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<p>In robotics, there exists drive-trains that can move omni-directionally (that is in any direction). These come in many shapes and sizes, but most come in a three wheel or four wheel configuration, to keep things simple lets assume three wheels. </p>
<p><img src="http://i.stack.imgur.com/Pt7ZH.png" alt="Three wheel "kiwi" drive"></p>
<p>These drives are made possible by what is know as an omni-wheel, that is a wheel with casters along it which allows power to be applied in the direction the wheel spins, but not in the direction perpendicular to the wheel. </p>
<p><img src="http://i.stack.imgur.com/0MnOX.png" alt="enter image description here"></p>
<p>You can think of the "power" direction as having a lot of friction, and of course the part that rolls with the wheel, while the "slip" of the casters provides a very low friction, non-powered direction.</p>
<p>I come from a programming background, not physics but I want to better understand the physics at work here. So let's look at a simplified picture:</p>
<p><img src="http://i.stack.imgur.com/K8Z0e.png" alt="enter image description here"></p>
<p>I know that friction and the center of mass are important factors. That's about it.</p>
<p>So my question is, given a direction (green arrow), can a set of forces (red arrows) be calculated such that they equal the overall resulting force. IE, given blue and green can you calculate the various red forces?</p> | 5,739 | [
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<p>I wanted to know what the observable universe is so I was thinking and I thought, it must be age of the universe times 2.</p>
<p>Well I was wrong. I found on one website that it is 46B LY across in each direction. How does this make sense?</p>
<p>I get how the universe has expanded since then, but we should only be able to see light that is 13.7 billion LY old. Does this mean that the Universe is expanding faster than the speed of light? Or light from other objects is travelling to us faster than speed of light?</p> | 10 | [
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<p>In Polchinski's book "String theory", he is saying "Unlike the case of D-branes, there are no string modes tied to the orientifold plane to represent fluctuations in its shape. Our heuristic argument that a gravitational wave forces a D-brane to oscillate does not apply to the orientifold plane. Essentially, the identifications (8.8.3) become boundary conditions at the fixed plane, such that the incident and reflected waves cancel. For the D-brane, the reflected wave is higher order in the string coupling."
I want know how to check "For O-plane,the incident and reflected waves cancel. For the D-brane, the reflected wave is higher order in the string coupling." Is there any papers or books or Web sites calculating it?</p> | 5,740 | [
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<p>I know and understand why equation below holds. But i am new to operator thing in QM and would need some explaination on this. </p>
<p>$$\langle x \rangle = \int\limits_{-\infty}^\infty |\Psi|^2 x \, \mathrm d x $$</p>
<p><strong>Q1:</strong> Is it ok if i interpret the equation above like quoted:</p>
<blockquote>
<p>If i want to get expected value for position $\langle x \rangle$ i have to multiply <em>probability</em> $|\Psi|^2$ with a <em>position operator</em> $\hat x$ which in this case is equal to x.</p>
</blockquote>
<p><strong>Q2:</strong>
How can i derive a <em>momentum operator</em> $\hat p$ and an expectated value $\langle p\rangle$ using the equation above?</p> | 5,741 | [
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<p>How does a <a href="http://en.wikipedia.org/wiki/Fresnel_rhomb" rel="nofollow">Fresnel rhomb</a> work (half and quarter wave plate)?</p>
<p>I am aware of <a href="http://en.wikipedia.org/wiki/Birefringence" rel="nofollow">birefringence</a>, which creates a phase shift of $\Delta\phi=\dfrac{2\pi\Delta nL}{\lambda_0}$. But this doesn't explain how a plane polarised light shifts it's polarization angle after a half-waveplate or how a linearly polarised light becomes elliptical after a quarter-waveplate.</p>
<p>For the half one let's assume that wikipedia has an answer <a href="http://en.wikipedia.org/wiki/Waveplate" rel="nofollow">http://en.wikipedia.org/wiki/Waveplate</a> but why this electric field</p>
<p>$E(\hat{f}+i\hat{s})e^{i(kz-\omega t})$ describes an elliptical polarised light?</p>
<p>What I need is a mathematical description of how the retarders work...</p> | 5,742 | [
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<p>I don't believe the following is an accurate description of the physical but a homework problem to help understanding.</p>
<blockquote>
<p>A beam of electron of energy 0.025 eV moving along x-direction, passes
through a slit of variable width w placed along y-axis. Estimate the
value of the width of the slit for which the spot size on a screen
kept at a distance of 0.5 m from slit would be minimum.</p>
</blockquote>
<p>I have the following idea:</p>
<p>$$\theta \sim \frac{\lambda}{d}$$, where $\lambda$ is the wavelength, d is slit width, and $\theta$ represents diffraction spot size as an angle measure. For any $L$, the distance to the screen, this quantity goes minimum for $d$ tending to $\infty$ </p>
<p>I have an intuition that the uncertainty principle shall be used to get a upper bound on $d$ as effects of Heisenberg's uncertainty exceed effects of diffraction.</p>
<p>That way, I would have $$v_y = \Delta v_y = \frac{\hbar}{m_e \Delta y} = \frac{\hbar}{m_e d}$$</p>
<p>Taking $$v_x = \sqrt{\frac{2 K}{m_e}}$$,
we have the spot size in angles as </p>
<p>$$\frac{v_y}{v_x} = \frac{\hbar}{d \sqrt{2 K m_e}} = \frac{\lambda}{d}$$</p>
<p>That leaves me nowhere only when it starts seeming I have solved it.</p>
<p>Please help.</p> | 5,743 | [
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<p>Every kid who first looks into a telescope is shocked to see that everything's black and white. The pretty colors, like those in this picture of the <a href="http://apod.nasa.gov/apod/ap110612.html">Sleeping Beauty Galaxy (M64)</a>, are missing:</p>
<p><a href="http://apod.nasa.gov/apod/image/1106/m64_hst_897.jpg"><img src="http://i.stack.imgur.com/jCLh5.jpg" alt="M64"></a></p>
<p>The person running the telescope will explain to them that the color they see in pictures like those isn't real. They're called "false color images", and the colors usually represent light outside the visual portion of the electromagnetic spectrum.</p>
<p>Often you see images where a red color is used for infrared light and purple for ultraviolet. Is this also correct for false color astronomy images? What colors are used for other parts of the spectrum? Is there a standard, or does it vary by the telescope the image was taken from or some other factor?</p> | 5,744 | [
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<p>The lenses of my telescope and binoculars are dusty. What is the best way to clean them without damaging the optic coating?</p> | 5,745 | [
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<p>The Schwarzschild metric reduces to the Minkowski metric in the limit of vanishing $M$, but the Hawking temperature which is proportional to $1/M$ diverges in the same limit. This would imply that flat spacetime has infinite rather than zero temperature. What am I missing?</p>
<p>EDIT: This question is back up on the front page because I've posted an <a href="http://physics.stackexchange.com/questions/3698/flat-space-limit-of-the-schwarzschild-metric-and-hawking-temperature-review/10510#10510">answer of my own</a> - let me know if I'm thinking about this right.</p> | 5,746 | [
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<p>Imagine we have cup A with 50 g of water and cup B (smaller in width than A) with 100 g of water. Now put cup B into cup A. If the width of both cups are of comparable size then the cup with 100 g of water floats. It does not touch the bottom of cup B. </p>
<p>Now think about <a href="http://en.wikipedia.org/wiki/Archimedes%27_principle">Archimedes law</a> of flotation. It says that the weight of displaced liquid = weight of the floating object. However in this case the bottom cup has only 50 g of water. How can an object float without displacing water equal to its own weight? Am I not applying Archimedes principle correctly or because of both things beings of comparable size Archimedes principle does not apply?</p> | 151 | [
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<p>I have preformed the muon lifetime experiment at my uni's lab, and got the data. It's text file with 8190 numbers. My TDC unit was set so that the time gates were at 10 $\mu s$, and it has 8192 channels (first and last contained some kind of noise so they were not included in data analysis).</p>
<p>Now, I made a program in python that will sum every n data, and make a new list that I can then plot in bar plots and from that calculate the mean muon lifetime, because it decays by exponential law.</p>
<p>I also found that I can estimate the muon lifetime by using this formula:</p>
<p>$$<t>=\frac{\sum\limits_{i=n_l}^{n_u}N_i t_i}{\sum\limits_{i=n_l}^{n_u}N_i},$$</p>
<p>where $N_i$ is the number of counts in the bin, $t_i$ is the 'time bin', and the bins go from $n_l$ (lower) to the $n_u$ (upper) bin.</p>
<p>Using that as a guide, I found that for my data I get the best estimate if I set the bin count at 7 (7 bins), and if I put 10, I get higher result. Now if I set the bin number to 20, I get even higher number.</p>
<p>But that sounds wrong. I mean, there must be some kind of catch. If I divide the time I get for 20 bins by two I get the correct result (~2.5 $\mu s$).</p>
<p>The pictures are like this:</p>
<p><img src="http://i.stack.imgur.com/hZ1Z4.png" alt="enter image description here"></p>
<p><img src="http://i.stack.imgur.com/QgNQJ.png" alt="enter image description here"></p>
<p><img src="http://i.stack.imgur.com/eS8a0.png" alt="enter image description here"></p>
<p>So what am I doing wrong with interpreting this?</p> | 5,747 | [
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<p>I'm designing a copter and trying to decide if the propellers should be ducted or open axial flow. I've read some theory on ducted and open air flow but I can't find any where that compares the two. I would prefer to use ducted over open for safety (ie put a safety guard round the propellers so they can't be run into with fingers etc). So are there any major advantages or considerations I should take into account, with ducted vs open axial flow before ploughing ignorantly ahead with one or the other type of flow?</p> | 5,748 | [
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<p>I am trying to analyse the problem of sticking an aluminum piece to a stainless steel piece only with pressure. The holding capacity of the system is determined by the static friction between the two pieces.</p>
<p>To improve improve the holding capacity I would like to introduce a polymer layer between the two pieces.</p>
<p><strong>My problem is that I am having trouble finding reliable values for the coefficient of static friction for:</strong></p>
<ul>
<li>aluminum-steel</li>
<li>aluminum-rubber-steel</li>
<li>aluminum-silicone-steel</li>
<li>... and other polymer metal combinations</li>
</ul>
<p>An example of the problems with the reliability of the numerical values are the steel-on-steel coefficient of static friction. The value is reported as follows:</p>
<ul>
<li><a href="https://en.wikipedia.org/wiki/Coefficient_of_friction#Approximate_coefficients_of_friction" rel="nofollow">English language Wikipedia</a> gives 0.8</li>
<li>German language Wikipedia gives 0.15</li>
<li>French language Wikipedia gives 0.2(0.15)</li>
</ul>
<p>Similarly the English language Wikipedia gives steel on aluminum as 0.61 which seems equally outlandish.</p> | 5,749 | [
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<p>Linked to this question: <a href="http://physics.stackexchange.com/q/64780/">Comparing Static Frictions</a></p>
<p>Suppose there is a cuboidal vertical tunnel, and a cubical block in it such that all surface of the block are in contact with the four walls of the cuboidal tunnel respectively. All walls of the tunnel are rough with different $\mu_s$'s. And the weight of the block is $m$ and it is a point mass such that it has only 3 translational degree of freedoms.</p>
<p>The block is in mechanical-equilibrium.</p>
<p>Now the block is being pulled vertically downwards by gravity, whose force is $mg$ and upwards by the four walls of the tunnel which apply static friction.</p>
<p>How to measure static friction provided by four walls? Can we even measure?</p> | 5,750 | [
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<p>1) The bent rod $ABCD$ rotates about the line $AD$ whit a constant angular velocity of $90 rad / s$. Determine the velocity and acceleration of the vertex $B$ when the rod is in the position shown in Figure.</p>
<p>2) Determine the velocity and acceleration of the vertex $B$ assuming now that the angular velocity is of $95rad / s$ decreasing the rate of $380rad / s^2$.</p>
<p><img src="http://i.stack.imgur.com/bTPHf.png" alt="enter image description here"></p>
<p>My attempt was to apply the formula $\vec{v}=\vec{r}\times \vec{\omega}$ but I think it is not valid if the rotation axis is not aligned with any of the axes in the $xyz$ space. Maybe I'm talking nonsense. I was thinking about making a change coordinatas to the rotation axis coincide with the $z$ axis. But it seems very counterproductive and does not seem to be the purpose of the exercise.</p> | 5,751 | [
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<p>Suppose you have a particle of mass $m$ fixed to a spring of mass $m_0$ that, in turn, is fixed to some wall. I'm trying to calculate the effective mass $m'$ that appears in the law of motion of the particle (suppose the system is isolated):$$m'\ddot x=-k(x-x_0).$$
I've read somewhere that this should be $m'=m+m_0/3$, but I'm getting a different result.</p>
<p>My reasoning is as follows. Suppose the particle is at position $x$. The lenght of the spring is $x$ and we can suppose that its center of mass is at $x/2$. So the spring/particle center of mass is at: $$X= \dfrac {\frac{m_0}{2} + m}{m+m_0}x$$
Differentiating two times we get $$\ddot X = \dfrac {\frac{m_0}{2} + m}{m+m_0}\ddot x$$
Now, the only external force causing acceleration to the center of mass is the ceiling reaction to elastic force, that is exactly $-k(x-x_0)$. Thus:$$-k(x-x_0)=(m+m_0)\ddot X=(\frac{m_0}{2} + m)\ddot x $$and so I'm getting:$$m'=\frac{m_0}{2} + m.$$</p>
<p>Could you please point out where am I wrong (if I am) and possibly how is the result demonstrated?</p> | 5,752 | [
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<p>My question may not be suitable here, because it's more of astrobiology. Life as we know it is carbon based. </p>
<ol>
<li>Is life based on silicon possible?</li>
<li>What would the conditions for habitability for silicon based life be?</li>
<li>Can both carbon- and silicon-based life co-exist?</li>
<li>Is any silicon-based life found on Earth? Can any be grown in the laboratory?</li>
</ol> | 5,753 | [
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<p>I've searched various publication archives but I'm a signal processing engineer and not an applied physicist, so I might not be using the right terms. While I can find various average values for the flux of the sky during the day (no clouds) (in both psychometric and radiometric units), what I'm really interested in is the spatial variance. With sufficiently-high dynamic range imaging equipment, what short-time-scale variance would I see for a small patch of sky? How homogeneous is the scattering? What is the spectra? I imagine many of these experiments may have been done long ago. Any pointers would be greatly appreciated. </p> | 5,754 | [
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<p>I have always fancied building my own reflector telescope. I am wondering - For a certain budget, can you get better results with a home made rig? Or is this a folly, and really it would be better to buy a ready made unit?</p> | 5,755 | [
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<p>I have read that noise is a result of there being "no such thing as a perfect one-way valve". That energy transfer is never perfectly one-way; there will always (at some level) a finite flow of energy from output to input.</p>
<p>I understand this much, but what is the fundamental reason for this? Does it have its roots in thermodynamics?</p>
<p>Edit: The idea of a "one-way valve" is merely a construction in the mind, it doesn't refer to an actual valve, like in a car engine or something. It's just a general way of describing 'what connects two systems'.</p>
<p>Additionally, although I am looking for the <em>fundamental</em> reason for noise, it may be easier to let people know that I am referring specifically to <em>measurement noise</em>.</p> | 5,756 | [
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<p>Why, when one releases 2 balls in Newton's cradle, two balls on the opposite side bounce out at approximately the same speed as the 1st pair, rather than one ball at higher speed, or 3 balls at lower speed?</p> | 5,757 | [
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<p>This is a simple question but it seems to me that both the explanations are acceptable.</p>
<p>Let say the least distance of least distinguished vision is 25 cm.</p>
<p>An object is placed 12.5 cm in front of a plain mirror. Where should my eyes be placed such that I can see the image clearly?</p>
<p>Actually, there are only two options:-</p>
<p>A) My eye should be placed 12.5cm from the mirror.</p>
<p>----[Because object distance = 12.5 cm and image distance = 12.5cm .]</p>
<p>B) My eye should be placed 25cm from the mirror.</p>
<p>----[Because the image is “printed” on the mirror. It is just like looking at a picture on the wall.]</p>
<p>So, which is correct?</p> | 5,758 | [
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<p>This question was triggered by a comment of Peter Shor's (he is a skeptic, it seems.) I thought that the holographic principle and AdS/CFT dealt with that, and was enough for Hawking to give John Preskill a baseball encyclopedia; but Kip Thorne is holding out, as are others. Why?</p> | 207 | [
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<p>Does the expectation value of an observable must be equal to an eigenvalue of the corresponding operator?
I already know that 0 is not an eigenvalue, but is there any other examples?</p> | 5,759 | [
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] |
<p>This question is in reference to <a href="http://arxiv.org/abs/arXiv%3ahep-th/0112258" rel="nofollow">this</a> very famous paper of Witten. </p>
<ul>
<li><p>In general through the whole paper why is the author able to just focus on the scalar field propagating in the bulk and not need to take into account all the other fields and the complicated Lagrangian in the bulk (a Type IIB superstrings?) </p></li>
<li><p>To construct the example in equation 4.1 (middle of page 6) why did the author choose half-BPS operators and is there a simple way to see that an example of ${\cal O}$ written down is a half-BPS operator? (..what are other such?..is there a classification?..) </p></li>
<li><p>How generic is the argument in equation 4.8 (top of page 8) to get the RG flow equation?
Or is this a special case which works here for some special reason? </p>
<p>With a change in the mass/renormalization scale/cut-off one usually asks for the connected n-point functions or the effective potential to be invariant - but here the author seems to want to have the scalar field's boundary asymptotics to be invariant - I found this renormalization condition very new and mysterious. </p></li>
<li><p>I guess the most exciting analysis in this paper is the argument in the first paragraph on the top of page 9. Can someone help understand that? </p>
<ul>
<li><p>To start off how one know that the operators ${\cal O}_1$ and ${\cal O}_2'$ related to the boundary values of the two scalar fields are actually (super?)conformal primaries of the boundary (S?)CFT? </p></li>
<li><p>I did not understand how one sees that the deformation as stated in equation 4.12 (and the line before it) preserves quantum conformal invariance. </p></li>
<li><p>and the main point about the structure of equation 4.12 and the conformal invariance of the boundary being maintainable for $f \neq 0$.. </p></li>
</ul></li>
</ul> | 5,760 | [
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<p>I was recently driving behind a car that had a bicycle mounted on a carrier over the rear bumper.</p>
<p>The bicycle wheels were not bound so they were rotating in the slip-stream of the car.
I wonder, the fact that the wheels are turning; does this theoretically increase or decrease the drag on the car?</p> | 5,761 | [
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<p>I am trying to explain to myself the motivation behind studying the asymptotic structures at null, time-like and space-like infinities (For the purposes of this post, I will stick to four dimensional Minkowski spacetime). I think I have the motivation behind the first two of them down, but not the third.</p>
<ol>
<li>Null Infinity is interesting since radiation and fluxes of massless particles can reach it. One must understand these in order to answer questions such as energy loss due to gravitational radiation, or charge fluxes. Further, information can travel from the bulk to null infinity only, and is therefore an important thing to study in that context as well. Geroch's paper notes that</li>
</ol>
<blockquote>
<p>At null infinity, one's description involves "what the system is doing through time", i.e. the dynamics of the system.</p>
</blockquote>
<p>I agree with this.</p>
<ol>
<li><p>Time-like infinity corresponds to the starting and ending points of massive particles travelling along geodesics. I would presume that these would be of importance when describing "In" and "Out" states of a quantum theory.</p></li>
<li><p>Space-like infinity is a little harder for me to motivate. In Geroch's paper, he notes</p></li>
</ol>
<blockquote>
<p>At spatial infinity, "what the system does" is never even recorded. Rather, one sees only the state of the system", once and for all.</p>
</blockquote>
<p>Here are my questions:</p>
<blockquote>
<ol>
<li><p>Are the motivations I provided above for null and timelike infinity correct? Is there anything more to add?</p></li>
<li><p>What does Geroch's comment mean? Surely, in general, a system will evolve in time. What then describes the "state of a system"?? Is it the initial conditions? What, precisely is the information about the system that is stored at space-like infinity? </p></li>
</ol>
</blockquote>
<p>I have often seen null and spacelike infinities being discussed thoroughly without much literature on time-like infinity. While the extensive study of null infinities does not surprise me, the study of spacelike infinity does. I therefore feel like I am missing something quite important here. </p>
<p>EDIT: I just had a thought. It seems to me that both null and spatial infinity can be reached in a finite value of coordinate time (appropriately defined). On the other hand, time-like infinity can only be reached asymptotically. This implies that study of time-like infinity requires taking two limits $r \to \infty$ and $t \to \pm \infty$, whereas, study of null and spacelike infinity involve single limits like $r \to \infty,~v$ fixed and $r \to \infty,~t$ fixed respectively. Maybe this has something to do with my answer, but I'm not quite sure how.</p> | 5,762 | [
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<p><strong>Problem.</strong> I know that the two wave functions $\Psi_1$ and $\Psi_2$ are all normalized and orthogonal. I now want to prove that this implies that $\Psi_3=\Psi_1+\Psi_2$ is orthogonal to $\Psi_4=\Psi_1-\Psi_2$. </p>
<p><strong>My naive solution.</strong> From the premises, we know that
$$\int_{-\infty}^\infty \Psi_1^*\Psi_1 dx=\int_{-\infty}^\infty \Psi_2^*\Psi_2 dx=1$$
and
$$\int_{-\infty}^\infty \Psi_1^*\Psi_2 dx=\int_{-\infty}^\infty \Psi_2^*\Psi_1 dx=0$$</p>
<p>We also have $(z_1+z_2)^*=z_1^*+z_2^*$</p>
<p>$$\int_{-\infty}^\infty \Psi_3^*\Psi_4 dx = \int_{-\infty}^\infty (\Psi_1+\Psi_2)^*(\Psi_1-\Psi_2)dx \\
=\int_{-\infty}^\infty(\Psi_1^*+\Psi_2^*)(\Psi_1-\Psi_2)dx\\
=\int_{-\infty}^\infty(\Psi_1^*\Psi_1-\Psi_1^*\Psi_2+\Psi_2^*\Psi_1-\Psi_2^*\Psi_2)dx\\
=1-0+0-1=0\,,$$</p>
<p>which is equivalent with what we wanted to prove. Is this a legitimate proof? Is there any simpler way to do this? I am afraid I still haven't grasped how wave functions behave mathematically, so I may have missed somethings very obvious here.</p>
<p><em>Edit</em>: The solution manual somehow uses normalization factors for $\Psi_3$ and $\Psi_4$. How are these factors when you don't actually know the exact functions? And how does this relate to the concept of orthogonality?</p> | 5,763 | [
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<p>Recently I learned functional integral's formalism in QFT. I have realized that I don't understand why exactly do we introduce it. We have the expression for $S$-matrix, then we may rewrite it in through functional integral. </p>
<p>But why do we use functional integration? I read that it helps to "conserve" gauge symmetries for massless fields and that it avoids infinite results after integration through integnal lines. But these are formal arguments. I don't understand it clearly. </p>
<p>Can you add some information which can help me?</p> | 5,764 | [
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<p>This is a thing Iïve seen on many papers dealing with Warped Extra Dimensions, specifically on slices of AdS5. But the one where it appears more clearly is a lecture by Tony Gherghetta:</p>
<p><a href="http://arxiv.org/abs/hep-ph/0601213" rel="nofollow">http://arxiv.org/abs/hep-ph/0601213</a></p>
<p>Essentially what is done is that one builds a 5-dimensional theory in a slice of a Anti-De-Sitter space, the slice meaning that the fifth dimension has a small size and ends with a 4D brane at each extreme. </p>
<p>The principle of least action applied to the 5D action leads us to two terms (eq. 7 in the paper above), one on the bulk (the vanishing of this one leads us to the bulk equation of motion) and one on the branes. The vanishing of this second term can be accomplished in two ways:</p>
<p>1) the variation of the field on the branes is zero</p>
<p>2) the term multiplying the variation is zero on the branes (let?s call it B)</p>
<p>Now, all authors affirm that, in absence of extra terms on the brane, this two conditions lead to Neumann or Dirichlet conditions for the field on the branes (in the paper above this is said just after equation 10). My question is: why is that?</p>
<p>For scalar fields it can be shown that B equals the derivative of the field, so I can see the Neumann condition there (am I wrong?). But the 1st condition says the variation of the field is zero not the field itself. Clearly I am misunderstanding something here...</p> | 5,765 | [
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<p>I am working through Yale's Physics 201, and on the first problem set there is a question that is puzzling me. From my reasoning, I think the answer they provided is incorrect hence why I am here. The question is as follows: </p>
<blockquote>
<p>Two sphere of mass m and negligible size are connected to two identical springs of force constant k as shown in Figure 1. The separation is a. When charged to q Coulombs each, the separation doubles. (i) What is k in terms of q,a,and ε0? (ii) Find k if the separation goes to a/2 when the charges are ±q. (iii) In case (i) suppose the charge on the right is held fixed while that on the left is displaced by a tiny amount x and released. Find the resorting force F = −kex and the (angular) frequency ω of small oscillations. (I call the effective force constant for oscillations as ke to distinguish it from the k for the springs.)</p>
</blockquote>
<p>For part (ii) they provided the answer:</p>
<p>$$
k = \frac{4q^2}{\pi\epsilon_0a^3}
$$</p>
<p>Their reasoning is that because the separation was reduced to $$\frac{a}{2}$$ the length of the spring was in turn reduced by $$\frac{a}{4}$$ and that's how they arrived at the answer. However, I thought that of a system where the length of both springs is 1 and the separation is also 1, so the system is 3 units long. If I were to reduce the separation by half (like the question), each spring would increase its size by a fourth, so that it would be $$\frac{5a}{4}$$ equalling to 1.25 for one spring, 1.25 for the other, and .5 for the separation, giving the total 3 units. </p>
<p>So my question is: is my reasoning incorrect? If so, how do I make sense of their answer?</p>
<p>Edit: My answer is $$k = \frac{4q^2}{5\pi\epsilon_0 a^3}$$</p>
<p>Edit2: </p>
<p><img src="http://i.stack.imgur.com/uQe0w.jpg" alt="Figure 1"></p> | 5,766 | [
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<p>I always thought of current as the time derivative of charge, $\frac{dq}{dt}$. However, I found out recently that it is the Ampere that is the base unit and not the Coulomb. Why is this? It seems to me that charge can exist without current, but current cannot exist without charge. So the logical choice for a base unit would be the Coulomb. Right?</p> | 869 | [
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<p>My question is: Can an electron volt be considered an alternate unit for electric potential? </p> | 5,767 | [
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<p>What i always thought that air conditioners blow the cool air without knowing that they actually take the warm air from indoor and from outdoor. But whats the point of taking air from indoor and outdoor, Shouldn't they just be blowing the cool air inside and out air outside? How does it really make the difference?</p>
<p>Also, Does all Air conditioners function the same way as in shown in this figure, even split A.Cs?</p>
<p><img src="http://i.stack.imgur.com/iXrvX.png" alt="enter link description here"></p>
<p>I've run out of <strong>tags</strong> for questions :\</p> | 5,768 | [
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<p>It is posible to define a good variational principle to describe Fluid Mechanics? if so, wath is the correct tratement of the issue. I guess something like:</p>
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<p>I am trying to do an aerodynamic drag equation on a descending parachute (the round variety) and have no idea what the reference area on one would be. I know for a sphere, you can use radius*radius*PI to get the reference area. Is that the same for a parachute?</p> | 5,770 | [
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<p>Hubble's law says that the universe is expanding.How come the milky way and the andromeda are on a collision course?How will they end up colliding with each other?</p> | 5,771 | [
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<p>When Yang-Mills field theory was introduced, a problem is that the gauge invariance can not allow mass for the gauge field. Later people invented spontaneous symmetry breaking and Higgs mechanism to give the gauge field mass. The Higgs particle is almost confirmed at LHC.</p>
<p>My question is, since there is symmetry non-conservation (P/CP) in nature, why not simply put a mass on Yang-Mill's theory directly, got a non-abelian Proca action, say gauge symmetry breaking (although gauge may not be a symmetry actually <a href="http://physics.stackexchange.com/questions/13870/gauge-symmetry-is-not-a-symmetry">Gauge symmetry is not a symmetry?</a>; actually this point is more subtle, one can also take a Stueckelberg action then fixing the gauge, it leads to the same Lagrangian)? Is there any theoretical reason for not adding mass by hand on Yang-Mills theory? Or just because Higgs particle was found, it works, that's it.</p>
<p>My friend has a guess, that gauge invariance implies BRST symmetry, which restricts the possible form of Lagrangian. If one did a renormalization flow transformation to lower energy scale without BRST symmetry, there will be other coupling in the effective Lagrangian at lower energy scale. I am not sure about this reasoning, because BRST symmetry can restrict the possible counter terms, can it also restrict the possible terms in the effective Lagrangian? </p> | 5,772 | [
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<p>Is Pluto made out of ice--as in frozen water?
If so, wouldn't that provide evidence for alien life in the cosmos?</p> | 5,773 | [
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<p>See <a href="http://peswiki.com/index.php/Directory:Milkovic_Two-Stage_Mechanical_Oscillator" rel="nofollow">http://peswiki.com/index.php/Directory:Milkovic_Two-Stage_Mechanical_Oscillator</a> </p>
<p>The Two-Stage Mechanical Oscillator Pendulum-Lever System is very simple, yet very puzzling because it appears that more energy is going out (since it is able to pump water) than is put in (only lift one lift of the pendulum and an occasional push). </p>
<p>Where is the apparent excess energy coming from? The reason I am asking this question is because all of the explanations I have seen claim that somehow Newton's Laws are violated, which I cannot believe. </p> | 5,774 | [
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<p>Liquid electrolytes ionize and hence a current can pass through them. So if a gas can ionize, can it conduct electricity too? If so, what are a few such gases?</p> | 5,775 | [
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<p>I have just started studying Microphones and Loudspeakers. I need a good text to refer which can explain their mechanical analogies with simplicity and basics too. </p> | 5,776 | [
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<p>A stack of N D-branes can have open strings ending on them. There is a U(N) brane gauge field, and r adjoint Higgs fields, with r equal to the number of transverse spatial dimensions. The eigenvalues of the Higgs fields gives the (noncommutative) brane displacements.</p>
<p>Surprisingly, this is equivalent to an extremal black brane with a p-form flux of N, with only closed strings. No gauge symmetry is manifest. Why are they equivalent? What is the exact nature of this correspondence? </p>
<p>It's true we can have "closed" strings which stretch all the way to the horizon an infinite distance away. Their total energy is actually finite though, because of gravitational time dilation. However, no trace of any Chan-Paton factors appear anywhere.</p> | 5,777 | [
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<p>I've read that light travels in a straight line and has a wavelength of 400nm to 700nm. But I don't understand why does it have a wavelength and what creates its wavelength? I agree with the concept of sound which also has wavelength, thus called sound waves which are created by the vibrational movement in air. I'm not aware of calling light a wave. Does the light vibrate too? If so, then how?</p> | 5,778 | [
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<p>Why is a white cloud is formed in the hot water from the pot when salt (NaCl) is added? </p> | 5,779 | [
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<p>A non-spinning, equally massive black hole and white hole experience a direct collision. </p>
<p>What shall happen? What shall be the result of such a collision?</p> | 5,780 | [
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<p>I'm reading a book on electromagnetism and I am a bit confused about some things in Maxwells equations. This is what I don't like about many physics books: they are very wordy, but at the end you don't know what is an experimental fact, what is a "theorem", what is an assumption and so on,...</p>
<p>Anyway the questions are:</p>
<ol>
<li><p>Is Faraday's law a consequence of Coulomb's and Biot-Savart's (or Ampere's) law?</p></li>
<li><p>Is divB = 0 consequence of Biot-Savart's law i.e. if we found the magnetic monopole, would that mean that Biot-Savart's law is not true? (I think so, but I am not 100% sure)</p></li>
</ol> | 5,781 | [
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<p>Theoretically if I were able to build some sort of device that let me sit 1 foot away from the surface of the Sun (or any star for that matter) without being vaporized, would a star produce any sort of audible sound at that distance? From my understanding, sound waves need a medium to propagate through (air, water, etc). Does this mean that all stars are completely silent (in the audible spectrum of frequencies at least) as they fuse hydrogen into helium? I know stars emit radio, xray, infrared, ultraviolet electromagnetic waves, but how about audible (20 Hz to 20Khz) frequencies? For some reason it seems like this process would produce some sort of sound. </p> | 882 | [
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<p>I have a question regarding, as the title says, this equation: $\nabla \times \textbf{E}=-\frac{\partial \textbf{B}}{\partial{t}}$</p>
<p>So, the above equation says that the curl of an electric field is proportional to the rate of change of a magnetic field. However, since electric fields have a potential associated with them (i.e. voltage), they should be conservative and thus have 0 curl.</p>
<p>Then this would mean that all magnetic fields are constant in time. This seems like a very strong result, though, and one that I never hear mentioned at that. So is my assumption above wrong? Or am I just misunderstanding the equation?</p> | 5,782 | [
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] |
<p>If you have a uranium atom of radius $10^{-14}m$ that undergoes fission into two fragments each with 46 protons and radius $8\times10^{-15}m$. There are two ways you can calculate the electrostatic energy realised (according to my textbook):</p>
<ol>
<li>Find the electric potential energy of the two fragments as if they where point charges at a distance of $2\times8\times10^{-15}m$ apart (i.e. find the potential energy one of these charges would have in the field of the other). This gives an answer of around 180 MeV.</li>
<li>Find the electrostatic energy of the original nucleus by imagining the nucleus was built layer by layer each of thickness $dr$ and then integrating over it to find the energy of the whole nucleus. If this is then repeated for the two fragments and the sum of the electrostatic energies of the fragments is taken from that of the original nucleus that is the electrostatic energy released. But this gives an energy of around 270MeV.</li>
</ol>
<p>Here is my question, why is there a difference between these two values (the 180MeV and the 270MeV)? </p> | 5,783 | [
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<p>I'm currently studying Goldstein's Classical Mechanics book and I can't get my head around his reasoning in section 2.4. (Extending Hamilton's principle to systems with constraints).
I'd like to understand the example he gives. Here it comes:</p>
<hr>
<p>Consider a smooth solid hemisphere of radius $a$ placed with its flat side down and fastened to the Earth whose gravitational acceleration is $g$. Place a small mass $M$ at the top of the hemisphere with an infinitesimal displacement off center so the mass slides down without friction. Choose coordinate $x, y, z$ centered on the base of the hemisphere with $z$ vertical and the $x$-$z$-plane containing the initial motion of the mass. </p>
<p>Let $\theta$ be the angle from the top of the sphere to the mass. The Lagrangian is $L = \frac{1}{2}\cdot M \cdot (\dot x^2 + \dot y^2 + \dot z^2) - m\cdot g\cdot z$. The initial conditions allow us to ignore the $y$ coordinate, so the constraint equation is $a - \sqrt{x^2 + y^2} = 0$. Expressing the problem in terms of $r^2 = x^2+z^2$ and $x/z = \cos(\theta)$, Lagrange's equations are
$$ Ma\dot\theta^2 - M g \cos(\theta) + \lambda = 0$$
and
$$ Ma^2\ddot \theta + M g\, a \sin (\theta) = 0$$</p>
<p>Solve the second equation and then the first to obtain
$$ \dot\theta^2 = -\frac{2g}{a}\cos(\theta) + \frac{2g}{a}$$
and
$$ \lambda = M g (3\cos(\theta)-2)$$
So $\lambda$ is the magnitude of the force keeping the particle on the sphere and since $\lambda = 0$ when $\theta = \cos^{-1}(\tfrac{2}{3})$, the mass leaves the sphere at that angle.</p>
<hr>
<p>I have the following questions:</p>
<ol>
<li><p>Shouldn't it be $x/z = \tan \theta$?</p></li>
<li><p>Could it be that he's mixing up $r$ and $a$? My guess is that from "Lagrange's equations are" it should say $r$ instead of $a$. I get confused whether $a$ is a system parameter or a Lagrangian multiplier.</p></li>
<li><p>Could you give me a) an explanation or b) a good read on why setting $L' = L + \lambda\cdot f$ gives us an analogue of Hamilton's principle on constraint systems? I don't understand Goldstein's derivation. ($L$ is the original Lagrangian, $f$ is the constraint and $\lambda$ is the Lagrangian multiplier.)</p></li>
<li><p>Why can $\lambda$ be thought of as the constraint force?</p></li>
</ol>
<p>When I understand 3., I understand the example -- I reverse engineered the supposedly Lagrangian equations to see that $L'$ needs to be of form $$\frac{1}{2}M r^2 \dot\theta^2 - Mrg\cos(\theta) + \lambda \cdot f$$ with generalized coordinates $\theta$ and $r$. Then everything works out just fine.</p> | 5,784 | [
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<p>I have two laser beams with same polarization running parallel to each other. Will they interfere? If yes, then what are the conditions (perpendicular distance etc) and how can I observe the interference pattern? </p> | 5,785 | [
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<p><img src="http://i.stack.imgur.com/zH5ft.png" alt="I don't understand this proof"></p>
<p>Could anyone tell me $\hat {\bf n}\cdot{\bf \sigma}$ is defined in such way? In the book they have not defined what is $n_z,n_x,n_y$.</p>
<p>It is from <em>Quantum Computing: From Linear Algebra to Physical Realization</em> by Nakahara, First Chapter, page 23.</p> | 5,786 | [
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<p>The question for the full derivation of Probability Conservation -> Probability Current was already asked here:
<a href="http://physics.stackexchange.com/questions/15790/probability-current">Probability current</a>. I apologize for not retyping it out, but it's already beautifully done in latex which I have yet to become proficient at.</p>
<p>I can follow most of the derivation. However, I'm still confused as to why one can take the complex conjugate of the actual Schrodinger Equation in the second step. What allows one to do this? It's not as if it's a wave-function, which can have a complex conjugate because it describes a state. I guess my question refers to how one can take the conjugate of a (momentum) operator, because that appears to be what's happening.</p>
<p>And if one is allowed to take complex conjugates of operators, why isn't this prevalent anytime we do use Schrodinger or other Equations?? Shouldn't they also be part of the full solution?</p>
<p>Thanks in Advance!</p> | 5,787 | [
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<p>I may not understand osmosis very well. Let us suppose two compartments filled with water, separated by a semi-permeable membrane. At equilibrium, both levels are equals. Let us introduce now a given volume of solute in one of the compartments (say right). At first, the level of the right compartment will increase, to accommodate the extra volume of solute. Because of this, the concentration of water in the right compartment has decreased, and is no more at equilibrium with the left compartment. Thus a net flux of water from left to right occurs until equilibrium is reached.</p>
<p>That's what I understand from what I read so far, but I have a problem with this. In this particular example, it is true to say that the water concentration on the right side decreased when the solute is introduced, but because of the excluded volume (from the solute), the water pressure should remain the same, and thus I would not expect a net flux.</p>
<p>Unless there is other effect I don't consider?
Thank you for your explanation.</p> | 5,788 | [
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<p>I am in a low level physics class that is taught in high school. We were given a couple formulas as seen on the board, but I don't know what the variables stand for (their names), and what units correspond to each variable.</p>
<p>I have searched a few formulas on Google word for word (4), but nothing positive came up. Can any one help me layout what each Variable stands for and what units they correspond to? Or maybe a website? </p>
<p><img src="http://i.stack.imgur.com/zAnxN.jpg" alt="enter image description here"></p> | 5,789 | [
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<p>Let's say I shine a laser from a stationary medium into a moving medium (suppose the water is moving very quickly) perpendicular to the interface and back to a stationary medium like this:</p>
<p><img src="http://i.stack.imgur.com/SiL6P.png" alt="Scenarios"></p>
<p>(Note: left and right sides of the image are stationary mediums, center is a medium moving in the direction indicated by the arrow)</p>
<p>Which of the above scenarios (A, B, C, or "I'm way off") correctly reflects the path the light will take (even if the translation is incredibly small)?</p>
<p><strong>Edit:</strong></p>
<p>To answer some good questions (and things I left out of the original question):</p>
<ul>
<li>The center (moving) medium is water</li>
<li>The left and right medium (stationary) is air</li>
<li>The first angle of incidence (on the left) between the air and the water is perpendicular</li>
</ul> | 5,790 | [
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<p>I just found out that electrolytic capacitors hardly leak whereas non-electrolytic capacitors leak and at times, leak a lot. Questions: </p>
<p>(a) what causes capacitors to leak? </p>
<p>(b) why do electrolytic capacitors hardly leak when wired correctly whereas if they are reversed, they leak a fair amount?</p>
<p>(c) why do non-electrolytic capacitors leak more than electrolytic capacitors?</p> | 5,791 | [
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<p>Some time ago I came across this simple task. Instead of describing it a lot I will just post a picture and try to explain like that.</p>
<p><img src="http://i.stack.imgur.com/V9SuT.jpg" alt="enter image description here"></p>
<p>So the man and the weight on the other side of the rope have the same mass, everything is in perfect condition, meaning there is no traction on the roll (only traction needed for human to hold on to the rope).</p>
<p>So the question is what will happen when the human tries to climb up the rope? Where will the weight go? Up, down or remain still?</p> | 799 | [
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<p>I would like to minimize a functional of the type:
$$L[\gamma]=\int_a^b F(T(\gamma(t))dt$$
on the space of paths $\gamma$, where $T=T(\gamma,t)$. Now, usually I would simply apply Euler-Lagrange's equations, but in this case $T$ is defined implicitly by an equation of type:
$$f(t,\gamma,\dot\gamma,\ddot\gamma,T)=0$$
So, how can I do to find a path $\gamma$ minimizing $L$?</p>
<hr>
<p>A possible way could maybe be the following (I will skip all formality <em>à la</em> physicist):</p>
<p>Assuming $f$ to be well behaved (i.e. a small variation in the path $\gamma$ leads to a small variation of the solution $T$), take a variation $\delta\gamma$ of the path. It will lead to a variation of $T$ in the following way:
$$0=f(t,\gamma+\delta\gamma,\dot\gamma+\delta\dot\gamma,\ddot\gamma+\delta\ddot\gamma,T+\delta T)$$
And this (in first order in the variation) gives:
$$0=f+f_\gamma\delta\gamma+f_{\dot\gamma}\delta\dot\gamma+f_{\ddot\gamma}\delta\ddot\gamma+f_T\delta T$$
where $f_x=\frac{\partial f}{\partial x}$ and $f=f(t,\gamma,\dot\gamma,\ddot\gamma,T)$. Assuming $f_T$ to be invertible (in particular this gives us unicity of the solution, by the implicit function theorem), then we find:
$$\delta T = -f_T^{-1}(f+f_\gamma\delta\gamma+f_{\dot\gamma}\delta\dot\gamma+f_{\ddot\gamma}\delta\ddot\gamma)$$
And thus we obtain:
$$\begin{array}{ll}\delta L=L[\gamma+\delta\gamma]-L[\gamma]&=\int F(T+\delta T)-F(T)dt\\&=\int F_T(T)\delta Tdt\\&=\int -F_T(T)f_T^{-1}(f+f_\gamma\delta\gamma+f_{\dot\gamma}\delta\dot\gamma+f_{\ddot\gamma}\delta\ddot\gamma)\end{array}$$
Integrating by parts (and omitting boundary terms) the term in the integral becomes:
$$\left(-F_T(T)f_T^{-1}f+\frac{d}{dt}(F_T(T)f_T^{-1}f_\gamma)-\frac{d^2}{dt^2}(F_T(T)f_T^{-1}f_{\dot\gamma})+\frac{d^3}{dt^3}(F_T(T)f_T^{-1}f_{\ddot\gamma})\right)\delta\gamma$$
By variational principle, $\delta L=0$ if, and only if the term in the big brackets is $0$.</p>
<p>Is this correct? And does this give us an effective way to find $\gamma$ (at least numerically)?</p> | 5,792 | [
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<p>Why after the expansion, only 0-mode of bc-ghost contributes to the 4-points ghost function on a torus worldsheet?</p>
<p>$$<c(z_1)b(z_2)\tilde{c}(\bar{z}_3)\tilde{b}(\bar{z}_4)>_{T^2} ~\longrightarrow~ <c_0 b_0 \tilde{c}_0 \tilde{b}_0>$$</p> | 5,793 | [
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<p>I have always been excited by examples in which a few simple assumptions and first principles are used to characterize a system. For example, I did an exercise in which Crawford estimates a lake to be rectangular and then predicts the frequency of its waves through the harmonic motion of its center of mass.</p>
<p>I'd love to come up with questions and simple answers about natural phenomena, but I am not very good at coming up with either the questions or the answers, and I'd like to see lots of examples. What are some of the books you like on the subject?</p> | 5,794 | [
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<p>I wanted to calculate the moment of inertia of a ring about an axis at 45° to its normal outside the plane of the ring . </p>
<p>How do i calculate without using integration? I was thinking about using perpendicular axis theorom , but I need 2 axis in the plane of ring which I was unable to find . Please help</p> | 5,795 | [
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<p>When treating QFT we want our theory to be invariant under different symmetry groups, for example, the Standard Model is a non-abelian gauge theory with the symmetry group $U(1)×SU(2)×SU(3)$. Moreover, it is invariant under Lorentz transformations which form the Lorentz group $O(1,3)$ with the proper transformations subgroup $SO(1,3)$ In this example $S$ stands for Special, meaning that these transformations are represented by matrices with determinant $1$.</p>
<p>My questions are:</p>
<ol>
<li>why is this requirement so important? </li>
<li>In particular, what would happen if we allow the determinant to be any real (or complex) number? </li>
</ol> | 5,796 | [
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<p>I am preparing myself for an exam and I got stuck with the following problem.</p>
<p>If I wanted to calculate the vector potential $A$ on a sphere (not off or in), where some points are removed, how would I have to tackle this problem effectively?</p>
<p>If I remove a point on the $2$-plane the standard solution is given by $A \propto \frac{1}{r^2}(-y,x,0)$, which is closed but not exact (due to the removed point at the origin). Can I lift this to $S^2$ and argue similarly?</p> | 5,797 | [
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<p>Is $E=mc^2$ not just $E=m$. What does the speed of light have to do with this other than to give it a really big number so it looks cool? What spectrum of light is used? How can we test the speed of light with out a stationary point to test it from?</p> | 5,798 | [
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<p>It's pretty rainy in Boston now, and there are streams of water by the sides of the street.</p>
<p>Why do they have a 'woven' appearance? What's the formal name for this kind of fluid flow? With lots of diagonals criss-crossing each other. And is there a good way to simulate it?</p> | 5,799 | [
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<p>I've started today the book of Landau and Lifshitz Vol.2: <em>The Classical Theory of Fields</em> $\S 2$. They start from the invariance of the speed of light, express it as the fact that $$c^2(\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2=0$$ is preserved when we change inertial frame, so they consider $$ds^2=c^2dt^2-dx^2-dy^2-dz^2,$$ and say</p>
<blockquote>
<p><em>We have observed that if $ds=0$ in one frame then $ds'=0$ in another frame. But $ds$ and $ds'$ are infinitesimal of the same order. So it follows that $ds^2$ and $ds'^2$ have to be proportional that is $ds^2=ads'^2$...</em> </p>
</blockquote>
<p>and he goes on to prove that $a=1$.</p>
<p>How to translate this argument in a rigorous one? I'm really interested in this, both to understand this deduction and also to be able in future to make similar ones.</p> | 5,800 | [
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<p>Suppose we have several interacting particles in pure state $\left|\psi\right>$. For each of particles we can extract density matrices via</p>
<p>$$\rho_i(x_i,x_i^\prime)=\int \left<X_i,x_i\middle|\psi\right>\left<\psi\middle|X_i,x_i^\prime\right>dX_i,$$</p>
<p>where $x_i$ is coordinates of particle $i$ and $X_i$ is coordinates of the rest of system.</p>
<p>Can the original wave function $\left<X\middle|\psi\right>$ of the whole system be restored from all the $\rho_i$?</p> | 5,801 | [
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<p><img src="http://i.stack.imgur.com/UhwNG.jpg" alt="enter image description here"></p>
<p>Translation first!<br>
The figure shows an electric circuit.<br>
a) Use Kirchhoffs 2.law on the part-circuit abed and show that $I_3$ = 0,50A.<br>
b) Explain that $I_2 + I_5 = 2,0A$. Use Kirchhoffs 2.law on the part-curcuit befc to find another equation with $I_2$ and $I_5$ as unknows. Solve the set of equations.<br>
c) Find $I_1$ and $I_4$.<br>
d) Find the resitance of R.</p>
<p>a) Assume clockwise rotation of circuit. That makes 8v positive and 5,5v negative. 8-5,5=2,5v. This would be the V over the resitance at $I_3$. $\frac{2,5v}{5 ohm} = 0,5A$ I think this is correct. </p>
<p>b) <strong>My problem</strong> is that I can't seem to find the equation properly. If I assume a clockwise current, that makes V at $I_3$ negative. And V at $I_2$ positive and V at $I_5$ negative.<br>
$-2,5 = 5I_2 - 2,5I_5$<br>
Solution in the book is $-2.5 = - 5I_2 + 2,5I_5 $ with $I_2 + I_5 =2$ It gives $I_2=1$ and $I_5 = 1$<br>
My logic is wrong, but wrong how? </p> | 5,802 | [
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<p>If there where no friction at all, would a spinning wheel held up by one end of the axis spin precess forever without falling down? </p>
<p><img src="http://i.stack.imgur.com/S3f2n.gif" alt="Spinning wheel"></p>
<p>I just asked another question about the same problem:</p>
<p><a href="http://physics.stackexchange.com/questions/27810/direction-of-torque-precission-of-a-spinning-wheel">Direction of torque precession of a spinning wheel</a></p>
<p>Since it seems to be a good practice on stackexchange not to ask several questions in one post, I splitted them up into two questions. However if I am wrong, feel free to merge this questions.</p> | 5,803 | [
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<p>Consider a spinning wheel, which is held up by one end of it's axis like this: <img src="http://i.stack.imgur.com/bGYsu.gif" alt="Spinning wheel from hyperphysics"></p>
<p>To explain why the change of angular momentum is directed as shown in the figure above, one usually says that there is an applied torque $\vec{\tau} = \vec{r} \times \vec{F}$, where $\vec{r}$ is in this case the radius vector from the point where the string is attached to the center of mass of the spinning wheel and $\vec{F}$ is just the gravitational force which acts on the center of mass of the wheel. So it seems to be clear that it must point into the direction as shown in the figure.</p>
<p>However is it possible to see this without applying the Formula for the torque above. I am thinking of something like this:</p>
<p>By applying the gravitational force one would move the free end of the wheel axis down a bit for a very small time, so we get the beginning of a rotation which angular velocity vector points into the same direction as the $\Delta L$ on the picture. <strong>Thus</strong> we get an additional angular momentum $\Delta \vec{L}$ into this direction, which changes the direction of the entire angular momentum vector $\vec{L}$. </p>
<p>However I don't see why the last "<strong>thus</strong>" must be true, since in general the angular velocity doesn't have to point in the same direction as the angular momentum vector. </p>
<p>This is true for the rotation about the wheel axis since this is a principal axis of the inertia tensor (which is needed to justify that the axis of rotation must turn, when the angular momentum vector turns), but the (beginning/infinitesimal) rotation above doesn't seem to be around a principal axis.</p>
<p>So it would be great if someone could make the argumentation I tried more clear and more rigorous and clarify the problem described above.</p> | 5,804 | [
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<p>Trying to teach myself general relativity and have just hit yet another confusion. I'm reading that in curved spacetime the energy-momentum tensor has zero divergence, ie</p>
<p>$$\nabla_{\mu}T^{\mu\nu}=0.$$</p>
<p>But that this doesn't imply the total conservation of energy and momentum as there is an additional source of energy (the gravitational field) that isn't included in the EMT. If that's the case, and if the EMT doesn't describe the total energy of system, how is it valid to use the tensor to describe various systems. For example, $$T^{\mu\nu}=0$$
for the Schwarzschild solution, or assuming spacetime is a perfect fluid in cosmology? How are these assumptions valid if they don't include the energy contribution of the gravitational field? Seems a bit of an elephant in the room-type situation. Or is that energy so small it can be ignored? </p>
<p>Thank you</p> | 5,805 | [
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<p>Why is physical space equivalent to $\mathbb{R}^3$, as opposed to e.g. $\mathbb{Q}^3$?</p>
<p>I am trying to understand what would be the logical reasons behind our assumption that our physical space is equivalent to $\mathbb{R}^3$ or 'physical straight line' is equivalent to $\mathbb{R}$ . </p>
<p>The set of reals $\mathbb{R}$ is a basically an algebraically constructed set, which is nothing but the completion of $\mathbb{Q}$, the set of rationals. For reference see here <a href="http://en.wikipedia.org/wiki/Construction_of_the_real_numbers" rel="nofollow">http://en.wikipedia.org/wiki/Construction_of_the_real_numbers</a>. Now my question is what is the reason behind our approximation of the physical space by this abstract set. Why is this approximation assumed to be most suitable or good approximation?</p> | 5,806 | [
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<p>The energy eigenstates of the infinite square well problem look like the Fourier basis of L2 on the interval of the well. So then we should be able to for example make square waves that are an infinite linear combination of those energy eigenstates. But since linear superpositions of solutions are solutions, does this mean that this square function is an allowable wavefunction that a particle can be in in this problem? The issue I'm having is the fact that the Schrödinger equation involves the derivative of the wavefunction, but this function has none, so how can it be a solution to the equation?</p> | 5,807 | [
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<p>Klein-Gordon equation in quantum field theory is known to suffer from the possibility of negative probability. So, the question is, despite this, Klein-Gordon describes spin-zero field. So, how can negative probability and scalar field co-exist?</p> | 5,808 | [
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<p>A ball is shot directly upward, and then it comes back to the place where it was shot. Suppose we have air resistance. Suppose $t_1$ is the time period from the moment that the ball was shot to the moment that it reached its highest altitude, and $t_2$ is the time period from the moment it reached its highest altitude to the moment it reached its original position. Is $t_1=t_2$? Why or why not?</p> | 5,809 | [
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] |
<p>Just as a thought experiment. </p>
<p>One factor in the economies of space exploration is that of fuel. This may be split as </p>
<ul>
<li>MSL - Earth Orbit </li>
<li>Earth Orbit to Inner Solar System </li>
<li>Beyond.</li>
</ul>
<p>In each of these cases, the rocket demands an oxidizer to be loaded at the point of launch as neither oxygen, nor oxidizers exist in free space.</p>
<p>Why does space not have free oxygen/oxidizers?</p>
<p>p.s. According to <a href="http://en.wikipedia.org/wiki/Abundance_of_the_chemical_elements" rel="nofollow">wikipedia</a> Oxygen is the third most abundant element in the known Universe ... </p> | 5,810 | [
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<blockquote>
<p>Let us divide the time $T$ into $N$ segments each lasting $δt = T/N$. Then
we write $\langle q_F | e^{−iHT} |q_I \rangle = \langle q_F | e^{−iHδt} e^{−iHδt} . . . e^{−iHδt} |q_I \rangle $
Our states are normalized by $\langle q' | q \rangle = δ(q' − q)$ with $δ$ the Dirac
delta function. ("Dirac's formulation, in the book "Quantum Field Theory" by A. Zee)</p>
</blockquote>
<p>1) Why must $\langle q' | q \rangle = δ(q' − q)$ hold? </p>
<p>2) Is this related to $\int_{-\infty}^{\infty}\Psi^*\Psi dx = 1$?</p>
<p>Also,</p>
<blockquote>
<p>Now use the fact that $|q \rangle$ forms a complete set of states so that $\int dq |q \rangle \langle q| = 1$.</p>
</blockquote>
<p>3) Does this even make sense? </p>
<p>4) Shouldn't integral be $\int_{-\infty}^{\infty}$?</p> | 5,811 | [
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] |
<p>In 1 dimension what is the solution of the Schrödinger equation with potential</p>
<p>$$ V(x) = V_r + i V_i $$</p>
<p>Potentials are constant.</p> | 5,812 | [
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<p><strong>Question.</strong> <em>Are there any references or techniques which can be applied to obtain energy gaps for ferromagnetic XXX spin-1/2 Hamitlonians, on general interaction graphs, or <a href="http://en.wikipedia.org/wiki/Tree_%28graph_theory%29" rel="nofollow">tree-graphs</a>?</em> </p>
<p>I'm interested in frustration-free Hamiltonians such as the XXX ferromagnet on spin-1/2 particles, <em>i.e.</em>
$$ H \;=\; -\sum_{j=1}^{n} \Bigl( \sigma^{(x)}_j \sigma^{(x)}_{j+1} \;+\; \sigma^{(y)}_j \sigma^{(y)}_{j+1} \;+\; \sigma^{(z)}_j \sigma^{(z)}_{j+1} \Bigr) $$
acting on $n$ spins (where we identify the spins labelled $1$ and $n+1$ to obtain periodic boundary conditions). The ground-state manifold here is the entire symmetric subspace on $n$ spin-1/2 particles, which has dimension $n+1$. In the usual manner of speaking, it has no "eigenvalue gap". What I am interested in, however, is the gap between the two lowest distinct eigenvalues $E_0$ and $E_1$ for such systems. <strong>However</strong>: I'm interested in such systems where the pair-wise interaction may occur over an arbitrary undirected graph — such as a tree graph — and not just the line.</p>
<p>In my searches, I have only found articles addressing the problem for 1d spin chains or for rectangular lattices in 2,3 dimensions. The references I have found are:</p>
<ul>
<li><p>Koma, Nachtergaele (1997). <em>The spectral gap of the
ferromagnetic XXZ chain</em>.<br>
Lett. Math. Phys., <strong>40</strong>, pp. 1–16. [<a href="http://arxiv.org/abs/cond-mat/9512120" rel="nofollow">arXiv:cond-mat/9512120</a>]</p></li>
<li><p>Starr (2001, PhD thesis).<br> <em>Some properties for the low-lying spectrum of the ferromagnetic, quantum XXZ spin system</em>.<br>[<a href="http://arxiv.org/abs/math-ph/0106024" rel="nofollow">arxiv.org:math-ph/0106024</a>]</p></li>
</ul>
<p>where the first reference formally shows that for spin-chains of length $n$, the spectral gap is precisely $$(E_1 - E_0) \;=\; 1 - \cos(\pi/n) \;>\; \tfrac{\pi^2\!\!}{2} n^{-2} -\, \tfrac{\pi^4\!\!}{24} n^{-4}\,,$$ and the second reference discusses that result and others on rectangular lattices for spin-1/2 particles (for instance in its preliminary chapters), as well as for higher spins.</p>
<p>These references represent the simplest cases of the problem that I would like to consider. To consider arbitrary graphs, it would be sufficient for me to know what results apply for trees, which are a reasonable generalization from spin chains topologically, and which I can use to obtain a lower bound for all graphs. However, I'm only interested in spin-1/2 systems.</p> | 5,813 | [
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<p>There's something known as a "resonating valence bond" (RVB) state, which plays a role in at least some attempts to understand physics of high-$T_c$ superconductors. This, roughly, involves a state that's in a superposition (hence the "resonating" part of the name, if I understand correctly) of different ways to pair electrons into strongly-bonded spin singlets. My question is: what is a more precise definition of this sort of state? What's the underlying physics, when does it arise, and why is it interesting?</p>
<p>Points an answer might address: is there a simple toy model for which this is the ground state, that sheds light on what sort of system it could arise in? Is there an interesting continuum limit, in which we can characterize this state in a more field-theoretic language? Are there particular kinds of instabilities such a state tends to be subject to?</p>
<p>I think I know where I would start digging if I wanted to really understand this for myself, but mostly I'm asking it to probe the community and see what kind of expertise might be lurking here, since there haven't been so many condensed matter questions.</p> | 5,814 | [
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<p>I've seen a documentary, whose name I don't remember but I'm curious because it suggests that subatomic particles are able to "foresee the future".</p>
<p>I'll try to describe it here:</p>
<p>Some particles are shot through two slits in a plate, if the particles are not interfered by trying to measure its trajectory, they form an interference pattern on a wall where the beam is being projected, otherwise they just hit the wall in a single point according to the measured trajectory.</p>
<p>The following image depicts what is described above:</p>
<p><img src="http://i.stack.imgur.com/E27zL.png" alt="enter image description here"></p>
<p>The dotted red lines are where waves are canceled, blue ones are where waves overlap.</p>
<p>Then, they modify the experiment, so that they sort of turn on the detector after the particles have crossed the slit, where particles have allready being "forced" to behave as waves or single particles, to see if they change its behaviour after the measurement, just to find that the particles were measured as if they always knew that they would be measured, and they crossed the slit as particles and not waves.</p>
<p><strong>Update:</strong></p>
<p><a href="http://www.youtube.com/watch?v=Ezf8nL76tLw&t=17m45s" rel="nofollow">This is the link</a> to the documentary, it's titled "Microscopic Universe", is part of <a href="http://en.wikipedia.org/wiki/The_Universe_%28TV_series%29" rel="nofollow">this TV series</a>. They describe the experiment about 17 minutes in.</p> | 5,815 | [
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<p>This is a follow-up question to <a href="http://physics.stackexchange.com/questions/44120/what-are-wightman-fields-functions/44142#comment91472_44142">What are Wightman fields/functions</a></p>
<p>Ok, so based on my reading, the field operators of a theory are understood to be operator-valued distributions, that is, to be integrated over a smooth function: e.g. $\hat\phi(f)≡\int dx\, f(x) \hat\phi(x)$. But if the smooth function has support over such a large region in space-time that if I try to compute time-ordered products, I run into some ambiguity. How do I define the time-ordering symbol for Wightman functions?</p> | 5,816 | [
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<p>I run some battery powered test equipment in close proximity (~10cm) to a conductor carrying 25kV (50Hz AC). I find the battery life is much shorter than expected, having checked the current consumption of the equipment. Could the electric field adversely affect battery operation? The battery is a primary lithium thionyl chloride 3.6V D-cell, rated at 17Ah.</p> | 5,817 | [
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<p>Some time ago I posted a <a href="http://physics.stackexchange.com/questions/19356/have-experiments-ever-suggested-two-different-values-to-the-same-divergent-serie">question</a> here on this forum. I would like to ask some questions regarding the way the energy per unit area between metallic plates is calculated. The full calculation is on <a href="http://en.wikipedia.org/wiki/Casimir_effect">wikipedia</a>. </p>
<p>At some point in the calculation on the relevant wikipedia page (see the link above), we have the equation: $$\frac{ \langle E \rangle }{ A} = - \frac{ \hbar c \pi^2 }{6a^3}\cdot\zeta(-3) . $$</p>
<p>In the next step, it is written rather casually that $\zeta(-3) = - \frac{1}{120} \qquad (*) $. This is true when considering the analytic continuation of the riemann zeta function or the Ramanujan Summation <a href="http://en.wikipedia.org/wiki/Ramanujan_summation">method</a>. </p>
<p>Therefore, it is concluded, that $$\frac{ \langle E \rangle }{A} = - \frac{ \hbar c \pi^2}{720 a^3} . $$
I am wondering under which circumstances people decided to assume the $(*)$-marked equation is 'true'. I can think of a couple of scenarios: </p>
<ol>
<li>The formula for $\frac{ \langle E \rangle }{A} $ was already derived by means of another method which did not require the use of (regularised) divergent sums. Therefore, physicists could infer that $\zeta(-3)$ <em>had</em> to be equal to $ - \frac{1}{120} $, making the derivation of the formula by means of this method, which does use divergent series, correct. </li>
<li>The exact formula for $\frac{ \langle E \rangle }{A} $ was not already known. Physicist did have some data points that roughly showed them how the formula should look. Therefore, they tried some different constants for $\zeta(-3)$. At some point they guessed $\zeta(-3) = - \frac{1}{120} $, which yielded a formula that coincided with the known data points. They might have already known that $\zeta(-3) = - \frac{1}{120} $ by means of zeta function regularisation, making it easier to use this equation as a "guess" to find a suitable formula for $\frac{ \langle E \rangle }{A} $ . </li>
<li>Some other scenario. </li>
</ol>
<p>Which scenario roughly describes how the formula for $\frac{ \langle E \rangle }{A} $ came into existence? If it was scenario 1, which other method did physicists formerly employ to derive the formula? If it was scenario 3, how <em>did</em> this whole process unfold? </p>
<p>Thanks a lot, </p>
<p>Max </p> | 5,818 | [
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<p>As I'm very well convinced for now, the better understanding of energy we have today in Physics is that of a conserved quantity, i.e, a numerical valued associated with a system that remains the same whatever process the system undergoes.</p>
<p>This is all fine, but then we have areas in Physics where we talk of flux and density of energy in a way that seems to be like some really material thing, like a fluid for example. For instance, in electromagnetism one consider the electric energy density given by the $3$-form:</p>
<p>$$dU = \dfrac{1}{2}\epsilon_0 |E|^2 dV$$</p>
<p>So how can a "simply mathematical object" have a density like material objects as mass and charge? I know how to derive this expression, but what is the underlying idea here?</p>
<p>Also, we usually talk about energy flux, like when we deal with the energy-momentum tensor. In that case we are again in a certain sense imagining that energy in a certain sense flows like fluids and this seems a little strange if we consider energy simply a conserved quantity.</p>
<p>How those ideas are to be interpreted?</p> | 5,819 | [
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<p>I'm working on a problem where a particle of mass $m$ is confined to the surface of an inverted half cone (and is circling downwards due to gravity), with the cone's half angle $\alpha$. I chose to use cylindrical coordinates $(z,\phi,\rho)$ and I used the Lagrangian to solve this problem.</p>
<p>After going through some math, I find the equation of motion for $z$, from which I can write that </p>
<p>$$\ddot{z}\sec(\alpha) - \frac{p^2_{\phi}}{m^2z^3tan^2(\alpha)}+ g = 0$$</p>
<p>Here, $p_{\phi}$ is the angular momentum, which is conserved. Only $z$ depends on time, the other expressions are all constants.</p>
<p>At this point, I've been told that it 'can be seen' that one solution to this is given by a circular motion at constant height $z_c$. I am then asked to impose a small perturbation $z = z_c + \eta$, and (keeping only first order terms in $\eta$) find the period with which $z$ will oscillate around $z_c$. </p>
<p>Now I am pretty clueless how to do this. First of all, how can you see that there is a circular motion at constant height $z_c$? I mean, I can plug in $z = z_c$ and solve for it, but then I don't see how to find the period of something like that with the small perturbation. All the perturbation does is add some terms, but I don't see how they are time dependent and I certainly don't see how to extract a period from it. Could someone perhaps suggest a 'plan of attack'? </p>
<p>If I do simply plug in $z = z_c$ I find that</p>
<p>$$z_c=\left(\frac{p^2_{\phi}}{gm^2\tan^2(\alpha)}\right)^{\frac{1}{3}}$$</p>
<p>which at least has the right units.</p>
<p>Moreover, plugging $z = z_c + \eta$ into the first equation and keeping only first order terms of $\eta$, I find that </p>
<p>$$z = \frac{2z_c}{3} - \frac{p^2_{\phi}}{3z_c^2gm^2\tan^2(\alpha)}$$</p>
<p>But I don't see any period in that.</p> | 5,820 | [
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] |
<p>In first order of perturbation theory the S-matrix amplitude for electron scattering in the Coulomb field will be (up to normalization factors)
$$
S_{fi} = \frac{iZ q^2}{\sqrt{2E_{f}2E_{i}}}\bar {u}(p_{f}, s_{f})\gamma_{0}u (p_{i}, s_{i}) \int \frac{d^{4}xe^{i(p_{f} - p_{i})x}}{|\mathbf x|},
$$
where $f, i$-indices mark correspondingly final and initial waves, $s_{i, f}$ marks the polarization.</p>
<p>What is the difference for the case of positron scattering (the amplitude of scattering has different expression)?</p> | 5,821 | [
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<p>We say that sound waves need a medium to propagate and we know that light doesn't need such a thing. But is that really how that works? There's no such thing as "nothing" according to Quantum Mechanics, so I'm wondering if light can travel through "perfect vacuum"? A vacuum that there's no field, no lowest energy thing, nothing. Or does it really need something like "aether" to propagate? I'm not talking about "Luminiferous aether" by the way. A kind of aether Einstein thought existed. </p> | 623 | [
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<p>I have a very simple question. A motorboat directed upstream is seen to be at rest from the bank of a river. Is the engine doing any work? Is it right to say that since it is not causing any displacement, so it is not doing any work.</p> | 5,822 | [
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<p>I hope the experts of the field forgive me for this n00b questions, but I am just trying to understand physics. Assume the following function:</p>
<p>$$\phi(t)=\omega t+\cos(\omega t)$$</p>
<p>The above function has the property that:</p>
<p>$$\phi(t+T)-\phi(t)=\omega T=const.$$
Where $T=\frac{2\pi}{\omega}$.</p>
<p>Now the question is based on these facts, may I consider that $\phi$ actually represents the phase of an oscillator??</p>
<p>NOTE: I know my question is a bit silly because I am not modelling any kind of real physical phenomenon but only introducing a function, but I imagine the function I have written is simple case of an oscillator perturbed by means of a second oscillator.</p> | 5,823 | [
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<p>I can't think of a good title for my post, sorry about that. </p>
<p>I've got a lunch box (called a <a href="https://www.google.com/search?q=bento+box&hl=en&tbm=shop&prmd=imvnse&source=lnms&sa=X&ei=LgB3UPerHOrq0gHFpIDIAg&ved=0CA4Q_AUoBQ&biw=1343&bih=782" rel="nofollow">bento box</a>) Basically it's a plastic box with a plastic lid with a rubber rim around the lid to create an air tight seal with the top is on.</p>
<p>Now when I microwave my lunch (take the top off, stick it in the microwave to heat the contents, then take it back out and put the top on) it creates a vacuum such that sometimes I really need to pry the lid off with a fork. </p>
<p>How does this work? Does it have to do with the lid and base being different temperatures? </p>
<p>My very rudimentary understanding of physics is that the hot gases should be expanding and pushing outwards. I would think that would make it easier to open when the contents were hot.</p> | 5,824 | [
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<p>I remember reading a brief note in Scientific American years ago about a mathematician/physicist who had published a paper that formally stated that no entity could both participate in a given system and simultaneously know all information about that system. His conclusions were not particularly ground breaking for the physics/math community since he did little more than apply Godel's Theorem to physical systems but he did generalize his results for any conceivable entity in any conceivable system irrespective of number of dimensions etc.</p>
<p>After I realized that the article formally proved the impossibility of omniscience, I wanted to be able to refer to it. Sadly, I did not think to clip this little article from the magazine when I read it and I have never since been able to find anything on the subject.</p>
<p>Does anyone know of such a formalism or paper drawing this conclusion?</p> | 5,825 | [
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<p>I have come across many hand dryers that attempt to dry your hands really fast after you wash them. Here are two of them:</p>
<p>XLERATOR
<a href="http://www.exceldryer.com/" rel="nofollow">http://www.exceldryer.com/</a></p>
<p>Dyson Airblade
<a href="http://www.dysonairblade.com/homepage.asp" rel="nofollow">http://www.dysonairblade.com/homepage.asp</a></p>
<p>So I guess I have a ridiculously high standard cause I think even these are too slow.
Would it be possible to create a large static electric field to attract the water molecules off of your hand? Can someone offer some ideas that would remove the water off of your hand using some type of electric field generated by charged plates or something? </p> | 5,826 | [
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<p>I wonder, why the receiver of this solar tower is not black? Wouldn't a black receiver be more efficient? If not, why is it not white then, but is an intermediate color?</p>
<p><img src="http://i.stack.imgur.com/qrWIF.jpg" alt="enter image description here"></p>
<p>It seems that the receiver of other solar towers, such as Solar One is indeed black:</p>
<p><img src="http://i.stack.imgur.com/7sgPV.jpg" alt="enter image description here"></p>
<p>Does it mean that the color is not important at all?</p> | 5,827 | [
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<p>Why does the <a href="http://en.wikipedia.org/wiki/Principle_of_locality" rel="nofollow">principle of locality</a> have so such great importance in physics that theory should be consistent with it?</p> | 5,828 | [
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<p>The book <em>Commonly Asked Questions in Thermodynamics</em> states:</p>
<blockquote>
<p>When we refer to the passage of the system through a sequence of internal equilibrium states without the establishment of equilibrium with the surroundings this is referred to as a reversible change. An example that combines the concept of reversible change and reversible process will now be considered.</p>
<p>For this example, we define a system as a liquid and a vapor of a substance in equilibrium contained within a cylinder that on one circular end has a rigid immovable wall and on the other end has a piston exerting a pressure equal to the vapor pressure of the fluid at the system temperature. Energy in the form of heat is now applied to the outer surface of the metallic cylinder and the heat flows through the cylinder (owing to the relatively high thermal conductivity), increasing the liquid temperature. This results in further evaporation of the liquid and an increase in the vapor pressure. Work must be done on the piston at constant temperature to maintain the pressure. This change in the system is termed a reversible change. It can only be called a reversible process if the temperature of the substance surrounding the cylinder is at the same temperature as that of the liquid and vapor within the cylinder. This requirement arises because if the temperatures were not equal the heat flow through the walls would not be reversible, and thus, the whole process would not be reversible.</p>
</blockquote>
<p>But if the system and surroundings are in fact at the same temperature then why would this process occur at all?</p>
<p>My understanding is that in fact that they are <em>infinitesimally</em> different in temperature so I guess my question is why infinitesimality gets these processes "off the hook" for being irreversible. In other words, why do these infinitesimal changes not correspond to an infinitesimal increase in the entropy of the universe, rather than none at all?</p> | 5,829 | [
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<blockquote>
<p>A yacht on a lake drops its anchor overboard. What happens to the water level in the lake? </p>
<ol>
<li>It rises very slightly.</li>
<li>It falls very slightly.</li>
<li>It stays exactly the same.</li>
<li>It's impossible to say.</li>
</ol>
</blockquote>
<p>My understanding is that due to Archimedes principle, when the anchor is in the boat, it contributes to the mass of the boat, and thus the mass of water displaced. When it is thrown overboard, it is now the volume of the anchor which contributes to the amount of water displaced. Therefore, without knowing the density, mass and volume of the anchor, it is impossible to determine the effect on the water level, therefore the correct answer is option 4.</p>
<p>Can anyone tell me if my reasoning is correct as there is no mark scheme for the test where this question came from.</p>
<p>Thanks in advance!</p> | 5,830 | [
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<blockquote>
<p>The generator in a nuclear reactor delivers 580 MW electric power.</p>
<p><strong>a)</strong> How much TWh electrical energy is delivered yearly if the reactor runs only 70 % of the time?</p>
<p><strong>b)</strong> What flow of coolant (water) is required if the efficiency of the steam turbines is 35 % and the temperature increase of the water is 8 °C?</p>
</blockquote>
<p>a) was easily solved but I am still posting it for contextual purposes; fact is that I am struggling with b).</p>
<p>Since the reactor is only running $70\,\%$ of the time, the energy-output (per second) is $406 \cdot 10^6 \text{ J}$. The steam turbines are only able to harness $35\,\%$ of that energy i.e. $1.421\cdot 10^8 \text{ J}$. Now, using $mc\cdot\Delta \text{T}$, we arrive</p>
<p>$1.421 \cdot 10^8 = m \cdot 4180 \cdot 8 \iff m \approx 4249 \text{ kg}$</p>
<p>A mass of $4249\,\text{ kg}$ (of water) entails approximately $4000$ litres but the key in my book says $32 \text{ m$^3$/s}$. What went wrong?</p> | 5,831 | [
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