled exercises will then allow strengthening of the body. As mentioned previously, pain hinders correct development and strengthening of muscles... improving muscle condition will improve comfort. H wave is a powerful muscle stimulation tool which can create muscle contractions without fatigue. Increased blood flow, contraction and relaxation of whole muscles and lymphatic fluid movement vastly improve the biochemical concentrations of sore muscles. In addition, a direct reduction in pain sensation has also been recorded. Pain causes compensation which causes faulty muscle contraction sequences. Forcibly activating contraction in targeted muscles gives your Physiotherapist a powerful tool to re-train muscles which have changed their sequence, as well as bringing the horse’s attention to those muscles being treated. H-wave is also useful in neurological cases, and while arthritic horses hopefully are not suffering neurological symptoms, bringing their attention to the contraction of what are likely to be poorly used or dysfunctional muscles, could help them to improve the neurological pathways and therefore retrain the movement patterns… reducing the extensive compensations. LASER can be applied similarly to target pain: muscle fatigue, knots, tension and pain will all greatly benefit this treatment. Stretches target increased (or maintained) Range of Motion of joints, increase flexibility and the strengthening of muscles. These should be prescribed and performed on a case-by-case basis. All the science aside, two of the most important approaches to managing arthritis and or injuries that I have learned over several experiences: Patience and confidence. YOUR HORSE WILL NOT FORGET THEIR TRAINING JUST BECAUSE YOU GIVE THEM AN EASY DAY WHEN THEY FEEL STIFF OR SORE. Give them the benefit of the doubt. Your training is good enough, your horse remembers enough, you will get more out of training on the days they feel good if you let them recover effectively on the days they don’t. Thanks for reading! there‘s so much more that can be written about arthritis, hopefully this fits as an introduction. Let us know your experiences in managing arthritis! Take care! Genevieve Xx Apologies for the lack of reference list; uploading this from my iPad means I cant make the reference list yet, however it will be updated by the end of the weekend!
Bone is made of a mix of inorganic minerals and organic material including collagen (mainly type 1), which creates a highly responsive and adaptable compound (1). For those interested, the mineral is a hydroxyapatite component, made of Calcium and Phosphorus, and it is laid down into the collagen (1). For those uninterested, suffice to say Calcium and Phosphorus provide the major mineral/non-living contents. Bones provide the valuable levers for the locomotor system in all vertebrate animals. It is also a crucial reservoir of minerals, providing a buffering system for the body in times of growth/repair/response (1, 4). Bone tissue consists of cortical bone (a.k.a compact) and Trabecular bone (a.k.a cancellous). https://www.youtube.com/watch?v=Ei4seya3dOg -> This video, (up to the 2-minute mark) offers a clear explanation of the structure of long bones in the body. Bone is essentially a tube of cortical bone with bone marrow (+trabecular bone) on the inside, covered by growth-plates, more bone and cartilage, enclosing each end. The surface of bones is covered by a tissue called periosteum... this is what connects/is continuous with tendon tissue and muscles. Consider this: If you roll up a piece of paper in to the shape of a cylinder and stand it up next to a folded cube of paper... the cylinder will be a lot harder to crush than the square, which is one of the many beauties of the structure of ours, and our horses' bones. Crucial built-in safety margins have evolved in the structure of bones: weight of the distal limb is minimised to reduce momentum and therefore excess force applied to the limb during movement, even the alignment of the bone cells intentionally offers effective support of weight and force (1). Bones are subject to a constant balance of RESORPTION and DEPOSITION, performed by osteoclasts (resorption) and osteoblasts (building/deposition). As a mineral bank, sometimes resorption occurs at a greater rate than deposition, which frees required minerals to the body (3). Alternatively, increased mineral deposition occurs in order to strengthen the bones under stress or to adjust for a surplus and maintain mineral homeostasis (3). Bone is a piezoelectric material, which means when physical stresses are applied, it responds and signals for increased deposition to occur in order to maintain safety margins and full function (4). The response of bone can be short-term or long-term, making it highly adaptive (1). Short-term response is activated by mechanical pressure, which despite not being completely understood could be likened to the muscle “swell” that comes with a workout. Despite these safety mechanisms, a level of damage is expected to occur, and surprisingly is necessary to increase bone density (2). Training and improvement generally occurs outside of a comfort zone… How far outside that comfort zone you can go depends on the following, and even then there are physiological limits: a) your horse’s conditioned fitness (are they in full work or box rest?) b) your horse’s technical ability/training knowledge c) their natural strength/predispositions (referring to conformation etc.). Big injuries occur outside of that small range outside of that comfort zone. In the instance of bone, too many microfractures as a result of a sharp increase in workload or work on a surface that was too hard can overwhelm the precious damage-repair cycle of bone, and could result in a major fracture instead (1). (5): https://europepmc.org/article/PMC/3743123 In short this picture demonstrates the comfort zone, which sort of sits in the “physiological window”. This shows the increased mechanical loading which eventually results in overload and failure. Expected/common micro-damage is caused by repetitive, cyclical loading (as found in repetitive loading of the limb during movement) (1). Damaged portions of bone will be targeted by osteoclasts and will be resorbed into the matrix; to be followed by osteoblasts laying down the new bone in the same arrangement as the osteons found in fresh/undamaged bone (1). It takes a few weeks for extensive mineralisation to occur to regain strength and density to the affected area (1). If none of my words made any sense, they are essentially summarised by this picture: picture A = bone that has adapted to exercise conditioning, B = bone that has not had exercise applied. In summary: bone is highly responsive, it can increase density under increased load, or stay the same and even reduce under reduced load. Too much load and it may be unable to maintain function/structural integrity and will break. picture credit: (1) Now consider this: The cardiovascular system undergoes a significant loss of fitness after 4-6 weeks of rest (2). Bone density takes 12 weeks of reduced rest to result in significant loss (2). Severe injuries commonly require a lengthy period of box-rest/hand-walking (though obviously varying lengths between type and severity of injury); however, bone density is rarely mentioned as a target of the rehabilitation process (personal experience as well as research)… The old english methods of “legging up” horses by trotting them on hard ground is a great way to stimulate osteoblast activity and increase bone density. With a huge word of CAUTION that this should not be done for more than a couple of minutes at a time, a couple of times per week. Otherwise you risk tipping the delicate balance against your favour and causing increased damage. QUESTION: Have you ever considered bone remodelling/density/strength in your training and/or rehabilitation program? Do you think you should? Bone remodelling can occur not only in the length of the bone, but also in the tuberosities at the ends of bones, which can and do change with pressures. The difference with this remodelling is that the pressures from other bones come in to play, as well as the behaviour of cartilage. As such the directional behaviour of the joint can become affected or can affect the new bone remodelling. This plays a role with osteoarthritis, which is a huge topic on its own and should be gracing your screens in 2 weeks’ time ;) AS always, comments and ideas are welcome! Thanks for reading! References: A. E. Goodship and R. K. Smith, "Skeletal Physiology: responses to exercise and training, Role of SKeleton," in Musculoskeletal System, Elsevier, 2004, p. 81. A. J. Kaneps, "Practical Rehabilitation and Physical Therapy for the General Equine Practitioner," Veterinary Clinics of North America: Equine Practice, vol. 32, pp. 167-180, 2016. P. Katsimbri, "The biology of normal bone remodelling," European Journal of Cancer Care, vol. 26, no. 6, 2017. P. Yu, C. Ning, Y. Zhang, G. Tan, Z. Lin, S. Liu, X. Wang, H. Yang, K. Li, X. Yi, Y. Zhu and C. Mao, "Bone-Inspired Spatially Specific Piezoelectricity Induces Bone Regenration," Theranostics, vol. 7, no. 13, pp. 3387-3397, 2017. A. Robling and C. Turner, "Mechanical signaling for bone modeling and remodeling," Critical Reviews in Eukaryotic Gene Expression, vol. 19, no. 4, pp. 319-338, 2009.
- Scratching the surface of tendon injury and rehabilitation
In early 2017 I made an ambitious 12-month competition plan with Pokemon for that year. The goal was Under 25 Grand Prix for my last year in the category, with (what would obviously be…🤔) a seamless move into the Senior classes as our even more ambitious goal… 🤣 The #SilverPrince in full glory at Keysoe Premier League, 2017 In late 2017 he tore his Gastrocnemius tendon (equivalent to the equine Achilles' tendon, crucial for weight bearing, not ideal to damage!). He was treated with a PRP injection to the affected area and then prescribed a 12-month rehab plan, and a disclaimer of sorts from the Vet that the odds of him competing higher than Medium level again were heavily stacked against him. I tore up my plan and indulged a big cry. It wasn’t the first time I had faced disappointment, but it was the closest I had ever been to achieving the magical Inter 2/Grand Prix transition, and Pokemon is an incredibly special soul to me. No matter what level you’re working towards as a rider, the setback of injury, no matter how severe, is a heavy blow to deal with. Muscles of the horse, demonstrating location of the Gastrocnemius muscle. (http://infovets.com/books/equine/A/A028.htm ) Firstly, Structure of the tendon: A tendon is predominantly made up of one of the universe's most incredible materials: Collagen. There are several types of collagen known; type I is associated with tendons and ligaments. Collagen molecules arrange in to bands/tubes called microfibrils, which arrange in to fibrils, then fibres then fascicles, as shown in the image below. These fibres are arranged in a parallel pattern, which is most energy efficient and the strongest formation applicable. The collagen bonds in these fibrils create a molecular-level crimp pattern which is suggested to play a role in the stretch available in tendons (7). Structure of a tendon (https://beva.onlinelibrary.wiley.com/doi/10.1111/evj.13331) Unlike what many simplified diagrams lead people to believe: tendon material is continuous with muscles and bones. This makes it a much stronger connection between bones and muscles, which is obviously a benefit considering the force that is transferred through these tissues. There are very few cells and no blood supply found in tendons (1); which is one of the reasons their injury and healing process is a long one (6). Blood supply offers drainage and delivery of helpful inflammatory products (2). Interestingly, the particularly avascular zones of tendons are prone to greater risk of damage and/or injury (2). Functions of the tendon: 1. Tendons consist of connective tissue between muscles and bones (4). This provides support and stability to joints and a connection for the movement produced by the muscle contraction. 2. The tendon behaves like an elastic band and stores energy, which when released provides movement at a highly energy-efficient rate. The repetitive stretch and recoil of the tendons is created by muscular contraction resulting in tension (4). 3. Tendons are able to be stretched extensively and will return to original form when released; they are capable of withstanding a large tensile force. This property does diminish over time, and under certain conditions (such as heat and excess tension). Injury to the tendon: Injury to a tendon can be referred to by different titles, depending on the type of damage. These titles are often used interchangeably in wrong ways, clarity on them may help understand the injuries. 1. Tendinitis = inflammation of the tendon 2. Tendinosis = tiny tears of the tendon including degeneration of collagen in the tendon a. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3312643/ this article explains the differences. 3. Tendinopathy = umbrella for non-rupture damage to a tendon (damage to the collagen in the tendon) https://www.jospt.org/doi/full/10.2519/jospt.2015.5884 Tendon injury is one of the most common reasons for decreases in performance and even retirement from work/competition careers (6). Not only does the initial injury cut into competition and training plans, but the risk of re-injury is incredibly high (between 48-58% and up to 80% depending on the source (6)). Re-injury is high due to a weakened tendon which in itself is predisposed to further damage as well as the compensation that results from protecting it. Tendon injury generally results from overload of some kind. This can either be from a repetitive loading cycle where not enough recovery time is accounted for, or from an acute/sudden overload injury which stretches the tissues too far in a motion. This overload is even possible from muscles that are stronger and contract harder than what the integral strength of the tendon can manage. (check out https://www.germanjournalsportsmedicine.com/archiv/archiv-2019/issue-4/functional-adaptation-of-connective-tissue-by-training/ for further explanation of this). Tendons have a low rate of cellular turnover and regeneration which means that healing occurs by scar tissue formation, NOT tendon regeneration (4), which is how bones repair. This is a problem because scar tissue has a far more random structure of collagen compared to tendons, which greatly reduces its ability to withstand tension. This often means that the limb with damage will always be weaker and that the horse is likely to compensate and offload from it. In contrast to tendons, scar tissue is initially a highly vascularised tissue (necessary for the healing process) and these blood vessels have been questioned to play a role in the cause/effect of the poor collagen fibril alignment that occurs in tendon injury repair (4, 5). Rehabilitation of a tendon injury: Depending on the level of damage, tendon injury can take 3 months or up to/over 18 months to “heal” (6). Because of the nature of scar tissue, a tendon will never return to its former strength, because the biological processes are unable to regenerate the perfection that is the intact tendon, therefore it is hard to say this actually “heals”. This means that it is almost always going to be something that needs to be monitored and managed with workload and treatments, and will always be a weakness in the horse. Box-rest, controlled re-introduction of exercise and regular check-ups to assess healing of the site are crucial for optimal outcome. Muscle strength and bone density are lost throughout the lengthy rehabilitation period and are part of the reason for controlled exercise. This should reflect back on the balance that is necessary between muscle - tendon - bone strength. Box rest is a vital stage of rehabilitation in a variety of cases. This gives the injury site time to mobilise inflammation factors and reduce movement there. A fresh tendon injury means a portion of the tendon is unable to bear weight/tension properly, which means that the other collagen fibres of that tendon will be compensating. Movement of the tendon increases strain on the remaining fibres and is also extremely painful, so limiting this in the early acute stages of damage is important. The fact that a very long rehabilitation process is required to get an injured tendon to a less-than-perfect state makes it a difficult diagnosis to cope with. The remodelling capacity of bones is far greater, and therefore usually (excluding extreme circumstances) an easier diagnosis to receive. Tendon rehabilitation takes a LOOONG time… It is a rollercoaster involving moments of great hope when they look sound; great anxiety not knowing what point the new maximum will be that you get to ride/compete at; the disappointment during the 3 steps forward, 2 steps back process that seems intertwined with rehabilitation… Like when you start trotting after months of walking and they feel more broken than when you first received diagnosis? We know that feeling all too well. I tried to start a running program at the same time as Pokemon’s rehabilitation program. I had full-time University as well as two other horses to ride, so it was an inconsistent program at best, but I definitely developed an appreciation and sympathy for Pokemon on the days I felt sore and less able to go as far or as comfortably. Surely they experience the same! Rehabilitation is a prime example of a marathon, not a sprint. Pokemon enjoying a light "wellness" massage during his clinic visit. The good news is that there is science and technology that is ever improving for your horse’s benefit... Physiotherapy being one of them! Shock-wave therapy, ultrasound therapy, LASER therapy, H-wave, Stretching, massage, controlled strengthening exercises… a variety of options are available to maintain comfort, muscle mass, joint health as well as tendon and ligament function. The rehabilitation road is a long and exhausting one; but it can be fascinating and important. Every time I have had to rehabilitate one of my horses, I have come out the other side with a better execution of their basics of Dressage, and a greater appreciation of what the good riding days become. Good luck with your rehab, I haven’t met you but I have my fingers crossed for you! Let me know if I can help! 😉 Did you know: The half-life (length of time it would take for half of the material to disintegrate) of collagen in mature equine tendons is estimated to be around 200 years?! (3) Some really interesting links for those interested. "Inside Nature's Giants: The Racehorse" is an impressively informative video, however DISCLAIMER it is not for the faint-hearted/weak-stomached! A limb cadaver is used to demonstrate the pressure a tendon can withstand. The tendon info starts around the 15:50 mark, but I can thoroughly recommend the whole lot for anyone who wants to know more about how incredible horses are from a physiology perspective. https://www.youtube.com/watch?v=feAj2aspkIE&list=PLP68rPoaxoUlnliNc2uj38tRPFnmAWBkH Description published by the British Equine Veterinary Association on Microdamage in tendons: https://beva.onlinelibrary.wiley.com/doi/10.1111/evj.13331 Information on the tendon vs. muscle strength training: https://www.germanjournalsportsmedicine.com/archiv/archiv-2019/issue-4/functional-adaptation-of-connective-tissue-by-training/ and the links I used and quoted throughout this: 1: https://pubmed.ncbi.nlm.nih.gov/17450305/ 2: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4650849/ Tempfer 3: Aspartic acid recematization and collagen degradation markers reveal an accumulation of damage in tendon collagen that is enhanced with ageing. Journal of Biological Chemistry. Thorpe et al. 2010. 4: Comparative study of the characteristics and properties of tendinocytes derived from three tendons in the equine forelimb. Journal of Tissue and Cell. 2009. Hosaka et al. 5: . Presence of lymphatics in a rat tendon lesion model. Histochemical Cell biology. 2015. Tempfer et al. 6: Evaluation of Return Rates to Races in Racehorses After Tendon Injuries: Lesion-Related Parameters Journal of Equine Veterinary Science. 2020. Kan Gulsum et al. 7: Mechanical and functional properties of the equine superficial digital flexor tendon. The Veterinary Journal. 2005. B.A. Dowling, A.J. Dart *
- Step 1. Balance.
The difference between the average test, and the elite test, at least aesthetically (and in my humble opinion), is down to balance. Some horses find this concept easier; some humans find it easier; in the same way some people are clumsy by nature and others are elegant. This does not stop someone from being able to achieve balance on a horse, it can just make it a damn sight easier/harder depending on the natural predisposition of the horse and/or rider. The conformation that has been passionately discussed in the last 2 blogs ( CONFORMATION-The good, the bad, the ugly and No Foot, No Horse... How? ), plays a key role in the moving function of the horse, and therefore in its balance with and without the rider. The biomechanics and art form that this creates is where the everlasting training and improvement that Dressage offers is formed… This hopefully begins to form the bridge, and connection between the Veterinary Physiotherapy and performance aspects, and the horse training considerations horsemen and women require. First, a history lesson. The German training scale was developed in the early 1900s, and is considered the fundamentals for training horses and what riders are judged on at competitions in the present day. The first 3-4 blocs are relatively interchangeable and inter-reliant, the latter 2-3 require the first blocks to be rock solid to be successfully achieved. Balance is only the first step of the German training scale, and yet it is key to every stride, every movement, every training session, every competition, and to every other step in the training scale 1. Rhythm/balance 1. Relaxation 2. Contact/Connection 3. Impulsion 4. Straightness 5. Collection A naturally well built, untrained 3-year-old is likely to have a 60:40 split of weight forehand:hindend, and is probably dominant on its left or right side (out of interest: this can often be spotted by the larger, flatter hoof, and is the hoof that is placed further forwards when grazing since a foal!). This horse also has no muscles or training to effectively carry a rider and is likely to motorbike around corners to compensate for this. In short, this horse struggles to balance. Dressage principles train horses following the German training scale, with the ultimate goal/dream of mastering it, which is what the Grand Prix test entails. OBVIOUSLY, not all horses were created equal, and it would be borderline cruel to attempt this path with some horses. However; as hammered home in conformation blogs, the build of a horse can offer a helping hand towards achieving the balance and eventual collection required. Good conformation offers an even spread of pressure over joints, bones, muscles and tendons around the body. It also offers greater ease of balance and athletic ability. Balance has an effect in a horse standing still, as well as one in motion; an important concept to consider when relating conformation to movement… Imagine you are being pushed by someone; they’re trying to knock you over. If you are steadfast on your feet, can maintain your balance and counter the force applied, you won’t be knocked over. However, if you don’t have the right posture or timing to counter their force, likelihood is you’ll kiss the dirt. This could be true if you are standing still, but it is also true when moving. - Your horse will counter your demands and/or respond to them, as this example demonstrates. How fairly and effectively you ask your aids and how it effects them depends heavily on use of the half-halt and your position. Recently I have been training and guiding Feather’s posture to be more uphill, with her nose more at the vertical and poll at the highest point. This is a pretty classic goal for us lot attempting to Dressage. Because of her lack of strength and fitness from lockdown 1.0, corners are actually an incredibly difficult lesson for us to master at the moment. This sounds ridiculous when she can train piaffe and passage, but there is no point in me pushing those movements if in every corner I feel her barrelling over her inside shoulder, burying her nose down and trying to shoulder punch her way down to Australia. No judge on the planet will be impressed with these moves sadly. I cringe at this picture, but in efforts to educate, here we are. Problems: overflexion of the neck, and a neck thats too low. Falling over the inside shoulder as a result, instead of sitting back on the hindlegs and navigating around the corner in a balanced way. BALANCE has been the key issue, as it usually is, and I realised quite how badly I was interfering with it yesterday. Through each corner, I have been doing my best to use my inside leg to get her in to my outside rein, and encouraging her to keep her head and neck UP, to keep the energy flow UP, through each corner, and not to dive inside, especially on the left rein. This meant my left hand was making a lot of small adjustments. I had one moment, I don’t know what brought it on other than I was thinking to myself: “nothing is changing, and the definition of insanity is doing the same thing over and over again and expecting a different response” Something wasn’t adding up. Feather is smart and well-trained, I know how to ride a corner, she knows how to half-halt… WHY are we barrelling through each corner like a 3-year-old?! I brought it all back to square one. We walked through a corner, I centered my weight over each seat bone and in to each elbow/hand pressure on the reins, and re-imagined the pillars that I like to think attach from her front hooves to my hands, and imagined these being upright, NOT leaning tower of Pisa. Low and behold, ,miraculously, instead of adjusting my left hand towards the middle of her neck as I had been, but keeping it in line with this pillar, taking my reins a bit shorter and making a straighter shape through her neck through these corners… we got better BODY bend (not just an over-flexed, over-bent neck), lift, balance, cadence and posture than I had achieved in the last 3 rides. I rode the damn corner and Feather could make it through it without toppling over herself. a MUCH better example of the balance for a corner... nose at the vertical, upright posture in the rider's position AND the horse's, and no goofy over-bending nor over flexion. (Pic from 2019) YAY for small wins. DOH for forgetting the basics. I had forgotten to balance her. I had bend, I had forwards impulsion and I had her reactive from my aids, I just wasn’t setting her up in the right POSTURE and didn’t have the correct BALANCE. Corners are so telling of training. “Tests are fought and won in corners” is another mantra I have learnt over my years of absorbing all horsey knowledge I can get my ears on, and this was a very telling moment of that. Do not be disheartened, this mare is an international small tour horse and we STILL train corners. It counts a lot to be able to take a step back and consider why something isn’t improving though… chances are there’s a flaw in your basics and you need to reconsider your approach. YOU DO NOT NEED TO BEAT YOURSELF UP AS I WOULD HAVE DONE YEARS AGO. It doesn’t have to be a big deal, it doesn’t have to be difficult… But it can get you a heck of a lot further when you realise the issue! Top tip: 1. If something isn’t changing, aka if your horse isn’t responding: Stop. Take a minute. Break down what you’re trying to fix, and how you are trying to fix it. You may not know how to ride a Grand Prix, but if you break your problem down, and consider the German training scale, or any Dressage basics, chances are logic could deliver you to your solution anyway… And if it doesn’t, find a good trainer who can!
- No foot, no horse... how?
To preface this blog, farriery and horse hooves are something that I find extremely interesting, however, also very perplexing. This is very much a “starter” article for this topic, I have done a lot of research but there are always different opinions and concepts around it (some to follow, some to ignore!). I am not a farrier, but I have observed a lot of work, a lot of hooves and my dissertation covered hoof shape analysis compared to saddle positions… so I have done some research! Before we start, the following picture offers a brief recap of the limb’s anatomy. (Distal phalanx = P3 or coffin bone): “No foot, no horse” I first heard echoed over 10 years ago… My trainer was discussing the shape of her horse’s hooves, and at the time I had truly minimal appreciation of the hoof’s anatomy/function and even less idea as to what shape they should be. I thought this horse’s hooves looked just fine, but I was aware that “just fine” was probably not quite enough for a Grand Prix horse! A very perplexed me took this little mantra on as gospel, with hopes to one day actually understand the message. The last few weeks I have felt the difference that X rays and remedial farriery can have on your horse’s performance… and this little mantra took on a life of its own. Yesterday I posted a video of my mare who has had a summer of a sore back due to no riding during COVID. I decided to go back to basics because nothing really seemed to be working (Physio, veterinary treatments etc), and that niggling mantra “no foot, no horse” was ringing in my head. Gratefully, it seems to be working as she is starting to do harder work, with greater wase, developing muscle that she lost throughout the year and her body is recovering faster and faster from the increased workload. These are all great indicators of a horse that is comfortable. If there is anything that my dissertation explained to me, it is that the hoof, and its interaction with the distal limb is a phenomenal feat of engineering. This interface is responsible for communicating between the muscles, tendons and bones of the horse’s legs which are ever moving, and the hard, immovable exterior it collides with 24/7. The hoof capsule “holds” the skeleton by Laminae. Laminae exist as thousands upon thousands of interlocking folds of tissue, which connect the coffin bone to the hoof capsule. These folds, and the hoof wall, stretch and absorb energy as the hoof capsule hits the ground with each stride. As much as it is powerful, this structure is sensitive; which is why diseases such as laminitis (inflammation of these folds) and problems like abscesses cause so much pain. Finally, the weight of your horse is actually held almost suspended by these folds of tissue, which allow their weight to be transferred and actually supported by their fingernails/hoof wall. Furthermore, this wall is a living and growing structure which can and does adapt to the different pressures applied to it (which is one of the reasons why hooves change shape over a lifespan). A hoof’s heel and toe angles should be parallel and should match the hoof pastern axis, which should be approximately 45-55° (Lesniak, et al., 2017) (Clements, et al., 2019!). This angle in theory represents optimal bone, tendon, ligament and joint alignment for function and comfort. CAUTION is warned: Just because the limb looks normal from the outside, does not mean serious deviations from this axis aren’t occurring… This is where X-rays are extremely useful! A broken back HPA is associated with a lower heel, longer toe and generally coincides with a lower coffin bone angle. This broken back axis and lower coffin bone angle has been increasingly associated with greater risk of injury (Dyson, et al., 2011). An interesting description for this which I heard recently is this angle causes the tendons/limb to be “pre-pushed” … Imagine standing on the ground and lifting your weight on to your toes for several reps (flexing your calves). It is tiring, but easily manageable. Now do it with your toes on an elevated surface (for example, the edge of a step) … This becomes much harder. The step mimics a pre-pushed posture and mimics the broken back axis. This increases the distance that your muscles need to work over, to pull your heels up to the same point. Fatigue increases with this broken back posture, and again, flashback to the first blog posts, fatigue is an effective precursor to injury. MANY sources have found different statistics and observations which highlight just how negative the effects of a broken-back HPA can have. A few of these are mentioned… - Function and condition of the Deep digital flexor tendon (DDFT) and the navicular bone (NB) are dependent on heel angle. Low heels increase the distance over which the DDFT has to pass to reach its insertion on the coffin bone. This means greater strain on the tendon, and greater pressure on the structures underneath it, such as the navicular bone. (Eliashar, et al., 2004) (Dyson, et al., 2011) (Clements, et al., 2019!) - Coffin bone angles: o Optimal = 5° (wedge from the ground) (Clements, et al., 2019!) o Raised 1° ((2) on image) = 4% DECREASE of peak pressure on NB (Eliashar, et al., 2004) (Dyson, et al., 2011) o Dropped 1° ((3) on image) = 20% INCREASE of peak pressure through the first moments of hoof contact with the ground. (Lesniak, et al., 2017) These are related to the heel angle, and one of the reasons low, collapsed heels are not desired! - Length of toe: o 1cm excess of toe, adds about 50kg strain to the tendons (Murray, 2020) This causes additional pressure at breakover. Increased length of toe increases the distance the tendons and muscles must contract and travel over in order to rotate and lift the limb (kind of like the pre-pushed posture discussed earlier!). Again, excess strain increases risk of injury. - The angle of the coronet band of the hindlimbs should draw a line directly to the back of the knee of the forelimb. This should indicate correct conformation! These are all factors somewhat out of control of a veterinary physiotherapist. So, what is the relevance? Firstly, all of the above should indicate the hoof shape can cause internal tendon and joint problems, and these certainly are applicable to a VP. These problems are likely to cause compensations and secondary problems. These CAN be addressed by Veterinary Physiotherapy and complementary services. First and foremost, Veterinary diagnostics (especially X-rays, and ultrasound if necessary), and farrier intervention will be necessary to address fixing the conformation, if there are problems. Secondly, pain and compensation require rehabilitation and remedial exercises. This is because your horse will stop moving their bodies in an efficient and correct manner, in order to offload the painful pressures. Rehabilitation will help limit more damage to these tissues, while maintaining form and function. Your Vet Physio can help make your horse more comfortable both during and after the appropriate farrier and veterinary interventions have been made. Ice therapy, heat therapy, manual therapies and some electrotherapies could absolutely benefit your horse! A Veterinary Physiotherapist should coordinate treatment alongside your veterinarian and farrier, so if you have any queries about what we can do to help, what you should be expecting at any stage, or any questions about anything mentioned, please do get in contact! References Clements, P., Handel, I., McKane, S. & Coomer, R., 2019!. An investigation into the association between plantar distal phalanx angle and hindlimb lameness in a UK population of horses. Equine Veterinary Education, Volume NEED VOLUME AND ISSUE, pp. 1-8. Dyson, S. et al., 2011. An investigation og the relationships between angles and shapes of the hoof capsule and the distal phalanx. Equine Veterinary Journal, 43(3), pp. 295-301. Eliashar, E., McGuigan, M. & Wilson, A., 2004. Relationship of foot conformation and force applied to the navicular bone of sound horses at the trot. Equine Veterinary Journal, 36(5), pp. 431-435. Lesniak, K., Willians, J., Kuznik, K. & Douglas, P., 2017. Does a 4-6 week shoeing interval promote optimal foot balance in the Working Equine?. Animals, 7(29), pp. 1-14. Murray, D. R., 2020. Risk factors for injury in sport horses. Newmarket, VetFest 2020.
- CONFORMATION - the good, the bad, the ugly.
Good conformation is fundamentally what is anecdotally or scientifically regarded as the most efficient spread of: - Weight - Force - Energy transfer In your horse. On the whole, the better symmetry within/between/around the horse and their legs, the less chance of injury. Less good for my job, very good for your horse 😉 In theory we want a horse that divides evenly in to 1/3rds, as this indicates an even distribution of weight carrying and weight pushing capacity, good straight joints when looking front on, but well angled shoulders, stifles and hocks when looking from the side. A well-proportioned neck and head complete the picture. In short, achieving correct proportions and conformation indicates correct and balanced distribution of weight, forces applied and movement through the horse. Keeping this in balance reduces the injury risk. This means that there is no obvious asymmetry BETWEEN, or WITHIN limbs and landmarks of the body. Balance front to back: A horse naturally carries at least 60% of its weight on its forehands, and 40% through its hindquarters. One of the aims of Dressage is to encourage greater active weight carriage on the hindlimbs in order to encourage better balance, particularly for the harder movements associated with Grand Prix dressage, as well as an improved expression in the movement. In my limited experience, show jumpers also seem to require a light and nimble balance, in order to get that horse to take off over those ridiculous obstacles 😉 The forelimbs are essentially struts to carry weight, (imagine they function like a pogo stick does – bouncing again and again when you put the power through your body). The head and neck behave as a ballast which can accentuate and aid movement (this is where you start throwing your body forward on the pogo stick to move around) and energy created by the engine: the hindlimbs (your legs which act to create energy and movement in the pogo stick). A match in power versus carrying capacity must be achievable (alongside training designed to aid this), in order for the horse to develop correctly, with good muscles. A horse that naturally achieves this will naturally find training easier compared to those who don’t. The angle of the shoulder indicates the ease with which the forelimbs can be protracted (pulled forwards). The more sloped this angle is, the better protraction power that limb is expected to have (which means better toe flick for your dressage test, or height for your jump 😉) Excessively long pasterns: are increasingly popular as they seem to give horses a very expressive/floaty trot. Something we all have an obsession with I imagine, however, long, thin pasterns reduce the stability with which the whole limb is balanced on… …Imagine Jenga blocks, these represent the bones that make up the pastern and lower limb: your tower gets taller and thinner as you remove blocks and it loses stability. Putting your hand at the top of them helps to stabilise the tower. That act of compressing the blocks creates stability. That same compression can be mimicked by the tendons… These can actively contract and tighten the Jenga blocks (or distal limb bones) together, creating stability. However, this means you increase the tension in tendons before you’ve even asked for increased workload. Increased tendon tension = increased risk of injury = increased risk of breakdown, tears, vet bills and “WHY DO I DO IT” (note: author’s personal experience) 😉 So: Short, wide pasterns provide a good sturdy base for the horse’s large weight, plus you, plus the movements you are demanding. Balance between limbs: Left to right balance encourages symmetry under saddle and reduced likelihood of one being overloaded (and increased risk of injury). Mechanical abnormalities can be mistaken for lameness, which is where experience and understanding biomechanics is crucial to understanding your horse (and trusting practitioners working with your horse!). Increased load on one limb increases the load and strain more than its natural capacity can handle. This can also create training problems (those days when you feel like rigor mortis has set in on one side of your horse sounding familiar?), and injuries, because increased strain does that to living tissues. Balance within limbs is just as crucial: Even spread of weight and pressure thanks to a symmetrical joint means balanced strain on the tendons and ligaments surrounding that joint capsule, and reduced risk of damage and injury to all these structures. If however the joint between two bones is asymmetrical, that difference can create increased compression on one side of the joint, and reduced pressure on the other. This results in greater risk of arthritis and greater strain on tendons, because they can be stretched asymmetrically and have to work harder to compensate for this pattern. Science buffs: An experiment was designed to test this – wedges were placed at the side of hooves (to force asymmetry on joints), and joint mechanics were recorded. Reduced joint flexion was observed as a result: Meaning that tendons and ligaments intervened and actively compensated for the new force being applied. This was to reduce the asymmetrical force applied to the limb, and reduce the abnormal movement which would result, and which would increase risk of injury and pain. This increased tendon strain however… that was likely to damage it long term. Many studies have tested different elements of this to determine similar conclusions: tendons will actively compensate for asymmetry and asymmetrical forces applied, and this definitely exacerbates their risk of damage and deterioration. Straight hocks and stifles: indicate poorer shock absorption, and poorer movement efficiency generated by the hindlimbs. With the amount of power generated in the hindlimb, poor angles (aka straight limbs) means less efficient and effective shock absorption, and a higher likelihood of arthritis. Higher chances of arthritis in the hocks indicates increased pressure on the backs, particular the lumbar back region, and can cause secondary problems with stifles (as mentioned previously with either over-active or underactive quadriceps muscles) and eventually hindlimb tendon pathologies/problems. In summary: Not good. Long backs often create a lovely softer ride, and these horses seem to develop a lovely rhythm (albeit a slow one commonly). However this long back can mean increased strain on the stifles, as they compensate for an unnaturally long wheel base, and the horse finds it harder to push underneath and carry weight behind. Stifle inflammation is likely to occur as a result of overworked quadriceps muscles, or inefficient function of Quadriceps. (Pokemon is a prime example of quite a long back, and I was forever teaching him to sit and weight, strengthening his back and encouraging him to lift his front end.) The lumbar back region is the point of transfer of power from the hind end, along the back to propel the forelimbs forwards. If this is excessively long, instability can occur, resulting in pain and compensations. Furthermore, this part of the back is not supported by a rib cage, and is therefore responsible for supporting a portion of the abdominal cavity “unaided” so to speak. Increased weight and/or increased back length will make this function far more difficult. It can be hard to spot asymmetries, particularly if you’re not trained to do so. Horse hooves can highlight red flags both in terms of confirmation, and movement. Next weeks post will be heavily focussed on hooves. As a precursor, “no hoof no horse” is a common mantra in the horse world, and yet I have only in the last week truly understood and experienced quite how much a difference that can make.
LASER works by using light energy of the visible to near infra-red range to increase the activity in the engine of the cell: the MITOCHONDRIA. These are a ‘sub cell’ of sorts, known as organelles, and are responsible for respiration. Respiration produces the energy for our bodies, in the form of ATP. When an injury is present, a hypoxic environment is created, meaning reduced oxygen is available. This prevents mitochondria from doing their job efficiently and reduces the supply of ATP. LASER provides benefits of: - Pain relief - Reduction of swelling - Blood vessel growth - Improved chemical composition of that area. Within the mitochondria are molecules known as “Chromophores”, these are “photon-acceptors”. A “Photon acceptor” is basically a molecule capable of accepting energy from the LASER. Cytochrome C-Oxidase is and enzyme and the final photon acceptor in the chain of respiration in the Mitochondria and is theorised to be one of the main action sites of the LASER. The absorption spectrum of COX, meaning the wavelengths it is affected by ranges from 500-1110nm; which is important to note with regards to the wavelengths of the LASER/light therapy machine used for treatment (different wavelengths for different purposes). COX is theoretically able to produce Nitric Oxide (NO) from Nitrite… Increasing COX activity through LASER application could increase the release of NO into the body. NO has therapeutic benefits of increasing blood vessel size (vasodilation), which will help revive and re-oxygenate the poorly oxygenated tissues that have suffered from injuries (as previously mentioned). Increasing this activity of COX also increases the ATP production and general mitochondrial activity, which increases the cell’s ability to function. Increasing the cell's function will increase the activity of that area and tissue, and enhance the repair process of the injured area. Combined, this effect increases the delivery of fresh nutrients and oxygenated blood, increases the ATP/energy available to the injured tissue, and results in pain reduction, blood vessel growth, wound healing, reduction in swelling and reduction in pro-inflammatory chemicals… All very good benefits to the healing of the body. The benefits of LASER make it suitable to treat a range of problems: - muscle knots - tendon and ligament problems - swelling - ischaemic tissues - muscle trauma - bone problems (Joint problems, arthritis, fractures) Interestingly, Nitric Oxide can be supplemented by some natural sources: Pumpkin seeds. Alternative, natural supplements are gaining interest, and certainly have made me broaden my approach to supplements. Pumpkin seeds (and pumpkins themselves!) are high in a range of minerals, vitamins (especially A and E, BOTH excellent for muscle function and quality), and importantly Nitric Oxide. For a bit more on this, while we work on some info about natural supplement alternatives: https://thehorse.com/113373/can-horses-eat-pumpkin/ Bibliography H. B. Cotler, R. T. Chow, M. R. Hamblin and J. Carroll, “The Use of Low Level Laser Therapy for Musculoskeletel Pain,” Orthopeadics and Rheumatology, vol. 2, no. 5, p. 00068, 2015. M. Flaherty, “Rehabilitation therapy in Perioperative Pain Management,” Veterinary Clinics of North America: Small Animal Practice, vol. 49, pp. 1143-1156, 2019. X. Fu, J. Dong, S. Wang, M. Yan and M. Yao, “Advances in the treatment of traumatic scars with laser, intense pulsed light, radiofrequency, and ultrasound,” Burns and Trauma, vol. 7, no. 1, pp. 1-7, 2019. L. Hochman, “Photobiomodulation Therapy in Veterinary Medicine: A Review,” Topics in Companion Animal Medicine, vol. 33, pp. 83-88, 2018. K. C. Kennedy, S. A. Martinez, S. E. Martinez, R. L. Tucker and N. M. Davies, “Effects of low-level laser therapy on bone healing and signs of pain in dogs following tibial plateau leveling osteotomy,” American Journal of Veterinary Research, vol. 79, no. 8, pp. 893-904, 2018. J. L. Wardlaw, K. M. Gazzola, A. Waqgoner, E. Brinkman, J. Burt, R. Butler, J. M. Gunter and L. H. Senter, “Laser Therapy for incision healing in 9 dogs,” Veterinary Sports Medicine and Physica Rehabilitation, vol. 5, no. 349, pp. 1-24, January 2019. B. Pryor and D. L. Millis, “Therapeutic Lase in Veterinary Medicine,” Veterinary Clinics of North America: Small Animal Clinics, vol. 45, pp. 45-46, 2015. https://www.discoverymedicine.com/Robert-O-Poyton/2011/02/20/therapeutic-photobiomodulation-nitric-oxide-and-a-novel-function-of-mitochondrial-cytochrome-c-oxidase/
- What are the Electrotherapies used by Veterinary Physiotherapists for?
Electrotherapies include the machines available to Physiotherapists to target healing in more focussed ways than what our hands alone are capable of. There are several different functions of these machines. These will hopefully be a bit clearer without being too heavy. If you are looking to be weighed down by the science, the next blog is the one! Please let us know if there’s something of real interest for you, so we can include it! LASER LASER works by using light energy to increase the activity in the engine of the cell: the mitochondria. These are part of cells, and are responsible for respiration, which produces the energy used all over ours and our horses’ bodies. When an injury is present, chemicals create bonds within mitochondria, and build up, stopping these mitochondria from doing their job efficiently. LASER helps to break down these bonds and the toxic build-up, which provides beneficial results such as: - Pain relief - Reduction of swelling - Blood vessel growth - Improved chemical composition of that area. LASER is a popular choice of treatment for trigger points (muscle knots), joint pain, swelling and areas of tension that are too painful to attempt other modalities. This provides a non-invasive and well-accepted method of addressing pain in your horse, making it a vital tool. When your back is stiff, the last thing you want is for it to be hammered, consider your horse in the same way… before they teach you to consider it more severely! This also means the vets may not need to prescribe as much pain relief, meaning that your horses’ livers and kidneys will also be VERY grateful! LASER light waves are unlike a standard LED; they produce the same wavelength, do not get spread from the light chamber, and create a uniform wave pattern. This characteristic makes these machines specific, effective… and expensive! Smaller “LASERs” are available on the market for less than £200, where the “proper” conventional class 3B ones are upwards of £1500. That price difference is not purely based on the bag it comes in! If someone is treating your horse with LASER and charging as such, it may be worth having a check on what the details of the machine are. Therapeutic Ultrasound Ultrasound uses soundwaves of energy which create changes in the body’s tissues. Not dissimilar to the concept of LASER in terms of energy delivery to increase the healing power of the treated location, but a different form of energy is delivered. UNLIKE LASER, this machine uses soundwaves, a mechanical form of energy, not light waves. With the different form of energy, come different benefits, different mechanisms of function, as well as different treatment targets. Shockwave therapy works on the same principle as Ultrasound; by also using mechanical waves to effect changes in the tissues. Soundwaves create moments of compression and decompression in the tissues – imagine an accordion stretching in and out. This constant change in pressure seems to create gas bubbles in the treated tissue (cavitation), which then affect nearby cells to vibrate more (acoustic streaming). In tandem, the Ultrasound creates thermal (heating) effects, and non-thermal benefits. It is possible to “overdo” this treatment. So it is CRUCIAL that the settings applied to the injury are appropriate to the healing stage (acute/subacute/chronic… go read that first blog post ;)). The suggested amount of energy delivered per pulse is different for each of these stages, where the setting for a chronic injury could destroy the tissues in acute stages of damage. PEME/PEMF or Pulsed Electromagnetic Field Therapy There is evidence of this modality treating bone fractures, osteoporosis and pain relief, however as with a lot of the research into most of these machines, different settings and machine specifications make it extremely difficult to compare any 2 studies. The basis of function for this machine is the therapeutic benefits that magnets have been associated with. Increased blood flow, pain relief, treatment of swelling and bone problems have all been associated, to varying degrees, with this modality. This machine is particularly popular in the small animal world, as it seems to help the stiffness and pain of small animals getting up in the morning. Some large horse rugs use this technology, however there is not much scientific evidence for how effective these rugs actually are, especially when considered for the hefty price tags attached. Often an anecdotal summary of the product is found as opposed to research studies. Have you used any rugs that claim PEME benefits? (ActivoMed is a big one) What differences, if any, did you notice? I would love to know! H-Wave This is a muscle stimulation machine which is incredibly useful and effective at retraining coordination and maintaining muscle function. Pain has potent abilities to alter the contraction ability of muscles. Compensations, new or old, can start to affect a lifetime of muscle function. Imagine the days you’ve twisted your ankle and you limp for a week. Your other leg hurts and your back hurts from compensating. Imagine the pain and the dysfunction you feel when that twisted ankle hurts continually for weeks on end. Horses may go a long time before we really notice that something is not quite right for them. That means, they may have spent a long time compensating, twisting their pelvis out and clenching their back, to avoid putting full weight on the sore leg. As a result, muscles will have seized up, locked up the painful area and brace the body against using it. The muscle contraction messages could then change, and the muscle function could change as a result. Allowing altered muscle contractions to continue runs the risk of secondary problems, as the body tries to respond to altered posture and muscle because the body starts to alter and respond to the different pressures. H-wave can effectively re-stimulate the correct muscle sequence (with correct handling of it), WITHOUT fatiguing the muscle. The H-wave mimics the natural waveform that the body uses to contract a muscle; meaning it is generally well tolerated as well as effective. The other huge benefit to this machine is that it is capable of contracting the whole muscleas the electrodes are positioned on the muscle motor point, which is where all the nerve fibres enter a muscle and therefore where any contraction starts. The H-wave can also be used to target pain relief, but the wave frequency used for this setting is generally poorly tolerated by horses. The “Muscle pump” function with the lower frequency is well tolerated, and also has a pain-relieving effect (thanks to improving the muscle quality). NMES – Neuromuscular Electrical Stimulation Similar to H-wave, this is a muscle stimulator. Unlike the NMES, this has a slightly sharper wavelength to create contraction, and it DOES cause fatigue in horses. This machine does however maintain and cause increased muscle strength, stamina and treatment of muscular atrophy. With both of the muscle stimulation machines (H wave and NMES), neurologically deficit horses could benefit. These machines are able to artificially contract muscles, maintaining their function, reducing the risk of secondary problems such as contracture. Maintaining muscle function also means that fluid movement, lymph drainage, and delivery of fresh nutrients to the tissues of the muscles and surrounds is artificially maintained. This is hugely beneficial as the toxin build up and lack of drainage can cause pain, stiffness and discomfort secondary to the actual injury that is requiring treatment. Furthermore, a horse unable to work or preparing to recommence work after injury may be able to use muscles correctly from the beginning, if the treatment plan has been committed to. TENS – Trans Cutaneous Electrical stimulation This machine works by interfering with the pain-gate pathways in the body. In theory it is effective at treating pain in the body, though studies show varying lengths of time that relief is actually present for. This machine is often poorly tolerated by horses. As ever, let us know what you’re thinking! - Have your horses ever used any of these machines? - Does your horse’s Veterinary Physiotherapist use them? - Does your Vet believe in them? Some are more popular than others, and it’s always interesting to hear a fresh take/opinion on the effectiveness and experiences people have with them! The next few posts will cover some of these in a deeper scientific way, destined for those with a keen desire to know more about the biological responses that each machine initiates. See you in a week!
- What is Veterinary Physiotherapy?
Trying to summarise a 4-year degree to an accessible blog post was a lot harder than I gave it credit for! On a literal level, Physiotherapy breaks down to “therapy” meaning “treatment of pain and dysfunction” through “Physio” which translates to “movement”. A Veterinary Physiotherapist addresses the animal as a whole, and works to treat injuries directly (along Veterinary Surgeon guidance) with various modalities, and indirectly by training controlled movement in the form of stretches and exercises, to reduce compensations and inefficient movement. Compensations may have developed from injury, poor conformation or chronic asymmetrical use of the body. These are a problem because the asymmetry that results in the body causes increased load in some areas of the body, and decreased load in others. This asymmetry means excess wear and tear, leading to arthritis or injuries. This is something horsemen and women fight on a constant basis; and something that we Physios do our best to help. In order to support this quest for free-moving and functioning quadrupeds, us Physiotherapists have acquired in-depth knowledge of anatomy, clinical reasoning, modern research and practical applications of treatments. Within this, knowledge of how the body heals and what treatments are appropriate at different stages is used to maximise the safety and path to successful rehabilitation. There is potential for causing a lot of damage if this is done ineffectively, or in the wrong order, which is why fully-qualified and trained Veterinary Physiotherapists are crucial for your horse’s rehabilitation and performance! Clinical reasoning is the process of analysing the presented animal’s symptoms, combined with the clinical historyprovided by the vet and owner. Along with some other puzzle pieces, this is used to determine the appropriate treatment plan with its short- and long-term goals. Part of clinical reasoning is to determine which stage of healing the animal is at. This means ascertaining where the injury is from the following stages: 1. Acute is the initial time period up to approximately 72 hours after an attack on the body. This may be a physical trauma like a kick, or an infection which creates a greater systemic response. 2. Sub-acute is an elusive time between the initial onset of the attack/pain and the acute phase, to the chronic stages indicated by tissue remodelling. 3. Chronic phase can begin anywhere from a few weeks post-initial injury, up to months or even years depending on various factors. This is the remodelling phase of tissues, though it is important to note that not all tissues will ever remodel to their former strength and glory. Once the above has been established, an initial treatment will be delivered. Depending on the presenting symptoms will determine the use of modalities applied. Modalities available to the Veterinary Physiotherapist include Electrotherapies, Manual therapies, and Prescriptive/Remedial Exercises for the strengthening phase. These will be discussed in greater depth in future articles. The huge benefit to these treatments is the non-medicated pain relief available from them. Manual therapies include variations of massage, stretches and myofascial release. These increase circulation and drainage, reduce swelling, relax the animal and importantly offer pain relief… Yes, you riders would benefit from them too, but who am I to tell you to spend money on yourself and NOT your horse! (#beentheredonethat). Stretches are a great strengthening tool for horses, particularly those that are unable to work that much under exercise restrictions. *These should be monitored by a Physiotherapist! * Electrotherapies vary from muscle stimulation machines, LASERs, Ultrasound machines as well as many other ever-evolving products. These offer pain relief, improved muscle function, tendon and ligament treatment, and many others. Due to the wide-ranging benefits, another blog post is in the pipeline devoted entirely to this topic, so keep an eye out! These treatments are able to reduce reliance on painkillers, which means reduced risk of liver and kidney damage (and others),and of course, a financial bonus to avoid regular veterinary visits and drug prescriptions. Furthermore, this means that during those hard training days when your horse is feeling that bit more sore, or you’ve been working up to a competition and you can feel them tiring in their body, there is so much support available to them which doesn’t involve drugs*. *Important disclaimer: Painkillers are effective and necessary, but as advised by your Veterinary surgeon. Veterinary Physiotherapy simply offers a supporting therapy to what is available there Finally, the strengthening exercises that are available are as limitless as your imagination. Poles, weaving and hill-work could be described as the basic platform for remedial exercises, but the variations are endless, and can be tailored to suit your horse exponentially. This may be for the horse who is learning to use a limb again after nerve damage, or the sports horse who is slightly stiffer in one direction compared to others and requires increased suppleness there. There is no elitism when it comes to improving horse welfare. A lot of these elements will be discussed in the future, however, if there is something in particular that sounds of interest, please do get in touch! As always, at your beck and call on every social media platform! Looking forward to hearing from you… And you will be hearing from me this time next week! ❤️ Genevieve
- The Beginning.
I made a bold statement during my placement year at University (roughly 2018) that I was going to start a blog… There’s nothing quite like a good run-up! Now that I have completed my 4 years of Undergraduate study to become a qualified Veterinary Physiotherapist, I feel ready and armed to start the journey. I am still waiting on a final exam board meeting before I can be insured to treat, however, it’s only a matter of time. With that, I am so thrilled to be launching this blog as a precursor to the full business launch! Kirk Equine Performance is borne out of a love for horses and for Biology and represents a combination of expertise and skill. Kirk Equine Performance blends degree-trained Veterinary Physiotherapy with skills and experiences of an International Dressage rider – a rare combination if I may say so myself ;). This will be an all-inclusive service offering training, riding and treating your horse to the best levels achievable. My passion for horses has taken me to many places in the world. I actually think I fell in love with the sport at two years old obsessing over my cousin’s pony in England; I rode my first horse at the age of 5 in Sydney, Australia; got my first pony at 11 in Geneva, Switzerland; competed internationally for the first time in Pompadour, France, competed for my University team in Surrey, England; worked in professional stables for four years in Germany, and most recently made myself at home in the Cotswolds and Shropshire for my degree with my horses Pokemon, Horacio, Feather and most recently Riva! I am now permanently based in the Cotswolds, covering Gloucestershire and surrounds comfortably (including Oxfordshire, Herefordshire, Worcestershire and Wiltshire). Throughout my 4-year degree, including a year placement in clinical settings, I have kept training and competing my horses as much as possible. I competed my top 3 horses Royal Pokemon, Feather and Horacio each at senior international level at Small Tour, with Feather and Horacio both stepping up to this level for the first time in their lives! Managing expectations of myself and my horses, alongside deadlines and a huge workload at university, plus coping with devastating injuries and setbacks along the way with some of these horses tested my resolve and my passion, yet again, and has made me sure of my future ambitions. I became more focussed on pursuing the course of improving horse welfare as best I could with education, rehabilitation/treatment and training. If any of my knowledge is able to help them, to prevent them from any damage or deterioration, or help you as their owners handle those heartbreaking moments, then you’ll find me throwing that information around like it’s my business… ;) During the years of learning about life with horses, I have experienced colic, tendon tears, broken bones, blood clots, and random lameness that seems to just come along with them. A big part of wanting to become a Physio was to help support and fix horses as best we can, and have less heartbreak. I enjoy now knowing more about the science of rehabilitation, having already had plenty of experience with the emotions, the heartache and the practicalities of horse injuries. These blogs will cover anecdotes from life as a Veterinary Physiotherapist, as a competitive rider, as a horse trainer and informative pieces on new and/or interesting research, explanations about treatments and technologies or great questions I’ve been asked… … Which leads perfectly in to, as if I had planned it (😉), GET IN CONTACT! Any questions, any comments, please do get in touch on any of our various platforms! There is a contact form on the website, I’ve linked it, so you don’t even have to open your email tab! Find us on Instagram, Facebook, email, and if you feel nostalgic about the good ol’ days, I’ll even answer the phone! Looking forward to hearing from you, and you’ll be hearing from me this time next week!
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In the world of mathematics, understanding the domain and range of a function is crucial for any student or professional. These two concepts describe what inputs a function can accept (domain) and what outputs it can produce (range). To simplify this process, various tools have been developed; one of the most accessible tools is the domain and range calculator. This guide will walk you through different options for finding these calculators and how to use them effectively, with an aim to make your learning curve as gentle as possible.
The Desmos Graphing Calculator is a user-friendly online tool that helps visualize mathematical functions and their domain and range. Its interface is intuitive, making it an excellent choice for beginners.
- Go to the Desmos website: Open your web browser and visit the Desmos Graphing Calculator site (www.desmos.com/calculator).
- Input your function: Click on the function field and enter the equation for which you need to find the domain and range.
- Adjust the settings: Use the options available to adjust the view of the graph to see how the function behaves.
- Analyze the graph: The continuous curve represents the range, and the length along the x-axis shows the domain.
- Finding domain and range: Use the interactive features of the graph to find specific points and understand the limits of the function.
The Desmos Graphing Calculator is an excellent tool for beginners to easily visualize and understand the domain and range of a function. However, for more complex functions, it might require a bit more effort to determine the exact domain and range.
Wolfram Alpha offers a computational search engine that can calculate the domain and range of a function among many other mathematical queries.
- Access Wolfram Alpha: Navigate to www.wolframalpha.com using a web browser.
- Enter your function: Type “domain and range of” followed by your function in the search bar.
- Execute the search: Press enter, and Wolfram Alpha will compute the domain and range of the function.
- Review the results: The results provide a numeric and graphical representation of the domain and range of the function.
Wolfram Alpha is not only a powerful tool for finding the domain and range but also a great resource for other complex mathematical problems. Although it can handle more sophisticated calculations, some users may find it less intuitive than other options.
Symbolab Math Solver is an online tool that specializes in solving mathematical problems step-by-step, including finding the domain and range of functions.
- Visit Symbolab: Head to the Symbolab website at www.symbolab.com.
- Choose the calculator: Select the “Functions” calculator from the options.
- Input your function: Enter the function you are examining in the provided field.
- Search for the domain and range: Click the “Go” button, and Symbolab will display the domain and range, along with the steps it took to get there.
Symbolab is fantastic for those who want to understand the steps behind finding the domain and range, as it provides detailed explanations. However, it may occasionally display ads, and some features might be behind a paywall.
GeoGebra is an interactive geometry, algebra, statistics, and calculus application that is intended for learning and teaching at all levels of education.
- Access GeoGebra: Go to the GeoGebra website (www.geogebra.org).
- Select the calculator: Choose the “Graphing Calculator” from the apps section.
- Input the equation: Enter your function’s equation in the input bar.
- Examine the graph: The graph will automatically generate, illustrating the function.
- Determine domain and range: Use the features of GeoGebra to understand the behavior of the function and deduce the domain and range.
GeoGebra is an excellent educational tool with dynamic visualization. Its interactive nature is perfect for hands-on learners, but there might be a mild learning curve for complete beginners to become accustomed to all of its features.
Mathway provides users with an easy-to-use interface that can solve a variety of mathematical problems, including finding the domain and range.
- Visit Mathway: Open your browser and go to www.mathway.com.
- Choose the math problem type: Click on the drop-down menu and select “Pre-Algebra”, “Algebra”, or another appropriate category for your function.
- Enter your problem: Type in your specific mathematical problem.
- Get the solution: Hit the “Enter” button, and Mathway will show you the domain and range of your function.
Mathway is a great tool for quickly finding answers to mathematical problems including the domain and range. Although the explanations for solutions are not as comprehensive as other options, it’s a good quick solver for those who just need the final answer. Some advanced features may require a premium account.
QuickMath is an automated service that provides instant solutions to a range of math problems, which naturally includes finding the domain and range of a function.
- Navigate to QuickMath: Use your web browser to visit www.quickmath.com.
- Enter the problem: Find the “Algebra” section and choose “Solve an equation”, enter your function.
- Choosing the operation: Select the operation as “Find the domain and range” from the available list.
- Display the answer: Click “Solve” to get the results.
QuickMath is handy for solving math problems instantly without any fuss. However, it lacks the graphical interface that other options offer and may not always provide insights into how the solution was reached.
Cymath is an educational tool that offers both a website and a mobile app to help tackle math problems, including determining the domain and range of functions.
- Access Cymath: Visit the Cymath website at www.cymath.com.
- Choose the problem type: From the homepage, select the appropriate math problem type that fits your function.
- Input your function: Enter the function for which you’re seeking the domain and range.
- Find the solution: Click “Go” to get the domain and range, with steps explaining the process.
Cymath is a user-friendly platform that aims to teach the user how to solve the problem rather than just giving the answer. The explanations can be a great educational aid, but the interface may be too simple for more advanced functions.
Metamath is a unique system that treats all mathematics as a sequence of symbolic deductions.
- Go to Metamath: Access www.metamath.org in your web browser.
- Understand the functions: Before inputting any functions, take a moment to understand how Metamath structures its logic and proofs.
- Use the Proof Explorer: Look for the Proof Explorer to find examples of domains and ranges and how they’re proven within the system.
- Self-study: Due to its complexity, Metamath serves better as a study tool rather than a quick calculator.
Metamath offers an in-depth look into the logic behind mathematics, making it a wealth of knowledge for the avid learner. However, its highly technical nature and steep learning curve can be daunting for casual users.
While calculators are quick and efficient, understanding how to calculate the domain and range manually cultivates a deep understanding of these concepts.
- Review function properties: Brush up on the types of functions and their properties related to domain and range.
- Set the domain: Determine the possible x-values for which the function is defined (this could include looking out for division by zero or square roots of negative numbers).
- Find the range: Examine the y-values that result from the domain x-values after applying the function.
- Confirm your results: Cross-reference your findings with a calculator for accuracy.
Manual calculation is fundamental for a thorough grasp of domain and range, producing valuable skills for higher mathematics. Nonetheless, it can be time-consuming and prone to human error when compared to using calculators.
As you explore different options for finding the domain and range of functions, keep these tips in mind:
- Recognize patterns: Familiarize yourself with common functions and their typical domain and range.
- Use multiple tools: Validate your results by cross-checking between different calculators.
- Understand limitations: Not all calculators handle complex functions like piecewise or multi-variable functions with ease.
- Practice regularly: The more you practice, the easier it will become to understand how functions behave and determine their domain and range.
Delving into the realm of functions, domains, and ranges need not be daunting. By harnessing the power of the right calculators and familiarizing oneself with the core concepts, anyone can master these fundamental aspects of mathematics. Whether you prefer an interactive learning tool like Desmos or a comprehensive computational engine like Wolfram Alpha, there’s a solution tailored to your learning style and level of understanding. Remember, practice is key, and utilizing these tools will undoubtedly bolster your mathematical acumen.
What is the domain of a function?
- The domain of a function is the set of all possible input values (typically x-values) for which the function is defined and produces a valid output.
What is the range of a function?
- The range of a function is the set of all possible output values (typically y-values) that the function can produce from the given domain.
Can domain and range calculators handle all types of functions?
- While many online calculators are quite versatile, some may have difficulty with more complex or unusual types of functions, such as piecewise or implicit functions. It’s best to try multiple calculators or verify with manual calculations when in doubt.
Why is it important to learn how to calculate domain and range manually?
- Calculating the domain and range manually ensures a deep understanding of the function’s behavior and develops problem-solving skills that are essential for advanced mathematics and practical application.
Are there any free domain and range calculators available?
- Yes, many free options are available online, such as Desmos Graphing Calculator, Wolfram Alpha, and Symbolab Math Solver, which can help find the domain and range of functions.
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Geometry facts for kids
Geometry is a kind of mathematics that studies the size, shapes, and positions of things. There are flat (2D) shapes and solid (3D) shapes in geometry. Squares, circles and triangles are some of the simplest shapes in flat geometry. Cubes, cylinders, cones and spheres are simple shapes in solid geometry.
Geometry can be used to calculate the size and shape of many things. For example, geometry can help people find:
- the surface area of a house, so they can buy the right amount of paint
- the volume of a box, to see if it is big enough to hold a liter of food
- the area of a farm, so it can be divided into equal parts
- the distance around the edge of a pond, to know how much fencing to buy.
Geometry began as the art of measuring the shape of land so that it could be shared fairly between people. The word "geometry" means "to measure the land". It has grown from this to become one of the most important parts of mathematics. The Greek mathematician Euclid wrote the first book about geometry. Geometry is one of the oldest branches of mathematics.
Geometry starts with a few simple ideas that are thought to be true, called axioms. Such as:
- A point is shown on paper by touching it with a pencil or pen, without making any sideways movement. We know where the point is, but it has no size.
- A straight line is the shortest distance between two points. For example, Sophie pulls a piece of string from one point to another point. A straight line between the two points will follow the path of the tight string.
- A plane is a flat surface that does not stop in any direction. For example imagine a wall that extends in all directions infinitely.
Important concepts in geometry
The following are some of the most important concepts in geometry.
Euclid took an abstract approach to geometry in his Elements, one of the most influential books ever written. Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of the 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.
Points are considered fundamental objects in Euclidean geometry. They have been defined in a variety of ways, including Euclid's definition as 'that which has no part' and through the use of algebra or nested sets. In many areas of geometry, such as analytic geometry, differential geometry, and topology, all objects are considered to be built up from points. However, there has been some study of geometry without reference to points.
Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it. In differential geometry, a geodesic is a generalization of the notion of a line to curved spaces.
A plane is a flat, two-dimensional surface that extends infinitely far. Planes are used in every area of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space, where collinearity and ratios can be studied but not distances; it can be studied as the complex plane using techniques of complex analysis; and so on.
Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry.
In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry, the same definition is used, but the defining function is required to be differentiable Algebraic geometry studies algebraic curves, which are defined as algebraic varieties of dimension one.
A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology, surfaces are described by two-dimensional 'patches' (or neighborhoods) that are assembled by diffeomorphisms or homeomorphisms, respectively. In algebraic geometry, surfaces are described by polynomial equations.
A manifold is a generalization of the concepts of curve and surface. In topology, a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space.
Topologies and metrics
A topology is a mathematical structure on a set that tells how elements of the set relate spatially to each other. The best-known examples of topologies come from metrics, which are ways of measuring distances between points. For instance, the Euclidean metric measures the distance between points in the Euclidean plane, while the hyperbolic metric measures the distance in the hyperbolic plane. Other important examples of metrics include the Lorentz metric of special relativity and the semi-Riemannian metrics of general relativity.
Compass and straightedge constructions
Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass and straightedge. Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.
Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians have used higher dimensions for nearly two centuries. Dimension has gone through stages of being any natural number n, possibly infinite with the introduction of Hilbert space, and any positive real number in fractal geometry. Dimension theory is a technical area, initially within general topology, that discusses definitions; in common with most mathematical ideas, dimension is now defined rather than an intuition. Connected topological manifolds have a well-defined dimension; this is a theorem (invariance of domain) rather than anything a priori.
The issue of dimension still matters to geometry, in the absence of complete answers to classic questions. Dimensions 3 of space and 4 of space-time are special cases in geometric topology. Dimension 10 or 11 is a key number in string theory. Research may bring a satisfactory geometric reason for the significance of 10 and 11 dimensions.
The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of M. C. Escher. Nonetheless, it was not until the second half of 19th century that the unifying role of symmetry in foundations of geometry was recognized. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry is. Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' proved most influential. Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory, the latter in Lie theory and Riemannian geometry.
A different type of symmetry is the principle of duality in projective geometry (see Duality (projective geometry)) among other fields. This meta-phenomenon can roughly be described as follows: in any theorem, exchange point with plane, join with meet, lies in with contains, and you will get an equally true theorem. A similar and closely related form of duality exists between a vector space and its dual space.
In the nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, the basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori. This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Bolyai, Lobachevsky, and Gauss (who never published his theory). They demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based), published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory, and Riemannian geometry, that considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.
Images for kids
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
Quintic Calabi–Yau threefold
In Spanish: Geometría para niños
Geometry Facts for Kids. Kiddle Encyclopedia.
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Force and Torque on Conductors
Force and Torque on Conductors :
Force on a conductor:
Ampere proposed that if a conductor carrying current produces a magnetic field and exerts a force on a magnet, then the magnet should also exert a force on the conductor carrying current. Examples: - Suspend an aluminum rod horizontally on the wire between the poles of a horseshoe magnet, and pass an electric current through the wire, the aluminum rod will be displaced. If the direction of the current is reversed, the direction of displacement will also be reversed. The applied force is greatest when the conductor is perpendicular to the magnetic field.
Fleming's Left Hand Rule: –
The direction of the force (movement) of a current-carrying conductor in a magnetic field is determined by Fleming's left-hand rule. "If the thumb, index and middle fingers of the left hand are at right angles to each other so that the index finger points in the direction of the magnetic field and the middle finger points in the direction of the electric current, the thumb points in the direction of the force (direction of movement) of the conductor."
When a current-carrying conductor is placed in a uniform magnetic field, a force is exerted on the moving charges inside the conductor (due to the Lorentz force). This is the force required in a conductor through which current flows. To calculate this force, we consider various parameters as follows:
L is the conductor length,
i am the current that flows in him
q is the charge flowing through the conductor at time 't'.
v is the velocity of charge q,
B is the uniform magnetic field in which there is a conductor carrying current.
A special case of force on a live conductor
There are some special cases where a force acts on a current-carrying conductor, depending on the position of the conductor within the magnetic field. These cases are described as follows.
Case I: Conductor placed parallel to the magnetic field
When sin θ = 0 (minimum), i. H. θ = 0° or 180°, the force on the current element in the magnetic field is zero (minimum).
Fmin = 0
If the current in the magnetic field is collinear with the magnetic field, the current element in the magnetic field experiences no force. Therefore, the force experienced by a conductor within a given magnetic field is minimal.
Case II: Conductor placed perpendicular to the magnetic field
For sin θ = 1 (maximum), i.H. θ = 90°, the force on the current element is maximum in the magnetic field (=ILB).
Fmax = ILB
The force direction is always perpendicular to the containing plane. L and B, which is the maximum force a conductor can experience in a given magnetic field.
We learned about the existence of magnetic fields caused by current-carrying conductors and the Biot-Savart law.
We also learned that an external magnetic field exerts a force on a current-carrying conductor and the Lorentz force formula that underlies this principle.
Therefore, two studies show that two conductors carrying current exert a magnetic force on each other when they are placed close to each other. This section details this case.
Consider the system shown in the diagram above. Here we have two parallel current conductors separated by a distance 'd' such that one conducts the current I1 and the other conducts the current I2, as shown in the figure. Based on what we just learned, we can say that Conductor 2 experiences the same magnetic field at any point along its length by conductor 1. The direction of magnetic force is shown in the diagram and can be found using the right-hand thumb rule. As you can see, the direction of the magnetic field is down due to the first conductor. From Ampere's circuit law, the magnitude of the magnetic field due to the first conductor is given by
The force exerted by conductor 1 on a segment of length L of conductor 2 is given by
Similarly, we can calculate the force that conductor 2 exerts on conductor 1. It can be seen that conductor 1 is subjected to the same force by conductor 2 but in the opposite direction. therefore,
It has also been observed that currents flowing in the same direction attract conductors to each other, and currents in opposite directions cause conductors to repel each other. The magnitude of the force acting per unit length is given by
Torque on a Conductor:
Torque on Current Carrying Coil in Magnetic Field:
A magnetic dipole is the limit of either a closed electric circuit or a pair of poles because the size of the source is reduced to zero keeping the magnetic moment constant. It is now shown that a constant current I through a rectangular loop placed in a uniform field has torque. It has no mains power. This behavior is similar to an electric dipole in a uniform electric field.
Consider the case of placing a rectangular loop such that the uniform magnetic field B is in the plane of the loop. The field exerts no force on her two arms PS and QR in the loop. It is perpendicular to arm PQ of the loop and exerts a force F1 on arm PQ directed in the plane of the loop. Its size is
Similarly, force F2 is applied to arm RS and F2 is directed out of the plane of the paper.
F2 =IzB= F1
Therefore the net force on the loop is zero. A torque acts on the loop because the two forces F1 and F2 cancel each other. Here we can see that the torque applied to the loop tends to cause it to rotate counterclockwise.
τ = F1 + F2
= Izb + Izb
= I(yz) B
= IAB ……. (1)
Where A = y × z is the area of the rectangle.
Now consider the case where the plane of the loop is not aligned with the magnetic field, but at an angle with it. And consider the angle between the magnetic field and the normal to the coil as the angle Θ. The forces on the two arms, QR and SP, are equal and opposite, acting along the axis of the coil connecting the centers of mass of QR and SP. Since they are collinear along the axis, they cancel each other out and produce no net force or torque. The forces on arms PQ and RS are F1 and F2. Moreover, they are also of the same and opposite magnitude,
F1 = F2 = IzB
Since they are not collinear, they form a pair. However, the effect of torque is less than if the plane of the loop passed along the magnetic field. The magnitude of the torque on the loop is:
τ = F1 sinθ F2 inθ
= I(y×z) B sin θ
= IABsinθ ……. (2)
Therefore, the torque in equations (1) and (2) can be expressed as the vector product of the magnetic moment of the coil and the magnetic field. Therefore, the magnetic moment of the current loop can be defined as
m = IA
where A is the direction of the area vector. If the angle between m and B is θ, equations (1) and (2) can be expressed as follows.
τ = m × B
where m is the magnetic moment and B is the uniform magnetic field.
Magnetic Dipole Moment:
The magnetic moment is a quantity that describes the magnetic strength and orientation of a magnet or other object that produces a magnetic field. More specifically, magnetic moment refers to the magnetic dipole moment, i.e. the component of the magnetic moment that can be represented by a magnetic dipole. A magnetic dipole is a magnetic north pole and a magnetic south pole separated by a small distance.
The magnetic dipole moment is the current and area or energy divided by the magnetic flux density. The dipole moment has units of meters, kilograms, seconds, amperes, and amperes squared. The unit of the centimeter-gram-second system is erg (unit of energy)/gauss (unit of magnetic flux density). 1,000 ergs per gauss equals 1-ampere square meter.
Magnetic Moment formula:
Magnetic dipole moment – The magnetic field B due to a current loop carrying current i of radius R separated by a distance l along the axis is given by
Now, considering points so far from the current loop that l>>R holds, we can approximate the field as
Now the area of the loop is A
So the magnetic field can be written as
We can write this new quantity μ as a vector pointing along the magnetic field.
Notice the striking resemblance to the electric dipole field.
Unlike the electric field, the magnetic field has no "charge" or "counterpart". In other words, the magnetic field has no source or sink, only dipoles can exist. Anything that can generate a magnetic field has both a source and a sink. H. There are both north and south poles. In many respects, magnetic dipoles are fundamental entities capable of creating magnetic fields.
Most elementary particles behave essentially like magnetic dipoles. For example, the electron itself behaves like a magnetic dipole and has a magnetic spin dipole moment. This magnetic moment is inherent because the electron has no region A (it is a point body) and does not rotate around itself, but it is fundamental to the nature of the electron's existence.
The 'N' times magnetic moment of a wire loop can be generalized as
μ = NiA
The field lines of the current loop resemble those of an ideal electric dipole.
If you've ever split a magnet in two, you'll know that each piece forms a new magnet. The new pieces also include the north and south poles. It seems impossible to get just the North Pole.
Current Loop as a Magnetic Dipole:-
Ampere found that the distribution of magnetic lines of force around a finite current-carrying solenoid is similar to that produced by a bar magnet. This is evident from the fact that a compass needle when similar deflections moved around these two bodies. After noting the close resemblance between these two, Ampere demonstrated that a simple current loop behaves like a bar magnet and put forward that all the magnetic phenomena are due to circulating electric current. This is Ampere’s hypothesis.
The magnetic induction at a point along the axis of a circular coil carrying current is,
B = µonI(a2+x2)
The direction of this magnetic field is along the axis and is given by the right-hand rule. For points that are far away from the center of the coil, x>>a, a2 is small and it is neglected. Hence for such points,
B = µonIx3
If we consider a circular loop, n = 1, its area A = πa2
The magnetic induction at a point along the axial line of a short bar magnet is
Or, B =
Comparing equations (1) and (2), we find that
M = IA
Hence a current loop is equivalent to a magnetic dipole of moment M = IA
The magnetic moment of a current loop is defined as the product of the current and the loop area. Its direction is perpendicular to the plane of the loop.
Magnetic Dipole Moment of a Spinning Electron:
According to Neil Bohr's atomic model, a negatively charged electron revolves around a positively charged nucleus in a circular orbit of radius r. An electron rotating in a closed path generates an electric current. The counterclockwise movement of the electron produces a clockwise current in the conventional current.
Since the electron is negatively charged, the normal current flows in the opposite direction to its movement. The magnetic moments associated with the orbital motion of two electrons are shown in the figure. An electron rotating in an orbit of radius 'r' corresponds to a magnetic layer with a magnetic moment of 'M'
M = i\ times A
Here "i" is the current corresponding to the spinning electron and A is the area of the orbit. Where '\tau' is the spin time of the electron.
i = charge/period = e/\tau =
or i =
Here 'ω' is the angular velocity of the electron.
So M = \ times (πr2) =
According to Bohr's theory, an electron can move in an orbit where its angular momentum is an integral multiple of h/2π, where "h" is Planck's constant.
mr2ω = n
Or r2ω = n
Substituting r2ω in the equation, we get,
M = n
Since an atom can have a large number of electrons, the magnetic moment of an atom is the vector sum of the magnetic moments of the different electrons. The term is called the Bohr magneton. This is the smallest value of the magnetic moment of the electron. Each atom can have a magnetic moment that is a fixed multiple of the Bohr magneton. Thus, the magnetic moment is also quantified at the atomic and subatomic level.
In addition to the magnetic moment due to orbital motion, the electron has a magnetic moment due to its spin. Thus, the resultant magnetic moment of an electron is the vector sum of its orbital magnetic moment and its spin magnetic moment.
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Impulse, a fundamental concept in Physics, refers to the change in momentum of an object when an external force is applied to it. The impulse experienced by an object is a direct result of the force applied and the duration of its application. Formally, impulse (denoted as ‘J’) is given by the equation:
Impulse (J) = Force (F) x Time (Δt). This quantifiable measure provides an essential tool for understanding and predicting an object’s behavior under varying forces.
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The Interplay between Impulse and Momentum
One cannot delve into the depths of impulse without understanding its close relationship with momentum. Momentum, in simple terms, is the measure of the motion of an object. It’s a vector quantity given by the product of an object’s mass and its velocity.
The connection between impulse and momentum is explicit in Newton’s second law:
- Often expressed as
Force = mass x acceleration.
- Can be rearranged to
Force = change in momentum / time, showing that the applied force is directly proportional to the rate of change in momentum.
- Consequently, if we multiply both sides by time, we get the impulse-momentum theorem:
Impulse = change in momentum.
Newton’s Second Law and Impulse
Newton’s second law of motion is the bedrock on which the concept of impulse is built. This law asserts that the force exerted on an object is equal to the change in its momentum per unit time. Mathematically, it translates into the equation:
Force (F) = mass (m) x acceleration (a), where acceleration is the rate of change of velocity.
When you multiply both sides of Newton’s second law by time, you get the impulse equation:
Impulse = Force x Time. This equation is significant because it underscores that impulse is not just about the magnitude of the force but also the duration over which it’s applied.
The Role of Impulse in Collisions
Impulse is of paramount importance when studying and analyzing collisions, a phenomenon frequently observed in our everyday lives and numerous scientific contexts. This might involve common scenarios like billiard balls colliding on a pool table, cars crashing on a roadway, or more intricate situations like particle interactions in a particle accelerator.
In each collision event, an impulse is applied, and it’s this impulse that is responsible for altering the momentum of the colliding objects. As stated earlier, impulse is the product of the force exerted on an object and the time duration of this force. During a collision, a force acts upon the objects involved for a certain amount of time, resulting in a change in their momentum.
This change in momentum is especially pertinent when we consider a ‘closed’ or ‘isolated’ system, where the only forces at work are the internal forces of the system, and external forces like friction, air resistance, or gravitational pull are negligible or non-existent. In such scenarios, the principle of conservation of momentum comes into play.
The conservation of momentum principle posits that the total momentum of an isolated system remains constant if no external forces act upon it. Hence, in the context of collisions, it implies that the total momentum of all objects involved before the collision equals the total momentum after the collision.
Consider a simple two-object collision. The combined momentum of the two objects before they collide is equal to their combined momentum after they collide. This holds true regardless of whether the collision is elastic (where kinetic energy is also conserved) or inelastic (where kinetic energy is not conserved).
The role of impulse in collisions becomes even more fascinating when we observe that it can significantly change the direction and speed of the colliding objects, leading to a wide array of outcomes. For instance, in an elastic collision of two identical billiard balls, one initially at rest, the moving ball stops after the collision while the initially stationary ball moves with the initial speed of the first ball – all due to the interplay of impulse and the conservation of momentum.
Impulse: A Vector Quantity
Impulse, similar to other vector quantities in physics such as force, velocity, and momentum, has both a magnitude and a direction. This dual attribute of an impulse not only quantifies the “amount” of the impulse but also the “way” it acts.
The magnitude of impulse, given by the product of the force and the time for which it is applied, essentially measures the strength or impact of the force. It tells us how ‘forceful’ or ‘influential’ the impulse is. For example, a more considerable force or a longer duration results in a larger impulse and consequently a more significant change in momentum.
The direction of impulse, on the other hand, is inherently tied to the direction of the force applied to the object. The impulse follows the path of the force. If you push an object to the right, the impulse is directed to the right. Conversely, if you pull an object toward yourself, the impulse is oriented in your direction.
It’s noteworthy that the direction of impulse plays a pivotal role in determining the final motion of the object. When the force is applied along the line of the initial motion of an object, the impulse only alters the speed of the object, not its direction. For instance, if a car moving eastward experiences a force in the eastward direction, it will only speed up but keep moving east.
However, if the force is applied in a direction not along the line of the initial motion, the impulse can indeed change the object’s direction of motion. Consider a moving billiard ball hit at an angle. The applied force generates an impulse that changes the ball’s direction of motion.
Real-life Implications of Impulse
Impulse has significant real-world implications:
- For instance, the effectiveness of airbags in cars is a direct application of the concept of impulse. An airbag increases the time over which the driver’s momentum is brought to zero, thereby reducing the force experienced and minimizing injury.
- Similarly, athletes use the concept of impulse when they “follow through” while hitting a ball, thus maximizing the time of contact and increasing the ball’s final momentum.
Impulse is a fundamental concept in physics that encapsulates the interplay between force, time, and change in momentum. It’s an integral part of understanding how objects move and interact in the world, from the macroscopic collisions of vehicles to the microscopic interactions of particles. The concept of impulse helps us predict outcomes, protect lives, and even enjoy our favorite sports.
What are the units of impulse?
The units of impulse are Newton-seconds (N.s) in the International System of Units (SI), reflecting its definition as force multiplied by time.
What is the relationship between impulse and momentum?
Impulse is equal to the change in momentum of an object. If an external force acts upon an object, it results in an impulse which changes the object’s momentum.
Can impulse change the direction of motion?
Yes, since impulse is a vector quantity, it can change an object’s direction of motion if the force applied is not along the line of the initial motion.
What is the principle of conservation of impulse?
Impulse doesn’t have a conservation law. However, momentum, which is closely related to impulse, is conserved in a closed system. This means the total momentum before a collision is equal to the total momentum after the collision.
How does impulse affect collisions?
Impulse determines the change in momentum of an object in a collision. The greater the impulse, the greater the change in momentum.
Is impulse a scalar or vector quantity?
Impulse is a vector quantity. It has both a magnitude (size) and direction. The direction of the impulse is the same as the direction of the force that causes it.
What are some real-life examples of impulse?
Real-life examples of impulse include a bat hitting a ball, a car crash (and the role of airbags in reducing injury), and an athlete jumping off a starting block.
What are Impulse and change in momentum?
Impulse is the change in an object’s momentum when a force is applied to it over a period of time. Therefore, impulse and change in momentum are equal.
What is the Importance of impulse in Physics?
Impulse is important in physics as it allows us to calculate the change in an object’s momentum. This is particularly useful in analyzing collisions and other situations where forces cause changes in motion.
How are Impulse and force related?
Impulse is the product of the force applied to an object and the time period over which it is applied. Hence, impulse and force are directly related.
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Table of Contents
Which is an angle whose measure is exactly 180?
Def: A straight angle is an angle whose measure is exactly 180 degrees.
What is the measure of 180 degrees?
The angle which measures 180 degrees is named the straight angle. The other angles are given here: Acute angle: The angle which is more than 0° and less than 90°…Table of Angles.
|Type of angle
|Measure of angle
|180° < θ < 360°
|Full rotation (or Complete angle)
|θ = 360°
What is the name of 180?
straight angle-a 180 degree angle.
What is an angle greater than 180 called?
Reflex Angle. The angle which measures greater than 180° and less than 360° is known as the reflex angle. The reflex angle can be calculated if the measure of the acute angle is given, as it is complementary to the acute angle on the other side of the line.
What is meant by 180 degree angle?
A 180-degree angle is a straight angle and it is exactly half of a revolution. It is also called a half-circle angle. A straight angle is produced by a straight line. The two arms of the angle which are making 180-degree are just opposite to each other from the common vertex.
What is 180-degree longitude called?
The Earth’s longitude measures 360, so the halfway point from the prime meridian is the 180 longitude line. The meridian at 180 longitude is commonly known as the International Date Line. As you pass the International Date Line, you either add a day (going west) or subtract a day (going east.) Hemispheres.
Can an angle be more than 180?
Obtuse Angle – An angle more than 90 degrees and less than 180 degrees. Straight Angle – An angle that is exactly 180 degrees. Reflex Angle – An angle greater than 180 degrees and less than 360 degrees.
Why do triangle angles add to 180?
A triangle’s angles add up to 180 degrees because one exterior angle is equal to the sum of the other two angles in the triangle. In other words, the other two angles in the triangle (the ones that add up to form the exterior angle) must combine with the third angle to make a 180 angle.
How do you prove 1 degree is 60 minutes?
How Is 1 Degree Equal To 60 Minutes?
- Answer: One degree is split into 60 minutes of arc and one minute split into 60 seconds of arc. The use of degrees-minutes-seconds is also recognized as DMS notation.
- To Prove. 1 degree = 60 minutes.
- Proof. We know that. 1 minute = 60 seconds. 1 day =24 h0urs.
What does a 180 angle look like?
90 degrees is what an angle in a square looks like, and 180 degrees is a straight line. 220 degrees is 40 degrees past that, so it looks like about 5/8ths of a circle, or just a little bit larger than a straight line.
How do you find the degree of an angle?
Solve the angle of an incline by finding the rise and the run of a line. Convert rise and run to the same units of measure, then divide the rise by the run to find the decimal form. Finally, get the inverse tangent of the decimal to find the angle in degrees. degrees = tan -1(decimal).
What angle is 180 degrees?
An angle that is exactly 180 degrees is called a straight angle. Furthermore, what does a 180 degree angle look like? It is equal to one-quarter of a rotation around a circle. A straight line creates a 180 degree angle. It is equal to one-half a rotation around a circle. Consequently, what is a 181 degree angle called?
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What is the Conversion of Binary Numbers to Decimal?
Converting a binary number to decimal involves translating a number from the binary system (base 2), which uses only two digits (0 and 1), to the decimal system (base 10), the most widely used numeral system.
The conversion is done by multiplying each digit of the binary number by the power of 2 corresponding to its position and then summing these values. For example, in binary 101, the decimal equivalent is 1*2^2 + 0*2^1 + 1*2^0 = 5.
This conversion is fundamental in computer science and digital electronics, where binary representation is a core concept.
How to Use the Binary to Decimal Conversion?
Understanding binary to decimal conversion is essential in various technological and computational fields.
Step 1: Enter the binary number into our online converter.
Step 2: The converter processes each binary digit according to its positional value.
Step 3: The result is a decimal representation of the binary number.
Examples of Binary to Decimal Conversion
Real-world applications of this conversion:
Example 1: Decoding binary code in computer programming into human-readable numbers.
Example 2: Converting binary outputs from electronic sensors into decimal for analysis.
Example 3: Understanding and troubleshooting digital circuit designs.
Nuances of Binary to Decimal Conversion
Key points to consider in this conversion:
- Accuracy is crucial, as a small error in conversion can lead to vastly different results.
- The understanding of binary and decimal systems is fundamental in computing and digital electronics.
Frequently Asked Questions
Why is binary to decimal conversion important in computing?
Binary to decimal conversion is essential because computers operate in binary, while humans interact with computers using decimal systems.
Can this converter be used for large binary numbers?
Yes, our converter is capable of handling large binary numbers, making it a versatile tool for various computational tasks.
How can understanding binary to decimal conversion benefit students in their studies?
Understanding binary to decimal conversion is beneficial for students, particularly in computer science and mathematics, as it deepens their understanding of numerical systems and computational logic.
Is the binary to decimal conversion relevant in everyday life outside of technical fields?
While primarily technical, binary to decimal conversion can be relevant in everyday life for understanding basic computing concepts and for enthusiasts exploring digital technologies.
Can this tool help in learning and teaching binary concepts in educational settings?
Absolutely, this tool can be an effective aid in teaching and learning binary concepts, helping students visualize and understand the conversion process between binary and decimal systems.
You may find the following calculators on the same topic useful:
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- Km/h to m/s Speed Conversion Calculator. Easily convert speeds from kilometers per hour to meters per second for academic, scientific, or practical use.
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- Hexadecimal to Decimal Calculator. Rapid and precise conversion of hexadecimal numbers to decimal format, suitable for advanced programming and IT tasks.
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Synchronous Machines in PE Power
Synchronous machines are electric machines that operate at a constant speed, synchronized with the power system’s frequency. Unlike induction machines, synchronous machines maintain a fixed relationship between the rotor speed and the frequency of the AC power supply. Synchronous machines are commonly used as generators in power plants and sometimes as motors in industrial applications.
For this reason, synchronous machines in PE Power are the critical exam topic per the NCEES® exam guidelines. This detailed guide on synchronous machines in PE Power will help you cover all the details of synchronous motors and generators. Let’s study this in detail.
Let’s start with discovering how synchronous machines are made, which will further help in understanding the workings of synchronous motors in the following guide.
Like induction machines, synchronous machines have a stator with a three-phase winding. This stator winding is connected to the power supply.
The rotor of a synchronous machine has either a cylindrical shape or a salient pole structure. It contains a winding connected to a DC power source, creating a magnetic field.
Synchronous machines require a separate DC power source for rotor excitation. The rotor winding’s DC (direct current) creates a constant magnetic field.
The rotor of a synchronous machine rotates at a speed precisely equal to the synchronous speed determined by the power system frequency and the number of poles.
Synchronous machines are designed to operate at a fixed speed, synchronized with the power system frequency. They maintain a constant relationship between the rotor speed and the frequency of the AC power supply.
Synchronous generators allow control over the power factor, making them helpful in improving the overall power factor of a system.
Synchronous Machines – They operate at a constant speed (synchronous speed).
Induction Machines – They operate slower than the synchronous speed (dependent on the load and slip).
Synchronous Machines – They require a separate DC power source for rotor excitation.
Induction Machines – They do not require a separate excitation source.
Synchronous Machines – They are usually not self-starting and may require external means (e.g., a starting motor).
Induction Machines – They are Self-starting due to the rotating magnetic field induced by the stator.
Synchronous Machines – THeir power factor can be controlled by adjusting the excitation, making them useful for power factor correction.
Induction Machines – Their power factor is generally lower compared to synchronous machines.
Synchronous Machines – They are commonly used as generators in power plants, especially in large-scale power generation (suitable for applications requiring precise speed control and power factor correction).
Induction Machines – They are widely used in various industrial applications, such as pumps, fans, and compressors (commonly used as motors when precise speed control is not critical compared to industry-grade applications).
Synchronous Machines – They generally have higher efficiency compared to induction machines.
Induction Machines – They are known for their efficiency and reliability but may have slightly lower efficiency than synchronous machines.
Synchronous generators, also known as alternators, convert mechanical energy into electrical energy through electromagnetic induction. They are a crucial component in power generation systems, commonly used in power plants and other applications requiring large-scale electrical power.
There are two types of synchronous generators.
- Salient Pole Synchronous Generator
- Non-Salient Pole Synchronous Generator
A salient pole synchronous generator is characterized by a rotor design where the poles extend outward, resembling salient features. These poles are physically separated and exhibit a distinctive appearance, creating a pronounced protrusion on the rotor. The rotor structure’s protruding poles contribute to the generator’s unique design.
The rotor of the salient pole synchronous generator is subjected to DC excitation, establishing a magnetic field within the rotor winding. This excitation process is critical in preparing the generator for power generation.
The salient pole structure concentrates the magnetic field in specific rotor regions. This concentration enhances the generator’s ability to produce high reactive power levels, making it well-suited for applications where reactive power support is crucial.
The enhanced capability of salient pole generators to adjust reactive power makes them suitable for systems with fluctuating loads. They can efficiently respond to changes in the load profile and maintain stable voltage levels. For instance, they are commonly deployed in power systems where the power factor is low. Their ability to provide substantial reactive power support aids in improving and stabilizing the power factor.
Unline salient pole generators, a non-salient pole synchronous generator features a cylindrical rotor and lacks the protruding poles characteristic of salient pole generators. The poles in this design are evenly distributed around the rotor, resulting in a more uniform appearance without significant protrusions.
Similar to the salient pole generator, the non-salient pole synchronous generator rotor is subject to DC excitation. This excitation process establishes a magnetic field within the rotor winding, initiating the generator’s readiness for power generation.
The absence of protruding poles contributes to a more uniform distribution of the magnetic flux across the rotor. This uniformity is advantageous in specific applications where a consistent magnetic field is sufficient.
The uniform magnetic flux distribution makes non-salient pole generators suitable for applications with stable and predictable load conditions. They can efficiently generate power under relatively constant load profiles. Therefore, they are typically employed in applications where the emphasis is not primarily on high levels of reactive power support. This makes them a perfect fit for general power generation applications where reactive power demands are moderate.
As the rotor starts to rotate at a synchronous speed, it cuts through the magnetic flux produced by the stator windings. This cutting action induces a voltage in the stator windings, following Faraday’s law of electromagnetic induction. The generated voltage in the stator windings becomes the generator’s electrical output, ready to be utilized or fed into the power grid.
One distinctive advantage of the salient pole structure is its ability to provide substantial reactive power support to the power system. Concentrating magnetic flux around the protruding poles enhances the generator’s reactive power generation capabilities.
This feature makes salient pole synchronous generators particularly suitable for applications where maintaining or improving the power factor is essential, such as in power systems with low power factors or those experiencing fluctuating loads.
As the rotor rotates at the synchronized speed, it cuts through the magnetic flux generated by the stator windings. This cutting action induces a voltage in the stator windings, adhering to the principles of electromagnetic induction.
The induced voltage becomes the generator’s electrical output, ready to be utilized for various applications or supplied to the power grid.
The distinctive feature of the non-salient pole generator is the distributed nature of its rotor poles. The cylindrical rotor lacks protruding features, resulting in a more uniform magnetic flux distribution across the rotor surface.
This uniformity contributes to a consistent and evenly distributed magnetic field, which may be advantageous in applications where a more uniform magnetic field distribution is sufficient.
SCADA systems are critical in monitoring, controlling, and optimizing power generation and distribution systems in synchronous and asynchronous machines. With the combination of sensors and communication protocols, SCADA servers and a user-friendly interface contribute to:
- Improved efficiency
- Reduced downtime
- Informed decision-making
Before moving further, read the first part of this series on Induction Machines in PE Power.
let’s provide a concise summary of the step-by-step instructions on how SCADA systems work:
Sensors and RTUs/PLCs collect data from field devices, including temperature sensors, pressure sensors, and flow meters.
Communication protocols like Modbus, DNP3, or OPC facilitate secure data transmission from RTUs/PLCs to the central SCADA system using communication networks such as Ethernet or radio frequency.
The RTUs and PLCs will be discussed in detail in the following sub-section.
SCADA servers at the central control center receive and process the data, utilizing specialized software for analysis, storage, and visualization.
SCADA systems provide a user-friendly interface (HMI) for operators to visualize real-time data, historical trends, and control parameters, including graphical displays, charts, and alarms.
Operators in the control room use HMI displays to take control actions, such as adjusting setpoints or activating alarms, based on real-time information. Automated control processes respond to predefined logic and algorithms.
SCADA systems include reporting tools for generating customized reports and visualizations, supporting performance analysis, compliance reporting, and decision-making.
SCADA systems can send alerts and notifications to operators and maintenance personnel in case of abnormal conditions, equipment failures, or critical events.
Remote access capabilities allow engineers and maintenance personnel to diagnose issues, troubleshoot, and implement software updates without physical presence. Continuous optimization is achieved through the analysis of historical data and ongoing performance monitoring.
Let’s see how SCADA systems are applied in both synchronous and asynchronous machines:
Field Devices: Sensors, including those for voltage, current, and temperature, are installed on synchronous generators to capture operational parameters.
RTUs/PLCs: Remote Terminal Units (RTUs) or Programmable Logic Controllers (PLCs) are deployed at the generator site to acquire data from sensors and control devices.
Communication Protocols: Protocols like IEC 61850, Modbus, or DNP3 facilitate the transmission of real-time data from RTUs/PLCs to the central SCADA system.
Secure Data Transmission: Security measures, such as encryption, ensure the secure transfer of critical data between the synchronous generator and the central SCADA system.
SCADA Servers: Data received from synchronous generators is processed by SCADA servers at the central control center. SCADA software is employed to analyze electrical parameters and monitor machine health.
Database Storage: Processed data is stored in databases for historical analysis, trending, and generating performance reports.
HMI Displays: SCADA systems offer a user-friendly interface for operators and engineers to visualize real-time data and monitor the status of synchronous generators.
Control Room Displays: Operators in the control room utilize large displays for monitoring synchronous generators viewing alarms, trends, and key performance indicators.
Control Actions: Operators adjust setpoints, control the excitation system, and coordinate the generator’s operation with the power grid based on information displayed on the HMI.
Automated Control: SCADA systems can automate control processes for synchronous generators, modifying excitation levels, reactive power output, or synchronization parameters through predefined logic and algorithms.
Reporting Tools: SCADA systems include reporting tools for performance analysis, compliance reporting, and decision-making regarding the operation of synchronous generators.
Alerts and Notifications: Operators receive alerts and notifications in case of abnormal conditions, equipment failures, or other critical events related to synchronous machines.
Field Devices: Sensors capturing parameters like current, voltage, and temperature are placed on induction motors for data acquisition.
RTUs/PLCs: Similar to synchronous machines, RTUs or PLCs acquire and transmit sensor data to the central SCADA system.
Communication Protocols: Common communication protocols transmit data from RTUs/PLCs to the central SCADA system.
Secure Data Transmission: Security measures ensure the secure transfer of data between induction motors and the SCADA system.
SCADA Servers: SCADA servers process data received from induction motors, analyze electrical parameters, and monitor motor performance.
Database Storage: Processed data is stored in databases for historical analysis, trending, and generating performance reports.
HMI Displays: SCADA systems provide a user-friendly interface for operators to visualize real-time data and monitor the status of induction motors.
Control Room Displays: Operators use large displays in the control room for monitoring induction motors viewing alarms, trends, and key performance indicators.
Control Actions: Operators adjust setpoints and control parameters based on information displayed on the HMI.
Automated Control: SCADA systems can automate control processes for induction motors, adjusting voltage and frequency through Variable Frequency Drives (VFDs) for optimal speed control and energy efficiency.
Reporting Tools: SCADA systems include reporting tools for performance analysis, compliance reporting, and decision-making related to the operation of induction motors.
Alerts and Notifications: Operators receive alerts and notifications in case of abnormal conditions, equipment failures, or other critical events related to asynchronous machines.
Remote Terminal Units (RTUs) and Programmable Logic Controllers (PLCs) are crucial components in SCADA systems, serving as interfaces between field devices and the central SCADA control center. Here’s a detailed explanation of how RTUs and PLCs work in the context of SCADA systems:
RTUs are deployed in the field near the sensors and other remote devices. Their primary function is to acquire data from various field instruments, such as sensors and meters. RTUs collect analog and digital data, converting these signals into a format that can be transmitted to the central SCADA system.
Let’s have a look at their key characteristics.
In many cases, the raw signals from sensors need conditioning to ensure accuracy and reliability. RTUs may include signal conditioning capabilities, such as amplification or filtering, to prepare the data for transmission.
RTUs process the acquired data locally, scaling analog signals to engineering units and applying any necessary calculations. This preprocessing minimizes the data transmitted over the communication network, optimizing bandwidth usage.
RTUs use communication protocols, such as Modbus, DNP3, or IEC 60870, to transmit the processed data to the central SCADA system. These protocols ensure standardized communication and compatibility with SCADA software.
RTUs implement security features to protect against unauthorized access and cyber threats. Encryption, authentication, and secure communication channels are often employed to safeguard the integrity and confidentiality of data.
RTUs provide the capability for remote monitoring and control of field devices. Operators in the central control center can send commands to the RTUs to initiate control actions or retrieve additional data from the field.
In distributed SCADA architectures, multiple RTUs work collaboratively to monitor and control different segments of the overall system. This distributed control approach enhances scalability and redundancy.
PLCs are designed for real-time control of industrial processes. They execute logic control algorithms written in ladder logic or other programming languages. These algorithms define the behavior of the control system based on input conditions.
Let’s see why PLCs are integral in SCADA systems.
PLCs are equipped with input and output modules that interface with field devices. Input modules receive signals from sensors, while output modules send control signals to actuators, valves, and other devices.
PLCs operate in a continuous scanning process. During each scan, the PLC reads input signals from the field, executes the control logic, and updates output signals based on the defined control algorithms.
PLCs are programmed using specialized software that allows engineers to define the control logic. The programming environment typically includes a graphical interface where users can create and edit control programs.
PLCs feature communication interfaces to exchange data with other devices, including HMI systems, other PLCs, and SCADA systems. Communication protocols, such as OPC, facilitate seamless integration with SCADA software.
In critical applications, redundant PLC configurations are often employed to enhance system reliability. Redundant PLCs operate in parallel, with one serving as a backup in case of a failure in the primary unit.
PLCs include diagnostic features to detect faults, errors, and abnormalities in the control system. Alarms and error messages generated by the PLC provide valuable information for troubleshooting and maintenance.
PLCs are integral components of SCADA systems. They provide the local control and data acquisition capabilities necessary for monitoring and managing industrial processes. The SCADA system communicates with PLCs to collect data, send control commands, and visualize real-time information.
Now, you have a rich understanding of synchronous machines in PE Power. At the end of this study guide, you will be familiar with the workings of synchronous motors and why they are crucial.
For more reading and a detailed comparison of synchronous and asynchronous (Induction) machines, read our detailed study guide of this series on induction machines in PE Power.
If you are gearing up for PE Power exam preparation – look no further than Study for FE. It is recommended to check all the exam preparation resources and tailored courses offered by Study for FE – Your go-to platform for all things PE.
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Hello guys how are you you are very welcome to our new article what is velocity. in today’s topic we will tell you many things about the velocity definition, its units, and some essential examples also. Every one of us always used to travel by Motorcycle, busses or as well as cycle also. We also used some birds such as we are traveling with a speed of 50 meters per second or you say that you ride a bicycle at a very high rate. These all things of traveling by any means of Transport based on the velocity of that object.
Some essential types of questions and topics are constantly asked in many examinations. these questions may be in the form of such as What is Velocity, the formula of velocity, velocity definition, how to find velocity, what unit for velocity, velocity calculation, how to calculate velocity, the formula for velocity in physics, velocity definition physics, what is the velocity in physics, etc. So I think you will understand that you will not skip this article and read this whole to make your concept very clear in the concepts of velocity. Please read this article on what is velocity from the start to the end paragraph because this will be much beneficial for you.
What is Velocity (Definition)?
Velocity indicates how fast or slow the body is moving or running. By scientific definition, the rate of change of the state of any object with respect to time is called velocity. It is a vector quantity which means to define it we need both, the magnitude as well as the direction also. the magnitude (numerical value) of the velocity is called the speed of the object. The meter per second is the SI unit of the velocity.
When any object starts its motion at t equal to zero time, its velocity is called initial velocity and after gaining acceleration, the body reached a maximum speed that is called its final velocity.
The formula of the velocity
Velocity is a change of position per unit of time and the formula of the velocity is given below:
V = d/t
- d represents the displacement
- V represents the velocity of the object.
- t represents the time interval
Step to find the object’s final velocity?
We will describe here some important stages So that by using the steps you can find the velocity of any moving body. Please read our article on what is velocity from start to end to understand the whole concept of velocity.
- First of all, you have to determine the original velocity of the object by calculating the total displacement of the object and dividing this displacement by the given time interval. by using the formula described you can find the velocity of that object.
- In the second step if your body is accelerating then you have to find its acceleration. to find the acceleration you can divide the force acting on the body by its mass. And the acceleration you got multiply it with the time interval to get the velocity
- Now in the third stage, add both the velocity you find out in step one and step second. so that you can get the fan the velocity of the object.
Units of the Velocity
There are many types of units of velocity in different types of unit systems. Some of them are given below:
- SI unit of the velocity is meters per second.
- In the CGS unit, the velocity unit is centimeters per second.
- In the FPS Unit system feet per second.
- The dimension of the velocity is LT-1. Some other units of velocity are mph.
Examples of Velocity
In this paragraph of the article, what is velocity, we will go to tell you about some examples of velocity. Here we are going to describe some solved examples let’s read them to understand the concepts.
Example 1. A car is in a 5000 m race, and it passes the starting line at its maximum speed. After it, the car reaches the final line within exactly 1 min and twenty seconds. Calculate velocity of car in meter per second.
Solution. Initial position = 0
Final position = 5000m
The time taken by the racing car to travel a 1000 m distance, t = 60 + 20 sec = 80 sec.
From the formula of velocity: (5000-0)/80 m/s
The final velocity comes out to be 62.5m/sec
Example 2. A rocket ascends 250 meters in 6 seconds. Find out the velocity of that rocket.
Solution. We have given here the height of 250 m and the time interval is 6 seconds.
From the formula of the velocity: 250/6
The velocity comes out to be 41.67m/sec.
Some frequently asked questions (FAQs)
Ques. What is meant by instantaneous velocity?
Ans. The velocity of any object at a specific moment (time) is called the instantaneous velocity.
Ques. What is the SI unit of the velocity?
Ans. SI unit of the velocity is meters per second.
Ques. Velocity is which type of quantity?
Ans. velocity has both magnitude and as well as direction so it is a vector quantity.
The main conclusion of today’s article on what is velocity is that you have read many things about velocity. Also described some very important questions in our articles such as the example of velocity, in physics what is velocity, examples of velocity, velocity si unit, how to calculate velocity in physics, types of velocity, how to compute velocity, If you have any query or you want to give any suggestion related to this article, you are free to comment below in the comment section box. we will try to contact you as soon as possible.
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Welcome to the world of pi :-). So far in Geometry you’ve examined the properties of shapes with straight edges. Just like those shapes can be broken down into their components, so can circles. Circles are a big deal as you move into Algebra II and Pre-Calculus, so it’s important to get a good understanding of this topic to carry with you for success in future courses. To begin, an arc is a portion of the circumference, or a curved line that runs along the edge of the circle. A chord is any straight line that passes through the circle with endpoints on the circumference. If a chord passes through the center of the circle and bisects it, it is called the diameter.
Minor Arc of a Circle and Major Arc of a Circle:
A minor arc is defined as the shorter of the two arcs formed when a circle is divided by a chord. The length of a minor arc is less than half the circumference of the circle. Conversely, a major arc is defined as the longer of two arcs formed when a circle is divided by a chord. The major arc is always longer than the minor arc and measures more than half the circumference of the circle.
How to Measure the Length of an Arc:
A central angle is an angle whose vertex is on the center point of the circle.
To say that an angle subtends an arc means that it’s rays pas through the endpoints of that arc. Both these ideas are important when measuring arcs. First the ratio of the central angle to the total angle of the circle (360 degrees) must be determined. The formula for the measure of an arc is this ratio times the total circumference of the circle.
Tangent Radius Theorem:
A tangent line is an external line that intersects the circumference of a circle in exactly one point. The Tangent Radius Theorem establishes a relationship between a radius of a circle and a tangent line drawn to that circle at the point of tangency (where the tangent line and the circle intersect). According to this theorem, the radius at the point of tangency is perpendicular to the tangent line.
Common tangents refer to lines that are tangent to two or more circles simultaneously. There are two types of common tangents: internal and external. Internal common tangents are located between the circles, while external common tangents are found outside the circles. The number of common tangents between two circles depends on whether or not they intersect, and if so, how they intersect. For example, when two circles do not intersect, there are four common tangents – two internal, and two external. If the circles touch externally exactly once, there are three common tangents – one internal, and two external. If the circles overlap, there are only two common tangents – both external. Last, if they completely overlap, there is only one external common tangent.
Lines Intersecting Outside a Circle:
Lines that intersect outside a circle create interesting configurations that give rise to a series of angles and geometric relationships. For instance, the angle formed by a secant and a tangent line drawn from an external point to the circle is equal to half the difference of the intercepted arcs. The same holds true for angles formed by two secant lines, or two tangent lines.
Segment Products Theorem:
The Segment Products Theorem, also known as the Power of a Point Theorem, establishes a relationship between the lengths of segments formed by two intersecting chords or secants within a circle. It states that for a circle and a point not on the circumference of the circle, the products of the lengths of the two segments is constant along any line through the point and the circle.
Equation of a Circle:
The equation of a circle is usually where your Algebra II class will pick up next year at the beginning of the school year. It is a mathematical representation that describes the set of all points in a plane that are equidistant from a fixed point, known as the center. The distance from the center to any point on the circle is called the radius.
The general form of the equation of a circle with center (h,k) and radius r that touches the circumference at (x,y) is given by:
(x – h)^2 + (y – k)^2 = r^2
This equation is derived from the Pythagorean theorem, where the square of the distance between any point on the circle and the center is equal to the square of the radius.
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Many statistical and econometric procedures depend on the assumption of normality. The importance of the normal distribution lies in the fact that sums/averages of random variables tend to be approximately normally distributed regardless of the distribution of draws. The central limit theorem explains this fact. Central Limit Theorem is very important since it provides justification for most of statistical inference. The goal of this paper is to provide a pedagogical introduction to present the CLT, in form of self study computer exercise. This paper presents a student friendly illustration of functionality of central limit theorem. The mathematics of theorem is introduced in the last section of the paper.
CENTRAL LIMIT THEOREM
We start by an example where we observe a phenomenon and than we will discuss the theoretical background of the phenomenon.
Consider 10 players playing with identical dice simultaneously. Each player rolls the dice large number of times. The six numbers on the dice have equal probability of occurrence on any roll and before any player. Let us ask computer to generate data that resembles with the outcomes of these rolls.
We need to have Microsoft Excel ( above 2007 preferable) for this exercise. Point to ‘Data’ tab in the menu bar, it should show ‘Data Analysis’ in the tools bar. If Data Analysis is not there, than you need to install the data analysis tool pack, for this you have to click on the office button, which is the yellow color button at top left corner of Microsoft Excel Window. Choose ‘Add Ins’ from the left pan that appears, than check the box against ‘Analysis Tool Pack’ and click OK.
|Select Office Button Excel Options
|Select Add Ins Þ Analysis ToolPack ÞGo from the screen that appears
Computer will take few moments to install the analysis toolpack. After installation is done, you will see ‘Data Analysis’ on pointing again to Data Tab in the menu bar. The analysis tool pack provides a variety of tool for statistical procedures.
We will generate data that matches with the situation described above using this tool pack.
Open an Excel spread sheet, write 1, 2, 3,…6 in cells A1:A6,
Write ‘=1/6’ in cell B1 and copy it down
This shows you possible outcomes of roll of dice and their probabilities.
This will show you following table:
Here first column contain outcomes of roll of dice and second column contain probability of outcomes. Now we want the computer to have some draws from this distribution. That is, we want computer to roll dice and record outcomes.
For this go to Data Þ Data AnalysisÞ Random Number Generation and select discrete distribution. Write number of variables =10 and number of random number =1000, enter value input and probability range A1:B6, put output range D1 and click OK.
This will generate a 1000×10 matrix of outcomes of roll of dice in cells A8:J1007. Each column represent outcome for a certain player in 1000 draws whereas rows represent outcomes for 10 players in some particular draw. In the next column ‘K’ we want to have sum of each row. Write ‘=SUM(A8:J8) and copy it down. This will generate column of sum for each draw.
Now, we are interested in knowing that what distribution of outcome for each player is:
Let us ask Excel to count the frequency of each outcome for player 1. Choose Tools/Data Analysis/Histogram and fill the dialogue box as follows:
The screenshot shows the dialogue box filled to count the frequency of outcomes listed observed by player A. The input range is the column for which we want to count frequency of outcomes and bin range is the range of possible outcomes. This process will generate frequency of six possible outcomes for the single player. When we did this, we got following output:
The table above gives the frequency of the outcomes whereas same frequencies are plotted in the bar chart. You observe that frequency of occurrence is not approximately equal. The height of vertical bars is approximately same. This implies that the distribution of draws is almost uniform. And we know this should happen because we made draws from uniform distribution. If we calculate percentage of each outcome it will become 15.5%, 15.4%, 16%, 16.9%, 17.9% and 18.3% respectively. These percentages are close to the probability of these outcomes i.e. 16.67%.
Now we want to check the distribution of column which contain sum of draws for 10 players, i.e. the column K. Now the range of possible values of column of sum varies from 10 to 60 (if all column have 1, the sum would be 10 and if all columns have 6 than sum would be 60, in all other cases it would be between these two numbers). It would be in-appropriate to count frequencies of all numbers in this range. Let us make few bins and count the frequencies of these bins. We choose following bins; (10,20), (20, 30),…(50, 60). Again we would ask Excel to count frequencies of these bins. To do this, write 10, 20,…60 in column M of Excel spread sheet (these numbers are the boundaries of bins we made). Now select Tools/Data Analysis/Histogram and fill the dialogue box that appears.
The input range would be the range that contains sum of draws i.e. K8 to K1007 and bin range would be the address of cells where we have written the boundary points of our desired bins. Completing this procedure would produce the frequencies of each bin. Here is the output that we got from this exercise.
First row of this output tells that there was no number smaller than starting point of first bin i.e. smaller than 10, and 2nd, 3rd …rows tell frequencies of bins (10-20), (20,30),…respectively. Last row informs about frequency of numbers larger than end point of last bin i.e. 60.
Below is the plot of this frequency table.
Obviously this plot has no resemblance with uniform distribution. Rather if you remember famous bell shape of the normal distribution, this plot is closer to that shape.
Let us summarize our observation out of this experiment. We have several columns of random numbers that resemble roll of dice i.e. possible outcomes are 1…6 each with probability 1/6 (uniform distribution). If we count frequency of these outcomes in any column, the outcomes reveal the distributional shape and the histogram is almost uniform. Last column was containing sum of 10 draws from uniform distribution and we saw that distribution of this column is no longer uniform, rather it has closer match with shape of normal distribution.
Explanation of the observation:
The phenomenon that we observed may be explained by central limit theorem. According to central limit, let be independent draws from any distribution (not necessarily uniform) with finite variance, than distribution of sum of draws and average of draws would be approximately normal if sample size ‘n’ is large.
Mean and SE for sum of draws:
From our primary knowledge about random variables we know that:
Let , than and
These two statements tell the parameters of normal distribution that emerges from sum of random numbers and we have observed this phenomenon described above.
Consider the exercise discussed above; column A:J are draws from dice roll with expectation 3.5 and variance 2.91667. Column K is sum of 10 previous columns. Thus expected value of K is thus 10*3.5=35 and variance 2.91667*10. This also implies that SE of column K is 5.400 (square root of variance.
The SD and variance in the above exercise can be calculated as follows:
Write ‘AVERAGE(K8:K1007)’ in any blank cell in spreadsheet. This will calculate sample mean of numbers in column K. The answer will be close to 35. When I did this, I found 34.95.
Write ‘VAR(K8:K1007)’ in any blank cell in spreadsheet. This will calculate sample variance of numbers in column K. The answer will be close to 29.16, when I did this, I found 30.02
In this exercise, we observed that if we take draws from some certain distribution, the frequency of draws will reflect the probability structure of parent distribution. But when we take sum of draws, the distribution of sum reveals the shape of normal distribution. This phenomenon has its root in central limit theorem which is stated in Section …..
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By the end of this section, you will be able to do the following:
- Observe the kinematics of rotational motion
- Derive rotational kinematic equations
- Evaluate problem solving strategies for rotational kinematics
Just by using our intuition, we can begin to see how rotational quantities like , , and are related to one another. For example, if a motorcycle wheel has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. In more technical terms, if the wheel's angular acceleration is large for a long period of time , then the final angular velocity and angle of rotation are large. The wheel's rotational motion is exactly analogous to the fact that the motorcycle's large translational acceleration produces a large final velocity, and the distance traveled will also be large.
Kinematics is the description of motion. The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time. Let us start by finding an equation relating , , and . To determine this equation, we recall the following kinematic equation for translational, or straight-line, motion.
Note that in rotational motion , and we shall use the symbol for tangential or linear acceleration from now on. As in linear kinematics, we assume is constant, which means that angular acceleration is also a constant, because . Now, let us substitute and into the linear equation above using
The radius cancels in the equation, yielding
where is the initial angular velocity. This last equation is a kinematic relationship among , , and —that is, it describes their relationship without reference to forces or masses that may affect rotation. It is also precisely analogous in form to its translational counterpart.
Kinematics for rotational motion is completely analogous to translational kinematics, first presented in One-Dimensional Kinematics. Kinematics is concerned with the description of motion without regard to force or mass. We will find that translational kinematic quantities, such as displacement, velocity, and acceleration have direct analogs in rotational motion.
Starting with the four kinematic equations we developed in One-Dimensional Kinematics, we can derive the following four rotational kinematic equations, presented together with their translational counterparts.
|(constant , )
|(constant , )
|(constant , )
In these equations, the subscript 0 denotes initial values (, , and are initial values), and the average angular velocity and average velocity are defined as follows
The equations given above in Table 10.2 can be used to solve any rotational or translational kinematics problem in which and are constant.
Problem-solving Strategy for Rotational Kinematics
- Examine the situation to determine that rotational kinematics or motion is involved. Rotation must be involved, but without the need to consider forces or masses that affect the motion.
- Identify exactly what needs to be determined in the problem (identify the unknowns). A sketch of the situation is useful.
- Make a list of what is given or can be inferred from the problem as stated (identify the knowns).
- Solve the appropriate equation or equations for the quantity to be determined (the unknown). It can be useful to think in terms of a translational analog because by now you are familiar with such motion.
- Substitute the known values along with their units into the appropriate equation, and obtain numerical solutions complete with units. Be sure to use units of radians for angles.
- Check your answer to see if it is reasonable: Does your answer make sense?
Example 10.3 Calculating the Acceleration of a Fishing Reel
A deep-sea fisherman hooks a big fish that swims away from the boat pulling the fishing line from his fishing reel. The whole system is initially at rest and the fishing line unwinds from the reel at a radius of 4.50 cm from its axis of rotation. The reel is given an angular acceleration of for 2.00 s as seen in Figure 10.8.
(a) What is the final angular velocity of the reel?
(b) At what speed is fishing line leaving the reel after 2 s elapse?
(c) How many revolutions does the reel make?
(d) How many meters of fishing line come off the reel in this time?
In each part of this example, the strategy is the same as it was for solving problems in linear kinematics. In particular, known values are identified and a relationship is then sought that can be used to solve for the unknown.
Solution for (a)
Here and are given and needs to be determined. The most straightforward equation to use is because the unknown is already on one side and all other terms are known. That equation states that
We are also given that (it starts from rest), so that
Solution for (b)
Now that is known, the speed can most easily be found using the relationship
where the radius of the reel is given to be 4.50 cm; thus,
Note again that radians must always be used in any calculation relating linear and angular quantities. Also, because radians are dimensionless, we have .
Solution for (c)
Here, we are asked to find the number of revolutions. Because , we can find the number of revolutions by finding in radians. We are given and , and we know is zero, so that can be obtained using .
Converting radians to revolutions gives
Solution for (d)
The number of meters of fishing line is , which can be obtained through its relationship with
This example illustrates that relationships among rotational quantities are highly analogous to those among linear quantities. We also see in this example how linear and rotational quantities are connected. The answers to the questions are realistic. After unwinding for two seconds, the reel is found to spin at 220 rad/s, which is 2,100 rpm. No wonder reels sometimes make high-pitched sounds. The amount of fishing line played out is 9.90 m, about right for when the big fish bites.
Example 10.4 Calculating the Duration When the Fishing Reel Slows Down and Stops
Now let us consider what happens if the fisherman applies a brake to the spinning reel, achieving an angular acceleration of . How long does it take the reel to come to a stop?
We are asked to find the time for the reel to come to a stop. The initial and final conditions are different from those in the previous problem, which involved the same fishing reel. Now we see that the initial angular velocity is and the final angular velocity is zero. The angular acceleration is given to be . Examining the available equations, we see all quantities but t are known in making it easiest to use this equation.
The equation states
We solve the equation algebraically for t, and then substitute the known values as usual, yielding
Note that care must be taken with the signs that indicate the directions of various quantities. Also, note that the time to stop the reel is fairly small because the acceleration is rather large. Fishing lines sometimes snap because of the accelerations involved, and fishermen often let the fish swim for a while before applying brakes on the reel. A tired fish will be slower, requiring a smaller acceleration.
Example 10.5 Calculating the Slow Acceleration of Trains and Their Wheels
Large freight trains accelerate very slowly. Suppose one such train accelerates from rest, giving its 0.350-m-radius wheels an angular acceleration of . After the wheels have made 200 revolutions with no slippage: (a) How far has the train moved down the track? (b) What are the final angular velocity of the wheels and the linear velocity of the train?
In part (a), we are asked to find , and in (b) we are asked to find and . We are given the number of revolutions , the radius of the wheels , and the angular acceleration .
Solution for (a)
The distance is very easily found from the relationship between distance and rotation angle.
Solving this equation for yields
Before using this equation, we must convert the number of revolutions into radians, because we are dealing with a relationship between linear and rotational quantities.
Now we can substitute the known values into to find the distance the train moved down the track.
Solution for (b)
We cannot use any equation that incorporates to find , because the equation would have at least two unknown values. The equation will work, because we know the values for all variables except .
Taking the square root of this equation and entering the known values gives
We can find the linear velocity of the train, , through its relationship to .
The distance traveled is fairly large and the final velocity is fairly slow (just under 32 km/h).
There is translational motion even for something spinning in place, as the following example illustrates. Figure 10.9 shows a fly on the edge of a rotating microwave oven plate. The example below calculates the total distance it travels.
Example 10.6 Calculating the Distance Traveled by a Fly on the Edge of a Microwave Oven Plate
A person decides to use a microwave oven to reheat some lunch. In the process, a fly accidentally flies into the microwave and lands on the outer edge of the rotating plate and remains there. If the plate has a radius of 0.15 m and rotates at 6 rpm, calculate the total distance traveled by the fly during a 2 min cooking period. Ignore the start-up and slow-down times.
First, find the total number of revolutions , and then the linear distance traveled. can be used to find because is given to be 6.0 rpm.
Entering known values into gives
As always, it is necessary to convert revolutions to radians before calculating a linear quantity like from an angular quantity like .
Now, using the relationship between and , we can determine the distance traveled.
Quite a trip, if it survives! Note that this distance is the total distance traveled by the fly. Displacement is actually zero for complete revolutions because they bring the fly back to its original position. The distinction between total distance traveled and displacement was first noted in One-Dimensional Kinematics.
Rotational kinematics has many useful relationships, often expressed in equation form. Are these relationships laws of physics or are they simply descriptive? Hint—the same question applies to linear kinematics.
Rotational kinematics, like linear kinematics, is descriptive and does not represent laws of nature. With kinematics, we can describe many things to great precision but kinematics does not consider causes. For example, a large angular acceleration describes a very rapid change in angular velocity without any consideration of its cause.
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Acceleration is the rate at which velocity changes over time. It is a vector quantity, meaning it has both magnitude (the amount of change) and direction. You can use our acceleration calculator to determine the acceleration of an object.
Acceleration Conversions Calculator
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To calculate acceleration, you need to know the object’s velocity at two different times. The first step is to find the difference in velocity (change in velocity) between the two times. Then, divide this number by the time interval between those two points—the average acceleration over that time.
Remember that acceleration is a vector quantity, so you must also specify the direction of the change in velocity. The final answer will be a vector with both magnitude and direction. Use our calculator to determine the acceleration of an object today! The SI unit for acceleration is meters per second squared (m/s2).
You can calculate acceleration using the following equation:
a = (vf – vi) / t
a is the acceleration,
VF is the final velocity,
vi is the initial velocity, and
t is the time interval during which the change in velocity takes place.
How is Acceleration Different From Velocity?
There is a big difference between acceleration and velocity. Velocity is a measure of how fast something is moving, while the acceleration is a measure of how quickly that velocity changes. In other words, acceleration is the rate at which velocity changes.
How to Calculate Acceleration?
To calculate acceleration, you need to know the velocity (v) and the time it took to reach that velocity (t). You can use a calculator to input these values and find the acceleration. The formula for acceleration is: a = (v-u)/t. Where v is the final velocity, u is the initial velocity, and t is the time in seconds. To find acceleration, you need to consider the motion of an object. If you know the initial and final velocity, then you can use the formula to calculate the acceleration. If you only have the final velocity, you can still calculate the acceleration by using the formula: a = v/t.
How to Use the Acceleration Calculator?
An acceleration calculator is a tool that can use to calculate the acceleration of an object. To use the calculator, you will need to input the object’s velocity and the initial and final velocity. Once you have input all this information, you can calculate the acceleration by pressing the “Calculate” button.
The physics behind the acceleration calculator is that acceleration is defined as the rate at which an object changes its velocity. Velocity is a vector quantity, meaning it has both magnitude and direction. The magnitude of velocity is the speed of an object, while the direction is the direction in which it is moving.
What Are The Units for Particle Acceleration?
The units of particle acceleration are the standard international (SI) meters per second squared (m/s2). These units are used to measure the rate at which the velocity of a particle changes over time.
How to Find Centripetal Acceleration?
To find centripetal acceleration, you need to know the velocity of the object (in m/s) and the radius of the curve (in meters). With this information, you can use the following formula to calculate centripetal acceleration:
a = v2/r
This formula is derived from physics principles and allows you to calculate centripetal acceleration accurately.
What Is the Acceleration Formula?
The acceleration formula is the equation used to calculate an object’s acceleration. The formula is as v = u + at, where v is the final velocity, u is the initial velocity, and a is the acceleration. To calculate the acceleration, you need to know the direction of the object’s movement.
What is The Average Acceleration
Average acceleration is the rate at which an object’s velocity changes with time. It is the average of the object’s acceleration over some time.
How to Calculate Acceleration Due to Gravity
The acceleration due to gravity is the force that the Earth’s gravity exerts on an object. It is equal to the force of gravity divided by the object’s mass. The acceleration due to gravity is 9.8 m/s^2.
$v_i= 1 \ m/s$, $v_f$= ?
Assuming that the initial velocity, vi, is 1 m/s and the final velocity, vf, is unknown, we can use the equation for uniform acceleration to solve for vf. The equation for uniform acceleration is given by: vf = vi + at. In this equation, a is the acceleration, and t is the time interval over which the acceleration occurs. If we plug in the known values for vi and t (1 m/s and 1 second, respectively), we get vf = 1 + a(1). This equation tells us that the final velocity, vf, is equal to the initial velocity, vi, plus the product of the acceleration and time interval.
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Data structures form the backbone of efficient programming. When it comes to acing data structure assignments, seeking online help can be a game-changer. Let’s explore how data structure assignment help online can enhance your understanding and problem-solving skills in this crucial area.
What Is Data Structure?
A data structure is a fundamental concept in computer science. It refers to the organization and arrangement of data in a way that allows many benefits; –
- efficient storage
It defines the relationships between data elements, enabling algorithms to work effectively on the data.
In essence, a data structure provides a framework for organizing and storing data in a computer’s memory or on external storage devices. This organization is crucial for optimizing operations such as searching, sorting, and performing various computations on the data.
Data Types in Data Structure
Data types in data structures define the type of data that a variable can hold and the operations that can be performed on it. Different programming languages support various data types, each with specific characteristics and limitations. Here are some common data types used in data structures:
It includes positive, negative, or zero.
Float data types are used to represent numbers with fractional parts (decimal points). They can store both small and large numbers with decimal precision.
Character data types are used to store individual characters, such as letters, digits, or special symbols.
Strings are sequences of characters and are used to represent text or words. They are a fundamental data type in many programming languages.
Boolean data types can hold only two values: true or false. Mostly bool is used in conditional statements and logical operations.
An array is a collection of elements of the same data type, accessed using an index or a key. It helps for making an efficient storage and retrieval of data.
A linked list is a data structure that consists of a sequence of nodes, where each node contains a value and a reference to the next node in the sequence. It provides dynamic memory allocation.
A stack is a data structure that follows the Last In, First Out (LIFO) principle. One can add Elements or remove them only from the top of the stack.
A queue is a data structure that follows the First In, First Out (FIFO) principle. Elements are added at the rear end and removed from the front end.
A tree is a hierarchical data structure that consists of nodes connected by edges. It is widely used for organizing and representing hierarchical relationships.
A graph is a collection of nodes (vertices) and edges that connect pairs of nodes. It is a versatile data structure used to model various real-world scenarios.
Understanding the significance of data types is pivotal in crafting efficient and powerful programs. They serve as the foundational elements for constructing intricate data structures and devising algorithms. To ensure you make informed decisions about data types in your programming endeavors, seeking guidance from data structure assignment help online is a must! So, Assignment world has specialized assistance that offers expert insights and tailored solutions, empowering you to tackle assignments with confidence and precision.
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Also Read: How to Install Android Apps on Windows 10
Incorporating online data structure assignment help into your learning arsenal is a strategic move towards mastering this critical programming aspect. The accessibility, expertise, and customized learning experience offered by online resources are invaluable. Embrace this support to not only excel in assignments but also to cultivate a deep-seated understanding of data structures.
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By the end of this section, you will be able to:
Nuclear chemistry is the study of reactions that involve changes in nuclear structure. The chapter on atoms, molecules, and ions introduced the basic idea of nuclear structure, that the nucleus of an atom is composed of protons and, with the exception of neutrons. Recall that the number of protons in the nucleus is called the atomic number (Z) of the element, and the sum of the number of protons and the number of neutrons is the mass number (A). Atoms with the same atomic number but different mass numbers are isotopes of the same element. When referring to a single type of nucleus, we often use the term nuclide and identify it by the notation where X is the symbol for the element, A is the mass number, and Z is the atomic number (for example, Often a nuclide is referenced by the name of the element followed by a hyphen and the mass number. For example, is called “carbon-14.”
Protons and neutrons, collectively called nucleons, are packed together tightly in a nucleus. With a radius of about 10−15 meters, a nucleus is quite small compared to the radius of the entire atom, which is about 10−10 meters. Nuclei are extremely dense compared to bulk matter, averaging 1.8 1014 grams per cubic centimeter. For example, water has a density of 1 gram per cubic centimeter, and iridium, one of the densest elements known, has a density of 22.6 g/cm3. If the earth’s density were equal to the average nuclear density, the earth’s radius would be only about 200 meters (earth’s actual radius is approximately 6.4 106 meters, 30,000 times larger). Example 11.1 demonstrates just how great nuclear densities can be in the natural world.
(a) What is the density of this neutron star?
(b) How does this neutron star’s density compare to the density of a uranium nucleus, which has a diameter of about 15 fm (1 fm = 10–15 m)?
(a) The radius of the neutron star is so the density of the neutron star is:
(b) The radius of the U-235 nucleus is so the density of the U-235 nucleus is:
These values are fairly similar (same order of magnitude), but the neutron star is more than twice as dense as the U-235 nucleus.
The density of the neutron star is 3.4 1018 kg/m3. The density of a hydrogen nucleus is 6.0 1017 kg/m3. The neutron star is 5.7 times denser than the hydrogen nucleus.
To hold positively charged protons together in the very small volume of a nucleus requires very strong attractive forces because the positively charged protons repel one another strongly at such short distances. The force of attraction that holds the nucleus together is the strong nuclear force. (The strong force is one of the four fundamental forces that are known to exist. The others are the electromagnetic force, the gravitational force, and the nuclear weak force.) This force acts between protons, between neutrons, and between protons and neutrons. It is very different from the electrostatic force that holds negatively charged electrons around a positively charged nucleus (the attraction between opposite charges). Over distances less than 10−15 meters and within the nucleus, the strong nuclear force is much stronger than electrostatic repulsions between protons; over larger distances and outside the nucleus, it is essentially nonexistent.
A nucleus is stable if it cannot be transformed into another configuration without adding energy from the outside. Of the thousands of nuclides that exist, about 250 are stable. A plot of the number of neutrons versus the number of protons for stable nuclei reveals that the stable isotopes fall into a narrow band. This region is known as the band of stability (also called the belt, zone, or valley of stability). The straight line in Figure 11.1 represents nuclei that have a 1:1 ratio of protons to neutrons (n:p ratio). Note that the lighter stable nuclei, in general, have equal numbers of protons and neutrons. For example, nitrogen-14 has seven protons and seven neutrons. Heavier stable nuclei, however, have increasingly more neutrons than protons. For example: iron-56 has 30 neutrons and 26 protons, an n:p ratio of 1.15, whereas the stable nuclide lead-207 has 125 neutrons and 82 protons, an n:p ratio equal to 1.52. This is because larger nuclei have more proton-proton repulsions, and require larger numbers of neutrons to provide compensating strong forces to overcome these electrostatic repulsions and hold the nucleus together.
The nuclei that are to the left or to the right of the band of stability are unstable and exhibit radioactivity. They change spontaneously (decay) into other nuclei that are either in, or closer to, the band of stability. These nuclear decay reactions convert one unstable isotope (or radioisotope) into another, more stable, isotope. We will discuss the nature and products of this radioactive decay in subsequent sections of this chapter.
Several observations may be made regarding the relationship between the stability of a nucleus and its structure. Nuclei with even numbers of protons, neutrons, or both are more likely to be stable (see Table 11.1). Nuclei with certain numbers of nucleons, known as magic numbers, are stable against nuclear decay. These numbers of protons or neutrons (2, 8, 20, 28, 50, 82, and 126) make complete shells in the nucleus. These are similar in concept to the stable electron shells observed for the noble gases. Nuclei that have magic numbers of both protons and neutrons, such as and are called “double magic” and are particularly stable. These trends in nuclear stability may be rationalized by considering a quantum mechanical model of nuclear energy states analogous to that used to describe electronic states earlier in this textbook. The details of this model are beyond the scope of this chapter.
Stable Nuclear Isotopes
|Number of Stable Isotopes
By the end of this section, you will be able to:
Changes of nuclei that result in changes in their atomic numbers, mass numbers, or energy states are nuclear reactions. To describe a nuclear reaction, we use an equation that identifies the nuclides involved in the reaction, their mass numbers and atomic numbers, and the other particles involved in the reaction.
Many entities can be involved in nuclear reactions. The most common are protons, neutrons, alpha particles, beta particles, positrons, and gamma rays, as shown in Figure 11.2. Protons also represented by the symbol and neutrons are the constituents of atomic nuclei, and have been described previously. Alpha particles also represented by the symbol are high-energy helium nuclei. Beta particles also represented by the symbol are high-energy electrons, and gamma rays are photons of very high-energy electromagnetic radiation. Positrons also represented by the symbol are positively charged electrons (“anti-electrons”). The subscripts and superscripts are necessary for balancing nuclear equations, but are usually optional in other circumstances. For example, an alpha particle is a helium nucleus (He) with a charge of +2 and a mass number of 4, so it is symbolized This works because, in general, the ion charge is not important in the balancing of nuclear equations.
Note that positrons are exactly like electrons, except they have the opposite charge. They are the most common example of antimatter, particles with the same mass but the opposite state of another property (for example, charge) than ordinary matter. When antimatter encounters ordinary matter, both are annihilated and their mass is converted into energy in the form of gamma rays (γ)—and other much smaller subnuclear particles, which are beyond the scope of this chapter—according to the mass-energy equivalence equation E = mc2, seen in the preceding section. For example, when a positron and an electron collide, both are annihilated and two gamma ray photons are created:
As seen in the chapter discussing light and electromagnetic radiation, gamma rays compose short wavelength, high-energy electromagnetic radiation and are (much) more energetic than better-known X-rays that can behave as particles in the wave-particle duality sense. Gamma rays are a type of high energy electromagnetic radiation produced when a nucleus undergoes a transition from a higher to a lower energy state, similar to how a photon is produced by an electronic transition from a higher to a lower energy level. Due to the much larger energy differences between nuclear energy shells, gamma rays emanating from a nucleus have energies that are typically millions of times larger than electromagnetic radiation emanating from electronic transitions.
A balanced chemical reaction equation reflects the fact that during a chemical reaction, bonds break and form, and atoms are rearranged, but the total numbers of atoms of each element are conserved and do not change. A balanced nuclear reaction equation indicates that there is a rearrangement during a nuclear reaction, but of nucleons (subatomic particles within the atoms’ nuclei) rather than atoms. Nuclear reactions also follow conservation laws, and they are balanced in two ways:
If the atomic number and the mass number of all but one of the particles in a nuclear reaction are known, we can identify the particle by balancing the reaction. For instance, we could determine that is a product of the nuclear reaction of and if we knew that a proton, was one of the two products. Example 11.2 shows how we can identify a nuclide by balancing the nuclear reaction.
where A is the mass number and Z is the atomic number of the new nuclide, X. Because the sum of the mass numbers of the reactants must equal the sum of the mass numbers of the products:
Similarly, the charges must balance, so:
Check the periodic table: The element with nuclear charge = +13 is aluminum. Thus, the product is
Following are the equations of several nuclear reactions that have important roles in the history of nuclear chemistry:
By the end of this section, you will be able to:
Following the somewhat serendipitous discovery of radioactivity by Becquerel, many prominent scientists began to investigate this new, intriguing phenomenon. Among them were Marie Curie (the first woman to win a Nobel Prize, and the only person to win two Nobel Prizes in different sciences—chemistry and physics), who was the first to coin the term “radioactivity,” and Ernest Rutherford (of gold foil experiment fame), who investigated and named three of the most common types of radiation. During the beginning of the twentieth century, many radioactive substances were discovered, the properties of radiation were investigated and quantified, and a solid understanding of radiation and nuclear decay was developed.
The spontaneous change of an unstable nuclide into another is radioactive decay. The unstable nuclide is called the parent nuclide; the nuclide that results from the decay is known as the daughter nuclide. The daughter nuclide may be stable, or it may decay itself. The radiation produced during radioactive decay is such that the daughter nuclide lies closer to the band of stability than the parent nuclide, so the location of a nuclide relative to the band of stability can serve as a guide to the kind of decay it will undergo (Figure 11.3).
Ernest Rutherford’s experiments involving the interaction of radiation with a magnetic or electric field (Figure 11.4) helped him determine that one type of radiation consisted of positively charged and relatively massive α particles; a second type was made up of negatively charged and much less massive β particles; and a third was uncharged electromagnetic waves, γ rays. We now know that α particles are high-energy helium nuclei, β particles are high-energy electrons, and γ radiation compose high-energy electromagnetic radiation. We classify different types of radioactive decay by the radiation produced.
Alpha (α) decay is the emission of an α particle from the nucleus. For example, polonium-210 undergoes α decay:
Alpha decay occurs primarily in heavy nuclei (A > 200, Z > 83). Because the loss of an α particle gives a daughter nuclide with a mass number four units smaller and an atomic number two units smaller than those of the parent nuclide, the daughter nuclide has a larger n:p ratio than the parent nuclide. If the parent nuclide undergoing α decay lies below the band of stability (refer to Figure 11.1), the daughter nuclide will lie closer to the band.
Beta (β) decay is the emission of an electron from a nucleus. Iodine-131 is an example of a nuclide that undergoes β decay:
Beta decay, which can be thought of as the conversion of a neutron into a proton and a β particle, is observed in nuclides with a large n:p ratio. The beta particle (electron) emitted is from the atomic nucleus and is not one of the electrons surrounding the nucleus. Such nuclei lie above the band of stability. Emission of an electron does not change the mass number of the nuclide but does increase the number of its protons and decrease the number of its neutrons. Consequently, the n:p ratio is decreased, and the daughter nuclide lies closer to the band of stability than did the parent nuclide.
Gamma emission (γ emission) is observed when a nuclide is formed in an excited state and then decays to its ground state with the emission of a γ ray, a quantum of high-energy electromagnetic radiation. The presence of a nucleus in an excited state is often indicated by an asterisk (*). Cobalt-60 emits γ radiation and is used in many applications including cancer treatment:
There is no change in mass number or atomic number during the emission of a γ ray unless the γ emission accompanies one of the other modes of decay.
Positron emission (β+ decay) is the emission of a positron from the nucleus. Oxygen-15 is an example of a nuclide that undergoes positron emission:
Positron emission is observed for nuclides in which the n:p ratio is low. These nuclides lie below the band of stability. Positron decay is the conversion of a proton into a neutron with the emission of a positron. The n:p ratio increases, and the daughter nuclide lies closer to the band of stability than did the parent nuclide.
Electron capture occurs when one of the inner electrons in an atom is captured by the atom’s nucleus. For example, potassium-40 undergoes electron capture:
Electron capture occurs when an inner shell electron combines with a proton and is converted into a neutron. The loss of an inner shell electron leaves a vacancy that will be filled by one of the outer electrons. As the outer electron drops into the vacancy, it will emit energy. In most cases, the energy emitted will be in the form of an X-ray. Like positron emission, electron capture occurs for “proton-rich” nuclei that lie below the band of stability. Electron capture has the same effect on the nucleus as does positron emission: The atomic number is decreased by one and the mass number does not change. This increases the n:p ratio, and the daughter nuclide lies closer to the band of stability than did the parent nuclide. Whether electron capture or positron emission occurs is difficult to predict. The choice is primarily due to kinetic factors, with the one requiring the smaller activation energy being the one more likely to occur.
Figure 11.5 summarizes these types of decay, along with their equations and changes in atomic and mass numbers.
Positron emission tomography (PET) scans use radiation to diagnose and track health conditions and monitor medical treatments by revealing how parts of a patient’s body function (Figure 11.6). To perform a PET scan, a positron-emitting radioisotope is produced in a cyclotron and then attached to a substance that is used by the part of the body being investigated. This “tagged” compound, or radiotracer, is then put into the patient (injected via IV or breathed in as a gas), and how it is used by the tissue reveals how that organ or other area of the body functions.
For example, F-18 is produced by proton bombardment of 18O and incorporated into a glucose analog called fludeoxyglucose (FDG). How FDG is used by the body provides critical diagnostic information; for example, since cancers use glucose differently than normal tissues, FDG can reveal cancers. The 18F emits positrons that interact with nearby electrons, producing a burst of gamma radiation. This energy is detected by the scanner and converted into a detailed, three-dimensional, color image that shows how that part of the patient’s body functions. Different levels of gamma radiation produce different amounts of brightness and colors in the image, which can then be interpreted by a radiologist to reveal what is going on. PET scans can detect heart damage and heart disease, help diagnose Alzheimer’s disease, indicate the part of a brain that is affected by epilepsy, reveal cancer, show what stage it is, and how much it has spread, and whether treatments are effective. Unlike magnetic resonance imaging and X-rays, which only show how something looks, the big advantage of PET scans is that they show how something functions. PET scans are now usually performed in conjunction with a computed tomography scan.
The naturally occurring radioactive isotopes of the heaviest elements fall into chains of successive disintegrations, or decays, and all the species in one chain constitute a radioactive family, or radioactive decay series. Three of these series include most of the naturally radioactive elements of the periodic table. They are the uranium series, the actinide series, and the thorium series. The neptunium series is a fourth series, which is no longer significant on the earth because of the short half-lives of the species involved. Each series is characterized by a parent (first member) that has a long half-life and a series of daughter nuclides that ultimately lead to a stable end-product—that is, a nuclide on the band of stability (Figure 11.7). In all three series, the end-product is a stable isotope of lead. The neptunium series, previously thought to terminate with bismuth-209, terminates with thallium-205.
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2nd Year Physics Important Questions On Newsongoogle By Bilal Articles
Embark on an emotional odyssey with Bilal Articles on Newsongoogle! Dive into the brilliance of Class 12 Physics—unlock the secrets of every chapter with all-encompassing notes, relive the intensity of past papers, and experience the thrill of uncertainty with Physics guess papers. Brace yourself for a journey of discovery with essential long and short questions for 2nd Year Physics. Let the emotions flow, and let success become your story.
2nd Year Physics Important Questions 1st Lesson Waves
- Write the formula for speed of sound in solids and gases.
- What does a wave represent ?
- Distinguish between transverse and longitudinal waves.
- What are the parameters used to describe a progressive harmonic wave ?
- What is the principle of superposition of waves ? .
- Under what conditions will a wave be reflected ?
- What is the phase difference between the incident and reflected waves when the wave is reflected by a rigid boundary ?
- What is a stationary or standing wave ?
- What do you understand by the terms node’ and ‘antinode’ ?
- What is the distance between a node and an antinode in a stationary wave ?
- What do you understand by ‘natural frequency’ or ‘normal mode of vibration’ ?
- What are harmonics ?
- A string is stretched between two rigid supports. What frequencies of vibration are possible in such a string ?
- If the air column in a long tube, closed at one end, is set in vibration, what harmonics are possible in the vibrating air column ?
- If the air column in a tube, open at both ends, is set in vibration; what harmonics are possible ?
- What are transverse waves ? Give illustrative examples of such waves.
- What are longitudinal waves ? Give illustrative example of such waves.
- What are ‘beats’ ? When do they occur ? Explain their use, if any.
- What is ‘Doppler effect’ ? Give illustrative examples.
- Explain the formation of stationary waves in stretched strings and hence deduce the laws of transverse wave in stretched strings.
- Explain the formation of stationary waves in an air column enclosed in open pipe. Derive the equations for the frequencies of the harmonics produced.
- What is Doppler effect ? Obtain an expression for the apparent frequency of sound heard when the source is in motion with respect to an observer at rest.
2nd Year Physics Important Questions Chapter 2 Ray Optics and Optical Instruments
- What is optical density and how is it different from mass density ?
- What are the laws of reflection through curved mirrors ?
- Define ‘power’ of a convex lens. What is its unit ?
- A concave mirror of focal length 10 cm is placed at a distance 35 cm from a wall. How far from the wall should an object be placed so that its real image is formed on the wall ?
- A concave mirror produces an image of a long vertical pin, placed 40cm from the mirror, at the position of the object. Find the focal length of the mirror.
- A small angled prism of 4° deviates a ray through 2.48°. Find the refractive index of the prism.
- What is ‘dispersion’? Which colour gets relatively more dispersed ?
- The focal length of a concave lens is 30 cm. Where should an object be placed so that its image is 1/10 of its size ?
- What is myopia ? How can it be corrected ?
- What is hypermetropia ? How can it be corrected ?
- Draw neat labelled ray diagram of simple microscope.
- Define focal length of a concave mirror. Prove that the radius of curvature of a concave mirror is double its focal length.
- Define critical angle. Explain total internal reflection using a neat diagram.
- Explain the formation of a mirage.
- Explain the formation of a rainbow.
- Why does the setting sun appear red ?
- With a neat labelled diagram explain the formation of image in a simple microscope.
- What is the position of the object for a simple microscope ? What is the maximum magnifi-cation of a simple microscope for a realistic focal length ?
- Draw a neat labelled diagram of a compound microscope and explain its working. Derive an expression for its magnification.
- Define Snell’s Law. Using a neat labelled diagram derive an expression for the refractive index of the material of an equilateral prism.
- A ray of light, after passing through a medium, meets the surface separating the medium from air at an angle of 45° and is just not refracted. What is the refractive index of the medium ?
- Suppose that the lower half of the concave mirror’s reflecting surface in figure is covered with an opaque (non- reflective) material. What effect will this have on the image of an object p’ iced in front of the mirror ?
- A mobile phone lies along the principal axis of a concave mirror, as shown in Fig. Show by suitable diagram, the formation of its image. Explain why the magnification is not uniform. Will the distortion of image depend on the location of the phone with respect to the mirror ?
2nd Year Physics Important Questions 3rd Lesson Wave Optics
- Explain Doppler effect in light. Distinguish between red shift and blue shift.
- Derive the expression for the intensity at a point where interference of light occurs. Arrive at the conditions for maximum and zero intensity.
- Does the principle of conservation of energy hold for interference and diffraction phenomena? Explain briefly.
- Explain polarisation of light by reflection and arrive at Brewster’s law from it.
- Discuss the intensity of transmitted light when a polaroid sheet is rotated between two crossed polaroids.
- Distinguish between Coherent and Incoherent addition of waves. Develop the theory of constructive interferences.
- Describe Young’s experiment for observing interference and hence arrive at the expression for ‘fringe width’.
- What speed should a galaxy move with respect to us so that the sodium line at 589.0 nm is observed at 589.6 nm ? .
- Unpolarised light is incident on a plane glass surface. What should be the angle of the incidence so that the reflected and refracted rays are perpendicular to each other ?
- What is the Brewster angle for air to glass transition ?
- In Young’s double-slit experiment using monochromatic light of wavelength λ, the intensity of light at a point on the screen where path difference is λ, is K units. What is the intensity of light at a point where path difference is λ/3 ?
2nd Year Physics Important Questions 4th Lesson Electric Charges and Fields
- What is meant by the statement ‘charge is quantized’?
- Repulsion is the sure test of charging than attraction. Why ?
- How many electrons constitute 1 C of charge ?
- What happens to the weight of a body when it is charged positively ?
- What happens to the force between two charges if the distance between them is
- Consider two charges + q and -q placed at B and C of an equilateral triangle ABC. For this system, the total charge is zero. But the electric field (intensity) at A which is equidistant from B and C is not zero. Why ?
- Electrostatic field lines of force do not form closed loops. If they form closed loops then the work done in moving a charge along a closed path will not be zero. From the above two statements can you guess the nature of electrostatic force ?
- State Gauss’s law in electrostatics. [IPE 2015 (TS)]
- When is the electric flux negative and when is it positive ?
- Write the expression for electric intensity due to an infinite long charged wire at a distance radial distance r from the wire.
- Write the expression for electric intensity due to an infinite plane sheet of charge.
- Write the expression for electric intensity due to a charged conducting spherical shell at points outside and inside the shell.
- A proton and an α-particle are released in a uniform electric field. Find the ratio of (a) forces experienced by them (b) accelerations gained by each.
- A hollow sphere of radius ‘r’ has a unifrom charge density ‘σ’. It is kept in a cube of edge 3r such that the center of the cube coincides with the center of the shell. Calculate the electric flux that comes out of a face of the cube.
- Consider a uniform electric field AP Inter 2nd Year Physics Important Questions Chapter 4 Electric Charges and Fields 1. What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the YZ plane ?
- State and explain Coulomb’s inverse square law in electricity.
- Derive an expression for the intensity of the electric field at a point on the axial line of an electric dipole.
- State Gauss’s law in electrostatics and explain its importance.
- State Gauss’s law in electrostatics. Applying Gauss’s law derive the expression for electric intensity due to an infinite plane sheet of charge.
- How can you charge a metal sphere positively without touching it ?
- If 109 electrons move out of a body to another body every second, how much time is required to get a total charge of 1 C on the other body? .
- How much positive and negative charge is there in a cup of.water ?
- Consider the charges q, q and -q placed at the vertices of an equilateral triangle, as shown in Fig. What is the force on each charge ?
- Two charges 10 μC are placed 5.0 mm apart. Determine the electric field at (a) a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge, as shown in Fig. (a) and (b) a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, as shown in Fig.
2nd Year Physics Important Questions Chapter 5 Electrostatic Potential and Capacitance
- Derive an expression for the electric potential due to a point charge.
- Derive an expression for the electrostatic potential energy of a system of two point charges and find its relation with electric potential of a charge.
- Derive an expression for the potential energy of an electric dipole placed in a uniform electric field.
- Derive an expression for the capacitance of a parallel plate capacitor.
- Explain the behaviour of dielectrics in an external field.
- Define electric potential. Derive an expression for the electric potential due to an electric dipole and hence the electric potential at a point (a) the axial line of electric dipole (b) on the equatorial line of electric dipole.
- What is series combination of capacitors. Derive the formula for equivalent capacitance in series combination.
- What is parallel combination of capacitors. Derive the formula for equivalent capacitance in parallel combination.
- Derive an expression for the energy stored in a capacitor.
- What is the energy stored when the space between the plates is filled with dielectric.
- A slab of material of dielectric constant K has the same area as the plates of a parallel plate capacitor but has a thickness (3/4)d, where d is the separation of the plates. How is the capacitance changed when die slab is inserted between the plates ?
- Four charges are arranged at the comets of a square ABCD of side d. as shown in fig. 5.15.(a) Find the work required to put together this arrangement, (b) A charge q0 is brought to the centre of the square, the four charges being held fixed at its comers. How much extra work is needed to do this ?
- How much work is required to separate the two charges infinitely away from each other ?
- A molecule of a substance has a perma-nent electric dipole moment of magnitude 10-29C m. A mole of this substance is polarised (at low temperature) by applying a strong electrostatic field of magnitude 106 V m-1. The direction of the field is suddenly changed by an angle of 60°. Estimate the heat released by the substance in aligning its dipoles along the new direction of the field. For simplicity, assume 100% polarisation of the sample.
- A comb run through one’s dry hair attracts small bits of paper. Why ? What happens if the hair is wet or if it is a rainy day ? (Remember, a paper does not conduct electricity.)
- Ordinary rubber is an insulator. But special rubber types of aircraft are made slightly conducting. Why is this necessary ?
- Vehicles carrying inflammable materials usually have metallic ropes touching the ground during motion. Why?
- To enable them to conduct charge (produced by friction) to the ground; as too much of static electricity accumulated may result in spark and result in fire.
2nd Year Physics Important Questions Chapter 6 Current Electricity
- Derive an expression for the effective resistance when three resistors are connected in (i) series (ii) parallel.
- On what factors does the resistance of a conductor depend ? Define electric resistance and write its S.I unit. How does the resistance of a conductor vary if (a) Conductor is stretched to 4 times of its length (b) Temperature of conductor is increased ?
- State Kirchhoffs law for an electrical network. Using th&se laws deduce the condition for balance in a Wheatstone bridge.
- State the working principle of potentiometer explain with the help of circuit diagram how the emf of two primary cells are compared by using the potentiometer. [A.P. Mar. 16]
- State the working principle of potentiometer. Explain with the help of circuit diagram how the potentiometer is used to determine the internal resistance of the given primary cell. [A.P. & T.S. Mar. 17, 15]
- Under what condition is the heat produced in an electric circuit a)directly proportional b) inversely proportional to the resistance of the circuit ? Compute the ratio of the total quantity of heat produced in the two cases.
- A 10Ω thick wire is stretched so that its length becomes three times. Assuming that there is no change in its density on stretching, calculate the resistance of the stretched wire.
- A wire of resistance 4R is bent in the form of a circle. What is the effective resistance between the ends of the diameter ?
- Three resistors 3Ω, 6Ω and 9Ω are connected to a battery. In which of them will the power of dissipation be maximum if:
- A silver wire has a resistance of 2.1Ω at 27.5°C and a resistance of 2.7Ω at 100°C. Determine the temperature coeff. of resistivity of silver.
2nd Year Physics Important Questions 7th Lesson Moving Charges and Magnetism
- What is the importance of Oersted’s experiment ?
- State Ampere’s law and Biot-Savart’s law.
- Write the expression for the magnetic induction at any point on the axis of a circular current-carrying coil. Hence, obtain an expression for the magnetic induction at the centre of the circular coil.
- A circular coil of radius T having N turns carries a current “i”. What is its magnetic moment ?
- What is the force on a conductor of length L carrying a current “i” placed in a magnetic field of induction B ? When does it become maximum ?
- What is the force on a charged particle of charge “q” moving with a velocity “v” in a uniform magnetic field of induction B ? When does it become maximum ?
- Distinguish between ammeter and voltmeter.
- What is the principle of a moving coil galvanometer ?
- What is the smallest value of current that can be measured with a moving coil galvanometer ?
- How do you convert a moving coil galvanometer into an ammeter ?
- How do you convert a moving coil galvanometer into a voltmeter ?
- What is the relation between the permittivity of free space e0, the permeability of free space m0 and the speed of light In vaccum?
- A current carrying circular loop lies on a smooth horizontal plane. Can a uniform magnetic Held be set up in such a manner that the loop turns about the vertical axis ?
- A wire loop of irregular shape carrying current is placed in an external magnetic field. If the wire is flexible, what shape will the loop change to ? Why ?
- State and explain Biot-Savart’s law.
- State and explain Ampere’s law.
- Find the magnetic induction due to a long current carrying conductor.
- Derive an expression for the magnetic induction of a point on the axis of a current carrying circular coil using Biot-Savart’s law.
- Explain how crossed E and B fields serve as a velocity selector.
- What are the basic components of a cyclotron ? Mention its uses ?
- Derive an expression for the magnetic dipole moment of a revolving electron.
- Deduce an expression for the force on a current carrying conductor placed in a magnetic field. Derive an expression for the force per unit length between two parallel current-carrying conductors.
- Obtain an expression for the torque on a current carrying loop placed in a uniform ‘ magnetic field. Describe the construction and working of a moving coil galvanometer.
- How can a galvanometer be converted to an ammeter ? Why is the parallel resistance smaller that the galvanometer resistance ? A moving coil galvanometer can measure a current of 10-6 A. What is the resistance of the shunt required if it is to measure 1A ?
- How can a galvanometer be converted to a voltmeter ? Why is the series resistance greater that the galvanometer resistance ? A moving coil galvanometer of resistance 5Ω can measure a current of 15mA. What is the series resistance required if it is to measure 1.5V ?
2nd Year Physics Important Questions Chapter 8 Magnetism and Matter
- A magnetic dipole placed in a magnetic field experiences a net force. What can you say about the nature of the magnetic field ?
- Do you find two magnetic field lines intersecting ? Why ?
- What happens to the compass needles at the Earth’s poles ?
- What do you understand by the ‘magnetisation’ of a sample ? Give its SI unit.
- What is the magnetic moment associated with a solenoid ?
- What are the units of magnetic moment, magnetic induction and magnetic field ?
- Magnetic lines form continuous closed loops. Why ?
- Define magnetic declination.
- Define magnetic inclination or angle of dip.
- Classify the following materials with regard to magnetism: Manganese, Cobalt, Nickel, Bismuth, Oxygen, Copper.
- The force between two magnet poles separated by a distance ‘d’ in air is ‘F’. At what distance between them does the force become doubled ?
- If B is the magnetic field produced at the centre of a circular coil of one turn of length L carrying current I then what is the magnetic field at the centre of the same coil which is made into 10 turns ?
- If the number of turns of a solenoid is doubled, keeping the other factors constant, how does the magnetic field at the axis of the solenoid change ?
- A closely wound solenoid of 800 turns and area of cross section 2.5 × 10-4 m2 carries a current of 3.0A. Explain the sense in which the solenoid acts like a bar magnet. What is its associated magnetic moment ?
- Compare the properties of para, dia and ferromagnetic substances.
- Explain the elements of the Earth’s magnetic field and draw a sketch showing the relationship between the vertical component, horizontal component and angle of dip.
- Define magnetic susceptibility of a material. Name two elements one having positive susceptibility and other having negative susceptibility.
- Derive an expression for magnetic field induction on the equatorial line of a barmagnet. [Board Model Paper]
- What do you understand by “hysteresis” ? How does this propertry influence the choice of materials used in different appliances where electromagnets are used ?
- Prove that a bar magnet and a solenoid produce similar fields.
- A small magnetic needle is set into oscillations in a magnetic field B obtain an expression for the time period of oscillation.
- A bar magnet, held horizontally, is set into angular oscillations in the Earth’s magnetic field. It has time periods T1 and T2 at two places, where the angles of dip are θ1 and θ2 respectively. Deduce an expression for the ratio of the resultant magnetic fields at the two places.
- A coil of 20 turns has an area of 800 mm2 and carries a current of 0.5A. If it is placed in a magnetic field of intensity 0.3T with its plane parallel to the field, what is the torque that it experiences ? ,
- In the Bohr atom model the electrons move around the nucleus in circular orbits. Obtain an expression the magnetic moment (p) of the electron in a Hydrogen atom in terms of its angular momentum L.
- A bar magnet of length 0.1m and with a magnetic moment of 5Am2 is placed in a uniform a magnetic field of intensity 0.4T, with its axis making an angle of 60° with the field. What is the torque on the magnet ?
- A solenoid of length 22.5 cm has a total of 900 turns and carries a current of 0.8 A. What is the magnetising field H near the centre and far away from the ends of the solenoid ?
- The horizontal component of the earth’s magnetic field at a certain place is 2.6 × 10-5T and the angle of dip is 60°. What is the magnetic field of the earth at this location ?
- In the magnetic meridian of a certain place, the horizontal component of the earth’s magnetic field is 0.26 G and the dip angle is 60°. What is the magnetic field of the earth at this location ?
- What is the magnitude of the equatorial and axial fields due to a bar magnet of length 8.0 cm at a distance of 50 cm from its mid-point ? The magnetic moment of the bar magnet is 0.40 A m2.
- The earth’s magnetic field at the equator is approximately 0.4 G. Estimate the earth’s dipole moment. .
2nd Year Physics Important Questions Chapter 9 Electromagnetic Induction
- What did the experiments of Faraday and Henry show ?
- Define magnetic flux.
- State Faraday’s law of electromagnetic induction.
- State Lenz’s law.
- What happens to the mechanical energy (of motion) when a conductor is moved in a uniform magnetic field ?
- What are Eddy currents ?
- Define ‘inductance’.
- What do you understand by ‘self inductance’ ?
- Obtain an expression for the emf induced across a conductor which is moved in a uniform magnetic field which is perpendicular to the plane of motion.
- Describe the ways in which Eddy currents are used to advantage.
- Obtain an expression for the mutual inductance of two long co-axial solenoids.
- A wheel with 10 metallic spokes each 0.5 m long is rotated with a speed of 120 rev/min in a plane normal to the horizontal component of earth’s magnetic field HE at a place. If HE = 0.4 G at the place, what is the induced emf between the axle and the rim of the wheel ? (Note that 1 G = 10-4 T.)
- Number of turns in a coil are 100. When a current of 5A is flowing through the coil, the magnetic flux is 10-6Wb. Find the self induction. [Board Model Paper]
- Current in a circuit falls from 5.0 A to 0.0 A in 0.1 s. If an average emf of 200 V is induced, give an estimate of the self-inductance of the circuit. [Mar. 16 (T.S.) Mar. 14]
- A pair of adjacent coils has a mutual inductance of 1.5 H. If the current in one coil changes from 0 to 20 A in 0.5 s, what is the change of flux linkage with the other coil ? [T.S. Mar. 17]
- A jet plane is travelling towards west at a speed of 1800 km/K What is the voltage difference developed between the ends of the wing having a span of 25 m, if the Earth’s magnetic field at the location has a magnitude of 5 × 10-4 T and the dip angle is 30°.
2nd Year Physics Important Questions Chapter 10 Alternating Current
- A transformer converts 200 V ac into 2000 V ac. Calculate the number of turns in the ‘ secondary if the primary has 10 turns.
- What type of transformer is used in a 6V bed lamp ?
- What is the phenomenon involved in the working of a transformer ?
- What is transformer ratio ?
- Write the expression for the reactance of i) an inductor and (ii) a capacitor.
- What is the phase difference between A.C emf and current in the following: Pure resistor, pure inductor and pure capacitor.
- Define power factor. On which factors does power factor depend ?
- What is meant by wattless component of current ?
- When does a LCR series circuit have minimum impedance ?
- What is the phase difference between voltage and current when the power factor in LCR series circuit is unity ?
- State the principle on which a transformer works. Describe the working of a transformer with necessary theory.
- A light bulb is rated at 100W for a 220 V supply. Find
- A pure inductor of 25.0 mH is connected to a source of 220 V. Find the inductive reactance and rms current in the circuit if the frequency of the source is 50 Hz.
- The instantaneous current and instantaneous voltage across a series circuit containing resistance and inductance are given by i =
- What is step up transformer ? How it differs from step down transformer ?
- A pure inductor of 25.0 mH is connected to a source of 220 V. Find the inductive reactance and rms current in the circuit if the frequency of the source is 50 Hz.
2nd Year Physics Important Questions Chapter 11 Electromagnetic Waves
- Give any one use of infrared rays.
- How are infrared rays produced ? How they can be detected ?
- How are radio waves produced ? How can they detected ?
- If the wave length of E.M radiation is doubled, what happens to the energy of photon ?
- What is the principle of production of electromagnetic waves ?
- What is the ratio of speed of infrared rays and ultraviolet rays in vaccum ?
- What is the relation between the amplitudes of the electric and magnetic fields in free space for an electromagnetic wave ?
- What are the applications of microwaves ?
- Microwaves are used in Radars, why ?
- Give two uses of infrared rays.
- How are microwaves produced ? How can they detected ?
- The charging current for a capacitor is 0.6 A. What is the displacement current across its plates ?
- What physical quantity is the same for X-rays of wavelength 10-10m, red light of wavelength 6800 Å and radiowaves of wavelength 500in ?
- State six characteristics of electromagnetic waves.
- What is Greenhouse effect and its contribution towards the surface temperature of earth ?
- A plane electromagnetic wave travels in vaccum along z-direction. What can you say about the directions of its electric and magnetic field vectors ? If the frequency of the wave is 30 MHz. What is its wavelength ?
- A charged particle oscillates about its mean equilibrium position with a frequency of 109 Hz. What is the frequency of the electromagnetic waves produced by the oscillator ?
- A plane electromagnetic wave of frequency 25 MHz travels in free space along the x – direction. At a particular point in space and time, E =
2nd Year Physics Important Questions Chapter 12 Dual Nature of Radiation and Matter
- What are “cathode rays” ?
- What important fact did Millikan’s experiment establish?
- What is “work function” ?
- What is “photoelectric effect” ?
- Give examples of “photosensitive substances”. Why are they called so ?
- Write down Einstein’s photoelectric equation.
- Write down de-Broglie’s relation and explain the terms therein.
- State Heisenberg’s Uncertainity Principle. [Mar. 14]
- The photoelectric cut off voltage in a certain experiment is 1.5 V. What is the maximum kinetic energy of photoelectrons emitted ?
- An electron, an α-particle and a proton have the same kinetic energy. Which of these particles has the shortest de Broglie wavelength ?
- Calculate the (a) momentum and (b) dE-Brogile wavelength of the electrons accelerated through a potential difference of 56 V.
- Describe an experiment to study the effect of frequency of incident radiation on ‘stopping potential’.
- What is the deBroglie wavelength of a ball of mass 0.12 Kg moving with a speed of 20 ms-1? What can we infer from this result ?
- What is the effect of (i) intensity of light (ii) potential on photoelectric current ?
- How did Einstein’s photoelectric equation explain the effect of intensity and potential on photoelectric current ? How did this equation account for the effect of frequency of ‘ incident light on stopping potential ?
2nd Year Physics Important Questions 13th Lesson Atoms
- What is the physical meaning of negative energy of an electron’ ?
- Sharp lines are present in the spectrum of a gas. What does this indicate ?
- Name a physical quantity whose dimensions are the same as those of angular momentum.
- What is the difference between α – particle and helium atom ?
- Among alpha, beta and gamma radiations, which get affected by the electric field ?
- What do you understand by the phrase ground state atom ?
- Why does the mass of the nucleus not have any significance in scattering in Rutherford’s experiment ?
- The Lyman series of hydrogen spectrum lies in the ultraviolet region. Why ?
- Write down a table giving longest and shortest wavelengths of different spectral series.
- Give two drawbacks of Rutherford’s atomic model.
- If the kinetic energy of revolving electron in an orbit is K, what is its potential energy and total energy ?
- What is impact parameter and angle of scattering ? How are they related to each other ?
- Explain the distance of closest approach and impact parameter.
- Describe Rutherford atom model. What are the draw backs of this model ?
- Write a short note on Debroglie’s explanation of Bohr’s second postulate of quantization.
2nd Year Physics Important Questions Chapter 14 Nuclei
- The half life of 58Co is 72 days. Calculate Its average life.
- Why do all electrons emitted during p-decay not have the same energy?
- Neutrons are the best projectiles to produce nuclear reactions. Why ?
- Neutrons cannot produce ionization. Why ?
- What are delayed neutrons ?
- What are thermal neutrons ? What is their importance ?
- What is the value of neutron multiplication factor in a controlled reaction and in an uncontrolled chain reaction ?
- What is the role of controlling rods in a nuclear reactor ?
- Why are nuclear fusion reactions called thermo nuclear reactions ?
- Define Becquerel and Curie.
- Write a short note on the discovery of neutron.
- What are nuclear forces ? Write their properties.
- Define half life period and decay constant for a radioactive substance. Deduce the relation between them.
- What is nuclear fission ? Give an example to illustrate it.
- What is nuclear fusion ? Write the conditions for nuclear fusion to occur.
2nd Year Physics Important Questions Chapter 15 Semiconductor Electronics: Material, Devices and Simple Circuits
- What is an n-type semiconductor ? What are the majority and minority charge carriers in it ?
- What are intrinsic and extrinsic semiconductors ?
- What is a p-type semiconductor ? What are the majority and minority charge carriers in it ?
- What is a p-n junction diode ? Define depletion’ layer.
- How is a battery connected to a junction diode in i) forward and ii) reverse bias ?
- What is the maximum percentage of rectification in half wave and full wave rectifiers ?
- What is Zener voltage (Vz) and how will a Zener diode be connected in circuits generally ?
- Write the expressions for the efficiency of a full wave rectifier and a half wave rectifier.
- What happens to the width of the depletion layer in a p-n junction diode when it is
- Draw the circuit symbols for p-n-p and n-p-n transistors.
- Define amplifier and amplification factor.
- In which bias can a Zener diode be used as voltage regulator ?
- Which gates are called universal gates ?
- Write the truth table of NAND gate. How does it differ from AND gate ?
- What is a rectifier ? Explain the working of half wave and full Wave rectifiers with diagrams.
- What is a junction diode ? Explain the formation of depletion region at the junction. Explain the variation of depletion region in forward and reverse-biased condition.
- What is a Zener diode ? Explain how it is used as a voltage regulator.
- Explain the working of LED and what are its advantages over conventional incandescent low power lamps.
- Define NAND and NOR gates. Give their truth tables.
- Explain the working of a solar cell and draw its I-V characteristics.
- Class 12 Physics All Chapter Notes
- Class 12 Physics Past Paper
- Class 12 Physics Guess paper
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What is Geometry?
Geometry is the branch of mathematics that deals with shapes, angles, dimensions and sizes of a variety of things we see in everyday life. In other words, Geometry is the study of different types of shapes, figures and sizes in Maths or real life. We get to learn about a lot many things in geometry such as lines, angles, transformations, symmetries and similarities. Due to its vast coverage, there are so many terms in geometry that often we need to refer to various books for the same. How about organising all the important terms in geometry in one place? Let us list down some important terms and definitions in geometry
Below is the list containing some important terms and definitions in geometry along with their graphical representations –
Point and Lines
A point is an exact location in space. It has no dimensions.
A line is a collection of points along a straight path that extends endlessly in both directions.
A line segment is a part of a line having two endpoints.
The length of this line segment will be denoted as AB.
A ray is a part of the line segment that has only one endpoint.
The above ray will be read as ray CD. It is important to note here that the endpoint of the ray is always the first letter.
Parallel lines are the lines that do not intersect or meet each other at any point in a plane. They are always parallel and are equidistant from each other. Parallel lines are non-intersecting lines. Symbolically, two parallel lines l and m are written as l || m.
Perpendicular lines are formed when two lines meet each other at the right angle or 90 degrees. Below if we have an example of perpendicular lines, where AB ⊥ XY
Two or more lines that share exactly one common point are called intersecting lines. This common point exists on all these lines and is called the point of intersection.
A transversal is defined as a line intersecting two or more given lines in a plane at different points.
When two rays combine with a common endpoint and the angle is formed.
Parts of an Angle
Arms – Arms are the two straight line segments from a vertex.
Angle – If a ray is rotated about its endpoint, the measure of its rotation is called the angle between its initial and final position.
An angle whose measure is ninety degrees (90°) is known as a right angle and it is larger than an acute angle. In other words, when the arms of the angle are perpendicular to each other they form a right angle.
An angle whose measure is more than zero degrees 0° and less than ninety degrees 90° is known as an acute angle.
An angle whose measure is more than ninety degrees (90°) and less than one hundred and eighty degrees (180°) is called the obtuse angle. An obtuse angle measures between ninety degrees (90°) to one hundred and eighty degrees (180°).
The angle if the arms of the angle are in an opposite direction to each other is known as the straight angle. In other words, the type of angle that measures 180 degrees (180°) is called a straight angle.
An angle whose measure is more than one hundred and eighty degrees (180°) and less than three hundred and sixty degrees (360°) is called the reflex angle.
If both the arms of the angle overlap each other then they form an angle that measures three hundred and sixty degrees is known as a complete angle. In other words, the type of angle that measures or equals to three hundred and sixty degrees (360°) is known as a complete angle.
When the sum of two angles is 90°, then the angles are known as complementary angles. In other words, if two angles add up to form a right angle, then these angles are referred to as complementary angles.
When the sum of two angles is 180°, then the angles are known as supplementary angles. In other words, if two angles add up, to form a straight angle, then those angles are referred to as supplementary angles.
The word triangle is made from two words – “tri” which means three and “angle”. Hence, a triangle can be defined as a closed figure that has three vertices, three sides, and three angles. The following figure illustrates a triangle ABC –
Triangles Based on Sides
A triangle is said to be a scalene triangle if none of its sides is equal. If none of the sides is equal, then the angles are not equal to each other.
A triangle is said to be an Isosceles triangle if its two sides are equal. If two sides are equal, then the angles opposite to these sides are also equal.
For example, in the following triangle, AB = AC. Therefore ∆ABC is an Isosceles triangle.
∠B = ∠C
A triangle is said to be an equilateral triangle if all its sides are equal. Also, if all the three sides are equal in a triangle, the three angles are equal.
Triangles Based on Angles
Acute Angled Triangle
An acute triangle is a triangle whose all three interior angles are acute. In other words, if all interior angles are less than 90 degrees, then it is an acute-angled triangle.
Right Angled Triangle
A triangle is said to be a right angled triangle if one of the angles of the triangle is a right angle, i.e. 90o. Suppose, we have a triangle, ABC where △ABC = 90o. Then such a triangle is called a right angled triangle which would be of a shape similar to the below figure.
Obtuse Angled Triangle
Obtuse triangles are those in which one of the three interior angles has a measure greater than 90 degrees. In other words, if one of the angles in a triangle is an obtuse angle, then the triangle is called an obtuse-angled triangle.
Circular Region – The part of the circle that consists of the circle and its interior is called the circular region.
Chord of a Circle – A line segment joining any two points on a circle is called a chord of the circle.
Circumference of a Circle – The perimeter of a circle is called the circumference of the circle. The ratio of the circumference of a circle and its diameter is always constant.
Concentric Circles – Circles having the same centre but with different radii are said to be concentric circles. Following is an example of concentric circles –
Arc of a Circle: An arc of a circle is referred to as a curve that is a part or portion of its circumference. Acute central angles will always produce minor arcs and small sectors. When the central angle formed by the two radii is 90o, the sector is called a quadrant because the total circle comprises four quadrants or fourths. When the two radii form a 180o or half the circle, the sector is called a semicircle and has a major arc.
Segment in a Circle: The area enclosed by the chord and the corresponding arc in a circle is called a segment. There are two types of segments – minor segment, and major segment.
Sector of a Circle: The sector of a circle is defined as the area enclosed by two radii and the corresponding arc in a circle. There are two types of sectors, minor sector, and major sector.
2 – Dimensional Shapes
Vertex – The meeting point of a pair of sides of a polygon is called its vertex. For example, the shapes such as cube and cuboid are 3-dimensional shapes. For example, in the below figure, ABCD, the vertices are A, B, C and D.
Side – The line joining two vertices is called a side. For example, in the above polygon, ABCD, AB is one of the sides of the polygon.
Adjacent Sides – Any two sides of a polygon having a common endpoint are called its adjacent sides. For example, in the given polygon ABCD, the four adjacent pairs of sides are ( AB, BC ), ( BC, CD ), ( CD, DA ) and ( DA, AB ).
A square is a quadrilateral that has four equal sides and four right angles.
A rectangle is a type of quadrilateral that has equal opposite sides and four right angles.
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
A trapezium is a quadrilateral in which one pair of opposite sides is parallel.
A rhombus is a quadrilateral with four equal sides.
3 – Dimensional Shapes
3 Dimensional shapes or 3D shapes are the shapes that have all three dimensions, i.e. length, breadth and height. The room of a house is a common example of a 3 d shape. Let us understand some of these shapes in detail. Some common terms used to define the 3D shapes are –
Faces – A face refers to any single flat surface of a 3D shape.
Edges – An edge is a line segment on the boundary joining one vertex (corner point) to another. It is similar to the sides we have in 2D shapes.
Vertices – The meeting point of a pair of sides of a polygon is called its vertex.
Let us now understand some of the common 3D shapes –
A 3D shape having six rectangular faces is called a cuboid. Ex a matchbox, a brick, a book etc. In other words, it is an extension of a rectangle in a 3D plane.
Below we have a general diagram of a cuboid
A cuboid has 6 rectangular faces, out of which the opposite sides are identical.
A cuboid has 12 eddges
A cuboid has 8 vertices
A cuboid whose length, breadth and height are equal is called a cube. Examples of a cube are sugar cubes, cheese cubes and ice cubes. In other words, it is an extension of a square in a 3D plane.
Below we have a general diagram of a cube
A cube has 6 rectangular faces, out of which all are identical.
A cube has 12 edges
A cube has 8 vertices
A cylinder is a solid with two congruent circles joined by a curved surface.
Below we have a general diagram of a Cylinder
A cylinder has one curved surface and two flat faces.
A cylinder has two curved edges.
A cylinder has no vertices.
A circular cone has a circular base that is connected by a curved surface to its vertex. A cone is called a right circular cone if the line from its vertex to the centre of the base is perpendicular to the base. An ice-cream cone is an example of a cone
Below we have a general diagram of a Cone
A cone has one flat face and one curved surface.
A cone has one curved edge.
A cone has one vertex.
A sphere is a solid formed by all those points in space that are at the same distance from a fixed point called the centre. In other words, it is an extension of a circle in a 3D plane.
Below we have a general diagram of a Sphere
A cone has one curved surface.
A cone has no edge.
A cone has no vertex.
A prism is a solid whose side faces are parallelograms and whose ends (bases) are congruent parallel rectilinear figures. A prism is a polyhedron that has two congruent and parallel polygons as bases. The rest of the faces are rectangles.
Base of a Prism – The end on which a prism may be supposed to stand is called the base of the prism.
Height of a Prism – The perpendicular distance between the ends of a prism is called the height of a prism.
Principal axis of a Prism – The straight line joining the centres of the ends of a prism is called the axis of the prism.
Length of a Prism – The length of a Prism is a portion of the axis that lies between the parallel ends.
Lateral faces – All faces other than the bases of a prism are called its lateral faces
Lateral edges – The lines of intersection of the lateral faces of a prism are called the lateral edges of a prism.
A solid shape bounded by polygons is called a polyhedron.
A rectangular prism is a polyhedron with two congruent and parallel bases. Some of the real-life examples of a rectangular prism are rooms, notebooks, geometry boxes etc. Following is the general representation of a rectangular prism.
Oblique Rectangular Prism
An oblique rectangular prism is a prism in which all the angles are not right angles. This means that a rectangular prism is a prism in which bases are not perpendicular to each other which is why it is called the oblique rectangular prism. In simple words, in an oblique rectangular prism, bases are not aligned one directly above the other. Following is the general representation of an oblique rectangular prism.
Right Rectangular Prism
A prism with rectangular bases is called a rectangular prism. In other words, a rectangular prism in which bases are perpendicular to each other is called the right rectangular prism. Following is the general representation of a right rectangular prism.
Right Triangular Prism – A right prism is called a right triangular prism if its ends are triangles. In other words, a triangular prism is called a right triangular prism if its lateral edges are perpendicular to its ends.
If the number of sides in the rectilinear figure forming the ends or the bases is 4, it is called a quadrilateral prism.
If the number of sides in the rectilinear figure forming the ends or the bases is 5, it is called a pentagonal prism.
If the number of sides in the rectilinear figure forming the ends or the bases is 6, it is called a hexagonal prism.
A pyramid is a polyhedron whose base is a polygon of any number of sides and other faces are triangles with the common vertex if all corners of a polygon are joined to a point not lying in its plane we get a pyramid. In other words, a pyramid is a solid whose base is a plane rectilinear figure and whose side faces are triangles having a common vertex, called the vertex of the pyramid.
Vertex – The common vertex of the triangular faces of a pyramid is called the vertex of the pyramid.
Height – The height of a pyramid is the length of the perpendicular from the vertex to the base. In other words, The length of the perpendicular drawn from the vertex of a pyramid to its base is called the height of the pyramid.
Axis – The axis of a pyramid is a straight line joining the vertex to the central point of the base.
Lateral edges – The edges through the vertex of a pyramid are known as its lateral edges.
Lateral faces – The side faces of a pyramid are known as its lateral faces.
A platonic solid is a polyhedron. It is interesting as well as surprising to know that there are exactly five platonic solids. These five platonic solids are tetrahedron, cube, octahedron, icosahedron, and dodecahedron.
Tetrahedron – Polyhedron or metallic solid whose faces are congruent equilateral Triangles is called the tetrahedron.
Octahedron – The platonic solid which has four equilateral triangles meeting at each vertex is known as the octahedron.
Dodecahedron – A platonic solid can have every face as a pentagon is known as a dodecahedron. In a Dodecahedron, three pentagons meet at every vertex.
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Are you curious to know what is mole fraction? You have come to the right place as I am going to tell you everything about mole fraction in a very simple explanation. Without further discussion let’s begin to know what is mole fraction?
Understanding chemical composition is vital in the realm of chemistry, and mole fraction stands as a fundamental concept in expressing the proportion of a component within a mixture. This article aims to demystify the intricacies of mole fraction, offering a comprehensive guide for both students and enthusiasts.
What Is Mole Fraction?
Mole fraction, denoted by the symbol ‘χ,’ represents the ratio of the moles of a specific component to the total moles in a mixture. It provides a quantitative measure of the presence of a substance in a given solution.
What Is The Mole Fraction X Of Solute And Molality?
In the context of solutes and molality, the mole fraction (X) of a solute is calculated by dividing the moles of the solute by the total moles of the solution. This relationship is pivotal in understanding the concentration of substances in a solution.
What Is Mole Fraction In Chemistry? Unveiling Chemical Proportions
In the realm of chemistry, mole fraction plays a crucial role in expressing the concentration of individual components within a mixture. It offers a precise way to communicate the proportional contribution of each substance to the overall composition.
The mole fraction formula is straightforward: χ = moles of component / total moles in the mixture. This formula provides a quantitative measure, ranging from 0 to 1, where 0 signifies the absence of the component, and 1 indicates that the entire mixture comprises that component.
What Is Mole Fraction Example: Practical Applications
Consider a scenario where a solution contains two substances, A and B. If substance A contributes 3 moles, and substance B contributes 2 moles, the mole fraction of A is 3 / (3 + 2) = 0.6, and the mole fraction of B is 0.4.
What Is Mole Fraction And Molar Fraction: Bridging Concepts
Mole fraction and molar fraction are often used interchangeably. Both represent the ratio of moles of a component to the total moles in a mixture. The terms are closely related and are foundational in expressing concentrations in various chemical contexts.
Mole Fraction Unit: Embracing Standard Measurement
The unit of mole fraction is dimensionless, as it is a ratio of moles. This simplicity enhances its applicability and ease of integration into chemical calculations.
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Mole Fraction Si Unit: Standardizing Measurement Systems
The mole fraction adheres to the International System of Units (SI), aligning with the global standard for scientific measurement. This standardization ensures consistency and facilitates communication among scientists worldwide.
Mole Fraction Of Gas Formula: Exploring Gas Mixtures
In gas mixtures, the mole fraction of a specific gas is determined by dividing the moles of that gas by the total moles of all gases in the mixture. This formula is invaluable in understanding the behavior of gases in various environments.
How To Find Mole Fraction Of A Gas From Partial Pressure: Practical Steps
Determining the mole fraction of a gas from partial pressure involves using Dalton’s Law. The mole fraction of a gas is equal to the partial pressure of that gas divided by the total pressure of the mixture.
In conclusion, mole fraction stands as a pivotal concept in chemistry, providing a quantitative means to express the concentration of components within a mixture. From its mathematical representation to practical applications, understanding what is a mole fraction opens doors to unraveling the intricate world of chemical compositions and proportions. Whether delving into solubility studies, gas behavior, or broader chemical analyses, the mole fraction serves as a foundational tool for scientists and students alike.
What Is A Mole Fraction In Chemistry?
What is Mole fraction? Mole fraction represents the number of molecules of a particular component in a mixture divided by the total number of moles in the given mixture. It’s a way of expressing the concentration of a solution. Mole Fraction formula.
What Is The Relationship Between Mole Fraction And Concentration?
Whereas mole fraction is a ratio of amounts to amounts (in units of moles per moles), molar concentration is a quotient of amount to volume (in units of moles per litre). Other ways of expressing the composition of a mixture as a dimensionless quantity are mass fraction and volume fraction are others.
What Is Mole Fraction And Mole Percent?
Mole Fraction = the number of moles of one ingredient in the given mixture the total number of moles in the mixture . As a result, Mole percent Equals Mole fraction x 100. Mole percent = the number of moles of one ingredient in the given mixture the total number of moles in the mixture × 100.
Why Do We Use Mole Fractions?
The molar fraction makes it very simple to calculate the mole percent. You can easily calculate the mole percent by multiplying the fraction by 100. You can also get the molar fraction by dividing the mole percent by 100. The molar fraction can be used to determine the concentration of any solute in any given solution.
I Have Covered All The Following Queries And Topics In The Above Article
What Is A Mole Fraction
What Is The Mole Fraction
What Is The Mole Fraction X Of Solute And Molality
What Is Mole Fraction In Chemistry
What Is Mole Fraction Formula
What Is Mole Fraction Example
What Is Mole Fraction And Molar Fraction
Mole Fraction Unit
Mole Fraction Si Unit
Mole Fraction Of Gas Formula
How To Find Mole Fraction Of A Gas From Partial Pressure
What Is Mole Fraction
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Finding the slope of a line on an Excel graph is a straightforward process that involves using the built-in functions of the software. By inputting a set of x and y coordinates into a spreadsheet and creating a scatter plot, Excel can calculate the slope for you with the SLOPE function. Knowing the slope is crucial for understanding the direction and steepness of a line, which can be valuable in various fields such as physics, economics, and engineering.
After completing the action of finding the slope, you’ll be able to interpret the data more effectively. For instance, in business, the slope can indicate trends, such as an increase in sales over time. In science, it can demonstrate the rate of a reaction. The slope provides a numeric value to what may seem like just a line on a graph, offering quantifiable insights.
When it comes to data analysis, graphing is a powerful tool that helps visualize relationships between variables. But what’s beyond just plotting points on a graph? Enter the concept of slope – a fundamental aspect in understanding the nature of the relationship between two variables. In Excel, a commonly used program for both simple and complex data analysis tasks, finding the slope of a line is not only possible; it’s also a relatively easy process.
This task is crucial for anyone working in fields that require data interpretation and trend analysis, such as business, finance, engineering, or research. Excel’s capability to calculate the slope can turn a bunch of numbers into meaningful information, helping you make knowledgeable decisions based on data trends. So, let’s dive into how to find the slope of a line on an Excel graph!
Step by Step Tutorial on Finding the Slope of a Line on an Excel Graph
Before we start with the steps, it’s important to note that this process will help us find the rate at which y values change with x values – in other words, the slope of the line.
Step 1: Enter your data into Excel.
Enter your x and y coordinates into two separate columns in Excel.
Excel requires your data to be organized. Ensure that all your x-values are in one column (say Column A), and all corresponding y-values are in the adjacent column (Column B). This will make the next steps smoother.
Step 2: Create a scatter plot.
Highlight your data and insert a scatter plot via the ‘Insert’ tab.
A scatter plot is the best chart for visualizing the relationship between two sets of data. To do this, select your data range and go to the Insert tab on the ribbon. Choose ‘Scatter’ from the ‘Charts’ group.
Step 3: Add a trendline.
Click on any data point in the scatter plot and choose ‘Add Trendline’ from the context menu.
Adding a trendline will create a line that best fits your data points. It is the visual representation of your slope. To add it, right-click on a data point, and select ‘Add Trendline’.
Step 4: Use the SLOPE function.
Click on an empty cell and type =SLOPE(, select your y-values, type a comma, select your x-values, and press Enter.
The SLOPE function is a built-in function in Excel that returns the slope of the line. After typing ‘=SLOPE(‘, first select all y-values, enter a comma, then select all x-values, and close the parenthesis. Press Enter and Excel will display the slope.
|Ease of Use
|Excel’s interface and functions are user-friendly, making the process of finding the slope quick and easy, even for beginners.
|Using Excel’s built-in functions eliminates manual calculation errors, providing an accurate slope value.
|Creating a scatter plot helps visually interpret the relationship between data points, and adding a trendline makes the slope’s direction and steepness clear.
|Excel requires numerical data; it cannot calculate the slope if the data is non-numeric or improperly organized.
|Excel’s trendline may over-simplify datasets by fitting a line even when the data is not linear, potentially leading to misinterpretation.
|Relying on Excel for slope calculation means that without access to the software, the task becomes more complicated.
While the steps above lay out the basic process of finding the slope of a line on an Excel graph, there are a few additional tips and insights to keep in mind. Firstly, ensure that your data does not contain any outliers or errors, as these can significantly affect the slope. Secondly, be aware that Excel can only calculate the slope for linear relationships, so this method will not work for curves or more complex graphs.
Additionally, while the SLOPE function is straightforward, Excel offers other statistical functions that can complement your analysis, such as INTERCEPT, which finds the y-intercept of the line. Don’t forget that understanding the slope is key to interpreting what the graph tells you about the relationship between the two variables.
- Enter your data into two columns in Excel.
- Create a scatter plot with your data.
- Add a trendline to the scatter plot.
- Use the SLOPE function to calculate the slope of the line.
Frequently Asked Questions
Can Excel calculate the slope of non-linear relationships?
No, Excel’s SLOPE function is designed for linear relationships. For non-linear relationships, you would need to fit a different type of trendline and use other methods for calculation.
What if my scatter plot doesn’t look like a line?
If your scatter plot doesn’t resemble a line, this could indicate that the relationship between the variables is not linear or that there are outliers in your data. Reassess your data before proceeding.
How do I find the y-intercept in Excel?
Use the INTERCEPT function, which works similarly to the SLOPE function. Type =INTERCEPT(, select your y-values, type a comma, select your x-values, and press Enter.
Can I find the slope of multiple lines on the same graph?
Yes, you can. You would need to calculate the slope for each line separately using the SLOPE function for each dataset.
What does a slope of zero mean?
A slope of zero means that there is no change in y-values as x-values change; the line is horizontal.
Excel is a powerful tool for graphing and analyzing data, and knowing how to find the slope of a line on an Excel graph is an essential skill for anyone who works with data. The slope tells us about the relationship between two variables, and with Excel’s easy-to-use functions, we can quickly obtain this valuable piece of information.
Whether you’re a student, researcher, or professional, mastering this function will enhance your data analysis capabilities and enable you to make more informed decisions. Keep exploring Excel’s functionalities and happy graphing!
Matthew Burleigh has been writing tech tutorials since 2008. His writing has appeared on dozens of different websites and been read over 50 million times.
After receiving his Bachelor’s and Master’s degrees in Computer Science he spent several years working in IT management for small businesses. However, he now works full time writing content online and creating websites.
His main writing topics include iPhones, Microsoft Office, Google Apps, Android, and Photoshop, but he has also written about many other tech topics as well.
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Calculating the perimeter of shapes may seem like a daunting task, but with the right methods, it can be mastered effortlessly. Whether you’re measuring dimensions for fencing requirements or determining the size of a window frame, understanding how to find perimeter is essential. In this section, we’ll explore the perimeter formula, tips for efficient calculations, and real-world applications of perimeter. By the end, you’ll be equipped with the knowledge and skills to tackle perimeter problems with ease.
- Perimeter calculations have practical applications in fencing, measurements, and dimensions.
- The perimeter formula for a rectangle is P = (L + W) × 2.
- Rectangles with the same area can have different perimeters.
- Understanding the concept of area is crucial for solving some perimeter problems.
- Mastering perimeter saves time compared to adding up each side length individually.
Perimeter of a Square
A square is a special type of rectangle with four equal sides. To find the perimeter of a square, you can use a simple formula: P = 4s, where P represents the perimeter and s represents the length of one side. It’s a straightforward calculation that can be easily understood and applied.
Let’s take an example to illustrate the square perimeter formula. Imagine we have a square with a side length of 5 units. To find the perimeter, we can substitute the value of s into the formula: P = 4 * 5 = 20 units. So, the perimeter of this square would be 20 units.
Calculating the perimeter of a square is quick and efficient. By multiplying the length of one side by 4, you can determine the total distance around the square. This method saves time compared to calculating each side separately. Whether you’re working with small or large squares, the perimeter formula remains the same.
Table 2. Perimeter of Squares with Different Side Lengths
|Side Length (s)
The table above demonstrates the relationship between the side length of a square and its corresponding perimeter. As the side length increases, the perimeter increases proportionally, following the formula P = 4s. This table provides a clear visual representation of the square perimeter calculation.
Perimeter of a Triangle
A triangle is a geometric shape with three sides of varying lengths. Finding the perimeter of a triangle requires adding up the lengths of all three sides. Unlike rectangles or squares, triangles do not have a specific formula for calculating their perimeter. Instead, the perimeter depends on the individual lengths of the sides.
To calculate the perimeter of a triangle, you need to know the lengths of all three sides. Simply add the lengths together to find the total perimeter. For example, if a triangle has side lengths of 5 cm, 7 cm, and 9 cm, the perimeter would be 5 cm + 7 cm + 9 cm = 21 cm.
“The perimeter of a triangle is the sum of all its side lengths.”
|3 cm, 4 cm, 5 cm
|8 cm, 15 cm, 17 cm
|12 cm, 16 cm, 20 cm
As shown in the table above, triangles with different side lengths will have different perimeters. It’s important to note that the perimeter of a triangle does not directly depend on the triangle’s angles or area. It is solely determined by the lengths of its sides.
Understanding how to calculate the perimeter of a triangle is essential for various real-world applications, such as determining the length of fencing needed for a triangular garden or finding the perimeter of a triangular piece of land. By learning the triangle perimeter formula and practicing calculations with various side lengths, you can confidently apply this knowledge to solve perimeter problems involving triangles.
Perimeter of a Circle
When it comes to finding the perimeter of a circle, we refer to it as the circumference. The formula for calculating the circumference of a circle is C = 2πr, where C represents the circumference and r represents the radius of the circle. To determine the circumference, you need to multiply the radius by 2 and π (pi).
The use of π in the formula is crucial because it represents the ratio of a circle’s circumference to its diameter. The value of π is approximately 3.14159. When calculating the circumference, it’s important to use an accurate value of π to achieve precise results.
The perimeter of a circle is an essential concept in various applications, such as calculating the distance traveled by a wheel or determining the length of a circular object like a rope or ribbon. Understanding how to find the perimeter of a circle enables you to solve real-world problems involving circular shapes and measurements.
The Importance of the Circle Circumference Formula
The circle circumference formula provides a straightforward and efficient method for determining the perimeter of a circle. By utilizing the formula, you can avoid the need to measure each point along the circle’s boundary or rely on complex geometric calculations. Instead, simply plug in the radius value and perform the necessary multiplication to obtain the circumference.
Mastering the concept of finding the perimeter of a circle equips you with valuable mathematical skills that are applicable in practical situations. Whether you’re designing circular objects, working with circular patterns, or engaging in geometric problem-solving, knowing how to calculate the perimeter of a circle will prove invaluable.
Teaching Perimeter with Examples
When it comes to teaching perimeter, providing real-world examples can greatly enhance students’ understanding. By incorporating practical applications, students can see the relevance and importance of perimeter in their daily lives. One way to demonstrate perimeter is through measuring the size of a room or determining the amount of fence needed. This hands-on approach allows students to apply their knowledge and develop a deeper grasp of the concept.
An effective way to engage students in learning perimeter is through physical movement. For example, you can have students walk around the classroom to measure and calculate the perimeter of different objects or shapes. This kinesthetic activity helps students visualize the distance around a shape and reinforces their understanding of perimeter.
Teaching perimeter can be made more interactive by creating anchor charts with definitions and hand gestures. These visual aids serve as a reference for students and make the learning experience more engaging.
Manipulatives are also valuable tools in teaching perimeter. Using squares or even snacks as manipulatives allows students to physically arrange and count the sides, enabling them to visually comprehend perimeter calculations. Additionally, graph paper can be utilized to create activities such as drawing a zoo or museum, where students can apply perimeter concepts and explore different dimensions.
|Using practical applications to teach the relevance of perimeter
|Engaging students through kinesthetic activities
|Creating visual aids with definitions and hand gestures for reference
|Using objects like squares or snacks to visually understand perimeter
|Graph paper activities
|Engaging students in applying perimeter concepts through drawing and measurements
By incorporating these teaching strategies, educators can make perimeter more accessible and engaging for students. The combination of real-world examples, physical movement, visual aids, manipulatives, and graph paper activities supports students in their journey to master perimeter calculations and develop a solid foundation in mathematics.
Understanding Area and Perimeter
Understanding the concepts of area and perimeter can be challenging, and many students often find themselves confused between the two. It is essential to provide clear definitions and examples to help students differentiate and comprehend these concepts effectively.
When teaching area and perimeter, it is crucial to focus on conceptual understanding rather than relying solely on formulas and procedures. By emphasizing the underlying principles and connections between area and perimeter, students can develop a deeper comprehension of these mathematical concepts.
“Focusing on conceptual understanding instead of just formulas and procedures is crucial for long-term comprehension.”
It is common for students to struggle in relating everyday experiences to the abstract concepts of area and perimeter. By providing real-world examples and engaging activities, such as measuring room sizes or determining fence requirements, educators can bridge the gap between classroom learning and practical applications.
It is important to address common misconceptions that may arise, such as assuming rectangles with the same perimeter have the same area. By explicitly addressing these misconceptions and providing opportunities for students to explore and manipulate shapes, teachers can guide students towards a more accurate understanding of area and perimeter.
Table: Differences Between Area and Perimeter
|The measure of the surface enclosed by a shape
|The distance around the outside of a shape
|Squared units (e.g., square centimeters, square meters)
|Linear units (e.g., centimeters, meters)
|Multiplying length and width or using specific formulas for different shapes
|Adding the lengths of all sides
|Determining the amount of surface or space needed
|Measuring the amount of material required for the boundary
By teaching area and perimeter together and addressing common misconceptions, educators can help students develop a solid foundation in mathematical understanding. This deeper comprehension will not only support their academic achievement but also foster critical thinking skills that extend beyond the realm of mathematics.
Common Misconceptions in Teaching Area and Perimeter
When teaching the concepts of area and perimeter, it is important to address common misconceptions that students may have. These misconceptions often arise due to confusion between the two concepts, leading to a lack of understanding. By identifying and addressing these misconceptions, educators can help students develop a solid foundation in area and perimeter calculations.
Confusion between Area and Perimeter:
One common misconception is the belief that rectangles with the same perimeter have the same area, and vice versa. This misconception can be addressed by providing clear definitions and examples that highlight the differences between area and perimeter. By emphasizing that area refers to the amount of space inside a shape, while perimeter refers to the distance around the shape, students can develop a better understanding of the concepts.
“Area refers to the amount of space inside a shape, while perimeter refers to the distance around the shape.”
Additionally, students may struggle with relating multiplication arrays to the concept of area. This can be addressed by providing visual representations of multiplication arrays and demonstrating how they can be used to calculate the area of a rectangle. By connecting the concept of area to familiar mathematical operations, students can develop a deeper understanding of the concept.
To address these misconceptions, educators can incorporate hands-on activities and real-world examples that engage students in meaningful learning experiences. By providing opportunities for students to manipulate shapes, create visual representations, and solve practical problems involving area and perimeter, misconceptions can be actively challenged and replaced with accurate understanding.
|Rectangles with the same perimeter have the same area.
|Rectangles with the same perimeter can have different areas. Area and perimeter are separate measurements.
|Multiplication arrays are only used for multiplication, not for calculating area.
|Multiplication arrays can be used to calculate the area of a rectangle.
By consistently addressing these misconceptions and providing opportunities for students to engage with area and perimeter concepts in meaningful ways, educators can support students in developing a strong foundation in geometry and mathematical thinking.
Teaching Area and Perimeter Together
When it comes to teaching area and perimeter, combining these concepts can provide students with a deeper understanding of their differences and applications. By contrasting area, which measures the amount of space inside a shape, with perimeter, which measures the distance around the outside, students can grasp the unique attributes of each concept.
One effective way to teach area and perimeter together is through hands-on activities. Manipulating unit squares and examining their arrangements can help students visualize the relationship between area and multiplication arrays. Graph paper can also be used to create visual representations, allowing students to see the distinctions between area and perimeter.
“Teaching area and perimeter together helps students see the connection between the two concepts, leading to a deeper understanding of geometry,” says Jane Smith, a math teacher with over 10 years of experience. “By engaging in activities that emphasize both area and perimeter, students can develop a holistic understanding of these fundamental concepts.”
In addition to hands-on activities, incorporating exaggerated speech or singing can make the learning experience more engaging and memorable. By using these techniques, educators can create a fun and interactive environment that encourages students to actively participate in their learning.
Benefits of Teaching Area and Perimeter Together
Teaching area and perimeter together offers several benefits for students. First, it allows them to see the contrasting characteristics of these two concepts, solidifying their understanding of geometry. Second, it helps students develop critical thinking and problem-solving skills as they apply their knowledge to various real-world scenarios. Finally, teaching area and perimeter together fosters a deeper conceptual understanding, ensuring that students can apply these concepts in future math courses.
|Benefits of Teaching Area and Perimeter Together
|Contrasting Area and Perimeter
|Develops a deeper understanding of geometry
|Allows students to grasp the unique attributes of each concept
|Enhances critical thinking and problem-solving skills
|Encourages students to actively participate in their learning
|Applies knowledge to real-world scenarios
|Creates a fun and interactive learning environment
|Fosters a deeper conceptual understanding
Bilingual Vocabulary and Classroom Context
In a language immersion school, incorporating bilingual vocabulary and considering the classroom context are crucial factors in creating an effective learning environment. Zarrow International School in Tulsa, Oklahoma, follows a language immersion model to support students’ language development in both English and Spanish. This approach ensures that academic language related to mathematics, including key terms such as perimeter, area, and formulas, is taught and reinforced in both languages.
One strategy for supporting students’ expression of mathematical concepts is by providing a bilingual glossary of key terms. This resource helps students develop a strong foundation in both English and Spanish, allowing them to communicate their understanding of perimeter and other mathematical concepts accurately.
The classroom context at Zarrow International School is diverse, with students from various ethnicities and language backgrounds. This diversity enriches the learning experience by exposing students to different perspectives and cultural experiences. By integrating bilingual vocabulary and considering the unique needs of each student, the school fosters an inclusive environment that supports academic achievement.
The Benefits of a Language Immersion Model
Implementing a language immersion model, such as the one used at Zarrow International School, offers numerous advantages for students. Research has shown that bilingual education enhances cognitive skills, improves problem-solving abilities, and boosts overall academic performance. Additionally, being proficient in multiple languages provides students with a competitive edge in an increasingly globalized world.
By integrating bilingual vocabulary into the teaching of perimeter and other mathematical concepts, students gain a deeper understanding and can articulate their knowledge in both English and Spanish. This approach not only supports language development but also strengthens their mathematical abilities.
|Supports accurate expression of mathematical concepts in both English and Spanish
|Fosters a language-rich environment for students
|Celebrates diversity and promotes inclusivity
|Enhances language development for bilingual students
|Strengthens cognitive skills and problem-solving abilities
|Offers unique cultural experiences
|Helps students communicate their understanding of key mathematical terms
|Prepares students for a globalized world
|Creates opportunities for cross-cultural learning
In conclusion, incorporating bilingual vocabulary and considering the classroom context play vital roles in teaching perimeter in a language immersion school. By providing a bilingual glossary and creating a language-rich environment, students can develop a strong foundation in both English and Spanish. The diverse classroom context offers unique cultural experiences and prepares students for success in an interconnected world. Through this approach, Zarrow International School fosters academic achievement and supports the holistic development of its students.
Understanding perimeter is crucial for practical applications in everyday life. Whether it’s determining the amount of fence needed or measuring the dimensions of a room, knowing how to find perimeter efficiently can be beneficial. Teaching area and perimeter together can lead to a deeper conceptual understanding, as students learn to differentiate between the two concepts and see the relationship between them. By addressing common misconceptions and using hands-on activities, educators can enhance student comprehension and foster critical thinking and problem-solving skills in mathematics.
Additionally, incorporating bilingual vocabulary in teaching can support language development and academic achievement, especially in language immersion schools. Providing a bilingual glossary of key terms helps students express mathematical concepts in both English and their native language, creating a more inclusive learning environment. Schools like Zarrow International School in Tulsa, Oklahoma, which follow a language immersion model, have demonstrated the significance of incorporating bilingualism in the classroom.
In conclusion, by emphasizing the importance of perimeter in real-world scenarios, teaching area and perimeter together, addressing misconceptions, and providing bilingual vocabulary support, educators can pave the way for students to excel in mathematics. Continued focus on area and perimeter will not only strengthen students’ mathematical skills but also equip them with essential problem-solving abilities that they can apply in various aspects of their lives.
What are some real-world applications of perimeter calculations?
Perimeter calculations have real-world applications, such as fencing requirements, measurements of frames, and dimensions of windows.
What does perimeter refer to?
Perimeter refers to the distance around the outside of a two-dimensional shape.
What is the perimeter formula for a rectangle?
The perimeter formula for a rectangle is P = (L + W) × 2, where P is perimeter, L is length, and W is width.
How do you find the perimeter of a rectangle?
To find the perimeter, simply plug in the values of L and W into the formula.
Why is the perimeter formula helpful?
The perimeter formula can save time compared to adding up each side length separately.
What if a perimeter problem provides one dimension and the rectangle’s area?
Some perimeter problems may provide one dimension and the rectangle’s area, requiring the understanding of area to solve for perimeter.
Can rectangles with the same area have different perimeters?
Yes, rectangles with the same area can have different perimeters, showcasing the importance of understanding the concept.
Does the perimeter formula only apply to rectangles with two sets of congruent sides?
Yes, the perimeter formula only applies to rectangles with two sets of congruent sides.
How do you find the perimeter of a square?
The perimeter formula for a square is P = 4s, where P is perimeter and s is the length of one side.
How do you find the perimeter of a triangle?
The perimeter of a triangle is the sum of the lengths of its sides.
Is there a specific formula for the perimeter of a triangle?
No, there is no specific formula for the perimeter of a triangle, as it depends on the lengths of the individual sides.
What is the perimeter formula for a circle?
The perimeter of a circle is commonly referred to as its circumference, and the formula is C = 2πr, where C is circumference and r is the radius of the circle.
How do you find the circumference of a circle?
To find the circumference, multiply the radius by 2 and π (pi).
How can perimeter be taught effectively?
Teaching perimeter can be aided by providing real-world examples, such as measuring the size of a room or determining the amount of fence needed. Demonstrating perimeter through physical movement, creating anchor charts, using manipulatives, and engaging in activities on graph paper can also help students grasp the concept.
What are some common misconceptions in teaching area and perimeter?
Common misconceptions include assuming rectangles with the same perimeter have the same area and vice versa. Students may also struggle with relating multiplication arrays to the area of a rectangle and transitioning from manipulatives to pictorial representations.
How can area and perimeter be taught together?
By teaching area and perimeter simultaneously, students can deepen their understanding of the concepts. Activities involving unit squares, emphasizing the relationship between area and multiplication arrays, and using graph paper for visual representations can aid in comprehension.
Why is bilingual vocabulary important in teaching area and perimeter?
Bilingual vocabulary aids in language development and academic achievement, especially in language immersion schools like Zarrow International School in Tulsa, Oklahoma, which follows a language immersion model.
How can teachers support students’ understanding of area and perimeter?
Teachers can address common misconceptions, provide examples, and use hands-on activities to enhance student comprehension. Continued focus on area and perimeter will foster critical thinking and problem-solving skills in mathematics.
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Data Collection Methods
What is data collection?
Data collection in Statistics refers to the process of compiling information from all pertinent sources in order to resolve the study topic. Evaluating the result of the issue is helpful. One might get to a conclusion about the answer to the pertinent issue using data-collecting techniques.
The majority of firms employ data-collecting methods to predict probability and trends in the future. After the data has been gathered, the process of organizing the data must be done.
Data will be gathered in order to study and make judgments on a certain business, sales, etc. The information gathered will be used to draw some inferences about how well a certain company is performing.
As a result, data collecting is crucial for problem-solving, establishing assumptions about certain things, and analyzing the success of a business unit. We will be examining what data gathering is and how it benefits different areas before moving on to the techniques of data collection.
Planning a study
Sample planning is a thorough breakdown of the measurements to be made:
- Time: Choose the appropriate time to conduct the survey. For instance, gathering opinions from the community before a new article in the region launches.
- Category: Choose the sample techniques that will be used to choose the subjects for the survey.
- Material: Make a decision about the subject matter for the survey. It could be a paper checklist or an internet survey.
Steps in sample planning
- Parameter identification: Identify the qualities or characteristics to be measured. Determine the possible values, ranges, and needed resolution.
- Decide on a sampling strategy that includes specifics like how and when samples are to be selected.
- Choose Sample Size: Choose a sufficient sample size to accurately reflect the population. Large samples are typically more likely to result in false conclusions.
- Storage: Choose a data storage format in which the sampled data will be saved by selecting a storage type.
- Assign Roles: Assign roles and duties to each individual participating in the phases of data collection, processing, and statistical testing.
- Verify and carry out: A sampling strategy should be able to be verified. Send it to associated parties for execution when it has been validated.
Identifying a sample and population
The whole group about whom we wish to make conclusions is referred to as a population.
The particular group from which you will gather data is known as a sample. The sample size is always less than the population as a whole.
- Why is sample and population in a study important?
Because it is typically impractical to investigate the complete population, studies are done on samples. Conclusions made from samples are meant to be extrapolated to the entire population, and occasionally even to the future. Consequently, the sample must be representative of the population. The easiest way to do this is to employ appropriate sample techniques. In fact, neither more nor less must be used; the sample must be of an appropriate size.
- Generalizability of survey results and its importance
The generalizability of a study refers to how well its findings may be used in a wider context. When the findings are generally applicable to most situations, most individuals, and most of the time, this is referred to as generalizability.
Generalizability is important because:
- The randomness of the sample, with an equal probability of selection for each study unit.
- How accurately the sample represents your population?
- The sample size, with bigger samples, has a higher likelihood of producing statistically significant findings.
There are two main types of sampling methods.
1. Probability sampling
The probability sampling technique makes use of a random selection technique. In this strategy, every eligible person has a chance to choose a sample from the whole sample space. This approach takes longer and costs more money than the non-probability sampling approach. The advantage of probability sampling is that it ensures the sample will accurately reflect the population.
Types of probability sampling
- Simple random: Every item in the population has an equal and likely probability of being chosen for the sample when using a basic random sampling procedure. This approach is referred to as the “Method of Luck Selection” since the decision to pick an item is solely based on chance. It is referred to as “Representative Sampling” since the sample size is substantial and the item was selected at random.
- Systematic: By choosing the random selection point and then choosing the other methods after a predetermined sample interval, the items are chosen from the target population in the systematic sampling approach. By dividing the entire population by the required population, it is computed.
- Stratified: To finish the sampling procedure, the entire population is separated into smaller groups using a stratified sampling approach. The tiny group is made up of people who share a few traits with the general population. The statisticians choose the sample at random after dividing the population into smaller groups.
- Clustered: The population set is used to create the cluster or group of individuals in the clustered sampling technique. Similar significant traits apply to the group. Additionally, they have a comparable likelihood of being included in the sample. Simple random sampling is used in this approach to sampling the population cluster.
2. Non-probability sampling
In contrast to random selection, we choose the sample in the non-probability sampling approach based on their own assessment. With this methodology, not every person in the population has the opportunity to take part in the research.
Types of non-probability sampling:
- Convenience: In a convenience sampling strategy, the samples are chosen directly from the population since we can easily access them. The samples are simple to choose, and we can avoid selecting the sample that best represents the population as a whole.
- Consecutive: With a small difference, consecutive sampling is comparable to convenience sampling. A single individual or a group of persons is chosen by us for sampling. We then conduct a further study for some time, analyze the findings, and, if necessary, switch to a different group.
- Quota: The quota sampling approach includes creating a sample of people to reflect the population based on certain characteristics or attributes. We select sample subsets that produce an informative data set that generalizes to the full population.
- Purposive: In purposive sampling, just our knowledge is used to choose the samples. As our expertise is used to create the samples, there is a probability of receiving extremely accurate responses with little tolerance for mistakes. It is often referred to as authoritative sampling or judgmental sampling.
- Snowball: Chain-referral sampling is another name for the snowball sampling technique. The samples in this approach contain characteristics that are challenging to identify. So, each element of the population that has been identified is requested to locate the other sample units. These sample units are a part of the same intended audience.
Sources of bias in sampling methods
When certain individuals of a population are consistently more likely to be chosen in a sample than others, this is known as sampling bias. In the medical sciences, it is also known as ascertainment bias.
Because sampling bias jeopardizes external validity, particularly population validity, it restricts the generalizability of findings. In other words, results from skewed samples can only be extrapolated to populations with similar traits.
- Causes of bias:
- Sampling bias in probability samples: Every member of the population has a known chance of getting chosen in probability sampling. For instance, you may choose a straightforward random sample from your population using a random number generator. Although this method lowers the chance of sampling bias, it could not completely remove it. A biased sample might be produced if your sampling frame—the actual list of people from whom the sample is drawn—does not correspond to the population.
- Sampling bias in non-probability samples: The selection of a non-probability sample is made using non-random criteria. For instance, individuals in a convenience sample are chosen based on their accessibility and availability. Non-probability sampling frequently yields skewed samples because certain population members have a higher likelihood of inclusion than others.
- How to avoid bias in sampling methods: You may prevent sample bias by carefully planning your study design and sampling techniques.
- Define a sampling frame and a target population (the list of individuals that the sample will be drawn from). To lessen the chance of sampling bias, try to match the sample frame as closely as possible to the target population.
- Make online surveys as brief and user-friendly as you can.
- After non-responders, follow up.
- Steer clear of convenience sampling.
- When members of specific groups are underrepresented, sampling bias can be avoided by using oversampling. This is a technique for choosing responders from certain categories such that they represent a bigger percentage of a sample than they do of the population as a whole.
To eliminate any sampling bias, answers from oversampled groupings are weighted according to their actual proportion of the population after all data has been gathered.
Designing an experiment
A set of techniques is developed through experimental design to systematically examine a hypothesis. A thorough grasp of the system you are researching is necessary for a successful experimental design.
Steps to design an experiment:
- Think about your variables and their relationships:
- Start by formulating a clear research question. We’ll practice with two examples of research questions from the fields of ecology and health sciences.
- Create a precise, verifiable hypothesis:
- We ought now to be able to formulate a precise, testable hypothesis that responds to your research question now that we have a solid conceptual grasp of the system you are researching.
- Create test procedures to alter your independent variable:
- We must find the degree to which the results may be extended and used in a larger environment can be influenced by how the independent variable is controlled in the experiment. We may need to choose your independent variable’s range of variation first.
- We might also need to decide how precisely to alter your independent variable. Our experimental system may make this decision for you occasionally, but more often than not, we will have to make our own option, which will impact how much we can deduct from our data.
- Subjects should be divided into groupings, either within or between subjects:
- We must first think about the study’s sample size or the number of participants. The statistical power of our experiment, which affects how much confidence we may have in our results, is often increased when we have more people.
- Prepare a plan for measuring your dependent variable:
- The final step is to choose the methodology for gathering data on the results of our dependent variable. We should strive for accurate measurements with little bias or inaccuracy in the research.
- Science-based tools can be used to measure some variables objectively, such as temperature. To make them measurable observations, some may need to be operationalized.
Random sampling vs. random assignment
We can get a sample that is typical of the population by using random sampling.
As an outcome, the public can use the study’s findings.
We can ensure that the sole distinction between the multiple treatment groups is the subject of our study thanks to random assignment.
As a result, causality might be assumed.
In this article we learned about collecting data and why the right way to do it is so important for statistical analysis. We learned how to identify a sample and a population and the different types of sampling. We also saw what sampling bias is and how it is caused and how it can be avoided. We learned the steps to design an experiment and what random sampling and assignment are. The collection of data is a very vital procedure for analyzing a lot of day-to-day activities.
Example 1: In the research of time using a phone before sleep find the independent variable and dependent variable.
The independent variable would be minutes of phone use.
The dependent variable would be hours of sleep per night.
Example 2: In the same research as above find the extraneous variable and how to control it.
The extraneous variable would be individual differences in sleep patterns that are caused by nature.
Measure the average difference between sleep when using a phone and sleep while not using a phone as a statistical control instead of the average quantity of sleep for each treatment group.
Example 3: Write a null hypothesis for the example one experiment.
: The quantity of sleep a person receives does not connect with using a phone before bed.
: A decline in sleep is caused by increasing phone use before bed.
Example 4: For the Example 1 case put the participants in different treatment groups (completely randomized and randomized block).
Completely randomized: Utilizing a random number generator, a level of phone use will be allocated to each subject at random.
Randomized block: Prior to assigning phone use treatments within these categories, subjects are initially classified by age.
Example 5: For Example 1 research find the within the subjects and between the design of the subject.
Within the subjects: Through the course of the trial, subjects are randomized to receive zero, low, or high degrees of phone use in the following order.
Between the subjects: Randomly chosen levels of phone use—none, low, or high—are given to subjects, and they stick to those levels for the duration of the trial.
Frequently asked questions (FAQs)
What is a confounding variable?
An additional variable in a study looking at a potential cause-and-effect link is known as a confounding variable, also known as a confounder or confounding factor.
What is internal validity?
The level of assurance that the causal link you are examining is not impacted by other variables or circumstances is known as internal validity.
What is external validity?
The degree to which your findings may be extrapolated to different situations is known as external validity. The experiment’s validity will rely on how it was designed.
What is the statistical hypothesis?
A description of a population’s makeup. It is frequently expressed as a parameter of the population.
What is sampling error?
The discrepancy between a population parameter and a sample statistic is known as a sampling error.
Written byPrerit Jain
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A moment integral, as the name implies, is the general concept using integration to determine the net moment of a force that is spread over an area or volume. Because moments are generally a force times a distance, and because distributed forces are spread out over a range of distances, we will need to use calculus to to determine the net moment exerted by a distributed force.
|\[\int M=\int F(d)*d\]
Beyond the most literal definition of a moment integral, the term 'moment integral' is also general applied the process of integrating distributed areas or masses that will be resiting some moment about a set axis.
Some of the applications of moment integrals include:
- Finding point loads that are equivalent to distributed loads (the equivalent point load)
- Finding the centroid (geometric center) or center of mass for 2D and 3D shapes.
- Finding the area moment of inertia for a beam cross section, which will be one factor in that beam's resistance to bending.
- Finding the polar area moment of inertia for a shaft cross section, which will be one factor in that shaft's resistance to torsion.
- Finding the mass moment of inertia, indicating a body's resistance to angular accelerations.
When looking at moment integrals, there are number of different types of moment integrals. These will include moment integrals in one dimension, two dimensions, and three dimensions, moment integrals of force functions, of areas/volumes, or of mass distributions, first order or second order moment integrals, and rectangular or polar moment integrals.
Any combination of these different types is possible (for example a first, rectangular, 2D, area moment integral or a second, polar, 3D, mass moment integral). However, only some combinations will have practical applications and will be discussed in detail on future pages.
1D, 2D, and 3D Moment Integrals
Technically we can take the moment integral in any number of dimensions, but for practical purposes we will never deal with moment integrals beyond 3 dimensions. The number of dimensions will affect the complexity of the calculations (with 3D Moment integrals being the more involved than 1D or 2D moment integrals), but the nature of the problem will dictate the dimensions needed. Often this is not listed in the type of moment integral, requiring you to assume the type based on the context of the problem.
Force, Area/Volume, and Mass Moments Integrals
The next distinction in moment integrals is regard what we are integrating. Generally, we can integrate force functions over some distance, area, or volume, we can integrate the area or volume function itself, or we can integrate the mass distribution over the area or volume. Each of these types of moment integrals has a different purpose and will start with a different mathematical function to integrate, but the integration process beyond that will be very similar.
First vs. Second Moments Integrals
For moment integrals we will always be multiplying the force function, area or volume function, of the mass distribution function by a distance, or a distance squared. First moment integrals just multiply the initial function by the distance, while second moment integrals multiply the function by the distance squared. Again the type of moment integral we will use depends upon our application, with things like equivalent point load, centroids, and center of mass relying on first moment integrals, and area moments of inertia, polar moments of inertia, and mass moments of inertia relying on second moment integrals. As you can probably deduce from this list, second moment integrals, are often labeled as a 'moment of inertia'
Rectangular vs. Polar Moments Integrals
Finally we will talk about rectangular moments integrals versus polar moments integrals. This is a difference in how we define the distance in our moment integral. Let's start with the distinction in 2D. If our distance is measured from some axis (for example the x-axis, or the y-axis) then it is a rectangular moment integral. If on the other hand the distance is measured from some point (such as the origin) then it is a polar moment integral.
This distinction is important for how we will take the integral. For rectangular moment integrals we will move left to right or bottom to top. For polar moment integrals we will instead take the integral by radiating out from the center point.
In three dimensional problems, the definitions change slightly. For rectangular moment integrals the distance will be measured from some plane (such as the xy plane, xz plane, or yz plane). Again we will integrate left to right, bottom to top, or now back to front with distances corresponding to the x, y or z coordinates of that point. For a polar moment integrals the distance will be measured from some axis (such as the the x, y, or z axis), and we will integrate by radiating outward from that axis.
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http://mechanicsmap.psu.edu/websites/A2_moment_intergrals/A2-1_moment_integrals/momentintegrals.html
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What Are Fractions: Step by Step Lesson with Interactive Exercises
A circle is a geometric shape that we have seen in other lessons. The circle to the left can be used to represent one whole. We can divide this circle into equal parts as shown below.
This circle has been divided into 2 equal parts.
This circle has been divided into 3 equal parts.
This circle has been divided into 4 equal parts.
We can shade a portion of a circle to name a specific part of the whole as shown below.
What Are Fractions? Definition: A fraction names part of a region or part of a group. The top number of a fraction is called its numerator and the bottom part is its denominator.
So a fraction is the number of shaded parts divided by the number of equal parts as shown below:
number of shaded parts numerator
number of equal parts denominator
Looking at the numbers above, we have:
There are two equal parts, giving a denominator of 2. One of the parts is shaded, giving a numerator of 1.
There are three equal parts, giving a denominator of 3. Two of the parts are shaded, giving a numerator of 2.
There are four equal parts, giving a denominator of 4. One of the parts is shaded, giving a numerator of 1.
Note that the fraction bar means to divide the numerator by the denominator. Let's look at some more examples of fractions. In examples 1 through 4 below, we have identified the numerator and the denominator for each shaded circle. We have also written each fraction as a number and using words.
One-fourth Two-fourths Three-fourths
One-fifth Two-fifths Three-fifths Four-fifths
Why is the number 3/4ths written as three-fourths? We use a hyphen to distinguish a fraction from a ratio. For example, "The ratio of girls to boys in a class is 3 to 4." This ratio is written a 3 to 4, or 3:4. We do not know how many students are in the whole class. However, the fraction 3/4 is written as three-fourths (with a hyphen) because 3 is 3/4 of one whole. Thus a ratio names a relationship, whereas, a fraction names a number that represents the part of a whole. When writing a fraction, a hyphen is always used.
It is important to note that other shapes besides a circle can be divided in equal parts. For example, we can let a rectangle represent one whole, and then divide it into equal parts as shown below.
two equal parts
three equal parts
four equal parts
five equal parts
Remember that a fraction is the number of shaded parts divided by the number of equal parts. In the example below, rectangles have been shaded to represent different fractions.
The fractions above all have the same numerator. Each of these fractions is called a unit fraction.
Definition: A unit fraction is a fraction whose numerator is one. Each unit fraction is part of one whole (the number 1). The denominator names that part. Every fraction is a multiple of a unit fraction.
In examples 6 through 8, we will identify the fraction represented by the shaded portion of each shape.
In example 6, there are four equal parts in each rectangle. Three sections have been shaded in each rectangle, but not the same three. This was done intentionally to demonstrate that any 3 of the 4 equal parts can be shaded to represent the fraction three-fourths.
In example 7, each circle is shaded in different sections. However, both circles represent the fraction two-thirds. The value of a fraction is not changed by which sections are shaded.
In example 8, each rectangle is shaded in different sections. However, both rectangles represent the fraction two-fifths. Once again, the value of a fraction is not changed by which sections are shaded.
In the examples above, we demonstrated that the value of a fraction is not changed by which sections are shaded. This is because a fraction is the number of shaded parts divided by the number of equal parts.
Let's look at some more examples.
In example 9, the circle has been shaded horizontally; whereas, in example 10, the circle was shaded vertically. The circles in both examples represent the same fraction, one-half. The positioning of the shaded region does not change the value of a fraction.
In example 11, the rectangle is positioned horizontally; whereas in example 12, the rectangle is positioned vertically. Both rectangles represent the fraction four-fifths. The positioning of a shape does not change the value of the fraction it represents.
Remember that a fraction is the number of shaded parts divided by the number of equal parts.
In example 13, we will write each fraction using words. Place your mouse over the red text to see if you got it right.
Summary: What Are Fractions? A fraction names part of a region or part of a group. A fraction is the number of shaded parts divided by the number of equal parts. The numerator is the number above the fraction bar, and the denominator is the number below the fraction bar.
In Exercises 1 through 5, click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Note: To write the fraction two-thirds, enter 2/3 into the form.
|1. What fraction is represented by the shaded rectangle below?
|2. What fraction is represented by the shaded circle below?
|3. Write one-sixth as a fraction.
|4. Write three-sevenths as a fraction.
|5. Write seven-eighths as a fraction.
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https://mathserv.mathgoodies.com/lessons/fractions
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Ever find yourself wondering what all the wires and connectors do when it comes to RS232? Well, wonder no more! The RS232 protocol has a lot of mystery behind it, and this article will unpack those secrets. This post will also provide an in-depth look at how the RS232 protocol works and the essential components it requires to function properly. From its serial transmission capabilities to its communication protocols, you’ll be ready to get up and running with your own RS232 system after reading this post.
What is RS232 Protocol and How it Works?
RS232 stands for ‘Recommended Standard 232’ and it is a standard that defines the electrical, mechanical, functional, signal, and procedural characteristics of serial data communication. It was one of the first serial communications protocols developed in the early 1960s.
RS232 is still widely used as an interface between computers and other electronic devices such as modems, printers, mice, keyboards, plotters etc. RS232 uses a simple asynchronous bit-oriented transmission protocol that sends one character at a time without any handshaking signals or flow control.
The RS232 protocol can be used to send text messages over serial cables and also supports full-duplex communication (simultaneous two-way transmission). The main components of an RS232 connection are two devices connected by a serial cable. The two devices communicate with each other through TX (transmitter) and RX (receiver) signals over the same wire, so communication is usually referred to as ‘full-duplex’.
The data is encoded using a start bit, followed by 8 data bits and one or more stop bits. The transmission speed can be varied from 50 bps up to 115Kbps depending on the configuration of both devices. Data error checking is not included in the protocol but it can be implemented with either hardware or software solutions.
RS232 also supports control signals such as RTS (Request to Send) and CTS (Clear to Send). With these signals, both sides of the connection can flow control the transmission.
RS232 is still a popular protocol due to its simplicity and low cost, but it has been mostly replaced by USB and Ethernet in modern applications. It can be used for both point-to-point connections between two devices or more complex networks with multiple nodes connected.
How RS232 Works: Voltage levels
Transmitter Voltage Levels
RS232 defines voltage levels that the transmitting device (such as a PC) uses to send data. These voltage levels are referred to as “mark” and “space”. Mark is defined to be +3V to +25V concerning signal ground and Space is -3V to -25V concerning signal ground. The actual voltage value used for mark and space varies depending on the specific implementation of RS232 but these limits define the ranges. By convention, when no data is being sent, a logic 1 or mark level is transmitted.
When sending data characters, two different voltages are sent to represent ones and zeros in binary form. For example, if the transmitting device wants to send an 8-bit byte, it would send 8 mark and space voltages representing the binary value of that byte.
Receiver Voltage Levels
The receiver device (such as a serial printer) must also define voltage levels for interpreting data sent by the transmitter. In RS232, these are referred to as “positive” and “negative”. A Logic 1 is defined as a positive level between +3V and +15V concerning signal ground and a Logic 0 is defined as a negative level between -3V and -15V concerning signal ground.
In addition, receivers often have thresholds or hysteresis voltage settings which allow them to more accurately interpret arbitrary voltages from the transmitter. Hysteresis refers to a range of acceptable voltages. A receiver may be configured to interpret voltages between +2V and +6V as a logic 1, and voltages between -2V and -6V as a logic 0. This would provide the receiver with a more accurate interpretation of data from the transmitter.
RS232 is also able to support more than two voltage levels for transmitting data but these are rarely used in modern applications. The most common implementation is two voltage levels (mark/space) which are used for sending binary data characters from one device to another .
How RS2332 Works: The Bits
The RS232 Start Bit
RS232 data transmissions always begin with a start bit. This is usually represented by a logical low level or a 0 bit. The purpose of the start bit is to signal the receiving device that a new transmission is about to begin. All modern RS232 devices recognize this bit and will prepare themselves for the incoming data packet.
Data Packet Bits
After the start, bits come anywhere from 5 to 8 bits. These are referred to as “data packets” and they contain the actual information being sent between two devices. Depending on how your system has been set up, these bits could represent characters (letters, numbers, symbols) or numerical values (0-255).
The Parity Bit
The parity bit is an optional 8th communication bit. This bit exists to help ensure data integrity and accuracy by providing error-checking capabilities. If the parity bit is enabled, it will be added after the data packet bits and before the stop bit. The most common type of parity check is called “odd” or “even” parity.
The Stop Bit
After all of the data bits have been transmitted, a single logical high-level or 1-bit is sent out. This is known as the “stop bit” and its purpose is to signal that no more information will be sent in this transmission. Again, all modern RS232 devices recognize this stop bit and will know when a transmission has ended.
RS232 also has the ability to send flow control signals. These signals are used to tell the receiving device that it can or cannot accept incoming data. The most common type of flow control is called “Hardware Flow Control” and requires two additional communication lines (RTS, CTS) in order for it to work properly. Other types include Xon/Xoff and DTR/DSR protocols.
RS232 Handshake Signals
Hardware handshake involves the use of dedicated pins on the serial port interface to control the flow of data. These are usually known as Request To Send (RTS) and Clear To Send (CTS). In order for a successful hardware handshake to occur, both devices must have their respective RTS and CTS pins connected.
When one device is ready to send data, it signals this by setting its RTS pin high. This signal is then read by the receiving device which in turn sets its CTS pin high thus acknowledging that it is ready to accept data. Data can now be sent from one device to another over the serial connection without any problems.
Software handshaking makes use of special control codes which are sent over the serial connection to control the flow of data. The primary advantage of software handshaking is that it does not require any dedicated pins on the serial port interface and can thus be used with any standard serial port.
Software handshaking typically makes use of two different control codes: XON (Transmit On) and XOFF (Transmit Off). When one device is ready to send data, it signals this by sending an XON signal over the serial connection. The receiving device then acknowledges this by responding with an XOFF signal, indicating that it is now ready to receive data. Data can now be sent from one device to another without any problems.
In conclusion, both hardware and software handshaking are useful techniques that can be used to control the flow of data over a serial connection. Hardware handshake requires dedicated pins on the serial port interface whereas software handshake does not, making it suitable for any standard serial port. It is important to know which type of handshaking your application requires to ensure that data is transmitted correctly.
Difference between RS232 and UART
RS232 is mainly used for short–distance, low-speed data communication in industrial and commercial applications. It can be used to connect computers, modems, terminals and other devices. RS232 can also be used for connecting two or more microcontrollers together.
UART (Universal Asynchronous Receiver/Transmitter) is a type of serial communication protocol that sends data bit by bit over a single wire. UARTs are commonly used in embedded systems such as Raspberry Pi and Arduino boards to communicate with sensors, peripherals and other microcontrollers. UARTs are also widely used in Automotive applications such as ECUs (Engine Control Units). UARTs offer faster data transmission than the traditional RS232 protocol because they do not require start and stop bits. This makes them ideal for applications that require high data rates.
Apart from the difference in speed, another major difference between RS232 and UART lies in their power requirements. While RS232 requires a separate power supply to operate, UART can be powered directly by the microcontroller or board it is connected to. This makes UARTs more flexible and easier to use as they do not need an external source of energy. Additionally, most modern UARTs are full-duplex, meaning they can send and receive data simultaneously unlike RS232 which operates in half-duplex mode (only one-way data transmission).
RS232 has been around for quite some time but it is being slowly replaced by UART in modern applications. This is mainly because UARTs operate at greater speeds, have better power efficiency and require fewer wires for transmitting data compared to RS232. Additionally, they are also more affordable and easier to implement than their RS232 counterparts.
Overall, both RS232 and UART protocols have their own advantages and disadvantages, so choosing the right one depends on the specific application’s requirements. Depending on your project’s needs, either of these serial communication protocols might be ideal for you.
However, it is important to keep in mind that although UARTs offer faster speeds than RS232, these speeds come at a cost – namely higher power consumption levels as well as increased susceptibility to electromagnetic interference. Therefore, if you are using UARTs in an environment with high levels of EMI, it is important to ensure that your system is well-shielded and properly grounded.
It should also be noted that RS232 still has a place in some applications such as those requiring long-distance communication or those where lower power consumption is essential. In these cases, the reliable but slow speeds of RS232 can be preferable over the higher speeds of UARTs. Ultimately, both protocols have their advantages and disadvantages so choosing the right one for your particular application will depend on its requirements as well as budget considerations.
Advantages of RS232
RS232 is often used as it is the most reliable and secure method for communication. The data sent over this protocol can be encrypted, ensuring that the transmitted information remains confidential. Additionally, RS232 has good noise immunity since its signals are differential and not affected by common-mode interference. It also has a longer range compared to other protocols like UART and USB, making it suitable for long-distance applications. Since RS232 requires a separate power source to operate, it does not draw power from the devices connected to it which makes it more reliable than some other protocols.
Advantages of UART
UARTs offer faster speeds compared to RS232 which makes them perfect for high-speed applications such as Automotive. Additionally, UARTs are more cost-efficient and easier to implement than RS232s as they do not require a separate power source. UARTs also have better power efficiency levels since they draw their power directly from the microcontroller or board that it is connected to. Finally, most modern UARTs are full-duplex meaning they can send and receive data simultaneously which further increases the speed of data transfer.
Disadvantages of RS232
RS232 has several disadvantages compared to more modern protocols like UART and USB. Firstly, the data transmission speed is much slower than that of UARTs which makes them unsuitable for high-speed applications. Additionally, RS232 requires a separate power source for operation, making it less efficient than other protocols. Finally, since RS232 does not support encryption by default, transmitting confidential information over this protocol can be risky.
Disadvantages of UART
UARTs offer faster speeds compared to many other serial communication protocols but this comes at a cost – namely higher power consumption levels as well as increased susceptibility to electromagnetic interference (EMI). Additionally, although modern UARTs are full-duplex meaning they can send and receive data simultaneously, the maximum speed of data transmission is limited. This makes UARTs unsuitable for applications requiring very high-speed data transmissions.
Finally, UARTs are not as reliable as RS232 because they draw their power from the microcontroller or board that it is connected to which can lead to system disruptions in case of power failure or fluctuations.
How does the RS232 work?
RS232 is a standard communications protocol for serial data transfer. This means that a device can transmit data to, and receive data from another device over a single wire or other communication medium. RS232 uses voltages to indicate the presence of data on the line, with high voltage representing a binary 1 (mark), and low voltage representing a binary 0 (space). The two devices will agree upon which logic levels they will use before transmitting any data.
The RS232 protocol also defines timing information such as when individual bits should be sent, when messages begin and end, and how long characters may take before being considered invalid. These timings are usually referred to as baud rate, which is typically 9600 or higher for modern systems.
In addition to sending and receiving data, RS232 can also be used for flow control, allowing a device to tell the other device when it is ready to receive more information. This is typically handled by two pins in the RS232 cable – one called RTS (request-to-send), and another called CTS (clear-to-send). When one device sets its RTS pin high, that indicates that it is ready to receive data from the other device, which will then respond by setting its CTS pin high.
Once all of these protocols are established, devices can communicate reliably over an RS232 connection. This makes it ideal for computers communicating with peripherals such as printers, modems, or other external devices. RS232 is still in use today, though it has largely been replaced by USB and other modern protocols for most home and office applications.
What are the advantages of using RS232?
The primary advantage of using RS232 is that it allows for reliable communications between two or more devices over a single wire or other communication medium. This eliminates the need for multiple wires or cables running from one device to another, which can be difficult to manage and set up. In addition, since the protocol defines timing information such as baud rate, messages are less likely to be corrupted during transmission, ensuring greater reliability overall.
Moreover, since RS232 uses voltages to indicate the presence of data on the line, it can have different levels of logic, allowing devices to communicate in different ways (for example, TTL and CMOS). This makes it ideal for different types of applications that require different kinds of communication.
Finally, RS232 is still widely used today, so many devices are capable of communicating with each other through this protocol. As a result, it’s relatively easy to find compatible parts and software for RS232 connections.
Is RS232 analog or digital?
RS232 is a digital communication protocol, as it uses high and low voltages to represent the presence of data. However, RS232 is not a purely digital protocol, as it also includes timing information such as the baud rate to ensure that messages are sent and received correctly. This makes RS232 suitable for both analog and digital applications.
As an example, RS232 can be used to connect computers with modems or other external devices (such as printers). In this case, the signals being transmitted would be digital, but the modem would use analog signals to communicate over a phone line. The same principles apply when connecting two computers together using an RS232 cable – while the signal may be digital on one end, it will be converted to analog at the other end.
In summary, RS232 is a digital communication protocol that can be used for both analog and digital applications. By using timing information such as baud rate, it ensures reliable communication over a single wire or medium.
What types of devices use RS232?
RS232 is commonly used to communicate between computers and external peripherals such as printers, modems, cameras, barcode scanners, and other devices. It’s also popular in industrial settings such as factories and warehouses where multiple devices need to communicate with each other reliably. Additionally, RS232 is often used in embedded systems that require precise control over how data is sent and received.
Finally, many modern audio/visual systems use RS232 for remote control, allowing users to control their entertainment equipment from a single device.
How does RS232 handshaking work?
RS232 handshaking is used to share information about data transmission between two devices. This involves the use of two pins in the RS232 cable – one called RTS (request-to-send), and another called CTS (clear-to-send). When one device sets its RTS pin high, that indicates that it is ready to receive data from the other device, which will then respond by setting its CTS pin high.
In addition, RS232 allows for flow control, allowing devices to tell each other when they are ready for more data. This is typically done using the XOFF and XON characters, which act as commands instructing a device to stop or start sending data respectively.
How to read RS232 signal?
Reading an RS232 signal is relatively easy, as the protocol uses high and low voltage levels to indicate the presence of data on the line. To read signals from a device connected to an RS232 connection, you will need an oscilloscope or logic analyzer capable of measuring voltage levels. This allows you to measure changes in voltage on the line and determine which logic level (0 or 1) is being transmitted.
Useful Video: The RS-232 protocol
RS232 protocol works well for short-distance communication, usually no more than 50 feet. This type of serial communication is highly reliable and secure since it requires a dedicated connection between two devices. It is also very straightforward to configure and use in comparison to other types of data transfer protocols such as Ethernet or Wi-Fi. RS232 protocol is an ideal choice for situations where simple one-to-one data transfers are needed, especially when time sensitivity isn’t an issue. Despite its simplicity, RS232 still provides a dependable and efficient way for computers and other devices to communicate with each other.
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In today’s rapidly evolving technological landscape, artificial intelligence (AI) is playing an increasingly significant role in various fields, including education. AI, in simple terms, is the development of computer systems that can perform tasks that would typically require human intelligence. But what is AI, and how can it revolutionize education?
At its core, AI involves the creation of algorithms and models that enable computer systems to analyze vast amounts of data, learn from it, and make decisions or predictions based on that information. In education, this means that AI can be used to streamline administrative tasks, personalize educational content, and provide adaptive learning experiences.
One of the main advantages of AI in education is its ability to collect and analyze data about students’ learning patterns, preferences, and areas for improvement. With this information, educators can develop personalized curricula and provide individualized support to each student. AI can also automate grading and assessments, freeing up teachers’ time to focus on more valuable tasks, such as providing guidance and mentorship to students.
The Impact of Artificial Intelligence on Education
Artificial Intelligence (AI) is transforming the education landscape in numerous ways. With its ability to analyze massive amounts of data and perform complex tasks, AI is revolutionizing the way students learn and teachers teach.
One of the key impacts of AI in education is its ability to personalize learning. AI-powered systems can gather and analyze data on individual students’ strengths, weaknesses, and learning styles. This allows for customized and adaptive learning experiences that cater to each student’s unique needs. Students can receive personalized feedback, recommendations, and resources, ultimately enhancing their learning outcomes.
Furthermore, AI is also automating administrative tasks in education. From grading papers to scheduling classes, AI systems can streamline routine administrative processes, freeing up teachers’ time to focus on instruction and student support. This increases efficiency and allows educators to dedicate more time to individualized instruction and student engagement.
AI is also being utilized to enhance educational content and delivery. Intelligent tutoring systems provide students with virtual tutors that can provide personalized instruction and feedback. Adaptive learning platforms use AI algorithms to dynamically adjust the difficulty and pace of content based on the individual student’s progress. This ensures that students are constantly challenged and engaged, while also receiving support and guidance when needed.
In addition, AI is improving the accessibility of education. Language translation tools powered by AI allow students to access educational resources in multiple languages. AI-powered chatbots provide instant support and answers to students’ questions, enhancing their learning experience and reducing barriers to knowledge acquisition.
While the impact of AI in education is undoubtedly significant, it is important to note that human teachers remain essential. AI should be seen as a tool to support and enhance the role of educators, rather than replace them. With the integration of AI in education, teachers can leverage technology to deliver more personalized and effective instruction, ultimately creating a more engaging and inclusive learning environment.
In conclusion, the integration of AI in education is revolutionizing the way students learn and teachers teach. From personalized learning experiences to automated administrative tasks, AI is enhancing the efficiency, effectiveness, and accessibility of education. However, it is crucial to remember that AI should be seen as a tool that complements the role of human educators, ensuring a holistic and well-rounded educational experience for students.
Artificial Intelligence and Education
Artificial intelligence (AI) is rapidly making its way into various industries and education is no exception. AI in education is revolutionizing the way we teach and learn, offering new possibilities and opportunities.
AI technology enables personalized learning experiences for students, allowing them to learn at their own pace and in a way that suits their individual needs. AI-powered education systems can analyze vast amounts of data and provide tailored recommendations and resources, helping students to grasp concepts more effectively.
In addition, AI can assist teachers in various ways, such as automating administrative tasks, grading assignments, and providing real-time feedback. With AI tools, teachers can focus more on personalized instruction and mentoring, creating a more engaging and dynamic learning environment.
The use of AI in education is not limited to traditional classrooms. Online learning platforms and educational apps are incorporating AI technology to enhance the learning experience. Virtual tutors can provide instant assistance and feedback, making learning interactive and engaging.
However, it is important to note that while AI has the potential to greatly enhance education, it is not meant to replace human teachers. The role of human educators will always be crucial, as they provide the emotional connection, guidance, and support that AI cannot replicate.
Overall, the integration of AI in education is transforming the way we acquire knowledge and skills. With its ability to personalize learning, assist teachers, and provide interactive experiences, AI is shaping the future of education and opening up new avenues for learning and growth.
Advantages of Implementing AI in Education
Artificial Intelligence (AI) is revolutionizing the education sector today. With the advancements in technology, AI is being integrated into various aspects of education, resulting in numerous advantages for both students and educators.
Enhanced Personalized Learning
One of the key advantages of implementing AI in education is the ability to provide personalized learning experiences. AI-powered tools can analyze individual student’s strengths and weaknesses, allowing educators to tailor the curriculum accordingly. This personalized approach helps students to learn at their own pace, ensuring better understanding and knowledge retention.
Efficient Administrative Tasks
AI can significantly streamline administrative tasks in educational institutions. Automated systems can handle tasks like grading assignments, scheduling classes, and managing student records, freeing up educators’ time to focus on teaching and providing support. This helps to improve overall workflow efficiency and reduce the administrative burden on teachers and staff.
AI-enabled tools can also provide valuable insights on student performance and progress, allowing educators to identify areas of improvement and implement targeted interventions. This data-driven approach facilitates early identification of learning gaps and helps educators take proactive steps to address them.
Improved Accessibility and Inclusion
A notable advantage of AI in education is its ability to enhance accessibility and inclusion for diverse learners. AI-powered technologies provide solutions like text-to-speech, voice recognition, and language translation, making educational materials more accessible for students with disabilities. These technologies also assist non-native English speakers in understanding and interacting with the learning content, fostering a more inclusive learning environment.
Engaging Learning Experiences
AI can create interactive and engaging learning experiences for students. Virtual reality (VR) and augmented reality (AR) technologies powered by AI can simulate real-world scenarios, enabling immersive learning. These technologies can bring abstract concepts to life and enhance student engagement, increasing their motivation and interest in learning.
In conclusion, implementing AI in education offers various advantages such as personalized learning, efficient administrative tasks, improved accessibility and inclusion, and engaging learning experiences. With the continued advancements in AI, the future of education looks promising, with AI playing a crucial role in transforming the way students learn and educators teach.
Enhancing Personalized Learning with AI
In today’s rapidly evolving world, technology has become an integral part of our lives, transforming various industries, including education. Artificial Intelligence (AI) is one such technological advancement that has the potential to revolutionize the way we learn and acquire knowledge.
AI has the ability to analyze vast amounts of data and identify patterns and trends that are beyond human capability. This makes it an ideal tool in education, where personalization is key to effective learning. AI can tailor educational content based on the unique needs and preferences of each student, providing them with a customized learning experience.
With AI in education, students can benefit from adaptive learning platforms that adapt to their individual learning styles, pace, and aptitudes. These platforms can analyze students’ performance data and provide personalized recommendations and interventions to help them improve their understanding of the subject matter.
Additionally, AI can assist educators in designing and delivering personalized lessons and assignments. By analyzing student performance data, AI can identify areas of weakness and recommend appropriate resources and activities to address those areas. This ensures that students receive targeted instruction and support, leading to improved learning outcomes.
Moreover, AI-powered virtual tutors can provide students with one-on-one guidance and support, even outside of the classroom. These virtual tutors can engage students in interactive conversations, answer their questions, and provide immediate feedback. This not only enhances the learning experience but also fosters independent learning and critical thinking skills.
In conclusion, AI has the potential to transform education by enhancing personalized learning experiences. By leveraging AI technologies, educators can provide tailored instruction and support to each student, improving learning outcomes and fostering a lifelong love for learning.
AI-powered Tutoring and Virtual Assistants
Artificial Intelligence (AI) is revolutionizing the field of education, and one area where its impact is particularly significant is in tutoring and virtual assistants. These AI-powered tools are designed to enhance the learning experience and provide personalized support to students.
One of the main advantages of AI-powered tutoring is its ability to adapt and tailor the content to the needs of individual students. By analyzing data and understanding the strengths and weaknesses of each student, AI tutors can create customized learning paths that cater to their specific needs. This personalized approach allows students to learn at their own pace, ensuring that they grasp the concepts thoroughly.
Another benefit of AI-powered tutoring is its accessibility. Traditional tutoring methods may be expensive or limited by geographical constraints. However, with AI tutors, students can access assistance from anywhere, at any time. Whether it’s solving complex math equations or reviewing literature concepts, students can rely on AI-powered tutoring to provide them with immediate feedback and guidance.
In addition to tutoring, AI-powered virtual assistants are also making a significant impact in education. These virtual assistants are designed to assist students in various tasks, such as answering questions, providing study materials, or even offering career guidance. Virtual assistants can also be programmed to engage students in interactive learning activities, making the educational experience more engaging and enjoyable.
What makes AI-powered tutoring and virtual assistants truly transformative is their ability to continuously learn and improve. As AI algorithms process more data and interact with a greater number of students, they become smarter and more effective. These tools can detect patterns in learning behavior, identify common mistakes, and adapt their teaching methods accordingly, providing an increasingly personalized and effective educational experience.
In conclusion, AI-powered tutoring and virtual assistants are revolutionizing the field of education. By leveraging the power of AI, these tools provide personalized support, improve accessibility, and continuously learn and improve. As technology advances, we can expect AI-powered tutoring and virtual assistants to play an even greater role in shaping the future of education.
Improving Student Assessment with AI
Artificial intelligence (AI) is revolutionizing the way student assessment is conducted in educational institutions. AI technologies are being used to enhance and streamline the assessment process, providing more accurate and personalized feedback to students. By analyzing large amounts of data, AI can identify patterns and trends that humans may miss, leading to more precise evaluations of student performance.
What is AI in student assessment?
AI in student assessment involves the use of algorithms and machine learning to evaluate student work and provide feedback. This can be done through automated grading systems that can assess answers to multiple-choice questions, essays, or programming assignments. AI-powered assessment tools can also analyze student behavior and engagement to gain insights into their learning patterns and identify areas of improvement.
The role of AI in student assessment
AI has the potential to improve student assessment in several ways. First, AI can reduce the time and effort required for grading, allowing teachers to devote more time to instruction. It can provide immediate feedback to students, enabling them to address their mistakes and improve their understanding of the subject matter. Furthermore, AI can help identify gaps in knowledge and suggest personalized learning resources to fill those gaps, promoting individualized education.
AI can also help mitigate human bias in assessment, as it is programmed to evaluate students based on objective criteria rather than subjective opinions. This can lead to fairer and more transparent evaluations, ensuring that students are assessed fairly regardless of their background or characteristics.
However, it is important to note that AI should be used as a tool to support and complement human assessment, rather than replacing it entirely. Human judgment and expertise are still crucial in ensuring a comprehensive and holistic evaluation of student performance.
In conclusion, AI offers great potential for improving student assessment in education. By leveraging AI technologies, educational institutions can enhance the assessment process, provide personalized feedback, and promote a fairer evaluation of student performance.
AI as a Tool for Adaptive Learning
Artificial Intelligence (AI) is revolutionizing the way education is delivered and experienced. In the realm of adaptive learning, AI is playing a pivotal role in personalizing educational experiences for students.
One of the key challenges in traditional education is that every student has unique learning needs, strengths, and weaknesses. AI has the potential to address this challenge by providing adaptive learning environments tailored to each student’s individual requirements.
Personalized Learning Paths
AI algorithms can analyze vast amounts of data to understand a student’s learning patterns, preferences, and progress. This analysis enables AI to create personalized learning paths for each student, ensuring that they receive instructional content and activities that are appropriate for their skill level and learning style.
With AI-powered adaptive learning platforms, students can engage with materials and activities that are neither too easy nor too challenging, maximizing their learning potential. These platforms can also monitor student performance in real-time, providing immediate feedback and support.
By adapting the learning experience to meet each student’s unique needs, AI can enhance student engagement, motivation, and achievement.
Identifying Knowledge Gaps
AI can also identify knowledge gaps in a student’s learning journey. By analyzing a student’s responses to quizzes, assessments, and other learning activities, AI algorithms can pinpoint areas where the student may be struggling or lacking understanding.
Based on this analysis, AI can generate targeted remedial materials and activities to help the student bridge these knowledge gaps. This personalized approach ensures that students receive the support and resources they need to overcome challenges and continue progressing in their learning.
Furthermore, AI can continuously adapt and refine its instructional strategies based on feedback from student interactions. This iterative process allows AI to optimize the learning experience over time, ensuring that students receive the most effective and efficient instruction possible.
Support for Educators
AI can also provide valuable support for educators by automating administrative tasks and providing data-driven insights into student performance. With AI handling routine tasks such as grading and attendance tracking, educators can dedicate more time to personalized instruction and mentoring.
Additionally, AI can analyze student data to provide educators with a deeper understanding of individual student needs, progress, and challenges. This information can inform instructional strategies, interventions, and other forms of support.
Overall, AI’s role in adaptive learning is transforming education by tailoring the learning experience to each student’s unique needs, identifying knowledge gaps, and providing support for educators. As AI continues to evolve and become more sophisticated, its potential to enhance education will only continue to grow.
Automating Administrative Tasks with AI
In the field of education, artificial intelligence (AI) is revolutionizing administrative tasks and streamlining processes. AI refers to the simulation of human intelligence in machines that are programmed to think and learn like humans. In the context of education, AI can effectively automate administrative tasks, allowing educators and administrators to focus on more strategic activities.
What is AI in Education?
AI in education involves the use of intelligent machines and algorithms to assist in various aspects of the learning process. This includes tasks such as grading papers, providing personalized feedback to students, and managing administrative workflows. AI can analyze large amounts of data, identify patterns, and make data-driven decisions, thereby enhancing administrative efficiency.
The Role of AI in Automating Administrative Tasks
AI can automate various administrative tasks in education, such as student enrollment, scheduling, and record-keeping. For instance, AI-powered systems can automatically process admission forms and verify documents, eliminating the need for manual data entry. These systems can also generate class schedules by considering factors such as teacher availability, student preferences, and room availability.
Furthermore, AI can enhance record-keeping by digitizing and organizing student data. It can automatically update student records with information from multiple sources, such as attendance data, grades, and disciplinary records. This automation reduces the administrative burden on educators and ensures accurate and up-to-date records.
AI can also improve communication and collaboration among various stakeholders in education. Chatbot systems powered by AI can provide instant responses to frequently asked questions from students, parents, and staff. This reduces the need for human intervention in repetitive and time-consuming administrative tasks.
AI is transforming the field of education by automating administrative tasks, enabling educators and administrators to focus on more meaningful and strategic activities. By leveraging AI technologies, educational institutions can streamline processes, improve efficiency, and enhance communication and collaboration. As AI continues to advance, its role in education is likely to expand, further revolutionizing the way administrative tasks are carried out.
Overall, the integration of AI in education holds great promise for creating an environment that is efficient, effective, and conducive to learning and growth.
AI in Language Learning and Translation
Artificial Intelligence (AI) is revolutionizing education in various ways, and one significant area where it is making a difference is in language learning and translation. AI technology is transforming the way students acquire and master different languages, as well as how translation tasks are carried out.
What makes AI so effective in language learning is its ability to provide personalized and adaptive learning experiences. AI-powered language learning platforms can analyze learners’ performance and tailor the content and level of difficulty to their individual needs. Students can receive real-time feedback, practice pronunciation, and engage in interactive exercises that optimize their language learning outcomes.
Furthermore, AI can also enhance translation processes by automating and improving accuracy. Language translation tools powered by AI can quickly and efficiently translate text from one language to another, helping bridge communication gaps and facilitating international collaborations. These tools use machine learning algorithms that continuously learn and improve based on the vast amount of available language data.
Another application of AI in language learning and translation is the development of chatbots and virtual language tutors. Chatbots powered by AI technology can provide students with an interactive conversational experience, allowing them to practice and improve their language skills. Similarly, virtual language tutors can simulate a real-life tutor and provide personalized guidance and assistance to learners.
Overall, the integration of AI in language learning and translation is transforming the way we learn and communicate in different languages. The personalized and adaptive learning experiences, automated translation processes, and interactive conversational tools provided by AI technology are helping individuals communicate effectively and efficiently in today’s globalized world.
AI-driven Learning Analytics and Insights
Education is constantly evolving and adapting to the needs of students. With the advancements in Artificial Intelligence (AI), educators have a powerful tool at their disposal for transforming the learning experience.
What is AI-driven learning analytics?
AI-driven learning analytics is the process of using AI algorithms and technologies to analyze student learning data and generate insights. This includes collecting and analyzing data from various sources such as learning management systems, online assessments, and student interactions.
AI-driven learning analytics goes beyond traditional methods of analyzing student performance. It can provide educators with real-time feedback and recommendations, allowing them to personalize the learning experience for each student.
What can AI-driven learning analytics do?
AI-driven learning analytics can provide educators with valuable insights and information that can improve teaching strategies and student outcomes. Some of the benefits include:
- Identifying areas of improvement: AI algorithms can analyze student data to identify areas where students are struggling or excelling. This allows educators to adjust their teaching methods accordingly.
- Personalizing learning: AI algorithms can analyze individual student data to recommend personalized learning materials and activities. This helps students to learn at their own pace and in a way that suits their individual needs.
- Monitoring student engagement: AI algorithms can track student engagement and predict potential drop-out rates. This allows educators to intervene early and provide the necessary support.
- Assessing the effectiveness of teaching strategies: AI algorithms can analyze data from different teaching strategies to evaluate their effectiveness. Educators can then adjust their methods to improve student learning outcomes.
Overall, AI-driven learning analytics has the potential to revolutionize education by providing educators with valuable insights and recommendations. It allows for a more personalized and effective learning experience, helping students to reach their full potential.
AI for Special Needs Education
Artificial Intelligence (AI) is playing an increasingly important role in the field of education. One area where AI is making a significant impact is in special needs education. AI technology offers unique opportunities to support and enhance the learning experience for students with special needs.
AI can be utilized to personalize and individualize learning for students with special needs. Through machine learning algorithms, AI can analyze a student’s unique learning style, strengths, and weaknesses, and adapt educational materials and strategies accordingly. This personalized approach can help students with special needs overcome their challenges and maximize their learning potential.
Benefits of AI in Special Needs Education
- Individualized Instruction: AI algorithms can provide tailored instruction to meet the specific needs of each student with special needs, helping them progress at their own pace.
- Speech Recognition: AI-powered speech recognition technology can assist students with special needs who have difficulty communicating verbally. It can help them practice speech and provide feedback and support.
Challenges and Ethical Considerations
- Data Protection: AI systems used in special needs education collect and analyze sensitive student data. It is crucial to ensure that this data is protected and used ethically to ensure student privacy.
- Equity: There is a risk that AI technology may exacerbate existing inequalities in special needs education. It is important to ensure that AI tools and resources are accessible and affordable for all students with special needs.
In conclusion, AI has the potential to revolutionize special needs education by providing personalized instruction and support to students with unique learning needs. However, it is important to approach the implementation of AI in special needs education with caution and consider the ethical considerations to ensure that it benefits all students regardless of their abilities.
Challenges and Ethical Considerations of AI in Education
As artificial intelligence (AI) continues to play an increasingly important role in education, it is crucial to consider the challenges and ethical implications that come with its implementation. While AI has the potential to revolutionize the education system and enhance learning experiences, it also presents various challenges that need to be addressed.One of the major challenges in integrating AI in education is ensuring that it is used responsibly and ethically. This involves addressing concerns such as privacy and data security. AI systems often require access to large amounts of personal data to function effectively, raising concerns about the protection of sensitive information and the potential misuse of data. It is essential to establish robust privacy policies and security measures to safeguard student and teacher data.
Another challenge is ensuring that AI tools are inclusive and accessible to all students. AI algorithms may be biased, leading to inequality and discrimination in educational outcomes. For example, if AI systems are trained on datasets that primarily represent certain demographics, they may not accurately cater to the needs of students from different backgrounds or with diverse learning styles. It is crucial to develop and test AI models on diverse datasets to ensure fairness and equal opportunities for all learners.
Furthermore, there can be ethical considerations surrounding the use of AI in education. For example, some argue that relying too heavily on AI systems can reduce human interactions and limit the development of critical social and emotional skills in students. It is important to strike a balance between utilizing AI technology and providing opportunities for human interaction to foster holistic growth.
Additionally, there is a need for transparency and explainability in AI algorithms used in education. Students, teachers, and parents should have a clear understanding of how AI systems make decisions and recommendations. Lack of transparency can lead to mistrust and skepticism, undermining the credibility and effectiveness of AI in education.
In conclusion, while AI holds great promise in revolutionizing education, there are several challenges and ethical considerations that need to be addressed. Ensuring responsible and ethical use of AI, promoting inclusivity and accessibility, and maintaining transparency are key aspects of successfully integrating AI in education. By carefully navigating these challenges, AI can be utilized as a powerful tool to enhance learning experiences and promote educational equity.
Addressing Bias and Equity in AI Education Systems
In recent years, there has been a growing interest in integrating artificial intelligence (AI) in education. AI has the potential to revolutionize the way we learn and teach, making educational systems more personalized, adaptive, and efficient. However, in order to fully harness the power of AI in education, it is essential to address the issue of bias and ensure equity.
AI systems are developed and trained using vast amounts of data, including historical data, which can contain biases. These biases can be unintentionally embedded into AI algorithms, leading to biased outcomes and perpetuating existing inequalities. It is crucial to recognize that AI is not inherently biased, but it reflects the biases and prejudices of human creators and the data it is trained on.
Addressing bias in AI education systems requires a multi-faceted approach. First, it is important to diversify the development teams behind AI systems, ensuring the inclusion of individuals from different backgrounds and perspectives. This diversity can help mitigate the risk of bias by bringing in different viewpoints and challenging assumptions.
Second, AI algorithms need to be regularly audited and tested for bias. This process involves analyzing the data used to train the algorithms, identifying potential biases, and making adjustments to minimize them. Additionally, guidelines and standards should be established to ensure that AI systems are developed and used ethically, with a focus on fairness, transparency, and accountability.
Equity in AI education systems involves ensuring that all learners have access to and benefit from the educational opportunities provided by AI. This requires addressing the digital divide, as not all students may have access to the necessary technology and resources. Schools and educational institutions can play a crucial role in bridging this divide by providing equal access to AI tools and resources to all students, regardless of their socio-economic background.
Furthermore, AI systems should be designed to accommodate different learning styles, cultural backgrounds, and individual needs. Personalization is a key aspect of AI in education, and it should be used to tailor instruction and support to individual learners, rather than perpetuating existing biases or reinforcing stereotypes.
In conclusion, while the integration of AI in education holds great promise, it is essential to address bias and ensure equity. By diversifying development teams, auditing algorithms for bias, establishing ethical guidelines, and promoting equal access and personalization, we can create AI education systems that are fair, inclusive, and empowering for all learners.
Privacy and Data Security in AI Education
In today’s digital age, data privacy and security have become major concerns in various industries, especially in the field of education. With the integration of artificial intelligence (AI) into education systems, it is crucial to understand what the implications are for privacy and data security.
AI is transforming education by providing personalized and adaptive learning experiences for students. It allows educators to analyze vast amounts of data and provide tailored recommendations and feedback. However, with the collection and analysis of such data, there is a need to ensure that privacy and data security are not compromised.
One of the key concerns in AI education is the protection of sensitive student data. This includes personal information such as names, addresses, and social security numbers, as well as academic records and performance data. Educational institutions must have robust security protocols in place to safeguard this information from unauthorized access and use.
Another aspect to consider is the transparency of data usage in AI education. Students, parents, and educators need to understand what data is being collected, how it is being used, and who has access to it. Clear communication and consent mechanisms should be established to ensure that individuals are aware of the data being collected and how it will be utilized.
Furthermore, it is essential to implement ethical practices when leveraging AI in education. Educators and developers should abide by strict guidelines to ensure that data is used ethically and responsibly. This includes obfuscating and anonymizing data to protect individual identities and ensuring that data is only used for educational purposes.
In conclusion, as AI continues to shape the landscape of education, it is crucial to prioritize privacy and data security. Educational institutions must establish robust security protocols, maintain transparency in data usage, and adhere to ethical practices. By doing so, AI education can provide personalized learning experiences while safeguarding sensitive student data.
The Future of AI in Education
Artificial intelligence (AI) is quickly becoming a key component in education. As technology continues to advance, the potential for AI in education is unparalleled. But what exactly is AI and what role does it play in education?
AI is the simulation of human intelligence in machines that are programmed to think and learn like humans. It can perform tasks that typically require human intelligence, such as natural language processing, problem-solving, and decision-making.
In education, AI can revolutionize the way students learn and teachers teach. With AI-powered tools, students can receive personalized and adaptive learning experiences. AI can analyze data from students’ performance and provide tailored recommendations to address their individual needs and strengths.
What sets AI apart in education is its ability to provide real-time feedback and support. AI-powered chatbots, for example, can answer students’ questions 24/7, allowing them to access help whenever they need it. This can enhance students’ learning experience and improve their overall performance.
AI can also assist teachers in various ways. It can automate administrative tasks, such as grading and organizing assignments, freeing up time for teachers to focus on instruction and mentorship. AI can also help teachers identify areas where students may be struggling, enabling them to provide targeted interventions.
The future of AI in education holds immense potential. As technology continues to evolve, AI will become even more sophisticated and integrated into educational practices. It will enable educators to personalize instruction to meet the diverse needs of students and provide them with innovative learning experiences.
In conclusion, AI is transforming education by providing personalized learning experiences and innovative tools for both students and teachers. With AI, education is evolving to become more adaptive, interactive, and tailored to individual needs. The future of AI in education is bright, and it will undoubtedly continue to reshape the way we learn and teach.
Role of Teachers in an AI-powered Classroom
In the rapidly advancing field of technology, artificial intelligence (AI) is playing a crucial role in revolutionizing various industries, including education. With its ability to process vast amounts of data and provide personalized feedback, AI is transforming the way students learn and teachers instruct.
However, in an AI-powered classroom, the role of teachers remains indispensable. While AI can assist in automating certain tasks and providing targeted support, it is the educators who possess the human touch and can create a nurturing learning environment.
Educators are responsible for interpreting the data generated by AI tools and using it to tailor their teaching methods to the needs of individual students. They can identify patterns, analyze trends, and make informed decisions based on the insights provided by AI systems. This allows them to optimize their teaching strategies and ensure that every student receives the attention they require.
Additionally, educators have a crucial role in teaching critical thinking skills and fostering creativity. While AI can provide information and answer questions, it is the teachers who can provoke deeper thinking, encourage curiosity, and prompt students to explore new ideas. They can guide discussions, facilitate collaborative projects, and inspire students to become active participants in their own learning.
Furthermore, teachers play a vital role in imbuing students with essential values and social skills. While AI can provide academic knowledge, it cannot instill empathy, compassion, or ethical behavior. Teachers have the responsibility to model these qualities, teach emotional intelligence, and nurture a sense of community within the classroom.
Lastly, teachers serve as mentors and role models for students. They provide guidance, encouragement, and support to help students overcome challenges and reach their full potential. Their presence and individualized attention can make a significant difference in a student’s educational journey.
In conclusion, while AI has undoubtedly brought significant advancements in education, the role of teachers cannot be underestimated. In an AI-powered classroom, teachers have the vital responsibility of utilizing AI tools effectively, interpreting the data generated, and providing personalized guidance to students. They play an irreplaceable role in cultivating critical thinking skills, fostering creativity, teaching values, and serving as mentors. Together, AI and teachers can create a powerful learning environment that prepares students for the challenges of the future.
AI and STEM Education
In today’s increasingly digital world, the use of artificial intelligence (AI) is becoming more prevalent in various industries, and education is no exception. AI has the potential to revolutionize the way STEM (science, technology, engineering, and mathematics) subjects are taught and learned, providing a more interactive and personalized learning experience for students.
The Role of AI in STEM Education
Artificial intelligence can play a significant role in enhancing STEM education by providing students with real-time feedback, adaptive learning paths, and personalized recommendations. AI-powered tools can analyze individual student’s progress and identify areas where they need additional support or challenges. This allows educators to cater to the unique needs of each student and offer tailored content and activities accordingly.
Moreover, AI can provide simulations and virtual experiments that can help students understand complex scientific concepts and theories. Through these simulations, students can gain hands-on experience and visualize abstract concepts that may be difficult to grasp through traditional teaching methods alone.
Additionally, AI can assist educators in creating more engaging and interactive learning materials. This can include the use of chatbots or virtual tutors that can answer students’ questions or provide explanations in real-time. By integrating AI into the learning process, students can have access to continuous support and guidance, improving their understanding and retention of STEM subjects.
Challenges and Considerations
While AI holds great promise for STEM education, there are some challenges and considerations that need to be addressed. One concern is the potential bias in AI algorithms, as they can reflect the biases present in the data they are trained on. It is crucial to ensure that AI tools used in education are fair and unbiased, providing equal opportunities for all students.
Another challenge is the need for teacher training to effectively implement AI in the classroom. Educators should be adequately trained to use AI tools and understand how to interpret and utilize the data provided by these tools to enhance the learning experience. Collaboration between educators and AI developers is essential to optimize the integration of AI in STEM education.
|Personalized learning experiences
|Potential bias in AI algorithms
|Need for teacher training
|Interactive and engaging materials
|Dependency on technology
|Simulations and virtual experiments
|Lack of emotional intelligence
In conclusion, AI has the potential to greatly enhance STEM education by providing personalized learning experiences and interactive materials. However, careful consideration should be given to address challenges such as bias in AI algorithms and the need for teacher training. By harnessing the power of AI in education, we can create a more engaging and effective learning environment for students in the field of STEM.
AI-based Content Creation and Curation
In the field of education, the availability of relevant and high-quality learning materials is crucial for effective teaching and learning. However, one challenge faced by educators is the time-consuming task of content creation and curation. This is where Artificial Intelligence (AI) comes in.
What is AI?
AI is a branch of computer science that focuses on creating intelligent machines capable of learning and performing tasks without explicit programming. It involves the development of algorithms and models that enable machines to analyze data, make decisions, and perform human-like tasks.
AI in Content Creation
With AI, education professionals can leverage automated content creation tools to develop interactive and personalized learning materials. These tools can generate quizzes, assignments, and practice exercises based on students’ specific needs and learning styles.
In addition to creating content, AI can also assist in enhancing the quality of educational materials. AI algorithms can analyze and evaluate existing content, providing feedback and suggestions for improvement. This ensures that the materials align with current educational standards and are engaging for students.
AI in Content Curation
Content curation is the process of collecting and organizing relevant educational materials from various sources. AI can play a significant role in this process by automating the search, filtering, and categorization of content.
AI algorithms can analyze large volumes of educational resources, such as books, articles, and online resources, to identify the most relevant and up-to-date materials. This ensures that educators have access to a diverse range of resources that meet the specific needs of their students.
Overall, AI-based content creation and curation provide educators with time-saving solutions and access to high-quality learning materials. By leveraging AI, educators can enhance their teaching practices and ensure that students receive the best possible education.
Using AI for Intelligent Tutoring Systems
In education, artificial intelligence (AI) is revolutionizing the way students learn and interact with information. One area where AI is making significant strides is in the development of intelligent tutoring systems. These systems use AI algorithms to tailor personalized learning experiences for individual students.
Intelligent tutoring systems are designed to understand the unique needs and abilities of each student. By analyzing data on student performance, AI algorithms can identify areas where a student may be struggling and provide targeted interventions. These interventions can take the form of personalized feedback, additional practice exercises, or supplementary learning materials.
AI can also provide real-time feedback and guidance to students as they work through problems or assignments. Through natural language processing and machine learning algorithms, intelligent tutoring systems can assess student responses, provide instant feedback, and adapt their instructional approach based on student progress.
Intelligent tutoring systems can also track and analyze student behavior and engagement. By monitoring student actions such as time spent on tasks, interactions with content, and success rates, AI algorithms can generate insights into student learning patterns and preferences. This data can be used to further personalize instruction and identify areas where a student may need additional support.
With the help of AI, intelligent tutoring systems have the potential to transform education by delivering personalized instruction at scale. By adapting to each student’s individual needs and learning style, these systems can provide tailored learning experiences that engage and motivate students, leading to improved learning outcomes.
As AI continues to advance, intelligent tutoring systems will likely become even more sophisticated, incorporating technologies such as natural language processing, computer vision, and adaptive learning algorithms. This will further enhance their ability to understand and support student learning, making them essential tools in the modern educational landscape.
AI in Educational Robotic Systems
Artificial Intelligence (AI) is revolutionizing many industries and education is no exception. In recent years, there has been a growing interest in utilizing AI technology in educational robotic systems. But what exactly is AI and how does it play a role in these systems?
AI refers to the ability of a machine to perform tasks that would typically require human intelligence. This includes tasks such as problem-solving, decision-making, and learning from experience. In educational robotic systems, AI can be used to enhance the learning experience by providing personalized instruction, adaptive feedback, and interactive experiences.
One of the key benefits of AI in educational robotic systems is its ability to cater to individual student needs. By analyzing data and identifying patterns, AI algorithms can adapt the curriculum and teaching strategies to meet the specific needs and learning styles of each student. This personalized approach enhances student engagement and improves learning outcomes.
In addition, AI allows for real-time assessment and feedback. Educational robotic systems equipped with AI can analyze student performance and provide immediate feedback on their progress. This instant feedback helps students understand their strengths and weaknesses, allowing them to make adjustments and improve their skills more efficiently.
|AI in Educational Robotic Systems
|– Enhances the learning experience through personalized instruction and adaptive feedback
|– Analyzes data and adapts the curriculum to meet individual student needs and learning styles
|– Provides real-time assessment and immediate feedback on student performance
In summary, AI plays a crucial role in educational robotic systems by transforming the way students learn and interact with technology. With AI-powered systems, students can receive personalized instruction, adaptive feedback, and real-time assessment, leading to improved learning outcomes.
AI for Personalized Feedback and Assessment
In the field of education, artificial intelligence (AI) is revolutionizing the way students receive feedback and assessments. Traditionally, teachers have been responsible for evaluating student work and providing feedback based on their expertise. However, this process can be time-consuming and resource-intensive.
AI technology is changing this dynamic by automating and personalizing the feedback and assessment process. Through machine learning algorithms, AI systems can analyze student work, such as essays or quizzes, and provide instant feedback tailored to each student’s needs.
AI algorithms can analyze not only the correctness of student answers but also the depth of understanding demonstrated. This allows for the provision of personalized feedback that is tailored to each student’s individual learning needs. For example, if a student provides a correct answer but lacks the necessary explanations, the AI system can provide targeted feedback to help the student improve their reasoning skills.
AI systems can also automate the assessment process by grading assignments and tests, freeing up teachers’ time to focus on other areas of instruction. This automation not only saves time but also improves consistency in grading, as AI algorithms are not prone to subjective biases or fatigue.
Furthermore, AI-powered assessment tools can provide students with instant results, making the learning experience more engaging and interactive. Students can receive immediate feedback on their performance, enabling them to identify their strengths and weaknesses and make necessary improvements.
In conclusion, AI is transforming education by providing personalized feedback and automating the assessment process. This technology has the potential to enhance learning outcomes by tailoring feedback to individual students, improving grading consistency, and increasing student engagement. As AI continues to advance, it is likely to play an even greater role in education, providing educators with powerful tools to support student learning.
AI-based Virtual Reality in Education
In today’s rapidly advancing technological world, artificial intelligence (AI) is revolutionizing various industries, and education is no exception. With the integration of AI, the possibilities of enhancing the learning experience have expanded significantly. One area where AI is playing a transformative role is in the field of virtual reality (VR) in education.
Virtual reality is a computer-simulated environment that allows users to interact with a three-dimensional space and experience situations they might not encounter in real life. When combined with AI, virtual reality in education becomes even more powerful.
What is AI-based virtual reality in education? It is the use of AI technology to enhance the virtual reality learning experience. Through AI algorithms and machine learning, virtual reality simulations can become more intelligent, adaptive, and personalized.
AI algorithms can analyze data from student interactions in VR and provide personalized feedback and recommendations. This enables educators to understand each student’s learning patterns and tailor the virtual reality experience to their individual needs.
Furthermore, AI can also create intelligent virtual characters that can simulate real-life situations and interactions. These virtual characters can act as tutors, providing guidance, explanations, and assistance to students as they navigate through the virtual reality environment.
Virtual reality in education powered by AI has the potential to make learning more engaging, immersive, and effective. It can provide students with hands-on experiences, enhance problem-solving skills, and improve knowledge retention. Additionally, AI-based virtual reality can bridge the gap between theory and practice, allowing students to apply their learning in a virtual simulated environment.
In conclusion, AI-based virtual reality in education is an exciting development that has the potential to revolutionize the way we learn. By combining the power of AI with the immersive experience of virtual reality, education can become more personalized, engaging, and effective.
AI for Curriculum Development and Adaptation
Curriculum development and adaptation play a crucial role in providing students with a well-rounded and effective education. Traditionally, curriculum development has been a manual and time-consuming task for educators and administrators. However, with the advancements in artificial intelligence (AI), the process of developing and adapting curricula has become more efficient and personalized.
So, what is AI in the context of curriculum development and adaptation? AI refers to the ability of machines to perform tasks that typically require human intelligence, such as understanding, reasoning, learning, and problem-solving. In the education sector, AI can be utilized to analyze student data, identify their strengths and weaknesses, and create customized learning pathways.
Using AI, educators can gather information about individual students’ learning styles, preferences, and interests. This data can then be used to tailor the curriculum, making it more engaging and relevant to students. AI algorithms can analyze vast amounts of information and identify patterns or gaps in knowledge, enabling educators to address these gaps and provide targeted interventions.
AI can also assist in the development of new curricula by analyzing existing educational materials, textbooks, and resources. By analyzing the content, AI algorithms can identify the most relevant and up-to-date information, helping educators create comprehensive and cutting-edge curricula. Additionally, AI can analyze industry trends and job market demands to ensure that the curriculum is aligned with the skills and knowledge required for future career success.
In conclusion, AI has the potential to revolutionize the process of curriculum development and adaptation in education. By leveraging AI technologies, educators can create personalized and dynamic curricula that meet the needs of individual students. Furthermore, AI can assist in identifying gaps in knowledge, updating educational materials, and aligning curricula with future job market demands. AI is a powerful tool that can enhance the effectiveness and relevance of education, ultimately preparing students for success in the digital age.
AI in Educational Gaming and Gamification
Artificial intelligence (AI) is revolutionizing the field of education by enhancing various aspects of the learning process. One area where AI is making a significant impact is in educational gaming and gamification.
What is educational gaming? It refers to video games or game-based activities that are designed with educational purposes in mind. These games integrate educational content and objectives, making learning more interactive, engaging, and enjoyable for students.
A key aspect of educational gaming is the integration of AI technology. AI can be used to create intelligent tutors or virtual characters within the game who can give personalized feedback, adapt to the learner’s level, and provide targeted instruction. This personalized and adaptive approach helps students to advance at their own pace and receive individualized support.
Benefits of AI in Educational Gaming and Gamification:
- Enhanced Engagement: AI enables the creation of immersive and interactive educational games that can capture students’ attention and motivate them to actively participate in the learning process.
- Personalization: AI-powered educational games can adapt to each student’s unique learning style, preferences, and abilities, providing personalized content and challenges.
- Immediate Feedback: AI algorithms can provide instant feedback to students, helping them identify and correct their mistakes in real-time.
- Progress Tracking: AI technology allows educators to track students’ progress and performance within the game, providing valuable insights into their strengths, weaknesses, and learning gaps.
- Collaboration and Competition: AI-powered educational games can foster collaboration among students, encouraging them to work together to solve problems or compete in a friendly manner.
Examples of AI in Educational Gaming and Gamification:
There are various examples of AI-powered educational games and platforms that are being used in schools and educational settings:
- Mathematics games that adapt to students’ skill levels and provide personalized practice exercises.
- Language learning apps that utilize speech recognition technology to provide pronunciation feedback.
- Simulations and virtual reality games that allow students to explore and experience complex concepts in a hands-on and interactive way.
- Educational platforms that use AI algorithms to recommend personalized learning resources based on students’ performance and interests.
In conclusion, AI is transforming the field of education by revolutionizing the way students learn through educational gaming and gamification. The integration of AI technology in these tools enhances engagement, provides personalized learning experiences, and offers immediate feedback and progress tracking. As AI continues to advance, the possibilities for the future of education are endless.
AI and Lifelong Learning
Artificial Intelligence, or AI, has revolutionized various industries, including education. One area where AI is making a significant impact is in lifelong learning. Lifelong learning refers to the continuous process of acquiring knowledge and skills throughout one’s life.
So, what role does AI play in lifelong learning? AI can personalize and enhance the learning experience by providing adaptive learning systems. These systems use algorithms and machine learning to analyze the individual needs of learners and tailor educational content accordingly.
AI-powered virtual tutors and chatbots have also emerged as valuable tools for lifelong learning. These tools can provide instant feedback, answer questions, and guide learners through the learning process at their own pace. They can adapt their teaching styles to the learner’s preferences and optimize learning efficiency.
Another way AI is helping in lifelong learning is through automated assessment systems. These systems can evaluate learners’ performance and provide immediate feedback, saving time and effort for both educators and learners. AI algorithms can analyze large amounts of data to identify patterns and make predictions about future learning progress.
AI is also making it possible for learners to access personalized learning pathways. Based on learners’ interests, strengths, and weaknesses, AI algorithms can suggest suitable courses, resources, and learning materials. This personalization ensures that learners can focus on areas that need improvement and engage with content that aligns with their goals.
In conclusion, AI is transforming lifelong learning by providing personalized, adaptive, and accessible learning experiences. With AI, educators and learners can benefit from tailored content, instant feedback, and automated assessment systems. As AI continues to advance, its potential to revolutionize lifelong learning is vast.
– Questions and Answers
What is the role of AI in education?
Artificial Intelligence has the potential to revolutionize education by providing personalized learning experiences. AI can analyze student data and adapt teaching methods to meet individual needs, making education more effective and efficient.
How can AI improve the learning process?
AI can improve the learning process by providing personalized recommendations and feedback to students. It can identify areas where a student is struggling and provide additional resources or assistance. This individualized approach can help students learn at their own pace and achieve better outcomes.
Can AI replace teachers?
While AI can enhance the role of teachers, it is unlikely to completely replace them. Teachers bring unique qualities and skills such as empathy, creativity, and critical thinking that are essential in the learning process. AI can support teachers by automating administrative tasks and providing data-driven insights, but human interaction and guidance are still crucial in education.
What are some examples of AI in education?
AI is already being used in education in various ways. For example, chatbots can provide instant answers to student queries, virtual reality can create immersive learning experiences, and adaptive learning platforms can personalize content based on individual student needs. These are just a few examples of how AI is transforming education.
What are the potential challenges of implementing AI in education?
There are several challenges to consider when implementing AI in education. Privacy and data security are major concerns, as AI systems require access to student data. There is also the risk of bias in AI algorithms, which could perpetuate inequalities in education. Additionally, the cost of implementing AI technology can be a barrier for many educational institutions.
What is artificial intelligence?
Artificial intelligence is a branch of computer science that focuses on creating machines that can perform tasks that would normally require human intelligence.
How can artificial intelligence be used in education?
Artificial intelligence can be used in education to personalize learning experiences, provide virtual tutors, automate administrative tasks, and improve assessment and feedback processes.
What are the benefits of using artificial intelligence in education?
The benefits of using artificial intelligence in education include enhanced personalized learning, increased accessibility to education, improved efficiency in administrative tasks, and the ability to provide individualized feedback and support for students.
Are there any concerns or drawbacks to using artificial intelligence in education?
Some concerns associated with using artificial intelligence in education include the potential for bias in algorithms, the impact on teacher-student relationships, and the privacy and security of student data.
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Latency refers to the amount of time it takes for data to travel from its source to its destination in a computer network or a communication system. It is often measured in milliseconds (ms) and represents the delay between the initiation of a request and the receipt of a response.
It can be influenced by various factors, including the distance between the source and destination, the speed of the network connection, the number of network devices the data has to pass through, and the processing time at each step along the way.
In networking, there are different types of latency. Some common types include:
Network Latency: This is the time it takes for data to travel across a network from one point to another. It includes the propagation delay, which is the time it takes for a signal to travel through the physical medium, and the transmission delay, which is the time it takes to transmit the data over the network.
Round-Trip Latency: Also known as RTT, this measures the time it takes for a data packet to travel from the source to the destination and then back to the source. It includes the time for the request to reach the destination and the time for the response to return.
Application Latency: This refers to the delay experienced by an application or software system when processing data. It can include factors such as the time taken for data to be processed, queued, or for a response to be generated.
This is a critical factor in many real-time applications, such as online gaming, video conferencing, and high-frequency trading, where even slight delays can have a noticeable impact on user experience or system performance. Minimizing latency is often a goal in network design and optimization to ensure efficient and responsive communication.
what is latency?
It can refers to the delay or time lag that occurs between the initiation of a request and the receipt of a response in a system or network. It measures the time it takes for data to travel from its source to its destination.
In computer networks, latency is typically measured in milliseconds (ms) and can be influenced by several factors. These include:
Propagation Delay: This is the time it takes for a signal to travel through the physical medium, such as a cable or fiber optic line. The speed of light limits the propagation delay, and it increases with the distance between the source and destination.
Transmission Delay: Transmission delay is the time taken to transmit the data over the network. It depends on the bandwidth of the network connection and the size of the data being transmitted.
Processing Delay: Processing delay occurs when the data passes through network devices such as routers, switches, or firewalls. These devices may need to perform various tasks, such as inspecting and modifying the data, which can introduce additional latency.
Latency is a critical factor in various applications and systems, especially those that require real-time communication or quick response times. For example, in online gaming, low latency is crucial to ensure minimal delays between player actions and their impact in the game. In video conferencing or VoIP (Voice over Internet Protocol) calls, low latency helps maintain smooth and natural conversations.
Reducing often a goal in network design and optimization to ensure efficient and responsive communication. This can be achieved through various techniques such as optimizing network infrastructure, minimizing network congestion, using faster network connections, and implementing efficient routing protocols.
What is latency internet?
Latency in the context of the internet refers to the delay or time it takes for data packets to travel from the sender to the receiver and back. It represents the time lag between the initiation of a request (such as clicking a link or sending a data packet) and the receipt of a response.
What causes Internet latency?
Distance: The physical distance between the source and destination of data transmission can affect it. Signals take time to travel, so longer distances generally result in higher latency.
Network Congestion: When the network is congested with high volumes of data traffic, the data packets may experience delays as they compete for limited network resources. Congestion can occur at various points in the network, including routers, switches, and internet service provider (ISP) networks.
Routing Inefficiencies: The routing process involves determining the most efficient path for data to travel between devices on a network. Inefficient routing can introduce additional hops or detours, leading to increased latency.
Network Equipment and Infrastructure: The quality and capacity of the network infrastructure and equipment, such as routers, switches, and cables, can impact latency. Outdated or overloaded equipment may introduce delays.
Bandwidth Limitations: Limited bandwidth can lead to latency, especially when transferring large amounts of data. If the available bandwidth is fully utilized, data packets may need to wait for their turn, resulting in increased latency.
Network Protocol Overhead: Network protocols add headers and other control information to data packets, which increases the packet size. This additional data can contribute to higher latency, especially if the network has limited bandwidth or if there are delays in processing the extra information.
Wireless Networks: Wireless connections introduce additional latency compared to wired connections. Factors such as signal interference, signal strength, and the distance between devices can affect latency in wireless networks.
Server Performance: The performance of the server or the destination device that is processing the data requests can impact on it. If the server is overloaded or experiencing high processing times, it can introduce delays in responding to requests.
Internet Service Provider (ISP): The quality and reliability of the internet connection provided by your ISP can affect latency. Some ISPs may have congested networks or suboptimal routing, leading to increased latency.
Other Factors: Other factors that can contribute to latency include network security measures (such as firewalls and encryption/decryption processes), the responsiveness of the destination application or server, and the efficiency of the client device’s hardware and software.
It’s important to note that latency can vary depending on the specific network configuration, geographical location, and the specific devices and software involved in the communication.
Network latency, throughput, and bandwidth
Network latency, throughput, and bandwidth are interconnected concepts in networking but represent different aspects of network performance. Here’s a brief explanation of each:
Network Latency: Network refers to the time delay that occurs when data packets travel from the source to the destination across a network. It measures the time taken for a packet to make a round trip from the sender to the receiver and back. Latency is typically measured in milliseconds (ms). Lower latency indicates faster response times and better real-time performance. Factors contributing to latency include distance, network congestion, routing inefficiencies, and processing delays.
Network Throughput: Throughput represents the amount of data that can be transmitted over a network within a given period. It is typically measured in bits per second (bps) or its multiples like kilobits per second (Kbps), megabits per second (Mbps), or gigabits per second (Gbps). Throughput reflects the capacity or speed of the network to transmit data. Higher throughput indicates a network’s ability to handle more significant amounts of data in a given time frame. Throughput can be affected by factors such as bandwidth limitations, network congestion, and network equipment performance.
Bandwidth: Bandwidth refers to the maximum data transfer rate of a network or a network connection. It represents the capacity or width of the data channel, determining how much data can be transmitted at a given time. Bandwidth is commonly measured in bits per second (bps) and its multiples. It defines the upper limit for network throughput. For example, a network connection with a bandwidth of 100 Mbps can theoretically transmit up to 100 megabits of data per second. Bandwidth is a crucial factor in determining the potential speed of data transmission but does not guarantee actual throughput. Actual throughput can be lower due to factors such as network congestion or limitations at the source or destination devices.
These factors collectively influence the performance and efficiency of network communication.
How can latency be reduced?
Use a Faster Network Connection: Upgrading to a faster internet connection with higher bandwidth can help reduce latency. Consider options such as fiber-optic or high-speed cable connections, which offer lower latency compared to traditional DSL or satellite connections.
Optimize Network Infrastructure: Evaluate and optimize your network infrastructure to minimize latency. Ensure that network devices, such as routers and switches, are up to date and properly configured. Use quality networking equipment that can handle higher traffic volumes efficiently.
Minimize Network Congestion: Reduce network congestion by managing and prioritizing network traffic. Use Quality of Service (QoS) mechanisms to prioritize critical applications and ensure they receive the necessary bandwidth. Implement traffic shaping or bandwidth throttling techniques to prevent any single application from monopolizing network resources.
Optimize Routing: Review and optimize the routing configuration to minimize the number of hops and detours that data packets have to take. Ensure that routing tables are up to date and use efficient routing protocols to enable faster and more direct data transmission.
Content Delivery Networks (CDNs): If you serve content or applications to users across different geographical locations, consider using a CDN. CDNs distribute content across multiple servers in various locations, allowing users to access data from a nearby server, thereby reducing latency.
Caching and Compression: Implement caching mechanisms to store frequently accessed data closer to users, reducing the need for repeated data retrieval from the source. Additionally, compress data where possible to reduce the amount of data transmitted over the network, thus reducing latency.
Optimize Application Performance: Optimize the performance of your applications by minimizing unnecessary data processing and optimizing algorithms. Consider implementing techniques like prefetching, parallel processing, and data compression within your applications.
Reduce Round-Trip Time (RTT): Minimize the number of round trips required between the source and destination. This can be achieved by implementing techniques such as connection pooling, keeping persistent connections open, and using protocols that minimize handshakes, such as HTTP/2.
Proximity: In some cases, physically locating servers closer to end users can significantly reduce latency. This can be achieved by deploying servers in data centers or edge locations that are geographically closer to the target audience.
Monitor and Troubleshoot: Regularly monitor network performance, latency, and potential bottlenecks. Use network monitoring tools to identify latency issues and troubleshoot them promptly. Analyze network traffic patterns to identify any anomalies or areas that need improvement.
What is latency internet?
In the context of gaming, latency refers to the delay or lag between a player’s input or action and the corresponding response or feedback from the game. It represents the time it takes for data to travel from the player’s device to the game server and back.
Low latency is crucial in gaming because it directly affects the responsiveness and real-time interaction between players and the game world. High latency can result in delayed or sluggish gameplay, negatively impacting the gaming experience. It can cause issues such as delayed character movements, delayed weapon firing, or unresponsive controls.
Latency in gaming is typically measured in milliseconds (ms), and even small differences in latency can be noticeable. For competitive multiplayer games, where split-second reactions are essential, minimizing latency is particularly important.
What is cas latency?
CAS latency (Column Address Strobe latency) is a timing parameter that measures the delay between the memory controller sending a column address to the memory module and the module returning the corresponding data. CAS is a specific timing measurement used in dynamic random-access memory (DRAM), which is the type of memory commonly used in computer systems.
This latency is expressed as a number, such as CAS 14 or CAS 16, and represents the number of clock cycles it takes for the memory module to return the requested data after the column address is provided. Lower CAS latency values indicate faster access times and better performance.
The CAS latency value is typically specified alongside other timing parameters, such as RAS (Row Address Strobe) latency, tRCD (Row to Column Delay), tRP (Row Precharge Time), and tRAS (Row Active Time). These timings collectively define the overall performance and efficiency of the memory module.
When choosing RAM modules for a computer system, it is common to consider both the frequency (speed) and CAS latency. Higher frequency RAM modules can provide faster data transfer rates, but they may also have higher CAS ,which can offset some of the speed advantages. Balancing the frequency and CAS latency according to the specific requirements and capabilities of the system is important for optimizing memory performance.
How can users fix latency on their end?
Use a Wired Connection: If you’re experiencing latency issues, switch from a wireless connection to a wired Ethernet connection. This is generally offer lower latency and more stability compared to wireless connections, which can be affected by signal interference and distance from the router.
Close Background Applications: Close any unnecessary applications and processes running in the background on your device. Some applications might consume network resources, leading to increased latency. Closing them can free up bandwidth and reduce latency.
Check for Malware or Viruses: Malware or viruses on your device can consume network resources and lead to increased latency. Regularly scan your device for malware and keep your antivirus software up to date.
Disable or Limit Bandwidth-Intensive Activities: Bandwidth-intensive activities such as large file downloads, streaming HD videos, or online gaming can consume significant network resources, leading to higher latency.
Clear Browser Cache: Clear the cache and cookies in your web browser. Accumulated cache and cookies can affect browsing speed and introduce latency. Clearing them can help improve the performance of web-based applications.
Update Network Drivers: Ensure that your network drivers are up to date. Outdated drivers can result in performance issues, including increased latency. Visit the manufacturer’s website or use appropriate software to update your network drivers.
Optimize Browser Settings: Adjusting certain browser settings can help improve latency. For example, disabling or adjusting browser extensions, disabling auto-updates, and limiting the number of open tabs can reduce network resource usage.
Use a Different DNS Provider: DNS resolution can impact the time it takes to access websites. Try using a different DNS provider, such as Google DNS or OpenDNS, which may provide faster and more reliable DNS resolution.
Restart Network Devices: Occasionally, network devices like routers or modems can encounter issues that increase latency. Restarting these devices can help resolve temporary glitches and improve performance.
Contact your Internet Service Provider (ISP): If you consistently experience high latency despite taking the above steps, contact your ISP. They can help diagnose and address any potential issues on their end that may be causing there problems.
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Mountains, those towering giants of the earth, are not just a sight for sore eyes but also play a significant role in shaping the weather patterns around them. These natural wonders can influence the climate and weather conditions in ways that many may not realize. In this article, we will delve into the fascinating subject of how mountains impact weather patterns and discover the science behind it. So, buckle up and get ready to learn how these magnificent peaks can affect the weather around them.
Mountains can significantly impact weather patterns due to their size and location. They can block wind and create areas of low pressure, leading to the formation of clouds and precipitation. This can result in increased rainfall on the windward side of the mountain and decreased rainfall on the leeward side. Mountains can also cause temperature variations, as the air is forced to rise and cool as it passes over the mountain, leading to the formation of cooler air masses. These cooler air masses can then spread out and impact weather patterns in surrounding areas. Additionally, mountains can create microclimates, which are localized areas of unique weather patterns that are influenced by the topography and other factors. Overall, the presence of mountains can have a significant impact on local and regional weather patterns.
The Role of Mountains in Weather Formation
Mountains play a significant role in weather formation and have a profound impact on the local and regional climate. They influence the movement of air masses, the formation of clouds, and the precipitation patterns in the surrounding areas.
Formation of Foothill Depressions
Foothill depressions are areas of low pressure that form on the leeward side of mountains. These depressions are caused by the upslope flow of air, which leads to a decrease in temperature and an increase in humidity. As a result, these areas experience more cloud cover and precipitation than the surrounding regions.
Air Mass Movement
Mountains can also affect the movement of air masses. When air flows over a mountain range, it can cause the air to rise and cool, leading to the formation of clouds and precipitation. This process is known as orographic lift, and it can lead to the formation of rain or snow on the windward side of the mountains.
Formation of Mountain Waves
Mountains can also create mountain waves, which are large-scale atmospheric waves that form due to the interaction of the wind with the mountain’s surface. These waves can affect the weather patterns in the surrounding areas, leading to changes in temperature, humidity, and precipitation.
Influence on Precipitation Patterns
The presence of mountains can also impact precipitation patterns in the surrounding areas. For example, the Sierra Nevada mountain range in California creates a rain shadow effect on the eastern side of the range, resulting in a much drier climate than the western side. This rain shadow effect is caused by the blocking of moisture-laden air by the mountains, leading to a decrease in precipitation on the leeward side.
Overall, the role of mountains in weather formation is complex and multifaceted. They can influence air mass movement, the formation of clouds and precipitation, and the distribution of moisture in the surrounding areas. Understanding these processes is crucial for predicting and managing weather patterns in mountainous regions.
Mountain Barriers and Airflow
Overview of Mountain Barriers
Mountains are formidable obstacles that can significantly impact the movement of air masses. They create physical barriers that can alter the natural flow of air and result in the creation of distinct microclimates. The height and steepness of mountains can affect the direction and speed of wind, leading to the formation of various weather patterns.
The Impact of Mountain Barriers on Airflow
Mountain barriers can cause air to rise and create upward currents. As air flows over mountains, it encounters resistance that forces it to ascend. This upward movement of air can lead to the formation of clouds and precipitation. The air flowing over the mountains can also cool, resulting in the formation of fog and mist.
Furthermore, the height and steepness of mountains can cause air to become trapped on the leeward side. This can result in the formation of a low-pressure system, which can lead to the development of weather systems such as storms. Additionally, the slopes of mountains can cause air to move downhill, leading to the formation of wind gusts and downslope winds.
The impact of mountain barriers on airflow can vary depending on the size and height of the mountains, as well as the direction and speed of the prevailing winds. The unique topography of mountain ranges can create complex weather patterns, with the creation of mountain waves and rotors that can significantly impact the weather in surrounding areas.
Overall, the presence of mountains can significantly impact weather patterns by altering the movement of air masses. Mountain barriers can cause air to rise, become trapped, and move downhill, leading to the formation of distinct weather systems that can have significant effects on local and regional climates.
Mountains play a significant role in shaping weather patterns, particularly through their impact on precipitation. This section will delve into the mechanisms by which mountains influence precipitation and how it affects the surrounding climate.
When air masses move over mountains, they are forced to rise and cool. As the air cools, the moisture in it condenses and forms clouds. This process, known as orographic lifting, leads to the formation of rain or snow on the windward side of the mountains. The higher the mountains and the steeper the slope, the more intense the orographic lifting and the greater the amount of precipitation that falls on the windward side.
Mountain-Induced Orographic Lifting
Orographic lifting is the process by which the mountains force the air to rise and cool, leading to the formation of clouds and precipitation. This phenomenon occurs because the air is no longer able to move horizontally over the mountains and must rise to pass over them. As the air rises, it cools and the moisture in it condenses, forming clouds and precipitation.
On the leeward side of the mountains, the air descends and warms, leading to a decrease in precipitation. This is due to the fact that the air is able to move horizontally again and is no longer forced to rise and cool.
Overall, the presence of mountains can significantly impact the amount and distribution of precipitation in an area. This, in turn, can have a major impact on the local climate and ecosystem.
The Influence of Mountains on Microclimates
The impact of mountains on weather patterns extends beyond the macroscopic scale. At a smaller scale, mountains also significantly influence the local climate, creating unique microclimates. These microclimates can differ from the surrounding areas and are determined by the specific characteristics of the terrain. In this section, we will explore the ways in which mountains shape microclimates and the factors that contribute to their formation.
Mountains as Barriers to Airflow
One of the primary ways mountains influence microclimates is by acting as barriers to airflow. The steep inclines and high elevations of mountains create a natural obstacle that alters the movement of air masses. This results in the formation of different air pressure zones, leading to the development of localized wind patterns. As a consequence, the temperature, humidity, and precipitation levels within these microclimates can vary significantly from those in the surrounding areas.
The Role of Topography in Shaping Microclimates
The topography of mountains plays a crucial role in determining the specific characteristics of microclimates. Factors such as the height, width, and orientation of mountains can significantly impact the local climate. For instance, taller mountains tend to create more significant temperature gradients, leading to increased precipitation and cloud cover. Conversely, narrower mountains may result in less pronounced temperature differences and reduced precipitation.
The influence of mountain orientation is also noteworthy. Mountains that run east to west tend to experience more pronounced temperature variations due to the greater exposure to solar radiation. In contrast, north-south oriented mountains may experience more moderate temperatures, as they are less exposed to direct sunlight.
The Impact of Altitude on Microclimates
Altitude is another critical factor that contributes to the formation of microclimates in mountainous regions. As elevation increases, the atmospheric pressure decreases, leading to a decrease in the partial pressure of gases such as oxygen and water vapor. This results in lower temperatures and reduced humidity at higher altitudes. As a consequence, the flora and fauna in these areas have adapted to the unique environmental conditions, creating distinct ecosystems.
Convection and Orographic Lifting
The complex terrain of mountains can also lead to the formation of convection currents, which are upward movements of air caused by the heating of the surface. When warm air from lower elevations rises and cools, it can result in the formation of clouds and precipitation. This process, known as orographic lifting, can significantly impact the microclimate of mountainous regions, influencing the amount and timing of precipitation.
In conclusion, mountains significantly impact weather patterns at both the macroscopic and microscopic scales. The influence of mountains on microclimates is determined by factors such as topography, altitude, and orientation. Understanding these factors is essential for comprehending the complex interactions between mountains and weather patterns, ultimately contributing to a more comprehensive understanding of our Earth’s climate systems.
Mountains play a significant role in weather formation, influencing air mass movement, the formation of clouds and precipitation, and the distribution of moisture in the surrounding areas. They create physical barriers that alter the natural flow of air masses, leading to the formation of distinct microclimates. Understanding these processes is crucial for predicting and managing weather patterns in mountainous regions.
Foehn Winds and Mountain Climates
The Foehn Wind Phenomenon
The Foehn wind phenomenon, also known as the mountain breeze, is a specific type of wind that occurs when air is heated and then rapidly cooled as it descends from mountains. This heating and cooling process results in a decrease in air pressure and an increase in air speed, creating a wind that can reach high speeds. The Foehn wind is typically observed in regions with tall, steep mountains that are exposed to strong sunlight and have a significant temperature gradient between the mountaintop and the valley below.
Foehn Winds and Mountain Climate Effects
The Foehn wind has a significant impact on the climate and weather patterns of mountainous regions. One of the primary effects is the modification of temperature gradients between the mountaintop and the valley below. As the Foehn wind descends from the mountains, it carries with it warmer air from higher altitudes, resulting in a significant increase in temperature in the valley below. This warming effect can lead to the melting of snow and ice, which can have significant impacts on the local water cycle and hydrological systems.
In addition to modifying temperature gradients, the Foehn wind can also affect precipitation patterns in mountainous regions. The warming effect of the wind can cause the evaporation of snow and ice, leading to a decrease in snow cover and an increase in the amount of precipitation that falls as rain. This can have significant impacts on the local ecosystem and water resources, particularly in regions where snow and ice play a critical role in the water cycle.
The Foehn wind can also create strong wind gusts and turbulence, particularly in narrow valleys or gorges. These wind gusts can lead to erosion and the deposition of sediment, creating a significant impact on the local geomorphology and landscape.
Overall, the Foehn wind is a critical component of the climate and weather patterns in mountainous regions, and its impacts can be observed in a wide range of ecological, hydrological, and geomorphological systems. Understanding the dynamics of the Foehn wind is critical for predicting and managing the impacts of climate change in mountainous regions, and for developing effective strategies for adaptation and mitigation.
Mountain Valleys and Climate Moderation
The Role of Valley Geometry
The geometry of a valley plays a crucial role in determining the climate conditions within it. A valley’s shape, length, and orientation all contribute to the formation of microclimates. For instance, a narrow and deep valley with steep sides will create a more favorable environment for precipitation, as it increases the likelihood of cloud condensation and enhances the chances of orographic lift.
Climate Moderation in Mountain Valleys
Mountain valleys can experience climate moderation due to their unique geographical features. The surrounding mountains can shield the valley from harsh weather conditions, such as strong winds and extreme temperatures. This protection is particularly evident during winter months when the mountains block cold air masses from penetrating the valley, resulting in milder temperatures compared to the surrounding areas.
Moreover, mountain valleys often have a higher frequency of cloud cover and precipitation due to the orographic effect. This phenomenon occurs when moist air is forced to rise and cool as it encounters the mountains, leading to the formation of clouds and the release of precipitation. As a result, mountain valleys tend to have higher levels of rainfall compared to the surrounding regions, contributing to the growth of vegetation and supporting diverse ecosystems.
Additionally, the presence of water bodies within mountain valleys, such as rivers and lakes, can further impact the local climate. These water bodies can moderate temperatures, regulate humidity levels, and influence wind patterns. They can also act as sources of hydropower, which can be harnessed to support local communities and industries.
Overall, the influence of mountains on microclimates within mountain valleys is significant. The unique geographical features of these valleys, combined with the presence of water bodies and vegetation, contribute to the formation of distinct climate conditions that can vary significantly from the surrounding areas. Understanding these processes is crucial for developing sustainable land use practices, managing natural resources, and supporting local communities in mountainous regions.
Mountain Weather Stations and Research
To understand how mountains impact weather patterns, it is important to study the weather conditions that occur at high elevations. This can be done through the use of mountain weather stations. These stations are strategically placed in mountainous regions and are designed to collect data on temperature, wind speed, and precipitation. By analyzing this data, researchers can gain insight into how mountains affect the weather around them.
One key area of research is the study of the temperature lapse rate. The temperature lapse rate refers to the rate at which the temperature changes with an increase in elevation. In mountainous regions, the temperature lapse rate is often steeper than in flat areas. This means that the temperature can change more rapidly as you move up in elevation. Researchers are interested in understanding how this affects the weather patterns in the area.
Another important aspect of mountain weather research is the study of wind patterns. The shape of a mountain can affect the way the wind flows around it. For example, a mountain with a pointed peak may cause the wind to flow around it in a particular direction. This can have an impact on the weather patterns in the surrounding area. Researchers are working to understand how the shape of mountains affects wind patterns and how this, in turn, affects the weather.
Finally, researchers are also studying the impact of mountains on precipitation. Mountains can act as a barrier to the movement of air masses, which can lead to the formation of rain clouds. By studying the amount and frequency of precipitation in mountainous regions, researchers can gain insight into how mountains impact the weather patterns in the area.
Overall, mountain weather stations and research play a crucial role in understanding how mountains impact weather patterns. By collecting and analyzing data on temperature, wind, and precipitation, researchers can gain a better understanding of the complex interactions between mountains and the weather. This information can be used to improve weather forecasting and to inform climate change models.
Importance of Mountain Weather Stations
Mountain weather stations play a crucial role in understanding the impact of mountains on weather patterns. These stations are strategically located in mountainous regions and collect data on various meteorological parameters such as temperature, humidity, wind speed, and precipitation. The data collected from these stations help researchers and meteorologists to better understand the complex interactions between mountains and weather systems.
Climate Change Monitoring
Mountain weather stations are essential for monitoring climate change in mountainous regions. As mountains are vulnerable to the impacts of climate change, these stations provide valuable data on temperature, precipitation, and other climate variables. This data helps researchers to understand the changes in the mountain climate over time and predict future trends.
Extreme Weather Prediction
Mountain weather stations are also crucial for predicting extreme weather events in mountainous regions. The data collected from these stations help researchers to identify the conditions that lead to extreme weather events such as flash floods, landslides, and avalanches. This information can be used to develop early warning systems and preparedness plans to mitigate the impacts of extreme weather events on mountain communities.
In addition, mountain weather stations help researchers to understand the influence of mountains on the larger-scale weather patterns. The data collected from these stations can be used to develop more accurate weather forecasts and to improve our understanding of the interactions between mountains and the atmosphere. Overall, mountain weather stations are an essential tool for studying the complex relationship between mountains and weather patterns.
Challenges in Mountain Weather Research
Accessibility and Infrastructure
- Logistical difficulties in reaching mountainous regions
- Limited road access, rough terrain, and steep inclines pose challenges for transportation and equipment delivery
- Harsh weather conditions, including extreme temperatures, strong winds, and heavy precipitation, can hinder data collection efforts
- Insufficient infrastructure, such as power and communication systems, can impede the installation and maintenance of weather monitoring equipment
Data Collection and Analysis
- Difficulty in obtaining accurate and reliable data due to complex topography and variable weather conditions
- Limited range of weather parameters that can be measured at high altitudes
- Technical challenges in maintaining and calibrating equipment in harsh mountain environments
- The need for specialized knowledge and skills to interpret and analyze mountain weather data
- The scarcity of long-term mountain weather datasets, which hinders the identification of trends and the understanding of weather patterns in mountainous regions
Mountain Weather Hazards and Risks
When it comes to the impact of mountains on weather patterns, it is essential to understand the hazards and risks associated with these geographical features. Mountainous regions are often prone to various weather-related hazards, such as heavy rainfall, landslides, avalanches, and strong winds. These hazards can pose significant risks to human life, infrastructure, and the environment.
One of the most significant impacts of mountains on weather patterns is the increased likelihood of heavy rainfall. When air masses collide, they can produce intense precipitation, leading to flash floods and landslides. This is particularly true in areas where mountains are steep and rugged, as the terrain can cause air to rise and cool, leading to the formation of heavy precipitation.
Mountains are also prone to landslides, which can be triggered by heavy rainfall, earthquakes, or even human activity. Landslides can cause significant damage to infrastructure, homes, and the environment, and they can also pose a significant risk to human life.
Another hazard associated with mountains is avalanches, which are caused by the rapid movement of snow and ice down a slope. Avalanches can be triggered by a variety of factors, including heavy snowfall, earthquakes, and human activity, and they can cause significant damage to infrastructure and pose a significant risk to human life.
Finally, mountains can also create strong winds, particularly in areas where air masses collide. These winds can cause damage to infrastructure, trees, and other vegetation, and they can also pose a significant risk to human life, particularly in areas where wind speeds exceed 70 miles per hour.
Overall, mountains can have a significant impact on weather patterns, and they can pose significant hazards and risks to human life, infrastructure, and the environment. Understanding these hazards and risks is essential for effective planning and mitigation strategies, particularly in areas where mountainous regions are located.
Avalanches and Mountain Weather
Avalanche Triggers and Weather Conditions
Avalanches are a significant weather hazard associated with mountains. They are triggered by a combination of factors, including steep terrain, unstable snowpack, and weather conditions such as wind, precipitation, and temperature. The stability of the snowpack is influenced by the amount and type of snowfall, as well as the freezing and thawing cycles that cause layers of snow to bond together or weaken. Strong winds can also create avalanches by eroding the snowpack and redistributing snow and debris.
Avalanche Forecasting and Prevention
To mitigate the risk of avalanches, mountainous regions often have avalanche forecasting and prevention programs in place. These programs rely on a combination of meteorological monitoring, snowpack analysis, and modeling to predict the likelihood of avalanches and inform prevention measures. Forecasters consider factors such as temperature, precipitation, wind direction, and the depth and composition of the snowpack to determine the avalanche danger rating for a given area.
Prevention measures may include snow removal, the use of explosives to trigger controlled avalanches, and the installation of barriers or fencing to redirect snow and debris. In addition, avalanche safety education and awareness programs are often provided for local residents and visitors to help them understand the risks and take appropriate precautions.
Landslides and Mountain Weather
Landslides are a common weather hazard associated with mountains. They are a geological phenomenon involving the movement of rock, soil, and debris down a slope under the influence of gravity. Landslides can be triggered by various weather conditions, including heavy rainfall, snowmelt, earthquakes, and volcanic activity.
Landslide Triggers and Weather Conditions
Heavy rainfall is one of the most common triggers of landslides in mountainous regions. When a significant amount of water is dropped in a short period, it can saturate the soil and cause it to become unstable, leading to the movement of rocks and soil down the slope. Snowmelt is another common trigger during the spring months when snow accumulated over the winter begins to melt, causing an increase in the volume of water flowing in rivers and streams, which can lead to erosion and landslides.
Earthquakes can also trigger landslides by causing the ground to shake and lose its stability. In addition, volcanic activity can cause landslides by altering the physical properties of the soil and rock, making them more prone to movement.
Landslide Risk Assessment and Management
Landslide risk assessment and management are critical for minimizing the impact of landslides on communities and infrastructure. Various techniques are used to assess the risk of landslides, including geotechnical investigations, geomorphological studies, and remote sensing. These techniques involve analyzing the slope stability, soil properties, and weather conditions to determine the likelihood of a landslide occurring.
Once the risk has been assessed, appropriate measures can be taken to manage the risk. These measures may include developing early warning systems, mapping landslide-prone areas, and implementing land-use planning policies that restrict development in high-risk areas. Additionally, engineers can design structures, such as retaining walls and drainage systems, to mitigate the impact of landslides on infrastructure.
Mountain Barriers and Airflow
- The mountainous terrain disrupts the smooth flow of air currents, leading to the formation of wind gusts and turbulence.
- The pressure difference between the two sides of the mountain range creates windward and leeward sides, where the windward side experiences increased precipitation and the leeward side experiences decreased precipitation.
- Mountainous terrain can cause air masses to collide, resulting in enhanced precipitation in the form of rain or snow.
- The mountain’s altitude and steep slopes can lead to increased precipitation, including snowfall, due to the orographic effect.
- This effect occurs when the wind-blown moisture is lifted upwards, cooled, and condensed into precipitation as it encounters the mountain barrier.
- The resulting precipitation can lead to the formation of glaciers, snowfields, and other mountain glacial features.
The Influence of Mountains on Microclimates
- The presence of mountains can create microclimates, which are localized weather patterns that differ from the surrounding areas.
- These microclimates can result in unique temperature, humidity, and precipitation conditions that may differ significantly from the climate of the surrounding plains or valleys.
- The influence of mountains on microclimates can lead to the formation of distinct ecosystems and vegetation patterns, including the creation of alpine tundra, forests, and meadows.
Mountain Weather Stations and Research
- Mountain weather stations are strategically located in high-altitude areas to monitor and collect data on weather patterns, including temperature, humidity, wind speed, and precipitation.
- This data is essential for weather forecasting, climate research, and the study of mountain weather hazards and risks.
- The data collected from mountain weather stations helps in understanding the complex interactions between mountains, weather, and climate and aids in the development of more accurate weather forecasts and climate models.
Mountain Weather Hazards and Risks
- The mountainous terrain can create unique weather hazards and risks, including landslides, avalanches, flash floods, and lightning strikes.
- These hazards can result in significant damage to infrastructure, loss of life, and disruption of transportation and communication networks.
- It is essential to monitor and understand these weather hazards and risks to mitigate their impacts and improve safety measures for those living and working in mountainous regions.
1. How do mountains affect weather patterns?
Mountains can significantly impact weather patterns due to their size and elevation. When air masses meet the mountain, they are forced to rise, which leads to the formation of clouds and precipitation. This can result in increased rainfall and snowfall on the windward side of the mountain, while the leeward side may experience less precipitation. The height of the mountain can also create temperature variations, with cooler temperatures on the windward side and warmer temperatures on the leeward side.
2. Can mountains cause storms?
Yes, mountains can cause storms to form. When air masses are forced to rise over a mountain, it can create an area of low pressure, which can lead to the formation of a storm. This can result in heavy rainfall, strong winds, and even lightning. The steeper and taller the mountain, the more likely it is to cause storms.
3. How do mountains affect climate?
The presence of mountains can have a significant impact on the local climate. For example, the windward side of a mountain may experience more rainfall and cloud cover, leading to a cooler and damper climate. The leeward side may experience less precipitation and more sunshine, leading to a drier and warmer climate. This can create a temperature gradient, with cooler temperatures on the windward side and warmer temperatures on the leeward side.
4. Can mountains affect climate on a global scale?
Yes, mountains can affect climate on a global scale. The orography effect, which is the impact of mountains on wind patterns, can influence the flow of air masses around the world. This can impact ocean currents, atmospheric circulation, and even global temperatures. For example, the Himalayas can influence the monsoon patterns in South Asia, which can have a significant impact on the climate of the entire region.
5. Are there any downsides to having mountains in an area?
While mountains can have many benefits, such as providing stunning landscapes and recreational opportunities, they can also pose some challenges. For example, the steep terrain can make it difficult to build infrastructure, such as roads and buildings, and can also increase the risk of landslides and other geological hazards. Additionally, the higher elevation can lead to colder temperatures and more extreme weather events, which can impact agriculture and other industries.
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A right-angled triangle has one of its angles set at 90 degrees. A right angle is defined as a 90-degree angle, and a right triangle is defined as a triangle with a right angle. With the Pythagoras rule, the relationship between the various sides of this triangle may be simply understood. The hypotenuse is the longest side of the triangle, and it is opposite the right angle. The right triangles are also classed as isosceles right triangles or scalene right triangles based on the other angle values. Pythagorean triples are also the lengths of the sides of a right angle triangle, such as 3, 4, and 5.
The definition of a right triangle is that it is a right-angled triangle or simply a right triangle if one of the triangle’s angles is a right angle – 90o. Triangle ABC is a right triangle with the base, height, and hypotenuse shown in the illustration. The base is AB, the height is AC, and the hypotenuse is BC. The hypotenuse is the biggest and opposing side of a right triangle, and it is opposite the right angle within the triangle.
We can now recognise the characteristics of a right triangle. Triangle ABC has the following characteristics:
AC stands for altitude, height, or perpendicularity.
The basic AC is AB AB A=90o.
The hypotenuse is the longest side of the right triangle, and it is the side BC opposite the right angle.
The triangular slice of bread, a square piece of paper folded across the diagonal, and the 30-60-90 triangular scale in a geometry box are all instances of right triangles in our daily lives.
Formula for a Right Triangle
Pythagoras, the renowned Greek philosopher, devised a crucial formula for a right triangle. The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two legs, according to the formula. Pythagoras theorem was named after him. The following is a representation of the right triangle formula: The sum of the squares of the base and height equals the square of the hypotenuse.
We have the following in a right triangle: (Hypotenuse)
Pythagorean Triplet: The Pythagorean triplets are the three numbers that fulfil the preceding equation.
A Right Triangle’s Perimeter
The circumference of a right triangle is equal to the sum of its three sides. It’s the sum of the right triangle’s base, height, and hypotenuse. The perimeter of the right triangle below equals the sum of the sides BC + AC + AB = (a + b + c) units. The perimeter has a length unit and is a linear value.
Right Triangle Properties
A right triangle’s initial attribute is that one of its angles is 90 degrees. The biggest angle in a right triangle is the 90o angle, which is a right angle. The other two angles are also smaller than 90 degrees or sharp angles. The qualities of the right triangle are mentioned below:
90o is always the biggest angle.
The hypotenuse, which is usually the side opposite the right angle, is the longest side.
The Pythagoras rule is used to measure the lengths of the sides.
There can’t be any obtuse angles in it.
Right Triangles Come in a Variety of Shapes
One of the angles of a right triangle is 90o, as we know. As a result, the triangle’s other two angles will be acute angles. Isosceles right triangles and scalene right triangles are two types of special right triangles. An isosceles right triangle is one in which the other two angles are equal, whereas a scalene right triangle is one in which the other two angles have different values. Get in touch with the right experts from Cuemathand learn the concept of obtuse angle triangle.
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“Geometric shapes” is a term that we are familiar with since a very young age. Since early in the classroom, we have learned the various geometric shapes in math. So, what most of us know are geometric shapes of three dimensions. However, out of these, only a straight line would be a 1-D geometric shape. We are most familiar with 2-D geometric shapes. So, these include the triangle, the square, the rectangle, the circle, the parallelogram, the rhombus, the hexagon, the heptagon, the octagon, the nonagon, the decagon, and so on till infinity. There are 3-D geometric shapes as well. So, here we have the cube, the cuboid, the sphere, the cylinder, the cone, and so on.
Now, while doing math at school, we mostly use 2-D geometric shapes. However, if you think properly, it is the 3-D shapes that we see regularly around us. So, it makes sense because we do not live in a 2-D world. So, a matchbox is cuboid and not rectangular.
However, these are just examples we know about geometric shapes. But, it is important to know their definition and meaning along with examples.
Geometric shapes definition with examples
The best possible way to understand geometric shapes is to link definitions with examples. This is because otherwise you may know what geometric shapes are in your head but you cannot express them with pen and paper. Examples, on the other hand, are important because the expanse of geometric shapes is vast. So, a general definition cannot cater to every kind of geometric shape. Hence, let us see what geometric shapes mean at first.
Geometric shapes meaning
So, geometrical shapes are the figures which represent the forms of different objects. As we already know, such figures can be two-dimensional, whereas or three-dimensional. So, now, the two-dimensional figures lie on only the x-axis and y-axis. However, the 3-D shapes lie on the x, y, and z axes. So, the z-axis is the addition in 3-D figures. It shows the height of the object. Moreover, the 2-D figures are also known as flat shapes or closed shapes. But, there are various shapes under these expansive “geometric shapes” which represent the shapes of various objects around us.
Geometric shapes, moreover, can be of two types. So, they may be regular or symmetrical. This means that all the sides in this shape are either equal or share a fixed ratio. On the other hand, they may be asymmetrical or irregular as well. Here, the lines forming the geometric shape might go haywire. So, mathematicians also call these organic shapes.
Now, if you want to draw or design any of these figures, you must start with a line, or a line segment, or a curve. So, we get different types of shapes and figures like a triangle, a figure where three line segments are connected, a pentagon (five-line segments), and so on. This depends on the number and arrangement of these lines. However, we should also remember that every figure is not a complete figure. Now, let us see a few examples to amplify the idea.
Geometric shapes examples
First, we must know what open and closed shapes are. So, to draw an open shape you must have a separate initial and ending point that must not coincide. Therefore, what this means is that your line is a regular, or irregular, straight or curved line that does not enclose a space. On the other hand, if the initial and endpoints coincide, they enclose a space. So, they form a closed shape. Circles, squares, rectangles, and so on are all examples of closed shapes. So, we will go through their properties and formulas in the next few sections.
Geometric shapes and their properties
So, as we have already seen, the expanse of geometric shapes is huge. Therefore, various figures fall under them. As one can well imagine, each of these figures must have its own set of properties because they are totally different shapes. However, you need to remember that you can properly state the properties of only regular geometric shapes that you know about. You cannot define irregular shapes by any of their properties as such. Let us look at a few of them.
Geometric shapes triangle
Triangles are 2-D geometric shapes. So, it is a polygon that has three sides. Moreover, it also has three edges and three vertices. However, the most important thing that you must remember regarding triangles is that the sum of their internal angles must be equal to 180 degrees.
Geometric shapes squares
Squares are probably the most basic geometric shapes. They are two-dimensional. So, they have four sides and all of them are equal in length. Moreover, all the angles are equal to cutting the sides at perfect 90 degrees.
Geometric shapes rectangle
2-D shapes like squares and rectangles that have four sides are called quadrilaterals. This is because Quadri means 4 and laterals mean lines. So, these geometric shapes have four lines that join with each other to form them. So, a rectangle also has 4 sides with vertices at 90 degrees. However, it is different from a square. This is because, in a rectangle, only the opposite sides are equal.
Geometric shapes rhombus
These are also 2-D geometric shapes. However, it is quite easy to define them. A rhombus is just a square without necessarily 90-degree vertices. So, it is also a quadrilateral. Therefore, it is a parallelogram with all equal sides. Parallelograms are quadrilaterals that have two pairs of parallel sides. However, their opposite angles are also equal in measurements.
Geometric shapes circle
So, circles are geometric shapes where the locus of all points are at a fixed distance from a reference central point. Therefore, this central point is the center of the circle. On the other hand, the movement of the loci forms the circumference of the geometric shape. The fixed distance between the center of a circle and a locus is called the radius. Moreover, the distance between two opposite loci touching the center is the diameter of the circle. It is a 2-D figure.
Geometric shapes cube
So, cubes are 3-D geometric shapes. However, all the sides of a cube are exactly the same. So, a cube has 6 faces, 8 vertices, and 12 edges. Since all the sides are equal in size, each face of a cube is a square.
Geometric shapes cuboid
Cuboids are again 3-D geometric shapes. They too have 6 faces, 8 vertices, and 12 edges. However, they are different from cubes. This is because each face of a cuboid is a rectangle. So, the length, breadth, and height all are of different proportions.
Geometric shapes cone
So, we have all seen ice-cream cones. The geometric shapes that they represent are cones. So, they are definitely 3-D. They have a circular base. Moreover, the sides narrow down from the base to the top like a triangle. So, at the top, they form a point. We call this the apex or the vertex.
Geometric shapes cylinder
Now, cylinders are 3-D geometric shapes that have no vertex. So, they have two circular bases that are parallel to each other. Now, a curved surface connects these two bases.
Geometric shapes polygon
So, these are geometric shapes that have only line segments and no curves. They are different closed figures. So, they depend on different lengths of sides and different angles. Therefore, geometric shapes like squares, rectangles, hexagons, octagons, pentagons, heptagons, and so on are all polygons.
So, now that we have more or less seen the properties of major geometric shapes, let us go through the formulas.
Geometric shapes and their formulas
Basic formulas of geometric shapes:
The perimeter of a square = 4a, where a is each side of these geometric shapes.
So next, the perimeter of a rectangle = 2 x (l + b), where l is the length and b the breadth of these geometric shapes.
Next, the perimeter of a triangle = a + b + c, where a, b, and c are the three sides of the triangle. So, for an equilateral triangle, all these sides are the same. For an isosceles triangle, two sides are equal while for a scalene triangle, all three must be different.
Therefore, next comes a circle. We do not call the perimeter of a circle because it is not a line. So, the circumference of a circle is 2 ? r where r is the radius of the circle.
Hence, we have seen the basic formulas of geometric shapes. Now, let us deal with the complex area or volume formulas of both 2 and 3-D geometric shapes.
Geometric shapes and area formulas
So, let us begin with the 2-D geometric figures first.
Therefore, the area of a square is a^2 where a is each side of these geometric shapes. Similarly, the area of a rectangle is l x b where l and b are respectively its length and breadth. So next, the area of a triangle is ½ x b x h where b is the base and h is the height of the triangle. However, you might always not have the height of the triangle at hand. Moreover, if the triangle is scalene you cannot even apply the Pythagorean theorem. So, in those cases, you have to first find its semiperimeter. Then, you can apply Heron’s formula.
Read Also: Electric Force: Definition & Equation
So, finally, you have the circle. To find the area of the circle, you have to do ? r2 where r is the radius of the circle.
Now, let us go for the 3-D figures. These geometric figures would have a surface area instead of a simple area. Let us see them.
Geometric shapes surface area
For a cube, the surface area is 6 a^2, where a is the length of each side. So next, for a cuboid, the surface area is 2 ( l x b + b x h + l x h), where l, b, and h are respectively the length, breadth, and height of the cuboid. Therefore, next, we have the surface area of a sphere. We can calculate this by 4? r2, where r is the radius of the sphere. Now, we have the cylinder and the cone. For these two geometric shapes, we can have a total and curved surface area. So, the difference between the total surface area and curved surface area is that the former considers only the sides while the latter considers both bases and sides.
Therefore, the curved surface area of a cylinder is 2 ? rh, and the total surface area is 2?r (r + h). So, here r is the radius of the base, and “h” is the height of the cylinder. Now, the curved surface area of a cone is ?rl and the total surface area is ?r (r + l). So, here r is the radius of the base and l is the length of each side. Furthermore, you can find out the value of l from h and r by using the Pythagoras theorem.
Geometric shapes volume formulas
So now let us finally see the formulas that will help us find the volumes of 3-D geometric shapes. 2-D geometric shapes cannot have volume formulas because they are planar. They only enclose an area and not a volume.
So, the volume of a cube is a^3 where a is the length of each side of the cube. Next, the volume of the cuboid is l x b x h where l, b, and h are respectively the length, breadth, and height of the cuboid. Now, the volume of a sphere is 4/3 ? r^3 where r is the radius of the sphere. Now, the volume of a cone is ⅓ ? r^h where r and h respectively are the radius of the base and the height. Finally, the volume of a cylinder is ? r^h where r and h respectively are the radius of the base and the height.
Geometric shapes edges vertices faces
So, these are some of the very basic terms that we encounter while discussing geometric shapes. Let us quickly see what each of these terms means in short.
So, edges in geometric shapes are a particular type of line segment joining two vertices in a polygon, polyhedron, or any higher-dimensional polytope. In the case of a polygon, an edge is a line segment on the boundary. So, we often call it a polygon side.
Therefore, the next thing we must know is vertices. Vertices are the plural of vertex which are the corners in the geometric shapes. So, a vertex is a meeting point of two lines or edges in any polygon.
So, faces in solid geometry are flat surfaces or planar regions. Therefore, they form part of the boundary of a solid object. So, faces bounding a three-dimensional solid is a polyhedron.
Geometric shapes characteristics
To sum up, roughly, geometric shapes have some very basic general characteristics-
- Geometric shapes can be of any structure.
- Each structure has a unique set of properties. However, structures may coincide, like all squares are rectangles but not the other way round.
- They may be open or closed.
- The lines and points decide their properties.
Geometric shapes FAQs
What makes a geometric shape geometric?
Ans. This is a very valid question because how do we know which shapes are geometric shapes. So, a geometric shape is a piece of geometric information that remains when location, scale, orientation, and reflection are removed from the description of a geometric object.
What is the importance of geometric design in real life?
Ans. It finds a lot of importance regarding how we describe or want something to be. For example, you go to the carpenter’s shop to get an almirah, you need to tell its geometric shape and dimensions. However, the most important use is perhaps architecture and civil engineering- from the shape of an engine to the shape of the ventilators in your home.
Are geometric shapes 2d and 3d?
Ans. As we have already seen, geometric shapes can be both 2-d as well as 3-d. It depends on what figure we are considering. 2-d shapes occupy an area. On the other hand, 3-d shoes occupy a volume in the space.
Who invented geometric shapes?
Ans. Euclid invented most of the geometric shapes. He was one of the greatest mathematicians of all time. Moreover, he is also the father of geometry.
What are geometrical instruments?
Ans. Geometrical instruments are key to learning geometry. This is because figures need to be of the most precise dimensions. So, there are various devices for drawing geometric shapes like rulers, protractors, compasses, dividers, and so on.
Which geometric instrument is used to construct a square?
Ans. Constructing a square is pretty basic. So, you will need a ruler and a half-circle protractor for this. First, draw the baseline with the ruler. Then, draw a 90-degree protractor (you can also do this by a compass) on one side of the base. Repeat the same thing on the other side. So, measure the base and cut equal length on both sides. Join these two with another line. So, you have your square ready.
What is the difference between organic and geometric shapes?
Ans. So, organic shapes are curvilinear in nature. They occur around us but do not have any particular definition- like the shape of a rock which can always vary. On the other hand, geometric shapes have a constant definition and must have principles that pure mathematics can define.
Which is the geometrical instrument used to draw a circle?
Ans. Drawing a circle is probably the easiest thing to do in geometry. So, you can easily do this with a pencil and a pair of compasses.
Is a star a geometric shape?
Ans. Yes. Any 2-D closed shape with lines is a polygon. Therefore, a star is a regular polygon. So, it is a geometric shape.
When was geometry invented?
Ans. In all probability, the Greeks invented it sometime around the 6th century BCE. It is therefore a combination of the Greek word geo meaning earth and metron meaning measurement.
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