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Why don't we have underground thermal power plants? I learned from wikipedia that for the production of electricity, the temperature of geothermal sources must be at least 150 degrees Celsius.
Also wiki says that at a depth of 6 kilometers, the temperature of the earth is about 270 degrees Celsius, that is more than enough to generate electricity. And this heat is not exhausted, because it comes from the core of the earth. So this energy can be called conditionally "infinite".
Why do we not have underground thermal power plants all over the world?
We have robots, rovers, drones and we know how to extract minerals... And also we can send robots to Mars! But here you need only 6 kilometers underground.
We can dig a separate hole in each area, 6 km deep, and have local free energy forever, or at least hundreds of years ahead.
Why doesn't anyone do this?
| The physics reason is that energy isn't generated directly from heat. It's generated from heat reservoirs that are different in temperature*. So if you dig a 6km deep hole and lower a thermal engine down into it, you now have to transport heat from your thermal engine to the surface to cool it.
The practical reason is that since you need the piping anyway, you may as well just have a heat exchanger down there.
The economic reason is that a 6km deep hole that's big enough, and the requisite insulated pipes and whatnot, costs a lot of money. In order to be economically feasible you need to earn your money back before the plant wears out. In order to be environmentally beneficial, you need to "earn back" whatever environmental damage you caused digging the hole and building the plant, again, before it all wears out.
* This is basic thermodynamics. Try searching on "thermodynamics", "heat engine", or "Carnot cycle".
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Diy Water park tipping bucket physics So I'm building a tipping water bucket like the ones at a water park, just much smaller. I want the bucket to tip at 500mL. I'm currently using 3 inch PVC pipe cut 11 cm down crosswise. I can have any amount of weight at the bottom, so I was planning on 50 grams. I want to know, where can I place my 8mm diameter rod to get a quick and instantaneous dumping motion? I'm currently placing the rod 5cm from the bottom and I've tried multiple tries from 2 to 5mm off center. These first trials were unsuccessful because the bucket would tip just enough water out to stabilize it and it'd stay in that limbo. Where should the rod be?
| To get a quick "dumping" motion you need to place your tipping axis below the centre of gravity of a full bucket. As the bucket fills, the centre of gravity rises, and once it is above the axis the bucket becomes unstable and will tip over. Once the water has emptied the weight at the bottom of the bucket will bring it upright again.
You may also get better results if you use asymmetric bucket shapes, like this Bucket Fountain kinetic sculpture in Wellington, New Zealand.
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Can we say that the center-of-momentum frame is the frame in which the center of mass is at rest? Isn't the center-of-momentum frame is same as the frame in which the center-of-mass is at rest? Since the position of the center-of-mass of a system of particles is defined as $\vec R=\sum_i m_i \vec r_i/M\Rightarrow\frac{d\vec R}{dt}=\sum_i m_i\vec v_i/M=\sum_i\vec p_i/M$. Therefore, the frame in which the center of mass is at rest, $d\vec R/dt=0\Rightarrow \sum_i \vec p_i=0$. Now, by definition, the center-of-momentum frame is one in which the total linear momentum vanishes. So am I right in saying that the "center-of-momentum frame is the same as the frame in which the center-of-mass is at rest."?
| In newtonian mechanics, center-of-momentum frame is the same as the frame in which the center-of-mass is at rest as showed in your derivation.
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Is an electron in an hydrogen atom being measured by the nucleus? In an hydrogen atom, the electron interacts with the nucleus by multiple forces, for example the Coulomb force. Does that mean that the nucleus makes quantum measurements of the electron?
EDIT: I became aware that the word ‘measurement’ is not present in all theories of quantum mechanics. Therefore this question is ill-defined. Instead I would like to know whether the wave-function of the electron is influenced by the interactions with the proton, in a way that is not part of the Schroedinger equation.
|
I would like to know whether the wave-function of the electron is
influenced by the interactions with the proton, in a way that is not
part of the Schroedinger equation.
In all interpretations without wavefunction collapse the wavefunction can be influenced only by Schroedinger's equation. Only two things affect the wavefunction: Schroedinger's equation and collapse (in those interpretations where it exists). Note though that a prolonged in time process similar to wavefunction collapse is often called "decoherence". It can be understood as "smooth collapse", as the system interacts with multiple low-energy quanta which measure only part of the information about the system's wave function and interact not with the detector directly but with environment. Under decoherence, the wavefunction reduces gradually.
In Copenhagen interpretation you definitely can use a proton to perform measurement of the position of electron (would be difficult as proton is more massive, but nevertheless). This will make the wavefunction of electron to collapse.
In atom the proton's fields, electromagnetic, and theoretically, gravitational (but to a much lesser degree) affect the form of electron's wave function, but this is described by Schroedinger's equation.
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Different definitions for effective potential in static spherically symmetric spacetimes – which is right? In the paper Existence and stability of circular orbits in general static and spherically symmetric spacetimes authors define the effective potential as
$$ V\equiv \frac{1}{g_{rr} g_{tt}}~[E^2-g_{tt}~(1+\frac{L^2}{r^2})] \tag{1}$$ whereas in Boundary Orbits: 1 Static Spacetimes authors define it as $$V \equiv g_{tt}~(1+\frac{L^2}{r^2 }) \tag{2}$$ and in Theoretical Search for Gravitational Bound States of Tachyons $$V \equiv \frac{E^2}{g_{tt}}-~(1+\frac{L^2}{r^2 }). \tag{3}$$ Which definition is correct, or rather, are all three definitions admissible?
| The short answer is yes, all above definitions are admissible!
To explain it I would like to quote professor Tiberiu Harko, who kindly answered my question in private communication as follows:
"The potential in static general relativity is rather arbitrary. As opposed to general relativity, the potential is an effective quantity, which does not have a direct physical meaning, like in Newtonian gravity. If you can write the equation of motion for $r$ in the form $\frac{\it{1}}{\it{2}}~\dot{r}^2+something=constant$, you can define something as an effective potential, by analogy with classical mechanics, $V\equiv something$. This would be a kind of "standard" definition. But, if for mathematical or other reason, it is more convenient to define the effective quantity in another way, I don't think any problem in this. The effective potential in general relativity is mostly a mathematical tool. However, the definition may be important, because some physical
quantities, like the marginally stable circular orbits, are obtained from it. But even in this case, once you impose the condition $\dot{r}=0$, you can handle the problem in various ways."
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Gauge symmetry and Gauge Transforms In QFT or CFT, say the action is invariant under some local transformation. Can we call that transformation a Gauge transform?
There is a specific notion of gauge transform in math which is defined as $G$-equivariant diffeomorphism from some principal bundle to itself with some specific properties.
Are physicist gauge theory and mathematicians' gauge theory the same thing?
| No, what physicists and mathematicians mean by gauge theory are not "the same thing", but of course there is a reason the mathematical subfield is named after the physical subfield:
You can phrase many physical gauge theories in terms of the language of principal bundles and mathematical gauge theory more generally. For instance, you can read me doing this here for generic Yang-Mills theories, here specifically for the notion of large gauge transformations, here for general relativity and here for Donaldson invariants.
Most "normal" physical texts will not talk about gauge theories in this manner, choosing to work only locally and always in a single coordinate patch. There are no bundles there, and none of the global constructions and invariants mathematicans care about play any role at all at the introductory level. An understanding of mathematical gauge theory does not automatically empower anyone to understand physics texts about gauge theories and vice versa.
Additionally, while the Lagrangian formalism at least in principle directly corresponds to the mathematical idea of gauge theory, the Hamiltonian formalism of gauge theories is much unlike that even though it should be physically equivalent. Here the kind of mathematics that matters is instead symplectic geometry and in particular symplectic reductions, not "gauge theory", see e.g. this answer of mine on the relation between Lagrangian and Hamiltonian gauge theories and this answer of mine on the physical relevance of symplectic reduction.
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Are there everyday materials that change color, depending on illumination spectrum? Initially I wanted to ask this exact question about color change due to "white" light source spectrum change, and the accepted answer satisfies me fully.
However the answer says that most of objects have "rarely a nice clean notch" in the reflectance spectrum. But I wonder where I can find a list of accessible materials, that could create such mind-boggling effect.
I want to make a scientific show for kids, and my goal is to explode their brains.
| Here is how to do it.
There are certain semi-precious gemstones which, when illuminated with a certain color of incident light, appear a different color from the incident color- or from their color when illuminated with full-spectrum light. When I remember the name of the stone I will edit this response, but it IS mind-blowing! See https://www.google.com/search?rlz=1C1CHBF_enUS746US748&lei=5YXHY6XVIvza0PEPs-Ck0A0&q=list%20of%20color%20changing%20gemstones&ved=2ahUKEwjl2-jWs9D8AhV8LTQIHTMwCdoQsKwBKAF6BAh-EAI.
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What is so significant about electron spins and can electrons spin any directions? I just want to know what is so significant of with direction electron is spinning. Does it have any effect on the element or on the atom?
Also, does electron must spin up or down or can they also spin sideways or vertically?
| the direction of electron spin is of great significance. For example, to fit two electrons into a electron orbital surrounding an atom, their spins must be pointing in opposite directions. In this sense, electron spin is at least partly responsible for the structure of the periodic table and for the manner in which chemical elements react with one another.
A free electron zooming through space is free to have its spin vector pointing in any random direction, but if you use a magnetic field oriented in some particular direction to set up the electron for a spin measurement, you will detect that electron spinning in either one direction (spin vector aligned with the field) or another (spin vector aligned opposite the field direction)- no inbetween values are allowed.
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Equivalence of various definitions of reversibility in classical mechanics I was reading Classical Mechanics: The Theoretical Minimum by Leonard Susskind, and the definition of reversibility in that was:
Given a state of a system, then we know exactly what state it came from, no ambiguity.
I have also heard three other definitions:
*
*If we reverse the film of a physical process, then if it's consistent with the laws of physics, then it's reversible.
*If we make the transformation $t \to -t$ then the form of physical laws remain the same.
*If we reverse the velocities of all particles, then the system retraces it steps.
I don't which of them are correct and whether they are equivalent and if some of them are equivalent then I have no idea how to prove that. Can anyone help?
| Your quoted definition is not about reversibility, rather about determinism.
1 and 2 are the same. 3 is a specific case to classical mechanics and Newton’s law.
To make things clearer, in a classical setting, you are typically interested in a certain number of observables, that I will regroup as one entity $O$, that depend on time. The “physical laws” are then the ordinary differential equation in time coupling your observables.
1 and 2 equivalently state that if $O(t)$ is a solution to the ODE, then so is $O(-t)$. This is reversibility. Due to the unicity of the solution given the initial conditions, this defines a transformation on the initial conditions, write it $T$. When applied to the the initial conditions of $O(t)$, it gives the initial conditions of $O(-t)$.
3 is the special case when the observables are positions $x$ and the ODE is Newton’s 2d law, namely:
$$
\ddot x=F(x)
$$
in this case, you can check that the system is reversible and that the transformation $T$ is given by fixing position and flipping the sign of velocity.
Hope this helps.
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Electric field inside open conductor Is the electric field zero inside a metal conductor whose surface doesn't enclose a volume ideally, that is, a conductor that is almost a closed surface, except for a small hole?
|
Is the electric field zero inside ... a conductor that is almost a closed surface, except for a small hole?
No. Consider for example a hole that is $5 \mathrm{\ \mu m}$ in diameter. This is “small” but it is still large enough for visible light to pass through. When visible light is inside then the E field is non-zero.
In principle, for any size hole there are wavelengths small enough to pass through, speaking classically and neglecting any quantum effects.
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Is it possible to statically generate lift with the difference in pressure like wings? If I understood it correctly, the shape of the wings and/or propellers generates lift/thrust with the difference in pressure in both sides of the wings/propellers; where the lower side has higher pressure airflow and the uper side has low pressure airflow.
With this in mind, I was wondering if it is possible to generate an area of low pressure around the upper part of the an aircraft without the moving balloons, wings or propellers/rotors.
A "static lift" is the best way I could put it.
So, would such thing be possible? Or lift would only be achieved with the airflow that wings already work around?
| The cartoon is missing a key feature: the flow beyond the wing is downward. This is necessary to create lift. The lift force is balanced by a force on the air, Newton's third law in action. This force accelerates the air downward. So, no, you cannot cannot generate lift statically.
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What causes blood flow to become turbulent after a constriction? If a blood vessel is narrowed from the middle, the blood flow turns turbulent (see image below).
Why does this happen? Does it have to do with Reynolds number becoming high because velocity has increased since $Re=\frac{\rho v D}{\eta}$ the higher Re is, the more likely the flow becomes turbulent, or is there another explanation?
| It depends a bit on what level you are seeking for an answer:
*
*One can say: Because a simulation says so. After all, the simulation is nothing but a bunch of calculations, which you could theoretically do by hand. (Mind that I have no idea whether your particular statement is based on a simulation.)
*One can consider the Reynolds number and the Hagen–Poiseuille equation. However, bear in mind that the Reynolds number is only a rough heuristics. You can make robust predictions for very high and low Reynolds numbers, but in between, the type of flow depends on the exact shape of the obstruction.
Thus, one can say that a narrower segment is faster and flow speed begets turbulence. Additionally irregularities tend to beget turbulence. Thus, if you are at the border between laminarity and turbulence, these things can make a crucial difference.
*One can answer from a mesoscopic point of view and observe: If a fluid needs to go around a corner (such as after an obstruction), it’s already doing half an eddy and thus more likely to form one.
Neither of these answers is wrong, but they address different aspects and have different usefulness depending on what you want to know. For example, a simulation probably gives you the most robust result for a given situation, but it won’t give you conceptual understanding or yield general patterns such as “bottlenecks are potential sources of turbulence”.
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Proof for equal eccentricity in a binary star system What is the proof that the orbits of two stars orbiting around a common center of mass have equal eccentricities?
You can use: $m_1r_1 = m_2r_2,$ then say that $r_1= a(1+e_1)$, $r_2=a(1-e_2)$ and from the condition of the center of mass: $m_1a_1=m_2a_2.$
The problem is that the proof for the condition of the center of mass requires knowing that the eccentricities are equal, so you're kind of spiralling around the same thing.
What is the actual, complete proof?
| Crudely you can transform this type of problem into the center of mass frame then the total $r$ can be written as $m_1r_1+m_2r_2$ where $r_1=\frac{m_2}{\sum m}$ and similar for $r_2=\frac{m_1}{\sum m}$, hence you will get a definite value of $e$ in the com frame since total energy and angular momentum is conserved
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Does rate of acceleration change as object gets closer or further to center of a mass? I learnt that newton's law of universal gravitation F = G(m1m2)/R^2, and thought if the R is distance and determined gravitational strength, why do we use 9.81 as default acceleration of earth's gravity when it is not even constant at different heights?
| Who is "we"? If we're students solving physics problem, $9.81 \,\rm{m/s^2}$ (at the surface) is close enough to the nominal average of
$9.80665 \,\rm{m/s^2}$.
If we're a hydrologist, cartographer, or geodesy-interested person we may use a geoid, such as EGM96 (or EGM08, or the new one), e.g.:
https://cddis.nasa.gov/926/egm96/egm96.html
Here you get a potential surface represented tide-free mean sea level (MSL) for the Earth's surface represented as degree 2500+ spherical harmonic expansion. That gives the height of MSL relative to some ellipsoid (usually WGS84), not the strength of gravity.
The local value of $g$ can be computed from the gravitational anomaly (https://www.ngdc.noaa.gov/mgg/gravity/), which is the difference from the global mean value.
Gravity doesn't always point straight down, the deviation is called the vertical deflection, and is available publicly (e.g. https://beta.ngs.noaa.gov/GEOID/xDEFLEC18/index.shtml).
There are both global models and local models to the Earth's shape and/or gravity. For instance, some well known local ellipsoids are "Everest 1830 India", "Indonesian 1984", South American 1969".
If we're submariners or we're targeting ICBMs, we use classified data sets.
Finally, if we're Earth scientists using gravity in LEO to measure dynamic processes down below, we compute the changing gravity from the differential orbits of two linked spacecraft (https://www2.csr.utexas.edu/grace/gravity/).
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Linear Harmonic motion (simple oscillator) We know that for a simple harmonic linear oscillator, the displacement is given by $x(t)=A\sin(\omega t + \phi)$, where $\phi$ denotes the phase angle. Now as per my understanding this $\phi$ is only significant when considering SHM in form of a sinusoidal wave. Is there any physical meaning in reality. Is there a way to measure the phase angle in reality just by virtue of the particle's (which is oscillating) position with respect to the mean position?
Some Assumptions
*
*I am considering SHM in only one axis.
*I am also considering SHM linearly for ex Spring Block System.
|
$x(t)=A\sin(\omega t + \phi)$, where $\phi$ denotes the phase angle... Is there any physical meaning in reality... Is there a way to measure the phase angle in reality just by virtue of the particle's (which is oscillating) position with respect to the mean position?
The mean position of what you wrote above is zero.
The phase angle $\phi$ (along with the other parameters $A$ and $\omega$) tells you the initial position and velocity.
You can re-write $x(t)$ as
$$
x(t) = A\left(\sin(\omega t)\cos(\phi)+\cos(\omega t)\sin(\phi)\right)\;,
$$
to see that the initial position is given by
$$
x(0) = A\sin(\phi)\;,
$$
and the initial velocity is given by
$$
v(0) = A\omega\cos(\phi)\;.
$$
You can solve for $\phi$ like:
$$
\phi = \tan^{-1}\left(\frac{\omega x(0)}{v(0)}\right)
$$
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How is the slit material not considered an observer? As far as I can tell, the essential process of "observation" is that there is an interaction with something else, providing a means by which any part of the universe noticed something about the thing being observed.
In the double-slit experiment it is absolutely obvious that the slits (and material the slit is in) affect the electron somehow. Why is this not considered an observation/interaction? Sure, the math works this way, but why can we act like one interaction doesn't matter while another does?
| Not only in the double-slit experiment, but for every edge, it is the edges that influence the electron. Behind each edge you find a wave-shaped distribution of electrons on a screen. And as you say, it is the perfectly functioning mathematical solution that seems to make deeper explanations superfluous.
If something influences the direction of movement of the electron - like the photon, by the way - then it is the surface electrons of the edges. There is the research field of phononic and other wave-like excitations in materials. It should not be difficult for a specialist in this field to make a theory of feedback on the movement path of the electron (photon) from this. And perhaps this even holds the chance to prove the passage of the electron (photon) through the slit.
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Wave operator in Kerr spacetime: change of coordinates The wave equation for a scalar field, in Kerr geometry and in Boyer-Lindquist coordinates, reads:
$$-\left[\frac{(r^2 + a^2)^2 }{\Delta} - a^2 \sin^2\theta \right] \partial^2_t \Phi - \frac{4Mar}{\Delta}\partial_t\partial_{\phi}\Phi + \left[\frac{1}{\sin^2 \theta} - \frac{a^2}{\Delta} \right]\partial^2_\phi \Phi + \partial_r(\Delta\partial_r \Phi) + \frac{1}{\sin\theta} \partial_\theta (\sin \theta \partial_\theta \Phi) = 0 $$
where $ \Delta = r^2 - 2Mr - a^2 .$
In many papers dealing with the time evolution of the scalar field, besides the tortoise coordinates $\frac{dr_*}{dr} = \frac{r^2 + a^2}{\Delta}$, a specific change of variables is introduced to cure unphysical pathologies near the horizon, which is
$$d\phi_* = d\phi + \frac{a}{\Delta} dr.$$
Assuming the ansatz:
$$\Phi = \Psi(t,r,\theta)e^{im\phi_*}$$
the wave equation in $(t,r_*,\theta,\phi_*)$ coordinates is:
$$-\partial^2_t \Psi - \frac{(r^2 + a^2)^2}{\sigma}\partial^2_{r*}\Psi + \frac{4imarM}{\sigma}\partial_t \Psi - \frac{2\left[r\Delta + iam(r^2 + a^2)\right]}{\sigma}\partial_{r*}\Psi - \frac{\Delta}{\sigma}\left[\partial^2_\theta \Psi - \cot \theta \partial_{\theta} \Psi + \frac{m^2}{\sin^2\theta}\Psi\right] = 0.$$
where $\sigma = -(a^2 + r^2)^2 + a^2\Delta \sin^2\theta$.
I tried to re-do the change of variables by myself but I got some "spurious" terms in the wave equation like for example a cross term $\partial_t\partial_{r*}$.
Can someone explicitly show how to get the last wave equation through the changes of variables shown above?
| It would have been good if you had shown some of your calculation in detail so that we could see where you might have followed the wrong path (which may also be helpful to others).
The solution to this problem comes back to expressing the partial derivatives in the old coordinates $(t, r, \theta, \phi)$ in partial derivatives of the new coordinates $(t, r_*, \theta, \phi_*)$.
Let's consider the differential of some function $f$ (which could be your scalar field $\Phi$) in the old coordinate basis
$$
df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial r} dr + \frac{\partial f}{\partial \theta} d\theta + \frac{\partial f}{\partial \phi} d\phi \qquad (1)$$
and in the new coordinate basis
$$
df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial r_*} dr_* + \frac{\partial f}{\partial \theta} d\theta + \frac{\partial f}{\partial \phi_*} d\phi_*.$$
With the two coordinate transformations you mentioned, i.e.
$$
dr_* = \frac{r^2 + a^2}{\Delta} dr\quad\text{and}\quad
d\phi_* = d\phi + \frac{a}{\Delta} dr,
$$
the latter becomes
\begin{align*}
& = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial r_*} \frac{r^2+a^2}{\Delta} dr + \frac{\partial f}{\partial \theta} d\theta + \frac{\partial f}{\partial \phi_*} d\phi + \frac{\partial f}{\partial \phi_*} \frac{a}{\Delta} dr \\
& = \frac{\partial f}{\partial t} dt + \left[\frac{\partial f}{\partial r_*} \frac{r^2+a^2}{\Delta} + \frac{\partial f}{\partial \phi_*} \frac{a}{\Delta} \right]dr + \frac{\partial f}{\partial \theta} d\theta + \frac{\partial f}{\partial \phi_*} d\phi. \qquad (2)
\end{align*}
As the 1-forms $dt, dr, d\theta, d\phi$ form a basis, their coefficients in both (1) and (2) must be identical; hence
\begin{align*}
\frac{\partial f}{\partial r} & = \frac{r^2+a^2}{\Delta} \frac{\partial f}{\partial r_*} + \frac{a}{\Delta} \frac{\partial f}{\partial \phi_*}, \\
\frac{\partial f}{\partial \phi} & = \frac{\partial f}{\partial \phi_*}.
\end{align*}
When you use those two expressions in your initial wave equation, you should reproduce the latter wave equations in a few steps.
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How is this child able to move their crib while standing inside of it? I was scrolling Instagram and saw this Reel which at first was normal but when I started to think how the child was able to move then I got confused. The video shows a child standing inside of a crib and repeatedly bouncing their body against the rail of the crib, which causes the crib to move.
Here is the logic behind the confusion:
The center of mass of any system if initially at rest will move only if there is an external force acting on the system.
In this case the child is applying a force on the bed in the forward direction by hitting it continuously so friction should be acting on it in the backward direction but if that's the case then they should move in the backward direction and not forward.
Can someone give an explanation for this kind of motion of the child?
Edit :-
As per the comment I am adding the screenshots of the child and the crib initially and how it ended up after some time (though it will not tell how the child make this happen)
| You're right that this would not work in space. However the crib is attached to the floor via static friction. When the crib is bumped, the floorboards are pushed in the opposite direction, which is eventually transferred to the house/earth.
| {
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Seeking expression for radial velocity of an object in an elliptical orbit (with focus at coordinate origin) as function of radius Consider a planar, elliptical orbit in a simplified two body, $\frac{K}{r^2}$ central attractive force problem (i.e. assume m1 >> m2 so focus $f1$ is effectively at m1, with m2 at point $p\left(x,y\right)$) and $\rho$ being the radius (green line) from m1 to m2 ($f1$ to $p\left(x,y\right)$), as indicated in the following plot ($f1$ at coordinate origin, periapsis at left blue hash, apoapsis at right blue hash):
I am trying to determine the expression for (what I am referring to as) the radial velocity $v_\rho=\frac{d\rho}{dt}$ along the direction of $\rho$ toward $f1$, at each point $p\left(x,y\right)$ on the orbital ellipse, strictly as a function of $\rho$. To be clear, I do not seek the expression for the velocity tangent to or normal to (red lines in plot) the orbital ellipse at point $p\left(x,y\right)$, but rather only the velocity along $\rho$ toward $f1$.
I have tried to come up with this expression using the very illuminating discussion in this item How do we describe the radial velocity in elliptical orbits?, but with no success yet. I also thought this might be readily found in a classical mechanics text (e.g. Symon), but haven't found (or recognized) such.
In line with How do we describe the radial velocity in elliptical orbits?, I expect the plot of this radial velocity to be structurally similar to the following but with a continuous, finite value - unlike the infinity exhibited in this plot - for the latus rectum at $\rho=4$ (note that $\dot{\rho}=0$ for periapsis at $\rho\approx+2.14359$ in the plot below, which differs from periapsis at $x\approx-2.14359$ in the elliptical orbit plot above):
Any reference to an existing solution, or advice on deriving one, would be greatly appreciated.
I think I have provided enough information to fully characterize the problem, but can certainly provide more info if I've missed something.
Thanks.
| The polar equation of the ellipse
$$\mathbf r=r(\theta)\begin{bmatrix}
\cos(\theta) \\
\sin(\theta) \\
\end{bmatrix}\quad,
r(\theta)={\frac {p}{1+e\cos \left( \theta \right) }}$$
thus $~\dot{\mathbf{r}}\cdot\mathbf e_r~$ equal to
$$\dot r(\theta)=\frac{\partial r}{\partial \theta}\,\dot\theta=\frac{\partial r}{\partial \theta}\,\frac{h}{r^2}={\frac {e\sin \left( \theta \right) h}{p}}$$
with:
$$p=a\,(1-e^2)\quad,h=\sqrt{\mu\,p}\quad,\mu=G\,(m_1+m_2)\quad\Rightarrow\\
\dot r={\frac {e\sin \left( \theta \right) \sqrt {\mu\,a \left( 1-{e}^{2}
\right) }}{a \left( 1-{e}^{2} \right) }}
$$
| {
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Wavefunction of distinguishable spin 1/2 fermions Does the total wave function for distinguishable (i.e. not identical) spin 1/2 fermions need to be anti-symmetric under particle exchange? Or does the Pauli exclusion only hold for indistinguishable fermions?
| Quoted from Pauli exclusion principle (emphasis by me):
A more rigorous statement is that, concerning the exchange of
two identical particles, the total (many-particle) wave function
is antisymmetric for fermions, and symmetric for bosons.
This means that if the space and spin coordinates of
two identical particles are interchanged, then the total wave
function changes its sign for fermions and does not change for bosons.
| {
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How would the universe look like without matter? I was wondering how the universe would look like if it would have been perfectly symmetrical in terms of matter and antimatter. If I understand correctly, there would be no "particle" but the energy released by matter-antimatter annihilation shouldn't just disappear, thus such universe probably wouldn't be an empty void. How does it differ from the current universe, apart from being deprived of persistent matter and related phenomena?
| Nothing new would replace the matter. Asymmetric matter was a tiny contribution to the energy density of the early universe, of order one part in a billion. Moreover, if we were to add the energy of the asymmetric matter to the primordial radiation bath, that would essentially just shift the exact same cosmic evolution to a slightly later time. An observer like us who arises from that cosmic evolution would arise at a correspondingly later time, so the added energy would make no difference to what they find.
So, would the universe be empty? Not quite. It would still contain the cosmic microwave and neutrino backgrounds, as well as a (potentially minuscule) primordial gravitational wave background. To the best of our knowledge, it would also still contain dark energy and dark matter. And indeed it would still contain essentially the same structure that we find today at scales larger than galaxies, except that structure would be composed of (almost) entirely dark matter.
Also, there would be some ordinary matter/antimatter in this universe. Its abundance would simply be low enough that a given particle is not expected to meet another particle to annihilate, even given the entire age of the universe. (In the field, we would say that these are thermal relics with an abundance fixed by freeze-out.) Specifically, the ordinary matter would be about a billion times less abundant than it is in our universe.
| {
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Why can you hear loud TV in the next room despite the wall and door? In a house, when two rooms are next to each other, why can you hear the loud TV on the next room, despite the wall between them and despite that their two doors are closed. (I don't know a lot on physics, but isn't there something like sound travelling, does the sound travel through the wall ? It must be 3 to 5 centimetres). The loud TV is especially annoying because I hear all the bass sounds (less of the higher-pitched sounds). And I read about how just bass sound makes you anxious (especially if you are trying to sleep).
| Sound can travel through walls and other solid objects because it is a pressure wave that travels through air, liquids, and solids. The sound from a loud TV can penetrate a wall and reach the adjacent room, especially if the wall is thin or has gaps in it.
Additionally, low frequency sounds like bass can travel further than high frequency sounds because they have longer wavelengths and are less easily absorbed by walls and other objects. This is why you may hear more bass from the loud TV in the next room. To reduce the noise, you could consider using soundproofing materials or adding an additional layer to the wall, adding weather stripping or a door sweep to the doors to reduce the gap between the doors and the floor, or using noise-cancelling headphones if you need to sleep.
| {
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Linear combination of atomic orbitals (LCAO) justification for molecular orbitals (MO) done in this MIT solid state chemistry lecture In this series on solid state chemistry by MIT, 11:00-13:00 in this lecture, LCAO for molecular orbitals is justified by the fact that Schrodinger's equation is linear (and therefore the superposition principle is used)
However, I have a vague recollection, either from high school or from reading somewhere that expressing the wave function of an MO as a linear combination of the atomic orbital MOs is an approximation. However, there's no such mention of it being so in the lecture. He seems to be implying that this type of a thing is exact.
Am mixing two different ideas up or misinterpreting something?
| LCAO is an approximation, partly because we usually only add up a finite number of orbitals. E.g. for the molecule LiH, we could combine the H 1s orbitals with the Li 2s and 2p to get some wavefunction. But we have ignored the Li 3s for example, because it's high in energy and won't overlap well with H 1s.
Another reason it's an approximation is we're using Hydrogen-like atomic orbitals (which are the exact solutions to the Hydrogen Schrodinger equation), and using a variational method to solve the actual Schrodinger equation which has more than one electron for which we don't have an exact solution.
The "exactness" here is just the fact that if 1s is a solution to the (Hydrogen) Schrodinger equation, and 2s is also a solution, then 1s+2s is a valid solution too.
| {
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How did Enrico Fermi compute when the Chicago Pile-1 nuclear reactor would become critical? I'm trying to understand the first nuclear reactor, the Chicago Pile-1, specifically the math Fermi did to figure out when the reactor would go critical. There's a nice report available from Fermi, where he tracks the value of $R_{eff}^2/A$, where $R_{eff}$ is the effective radius of the pile, and A is the measured neutron intensity at the center of the pile, some screenshots below.
Fermi then claims that when $R_{eff}^2/A$ reaches 0, the pile will become critical. This is where I get lost - I read through the report and some other sources, but I don's see where this math is coming from -> why should the pile become critical when $R_{eff}^2/A=0$?
| I cannot be sure I understand it correctly, but this is how it looks to me at the moment. The report that you quote says:
In a spherical structure having the reproduction factor 1 for
infinite dimensions the activation of a detector placed at the center
due to the natural [my emphasis] neutrons is proportional to the
square of the radius.
@Adam says in a comment:
I guess the question really is why he prefers to plot $R^2_{eff}/A$
instead of just $A$
According to the quote from the report, if you just have $A\to\infty$ when the radius increases, it is possible that you just have natural neutrons. If, however, $R^2_{eff}/A\to 0$, it means that the number of natural neutrons becomes negligible compared to the number of neutrons from chain reaction, therefore, the reactor is getting critical.
| {
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Coherent unpolarized laser light I notice that in semiclassical treatments of laser light absorption by particles, they treat the laser beam as a coherent oscillating electric field over the form $E_0\cos(kx-\omega t)$, sometimes with a factor added in to account for the Gaussian spread of wavelengths. However, my understanding is that most laser beams are unpolarized, so that they are composed of a practically infinite number of photons whose electric field are all oscillating at the same frequency but in different directions. So shouldn't the average of all these oscillating field be $0$ at all times, or at least vary randomly about $0$? How can we pretend that the atom is question is just exposed to a single linear oscillating electric field and expect to get good results?
| All lasers are polarized .... where did you hear otherwise?
We can never directly observe the superposition of EM waves in the EM field ..... we can only observe a photon when it excites an electron in a CCD or in your eye. It does not matter if the photons are out of phase (net zero E,M) .... the electrons are able to scatter/separate/absorb them.
The idea of waves of light cancelling is old (and misleading)... based on Fresnel Huygens (1818) ... but is still taught today! For example in the DSE (double slit experiment) all the high school physics formulas do work to calculate the bands ... but fundamentally photons are not cancelling ... it is likely that the virtual fields are cancelling even before the original photon takes flight.
| {
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Mass-Energy Equivalence and First Law of Thermodynamics Einstein showed mass can be converted into energy and vice versa.
$E=mc^2$
However, in school we are taught that according to the First Law of Thermodynamics, energy can neither be created nor destroyed.
Are they not contradicting each other? I already tried finding it on other sites but was surprised that there was little information regarding this.
|
Are they not contradicting each other?
Yes, there is a contradiction, but not between $E=mc^2$ and thermodynamics. The contradiction is between the actual meaning of $E=mc^2$ and its usual pop-science description. Unfortunately, although $E=mc^2$ is very famous, it is also very misunderstood. The usual English description of that equation is, as you stated, "mass can be converted into energy". However, that English description is obviously incorrect due to the conservation of energy, as you stated.
If scientists had wanted to write an equation that did state "mass can be converted into energy" then the equation would be $\Delta E = -c^2\Delta m$. This equation says that a negative change in mass gives a positive change in energy. That is what it would mean to convert between the two. So not only does $E=mc^2$ not describe a conversion, it is not intended to do so.
$E=mc^2$ is a statement at any given time. At that time a mass with no momentum has energy. There is no sense of conversion between the two. The mass and the energy are both present at the same time.
Note that I specifically said "a mass with no momentum". $E=mc^2$ only applies when $m$ has no momentum ($p=0$). In any scenario where $m$ has momentum then the more general formula is $m^2 c^2 = E^2/c^2 - p^2$.
| {
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What exactly does a Lorentz transformation provide?
The primed reference frame is moving relative to the unprimed frame. So if we were to take the lorentz transformation of point P from the unprimed to primed, would it be the point A or B that it returns ? Assuming that the Lorentz transformation is passive i.e. we are talking about the same event in two frames, my first guess was B, because only at B will the primed reference frame see P to occur. A on the other hand is what the unprimed reference frame sees when P occurs. Is this true ?
PA is parallel to the x axis and PB is parallel to the x' axis.
|
Look at this Minkowski diagram point P move parallel to the x‘ axis , the event point is then $~E(ct‘,x‘)$
| {
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Why does the double slit experiment not prove that the wave function is ontological? To me, it seems that the interference pattern is the evidence that the wave function is a physical aspect of reality, but people still seem to be trying to decide whether or not it's ontological or just a mathematical construct.
Why is the double slit experiment not considered proof that the wave function is ontological?
| The wave function is a projection of the more abstract state vector onto position space. If you want to say that the wave function is a "physical aspect of reality", then naturally you have to say that any other space one can project the state onto (momentum, energy, angular momentum, etc.) is also a "physical aspect of reality". Sometimes these spaces can even be discrete. What makes this even more interesting is that you can project state vectors onto spaces that don't represent physical observables as well.
I like to give the analogy of vectors in classical mechanics. We use vectors to describe many things in classical mechanics (position, velocity, force, etc.). We can use formalism that relies on vectors to make very nice predictions and verifications about how the world works. Does this mean that vectors are "a physical aspect of reality"? I think most people would relegate vectors to be "mathematical tools" rather than things that physically "exist".
In any case, not everyone even agrees on what "a physical aspect of reality" is. Even if we did, at best ask we can say is that Quantum Mechanics is a successful model that gives correct results. It doesn't give us a "physicality rating" of what is used, and the fact that we have many interpretations of Quantum Mechanics that model things differently yet give the same result indicates that maybe there is more we have yet to (or may never) learn and understand.
| {
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First mode of vibration for a glass window How to calculate the first mode of vibration for a glass window?
I have a window of the size 57 cm by 106 cm and 4 mm of width. I hear the loud noise from it on about 166 Hz and want to realize if it is window's self first mode of vibration that intensify this frequency (maybe even with resonance, etc.) or not.
I understand that this could depend on exact glass, fixing, etc., but is there a common idea to calculate it and have at least rough approximation?
If there is some software to calculate it (like shown in the Understanding the Finite Element Method), please advise.
| I am a mechanical engineer who does full time work with finite element analysis including modal analysis which involves finding the natural frequencies of structures.
Knowing the exact window dimensions is a good first step, but to calculate the exact first mode with FEM will require the two additional items listed...
*
*You need to know the density and stiffness of the glass. I don't work with glass and I'm not sure if all forms of glass have the same stiffness or not.
*You need to know the boundary conditions of the edges of the window. This is very important! If the window edges are held loosely, then the first mode frequency will be much lower than if the edges are held tightly.
You can likely conduct a test to find the first mode if you have a speaker and computer available. Place the speaker near the window and play a sine sweep (search for this on Youtube). Lightly place your finger on the window and you will feel vibrations when the sweep hits your first mode.
Good luck!
| {
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Why can we always find $\vec A$ such that it satisfies Coulomb (or Lorenz) gauge and Maxwell's equations? I have a short question about the Coulomb potential.
Let $\vec{E}$ and $\vec{B}$ be the electric field and magnetic field respectively.
The electric field and magnetic fields are described by the scalar potential $V$ and vector potential $\vec{A}$ respectively. As we know, there are a lot of scalar potentials and vector potentials describing the same electric field and magnetic field. My question is, can we $\textbf{always}$ find, in this case, the vector potential $\vec{A}$ that describes the given field $\vec{B}$ and also satisfy $$\nabla \cdot \vec{A} = 0~?$$
And is this also the case for Lorenz gauge, $$\nabla \cdot \vec{A} = -\mu_{0}\varepsilon_{0}\frac{\partial V}{\partial t}~?$$
| The short answer is yes.
The general proof of this does not make any assumptions about the fields and thus is valid in general. In particular, if $\vec{A}_1$ is a vector potential for $\vec{B}$, i.e. a potential such that $\nabla \times \vec{A}_1 = \vec{B}$, then $\vec{A}_2 = \vec{A}_1 + \nabla f$ describes the same magnetic field $\vec{B}$ since $\nabla \times \nabla f = \vec{0}$ for any function $f$. However,$$\nabla \cdot \vec{A}_2 = \nabla\cdot \vec{A}_1 + \nabla^2 f,$$
and thus by appropriately choosing the function $f$ we can set the divergence of $\vec{A}_2$ to anything we may want!
| {
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Why does magnetic force only act on moving charges? I don't understand why the magnetic force only acts on moving charges. When I have a permanent magnet and place another magnet inside its field, they clearly act as forces onto one another with them both being stationary. Also, I am clearly misunderstanding something.
| After reading some of the other answers I think there is a simpler more fundamental explanation.
According to Field Theory, rather than considering the interactions between particles and field, we can consider the interactions between the fields associated to the particles.
Under this understanding, Classical gravitation $F = G \frac{m_1 m_2}{r^2}$ is the interaction between 2 objects, each with it's own gravitational field. Similar Coulomb force $F = k\frac{q_1 q_2}{r^2}$ is the interaction between 2 charged particles, each with it's own electrical field.
Magnetic interactions are a bit more complicated, but we can also understand magnetism as interaction between objects that each have their own magnetic field.
Now to your question.
A static electric charge generates an electric field, but not a magnetic field, as per the relevant Maxwell equation. $\vec\nabla\cdot \vec E = \frac{\rho}{\epsilon_0}$.
A moving charge has a magnetic field, given by $\vec\nabla\times \vec B = \mu_0 ( \vec J + \epsilon_0 \frac{\partial \vec E}{\partial t})$. in which the current density $J = q\cdot \vec v$
Therefore a magnetic field does interact with a moving charge, because the moving charge also has a magnetic field, but not with a static charge, because the static charge does not have a magnetic field.
| {
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Why is it not advisable to jump onto the shore from a boat? Just to clarify, I have done this several times without facing any consequences.
But I have heard that we are not recommended to do so. Why? Is it because the boat may move backwards (water is fluid) and we will lose balance and fall into the water? Does this also explain why the boatman ties the boat before allowing the passengers to deboard?
My Theory: I think when I exert force on the boat in the backwards direction, the boat pushes me in the forward direction and I am able to jump onto the pier (Newton's third law).
| I think the problem is that the boat recoils: when jumping from a hard surface one's force goes fully into accelerating the one's body, which determines how far one jumps/lands. When jumping from a boat, one's force is expended on accelerating oneself and the boat, which moves in the opposite direction. In other words, one's intuition (based on jumping from the hard surface) seemingly predicting how far one lands from the point of the jump, actually tells us how far one lands from the boat (which moved the other way.) This makes it easy to miscalculate the length of the jump with various unpleasant consequences.
| {
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What is happening at the particle level in the Bernoulli Principle? One might think that increasing the speed of particles would increase pressure -- if I understand what the Principle states, it is very counterintuitive.
My guess is, the pressure has something to do with particles moving perpendicular to the fast motion and maybe the particles spend less time hitting the sides of the pipe but there are also more particles per unit time.
I have some other ideas but the above guess is the kind of explanation I am looking for.
| The Bernoulli principle comes from the second law of Newton. If we take a slice of fluid with cross section $A$ of a horizontal pipe, and with length $\Delta x$, the net force acting on it is:$$\Delta F = -\Delta P A$$ The minus sign indicates that the pressure must be decreasing along $x$, for a positive force on the slice. The mass is $\Delta m = \rho A \Delta x = \rho A v \Delta t$. The acceleration of the slice is $a = \frac{dv}{dt}$. When the deltas go to zero, $$dF = (dm) a \implies -dpA = \rho A v dv \implies -dP = d\left(\frac{1}{2}\rho v^2\right)$$
That means: an increase in velocity requires a decrease of pressure if the density is constant.
By the way, it has nothing to do with movement of particles, at least for liquids. The pressure is related to the repulsion of the electronic shell of the molecules.
| {
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Why is electromotive force in magnetohydrodynamics a vector quantity? In the mean-field dynamo theory in magnetohydrodynamics, I frequently came across a quantity;
$\langle v'\times B' \rangle$, which is termed as the mean electromotive force. I want to know that why is it termed as electromotive force, if it is a vector.
Everywhere else I have seen emf is just the potential difference and hence a scalar. Is this emf different than the emf used in mean-field dynamo theory?
| $\left\langle \mathbf{v}' \times \mathbf{B}' \right\rangle$ has dimensions of electric field, rather than potential. Therefore, it is different from the standard definition of electromotive force. In a highly conductive fluid it would be equal to $-\left\langle \mathbf{E}' \right\rangle$ (by Ohm's law). It could be considered the electromotive force per unit length in the direction parallel to the vector resulting from the motion of the fluid.
| {
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Uncertainty Calculation: Applying Product Rule instead of Power Rule I use $\delta$ to represent absolute uncertainty. The power rule for the calculation of relative uncertainty in $t^2$ is
$$\frac{\delta (t^2)}{(t^2)}=2\left(\frac{\delta t}{t}\right).$$
But if I treat the power as a product and apply the product rule, I have
$$\frac{\delta (t \times t)}{(t \times t)} = \sqrt{\left(\frac{\delta t}{t}\right)^2 + \left(\frac{\delta t}{t}\right)^2} = \sqrt{2\left(\frac{\delta t}{t}\right)^2} = \sqrt{2}\left(\frac{\delta t}{t}\right).$$
Am I making a mistake? If not, how is this inconsistency reconciled?
| The product rule assumes that the things being multiplied vary independently of one another, which is clearly not the case when multiplying something by itself. As such, the power rule is the correct one here.
| {
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Why do some objects tend to sink after some time in water even if they float at the start? I have observed this phenomenon in swimming pools: I have seen many dead insects floating on the surface, but after some time some they tend to sink down without any external influence. Why does this exactly happen? It even happens with paper: When it is fully immersed in water, after some time, it overcomes surface tension and buoyant force and sinks down. Do paper/insects gain more density?
| There are different mechanisms that can possibly be relevant here:
*
*The object is floating because its less dense than water on account of enclosed or separated air (or other gases).
*
*Due to contact with water it slowly fills up with water making it more dense and thus sinking.
Typical examples are a sponge or a boat with a small hole.
*Due to random interactions such as wind, the enclosed air gets reduced.
A typical example would be a boat which gets filled by waves.
*Due to decay processes the enclosure breaks.
A typical example would be a boat which rots until it just breaks.
*The object is floating on account of surface tension.
Typical examples are water striders.
Here, a slow wetting of the object or random interactions (e.g. due to waves and wind) can destroy the effect.
You can experience this yourself with the classical experiment with the floating paper clip.
Here, surface tension is the only thing keeping the clip afloat.
Usually, the clip will sink at some point – which is when the surface tension “breaks”.
In contrast to the above effects, this happens suddenly rather than gradually.
Often you have a combination of several mechanisms, e.g., an object that requires some enclosed air and surface tension to float.
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Invariant nature of mass and particle annihilation Since mass is a Lorentz invariant, it can never change to preserve the vectorial nature of the four-momentum and the other four vectors. Thus the only interpretation of the energy-mass equation that I can give is that in
$$E^2=m^2+p^2$$
the first term, $m^2$ is a constant energy, that can never change, but can affect and can be affected by gravity, whereas the second term of $p^2$ can change. But if we write this expression as
$$E=\gamma m$$
Then some say that mass changes, in the sense that energy and mass are interchangeable, but what changes, is of course the $\gamma$ and not $m$.
In accordance with all this, how do particle annihilation and creation fit in the picture? Since annihilation means a change in mass, how is mass invariant?
I have no knowledge of quantum theory but from a lecture that I watched on special relativity, the professor said that the conflict solved by QFT was that in QM the probability of the particle existing should always be 1, whereas in SR particles can be annihilated. So it seems to me that the particle annihilation and invariant nature of the mass can be understood directly from SR and hence the question.
|
Since annihilation means a change in mass, how is mass invariant?
In this context, mass being invariant means it is Lorentz invariant, which means it is a concept that has the same numerical value in all inertial frames of reference. It does not mean it cannot change in time. If a small part of a rocket (judging by mass) leaves the rocket, and by rocket we mean the mass that moves together with the tip of the rocket, then mass of the rocket decreased, while remaining Lorentz invariant at all times.
Annihilation does not necessarily mean change in mass. When the pair electron-positron disappears, EM radiation is created. Mass of this radiation, defined by
$$
m_{rad}^2 = E_{rad}^2 - p_{rad}^2,
$$
is the same as mass of the system just before the transformation. This however is somewhat less than $2m_e$, due to negative energy of interaction in the system.
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What is sound and how is it produced? I've been using the term "sound" all my life, but I really have no clue as to what sound exactly is or how it is created. What is sound? How is it produced? Can it be measured?
| Sound is basically produced by vibrating by either blowing through something like a flute or a recorder, hitting something like a drum, shaking something like maracas, or strumming something like a guitar when you do all sorts of these things the air goes through the instrument and hits each side of it to make it vibrate.
For example, when you talk, the oxygen goes through your larynx and makes it vibrate. Try this by talking and putting your hand on your larynx, and you can feel that it vibrate.
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Where do magnets get the energy to repel? If I separate two magnets whose opposite poles are facing, I am adding energy. If I let go of the magnets, then presumably the energy that I added is used to move the magnets together again.
However, if I start with two separated magnets (with like poles facing), as I move them together, they repel each other. They must be using energy to counteract the force that I'm applying.
Where does this energy come from?
| As is said in a comment, the reasoning in the first paragraph is correct but the one in the second paragraph is wrong.
If you apply a force on something without "moving" the work is null and there is no energy exchange involved (this is not the same thing than doing that with your muscles, but that's another story :p). ( Work = integral[a to b] of F dot dx ; so Work = 0 if there is not "circulation").
Thus the magnets do not need any energy to statically counteract the force.
However, if you do move the magnets, then you need to give some energy. This energy is stored in the system because you cause a variation of magnetic flux: magnet 1 moving induce a variation of flux seen by magnet 2, and this will change the state of magnet 2, increasing its potential energy.
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Mnemonics to remember various properties of materials I'm trying to figure out how to remember that
*
*hardness: how resistant it is to deformation
*toughness: how resistant it is to brittle failures
*stress: force on a surface area
*strength: ability to withstand stress without failure
*strain: measurement of deformation of a material
Does anyone know of a mnemonic or easy way? I only know these from googling them, and I'm finding it tricky to remember them.
| I always used to confuse stress and strain: most of my mnemonics involved making words out of initial letters.
When you're stressed, you show the strain.
Stress is what is applied to the material, strain is what it does in response - I always used to get these the wrong way around.
E equals Fl/ea
Young's Modulus = (force × length) / (extension × area)
Good luck with the others: I suggest imagining a hard man (grizzled veteran) who is actually secretly limp wristed and camp (he bends unlike hard materials), a tough guy (showy, probably with twin pistols) who literally goes to pieces in difficult situations (imagine his brittle bones snapping) and a strong man (lifting a dumbbell) who is crushed to a pancake by the weight.
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Is Newton's Law of Gravity consistent with General Relativity? By 'Newton's Law of Gravity', I am referring to
The magnitude of the force of gravity is proportional to the product of the mass of the two objects and inversely proportional to their distance squared.
Does this law of attraction still hold under General Relativity's Tensor Equations?
I don't really know enough about mathematics to be able to solve any of Einstein's field equations, but does Newton's basic law of the magnitude of attraction still hold?
If they are only approximations, what causes them to differ?
| May be the case that Gerber could not give an exact explanation for his formula, 18 years before GR, on the advance of Mercury's perihelium as we can see at mathpages. After reading the fine explanation on Lienard & Wiechert retarded potentials in the Hans de Vries online book I think that the treatment of the subject is not correct in the mathpages.
It appears to me that Walter Orlov, 2011 has a nice way to explain why Gerber's formula is correct to explain Mercury's orbit.
The answer is that they are mutually consistent because Gerber'gravity (post-Newtonian treatment with delayed potentials) is consistent with observations, the same as with GR's formulation.
Before I can ask 'Do I need GR to explain the observations?' I need to be sure that Orlov got it right.
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Book about classical mechanics I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical consideration related to these to formalism (like tangent bundle, cotangent bundle, 1-form, 2-form, etc.).
I have this book from Saletan and Jose, but I would like to go into more details about the [symplectic] geometrical and mathematical foundations of classical mechanics.
Additional note: A chapter about relativistic Hamiltonian dynamics would be a good thing.
| I can't believe nobody's mentioned Arnol'd's book "Mathematical Methods for Classical Mechanics" - it covers everything you ask for in the first paragraph quite elegantly (though sometimes somewhat tersely).
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Law for tap water temperature I was wondering if anyone put together a law to describe the rising temperature of the water coming out of a tap.
The setup is fairly simple: there's a water tank at temperature T, a metal tube of length L connected to it and a tap at the end where temperature is measured. The water flows at P l/s.
Given that the metal tube is at room temperature initially, what law describes the temperature of the water at any instant? What is the limit temperature of the water?
Thanks.
| We can consider the following model: a tube of constant temperature $T_e$ of lenght L, radius $r$ where water is flowing uniformly at a speed $v$ (that you can obtain from your flow $P$).
A "slice" of water travels an interval $dx$ in a duration $dt = \frac{dx}{v}$.
The tube will contribute to the "heating" of the water by $\frac{dQ}{dt} = (T-T_e) k 2 \pi r dx$ where $k$ is the conductivity and where we use a very simple model (in particular for the radius, we do not distinguish external and internal radii).
During this interval the temperature $T(x)$ of the water will vary by $dT = -\frac{dQ}{c \rho dV}$ where $C$ is the heat capacity at constant pressure of water, and where $dV = 2 \pi r dx$.
Replacing we have $\frac{dT}{T-T_e}=-\frac{k}{\rho C v} dx$ whose solution, if the temperature in the tank (ie x = 0) is $T_t$ :
$T(x) = (T_t - T_e) e^{(-\alpha x)}+T_e$ where $\alpha = \frac{k}{\rho C v}$.
Depending on the lenght of the tube you have the temperature at the tap.
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Why can you "suck in" cooked spaghetti? We all know that there is no "sucking", only pushing. So how are cooked spaghetti pushed into your mouth? The air pressure applies orthogonal on the spaghetti surface. Where does the component directed into your mouth come from?
| The component directed into your mouth comes form the different pressure between the outside and the inside of the mouth. If you create a difference in pressure of $\Delta P$ the force pushing the spaghetti in will be $\Delta P \cdot S$ where $S$ is the section of the spaghetto (or spaghetti.. depending how hungry you are;) )
If for example you create inside your mouth a void with a 10% efficiecy the difference in pressure will be:
$\Delta P\approx 1atm - 0.1atm= 10^4N/m^2$
If the cross section of your single spaghetto is $1mm^2$ the force pushing the food in your mouth will be 10 N: almost a kilogram!
You won't even need to be as efficient as 10% ;)
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Finding the volume of this irregular shape I have I have an approximately basketball-sized non-hollow piece of aluminum sitting in my house that is of irregular shape. I need to find the volume of it for a very legitimate yet irrelevant reason.
What is the best way I can do this? In fact, what are all the ways I could feasibly do this without going to a lab? (I don't live near any labs)
| Since the object is basketball size, it would displace a significant volume and weight of water when submerged. Weigh a container with some water and take a reading. Then attach a wire to the object, suspend it in the container of water till it's submerged without touching the container, and take a second reading. The difference between the two readings is the water weight that is displaced by the object. Volume (m^3) = weight difference (kg) / 1000 (kg/m^3).
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Why don't spinning tops fall over? One topic which was covered in university, but which I never understood, is how a spinning top "magically" resists the force of gravity. The conservation of energy explanations make sense, but I don't believe that they provide as much insight as a mechanical explanation would.
The hyperphysics link Cedric provided looks similar to a diagram that I saw in my physics textbook. This diagram illustrates precession nicely, but doesn't explain why the top doesn't fall. Since the angular acceleration is always tangential, I would expect that the top should spiral outwards until it falls to the ground. However, the diagram seems to indicate that the top should be precessing in a circle, not a spiral. Another reason I am not satisfied with this explanation is that the calculation is apparently limited to situations where: "the spin angular velocity $\omega$ is much greater than the precession angular velocity $\omega_P$". The calculation gives no explanation of why this is not the case.
| The quick answer is that, for the top to fall over due to gravity, each fragment of the top that is moving around the spin axis has to change its individual direction of movement.
They are already changing direction around the spin axis, due the rigidity of the top keeping them moving in a circle. But gravity is operating at 90º to their direction of movement, and its effect depends on the velocity or inertia of the fragment. For a fast rotating top, this slight change of direction is what causes the top’s precession. And as it slows down, the effect of gravity has more effect, and it falls over.
It’s a similar situation to changing the orbit of a satellite with side thrusters.
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Why does water make a sound when it is disturbed? When I disturb a body of water, what causes the familiar "water moving" sound?
| The major source of sound when water is disturbed is the creation of underwater air bubbles, which oscillate in shape and size, producing damped sinusoid sound waves. The resonant frequency of the bubble depends on its size, so many bubbles of different sizes and different resonant frequencies produce the "burbling" sound that we associate with water being disturbed.
*
*Bubble Resonance
*The Impact of Drops on Liquid Surfaces and the Underwater Noise of Rain
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Planet orbits: what's the difference between gravity and centripetal force? My physics teacher says that centripetal force is caused by gravity. I'm not entirely sure how this works? How can force cause another in space (ie where there's nothing).
My astronomy teacher says that gravity is (note: not like) a 3D blanket and when you put mass on it, the mass causes a dip/dent in the blanket and so if you put another object with less mass it will roll down the dip onto the bigger mass. Is this true and is this what causes the centripetal force.
| I think a key element that this student is missing is actually a very common misconception about centripetal force. We hear of various types of forces, like normal forces, frictional forces, and gravitational forces, and then we tack on "centripetal force" as if it was another type of force like that. But it's not-- those other forces are real forces with their own force laws and their own behaviors. Centripetal force is not so much a name for a force at is a name for the mass times acceleration of an object moving in a circle. So, centripetal force is not a kind of force, it is a net force that other forces have to add up to-- if and only if you already know the acceleration is that of an object moving in a circle. So for an orbiting body with only gravity on it, gravity must be the centripetal force, but if a body is orbiting with both gravity on it, and a stretched rubber band going around as well, then the force of gravity plus the force of the rubber band will be the centripetal force. In neither case is the centripetal force a type of force of its own. (Oops, just noticed the date on the question, perhaps this will be of help to someone else.)
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Home experiments to derive the speed of light? Are there any experiments I can do to derive the speed of light with only common household tools?
| I can't think of a way to do it with "common household tools" but if you have an oscilloscope, a laser diode, a couple of photo-sensors, a beam splitter, you can do it. All of these things are readily available from science supply/hobby stores online, but not usually in most homes.
Set up the laser diode to hit the beam splitter and be split into two beams. Set up the two beams so that they hit two photo-sensors, but make one of the photo-sensors exactly twice the distance from the beam splitter as the other. This will create two separate paths for the light, one twice as long as the other. Run the output of the photo-diodes into two channels of the oscilloscope. Switch on the laser diode, and you should see two pulses on the o-scope, one from each of the two laser diodes. The difference between them is the time it takes the light beam to travel the distance of the difference in the two paths.
The reason to do it this way is accuracy - if you only had one beam, and your photo-diode took, say, 1 microsecond longer to turn on than what was in the documentation, or your laser were slow to turn on, then you would get very inaccurate results. But with two beams, those errors cancel each other out, and so all you're left with is the time of the light.
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Advantages of high-energy heavy-ion collisions over proton-proton collisions? Some high-energy experiments (RHIC, LHC) use ion-ion collisions instead of proton-proton collisions. Although the total center-of-mass energy is indeed higher than p-p collisions, it might happen that the total energy per nucleon is actually lower. What are the advantages of using ion-ion collisions (e.g. gold-gold or lead-lead) instead of proton-proton collisions, considering the same accelerator?
| The main goal of accelerating heavy ions in the LHC is to produce the quark gluon plasma as a result of large energy concentration in a small volume in case many nucleons of the two nuclei interact producing a collective effect of hadronig heating to a temperaturę of milliards degrees which melts all hadrons. In heavuy ion collisions even if energy per nucleon is smaller the total energy released in case of many nucleons interactions makes these collisions so different from the standard proton proton collisions.
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Would a magnet attract a paperclip indefinitely? Let's say we have a magnet stuck to a metal bar, suspended above the ground. If I attach a paperclip to the magnet, where is the energy to hold the paperclip coming from (against the force of gravity), and for how long will the paperclip remain there - will it remain there forever?
| The paperclip is not moving relative to earth - this means that no energy is being spent by magnet to hold it, so it can hold the paperclip as long as the magnet has magnetic properties.
As a contrast, when you hold something with your hands, energy is being spent by your muscles not to hold the object, but to remain contracted against gravity. This is because natural state of muscle cells is to be stretched or contracted as gravity tells them to, not against it.
By analogy, electromagnet spends energy not to hold things, but to to have magnetic properties, because its natural state is not magnetic. So natural magnet does not spend any energy, since its natural state is being magnetic.
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Why is it thought that normal physics doesn't exist inside the event horizon of a black hole? A black hole is so dense that a sphere around it called the event horizon has a greater escape velocity than the speed of light, making it black. So why do astronomers think that there is anything weird (or lack of anything Inc space) inside the event horizon. Why isn't simple the limit to where light can escape and in the middle of event horizon (which physically isnt a surface) is just a hyper dense ball of the matter that's been sucked in and can't escape just like light. Why is it thought that the laws of physics don't exist in the event horizon?
| For large enough black holes, space is still weakly curved at the event horizon, so of course we should expect that normal physics still exists there. An infalling observer wouldn't experience anything out of the ordinary when crossing an event horizon.
What is true is that for an outside observer, it's impossible to probe what's happening inside the event horizon of a black hole. (The best you can do is wait a long time and collect the outgoing Hawking radiation.) So from the point of view of such an observer, you can't really tell the difference between living in a world where spacetime keeps going across the horizon, or living in a world where space just ends there and some radiation emerges. This might be the sort of idea your teacher was getting at. You might want to look up "black hole complementarity" to learn more.
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Notation of plane waves Consider a monochromatic plane wave (I am using bold to represent vectors)
$$ \mathbf{E}(\mathbf{r},t) = \mathbf{E}_0(\mathbf{r})e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}, $$
$$ \mathbf{B}(\mathbf{r},t) = \mathbf{B}_0(\mathbf{r})e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}. $$
There are a few ways to simplify this notation. We can use the complex field
$$ \tilde{\mathbf{E}}(\mathbf{r},t) = \tilde{\mathbf{E}}_0 e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)} $$
to represent both the electric and magnetic field, where the real part is the electric and the imaginary part is proportional to the magnetic. Often it is useful to just deal with the complex amplitude ($\tilde{\mathbf{E}}_0$) when adding or manipulating fields.
However, when you want to coherently add two waves with the same frequency but different propagation directions, you need to take the spatial variation into account, although you can still leave off the time variation. So you are dealing with this quantity:
$$ \tilde{\mathbf{E}}_0 e^{i\mathbf{k} \cdot \mathbf{r}} $$
My question is, what is this quantity called? I've been thinking time-averaged complex field, but then again, it's not really time-averaged, is it? Time-independent? Also, what is its notation? $\langle\tilde{\mathbf{E}}\rangle$?
| I'd call it the initial complex field since it's {E(t=0)}. No reference though.
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Why does kinetic energy increase quadratically, not linearly, with speed? As Wikipedia says:
[...] the kinetic energy of a non-rotating object of mass $m$ traveling at a speed $v$ is $\frac{1}{2}mv^2$.
Why does this not increase linearly with speed? Why does it take so much more energy to go from $1\ \mathrm{m/s}$ to $2\ \mathrm{m/s}$ than it does to go from $0\ \mathrm{m/s}$ to $1\ \mathrm{m/s}$?
My intuition is wrong here, please help it out!
| Throw 3 balls up with equal weight with no air friction.
The ball have up velocity
$$V_{1}$$
$$V_{2}$$
$$V_{3}$$
And let's for simplicity sake we have
$$V_{2}=2V_{1}$$
$$V_{3}=3V_{1}$$
Let's for simplicity sake ball 1 travels
$$S$$
And stop after time
$$T$$
How far will ball 2 travel?
Notice that $$V_{2}=2V_{1}$$
Also deceleration is constant.
So ball 2 will requires
$$2T$$ to deplete all it's velocity.
However the average speed of ball 2 will also double. So ball 2 will travel 4 times higher.
For the same reason, ball 3 will require
$$3T$$
And the average speed will also be triple.
So ball 3 will go up 9 times higher.
So amount of kinetic energy of the balls will be quadratic amount of the velocity.
In fact, we can measure it right now.
Total potential energy will be
$$mgh$$
The average up velocity will be
$$V_{average}=\frac{1}{2}{V_{0}}$$
Here, $$V_{0}$$ is the initial velocity.
So $$h$$ will simply be time to stop times average speed. Time to stop is simply
$$T=\frac{V_{0}}{g}$$
So,
$$h=T V_{average}$$
$$=\frac{V_{0}}{g} \frac{1}{2}{V_{0}}$$
Total kinetic energy of ball moving upward with speed $$V_{0}$$
will simply be the potential energy of the ball when it stops.
So it is
$$E_{potential when stop}=E_{kinetic energy at start}$$
$$mgh=m\frac{V_{0}}{g} \frac{1}{2}{V_{0}}$$
$$mgh=\frac{1}{2}m{V_{0}^{2}}$$
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Material resistency to lasers beam Keeping the average power constant, why some materials are more eager to be damaged by pulsed laser with respect of C.W. lasers, or viceversa?
When i talk about pulsed lasers i think for examples of duty cycles in the order of $10^5$.
For example optical elements (such as a vortex phase plate for donut-shaping the beam) have different tolleration regimes regarding the incident power not simple dependent on the average power but also on the peak power for pulsed beam.
| One thing that ought to matter is how much laser light gets reflected versus how much gets absorbed and transmitted.
I don't know much about this, but my naïve guess is that materials get damaged by lasers primarily because they are heated by the intensity of the light, and then they melt (or burn!). Therefore thermal conductivity or response properties of the material will come into play as well. For instance, if heat conducts well in a material, the local heating of the material by the laser will be able to diffuse to a larger region.
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Equilibrium and movement of a cylinder with asymmetric mass centre on an inclined plane A cylinder whose cross section is represented below is placed on an inclined plane. I would like to determine the maximum slope of the inclined plane so that the cylinder does not roll. The mass centre (CM) of the cylinder is at a distance r from the central axis. The cylinder consists of a cylindrical shell with mass $m_1$ and a smaller cylinder with mass $m_2$ placed away from the axis and rigidly attached to the larger cylinder. What is the influence of friction? Is it possible to establish the law of the movement? I think that the piece may roll upwards until it stops.
The figure was copied from Projecto Ciência na Bagagem -- Cilindro desobediente
EDIT: Depending on the initial conditions is it possible to find the highest point the cylinder rolls to, before stopping?
EDIT2: From Institute and Museum of the History of Science -- Cylinder on inclined plane [another cylinder]
"When placed on the inclined plane,
[another] cylinder tends to
roll upward, coming to a halt at a
well-determined position."
| The effect of friction is to make the cylinder roll down the ramp rather than slide.
To find an equilibrium angle, use virtual work.
If $\phi$ changes by a small amount $d\phi$, as the cylinder rolls, then everything goes down a little (neglecting at first the small interior cylinder's upward movement) because you're moving down the ramp. You move $R d\phi$ down the ramp, and lose elevation $\sin \Phi R d\phi$. The total work done by gravity is $(m_1 + m_2) g \sin\Phi R d\phi$
On the other hand, the interior cylinder rises with respect to the center of the big cylinder by an amount $(R - R_2) d\phi$. The work done by gravity on the little cylinder is $g m_2 (R-R_2) d\phi$.
Equilibrium is achieved when these are equal, so
$m_2 (R - R_2) = (m_1 + m_2) R \sin\Phi$
or
$\sin\Phi = \frac{m_2(R-R_2)}{(m_1+m_2)R} = \frac{r}{R}$
| {
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Evolution in the interpretation of the Dirac equation As I understand, Dirac equation was first interpreted as a wave equation following the ideas of non relativistic quantum mechanics, but this lead to different problems.
The equation was then reinterpreted as a field equation and it is now a crucial part of quantum field theory.
My question is: could you provide me a reference (paper, book) that explains this evolution, including the different historical steps, etc. ?
I have a good knowledge of QM and I studying field theories, but I would like to have a clearer view on this historical evolution.
| Although not directly relevant to your question, it is helpful to compare the different editions of Dirac's Principia. He revised his treatment of QED every time, and such an evolution must shed light on the evolution of the Dirac eq. from a one-particle eq. to its current status as a QFT equation. I cannot quite recommend any of Greiner's books, although i myself consult them all the time, I have the distinct impression ..... there is a kind of vagueness about his style which contrasts badly with the styles of the other texts that the posters here have recommended. I fear that that is a symptom of something.
| {
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Suggested reading for renormalization (not only in QFT) What papers/books/reviews can you suggest to learn what Renormalization "really" is?
Standard QFT textbooks are usually computation-heavy and provide little physical insight in this regard - after my QFT course, I was left with the impression that Renormalization is just a technical, somewhat arbitrary trick (justified by experience) to get rid of divergences. However, the appearance of Renormalization in other fields of physics Renormalization Group approach in statistical physics etc.), where its necessity and effectiveness have, more or less, clear physical meaning, suggests a general concept beyond the mere "shut up and calculate" ad-hoc gadget it is served as in usual QFT courses.
I'm especially interested in texts providing some unifying insight about renormalization in QFT, statistical physics or pure mathematics.
| Regarding "providing unifying insight about renormalization in QFT, statistical physics or pure mathematics", this is what I tried to do in my detailed answer to Wilsonian definition of renormalizability
| {
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Books that every physicist should read Inspired by How should a physics student study mathematics? and in the same vein as Best books for mathematical background?, although in a more general fashion, I'd like to know if anyone is interested in doing a list of the books 'par excellence' for a physicist.
In spite of the frivolous nature of this post, I think it can be a valuable resource.
For example:
Course of Theoretical Physics - L.D. Landau, E.M. Lifshitz.
Mathematical Methods of Physics - Mathews, Walker. Very nice chapter on complex variables and evaluation of integrals, presenting must-know tricks to solve non-trivial problems. Also contains an introduction to groups and group representations with physical applications.
Mathematics of Classical and Quantum Physics - Byron and Fuller.
Topics in Algebra - I. N. Herstein. Extremely well written, introduce basic concepts in groups, rings, vector spaces, fields and linear transformations. Concepts are motivated and a nice set of problems accompany each chapter (some of them quite challenging).
Partial Differential Equations in Physics - Arnold Sommerfeld. Although a bit dated, very clear explanations. First chapter on Fourier Series is enlightening. The ratio interesting information/page is extremely large. Contains discussions on types of differential equations, integral equations, boundary value problems, special functions and eigenfunctions.
| What is life? E. Schrodinger
The origin of life. F. Dyson
How nature works. P. Bak
Because physics is not only particle physics.
And the trilogy of Weinberg, that with the Landau course forms the holy bible of theoretical physicist.
| {
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Relativistic Cellular Automata Cellular automata provide interesting models of physics: Google Scholar gives more than 25,000 results when searching for "cellular automata" physics.
Google Scholar still gives more than 2.000 results when searching for "quantum cellular automata".
But it gives only 1 (one!) result when searching for "relativistic cellular automata", i.e. cellular automata with a (discrete) Minkoswki space-time instead of an Euclidean one.
How can this be understood?
Why does the concept of QCA seem more
promising than that of RCA?
Are there conceptual or technical barriers for a thorough treatment of RCA?
| Some cellular automata, like the basic rule 110 are universal, i.e., Turing complete. What this means is that you can simulate/emulate any mathematics on them, including any physical theory, including non-local ones. Many people make the mistake of thinking that because cellular automata have local and discrete rules they are limited to simulate only those behaviours, however, the correct point of view is to consider that a cellular automaton can simulate anything a regular computer can do, the software is on the initial conditions. The rules are strong enough to work as a microprocessor. You could ask if there is any physics that a computer simulations cannot grasp. But relativity is not on of them, at the most, people use to argue if QM is emulable with a cellular automata, with many thinking it is not. My personal believe is that using Bohm theory you can even simulate quantum mechanics at any level you want (shower of negative votes expected).
| {
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Lightning strikes the Ocean I'm swimming in - what happens? I'm swimming in the ocean and there's a thunderstorm. Lightning bolts hit ships around me. Should I get out of the water?
| Most probably the current just spreads in all directions and weakens quite fast (at least like $r^{-2}$, not counting resistance), so I don't think the hazard is much (in magnitude) larger than on land in similar conditions.
EDIT: In what I found in Internet salty water has only 10 times better conductivity than wet soil; yet on land the wet soil layer lays on insulating layer of dry soil, so the current is directed to rather "flood" than penetrate.
| {
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How relevant is LHC to quantum gravity? Premise: the LHC is obviously mapping unseen territory in high energies, and therefore it's always possible to imagine far out results.
Excluding completely unexpected outcomes - is the LHC performing any experiment that could help with string theory or m-theory? For example:
*
*direct super-strings or m-theory predictions to be tested or confuted
but also
*
*measurements that would help "shape" string/m-theory into something more concretely testable or practical than the current blurry incarnation?
| Well, there's no reason to believe in supersymmetry, beyond some theoretical niceness to it, so if they see THAT at the LHC, then string theory gets a big boost, as there is no way other than supersymmetry to produce fermions in string theory.
The other thing that might be relevant to quantum gravity is that if there are large extra dimensions (as in, large compared to the Planck length, but smaller than detectable by things like the Cavendish experiment). If that is the case, then the 'fundamental' gravitational constant may be much larger than Newton's constant (they differ by a factor of the volume of the large extra dimensions), and quantum gravitational effects would be accessible at the LHC.
| {
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Common false beliefs in Physics Well, in Mathematics there are somethings, which appear true but they aren't true. Naive students often get fooled by these results.
Let me consider a very simple example. As a child one learns this formula $$(a+b)^{2} =a^{2}+ 2 \cdot a \cdot b + b^{2}$$ But as one mature's he applies this same formula for Matrices. That is given any two $n \times n$ square matrices, one believes that this result is true: $$(A+B)^{2} = A^{2} + 2 \cdot A \cdot B +B^{2}$$ But eventually this is false as Matrices aren't necessarily commutative.
I would like to know whether there any such things happening with physics students as well. My motivation came from the following MO thread, which many of you might take a look into:
*
*https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics
| "Summer is when the Earth is closest to the sun, and winter is when it's furthest away."
It's true that the Earth's orbit is slightly elliptical, but the effect of this, as far as seasons, is very small. For one thing, this wouldn't explain why the sun rises and sets at different times in different seasons, and if this were true, the whole planet would have summer at the same time.
The seasons are actually caused by the tilt of the Earth relative to its orbit around the Sun.
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How do contact lenses work? I understand how telescope, microscope and glasses work.
But how do contact lenses work?
| Your question is really "how does the human eye work?", since the contact lens is designed to adjust the optics of the human eye.
This image from the wikipedia article on the anatomical lens shows how the cornea and lens focuses incoming light from the left onto the retina (right).
Previously, I'd written that the biological lens in the eye was responsible for the optics; however, as pointed out in the comment of David White below, the cornea actually performs most of the focusing; according to wikipedia, it contributes about 2/3 of the focusing power. Most of the rest is due to the lens, which also performs an important role in the process of accomodation, which is how we change our focus between objects at different distances.
For people who are nearsighted or farsighted, the light coming into the eye does not end up in focus. See this picture for the case of nearsightedness:
You should imagine the contact lens being placed over the cornea (left surface) and causing the rays to adjust so that the image ends up in focus.
| {
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Home experiment to estimate Avogadro's number? How to get an approximation of Avogadro or Boltzmann constant through experimental means accessible by an hobbyist ?
| Electrolysis. Run a current through a weak acid, and measure the current going in and coming out. Hydrogen ions in the acid will capture electrons and bond to each other to form hydrogen gas. If you accept a measured value for the charge of an electron, you can find the number of hydrogen molecules liberated. Then measure the volume of hydrogen formed at STP. Avogadro's number is the number of molecules in a gas at STP with a volume of 2.2*10^-2 m^3.
| {
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What's the difference between helicity and chirality? When a particle spins in the same direction as its momentum, it has right helicity, and left helicity otherwise. Neutrinos, however, have some kind of inherent helicity called chirality. But they can have either helicity. How is chirality different from helicity?
| At first glance, chirality and helicity seem to have no relationship to each other. Helicity, as you said, is whether the spin is aligned or anti aligned with the momentum. Chirality is like your left hand versus your right hand. Its just a property that makes them different than each other, but in a way that is reversed through a mirror imaging - your left hand looks just like your right hand if you look at it in a mirror and vice-versa.
If you do out the math though, you find out that they are linked. Helicity is not an inherent property of a particle because of relativity. Suppose you have some massive particle with spin. In one frame the momentum could be aligned with the spin, but you could just boost to a frame where the momentum was pointing the other direction (boost meaning looking from a frame moving with respect to the original frame). But if the particle is massless, it will travel at the speed of light, and so you can't boost past it. So you can't flip its helicity by changing frames. In this case, if it is "chiral right-handed", it will have right-handed helicity. If it is "chiral left-handed", it will have left-handed helicity.
So chirality in the end has something to do with the natural helicity in the massless limit.
Note that chirality is not just a property of neutrinos. It is important for neutrinos because it is not known whether both chiralities exist. It is possible that only left-handed neutrinos (and only right-handed antineutrinos) exist.
| {
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What is the difference between "kinematics" and "dynamics"? I have noticed that authors in the literature sometimes divide characteristics of some phenomenon into "kinematics" and "dynamics".
I first encountered this in Jackson's E&M book, where, in section 7.3 of the third edition, he writes, on the reflection and refraction of waves at a plane interface:
*
*Kinematic properties:
(a) Angle of reflection equals angle of incidence
(b) Snell's law
*Dynamic properties
(a) Intensities of reflected and refracted radiation
(b) Phase changes and polarization
But this is by no means the only example. A quick Google search reveals "dynamic and kinematic viscosity," "kinematic and dynamic performance," "fully dynamic and kinematic voronoi diagrams," "kinematic and reduced-dynamic precise orbit determination," and many other occurrences of this distinction.
What is the real distinction between kinematics and dynamics?
| Primarily, the distinction between kinematics and dynamics is one of causation. What do we mean by this? The etymology of the word kinematics is the Greek kinēma, which means motion. On the other hand, dynamics draws its origin from dunamis, meaning power (though we are better off thinking of it as a power in potentia, as in the ability to do something).
So? Well, we all know another word that is based of the Greek kinēma -- cinema. If we were to represent the trajectory of a particle as a function, as a computer simulation or as a movie, we would be providing a description of the particle's motion without explaining what caused it to move that way. But as physicists, we should not be merely content with painting a picture of a system's motion. That is the job for the artists.
In order to explain why our system of interest exhibits the motion that we observe, we identify forces and potentials that may have been responsible for triggering the motion. In classical mechanics, once we have identified the forces and the initial conditions of the system (the dynamics), we can then solve a differential equation to obtain a solution parameterized in time (the kinematics).
The study of dynamical systems is, after all, a branch of mathematics. I hope this helps. Moreover, I hope that you appreciate that the distinction is not at all arbitrary, as some people have been led to think.
| {
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Are gauge choices in electrodynamics really always possible? If $B$ is magnetic field and $E$ electric Field, then
$$B=\nabla\times A,$$
$$E= -\nabla V+\frac{\partial A}{\partial t}.$$
There is Gauge invariance for the transformation
$$A'\rightarrow A+{\nabla L}$$
$$V'\rightarrow V-\frac{dL}{dt}.$$
Now, we can write:
*
*Coulomb Gauge (CG): the choice of a $L$ that implies $\nabla\cdot A=0$.
*Lorenz Gauge (LG): the choice of a $L$ that implies $\nabla \cdot A - \frac{1}{c^2} \frac{\partial V}{\partial t}=0$.
Now, I'm trying to mathematically prove that it's always possible to find such an $L$ satisfiying $CG$ or $LG$.
| Since, my similar question was closed, I will answer here.
The gauge transformation
$$ \mathbf {A} \rightarrow \mathbf {A} +\nabla \lambda, $$
$$ \varphi \rightarrow \varphi - \frac {\partial \lambda}{\partial t}, $$
(where $ \lambda=\lambda(\vec {r},t) $ is an arbitrary scalar function of coordinates $ \mathbf{r} $ and time $ t $) do not change the form of Maxwell's equations, and hence are admissible from a physical point of view.
In practice, no one chooses a special function $\lambda(\vec{r}, t)$ per se, although one is always implicitly assumed. But the described ambiguity of the potentials from mathematical point of view tells us that one can always be chosen to satisfy one arbitrary additional condition. One, since we can arbitrarily choose only one function $\lambda(\vec{r}, t)$.
For example, one can always choose the field potentials so that the scalar potential $\varphi = 0$ (is equal to zero). To make the vector potential equal to zero, is impossible, since the condition $\mathbf{A} = 0$ is three additional conditions (for the three components of $\mathbf{A}$).
Another possible way is to choose one arbitrary additional is
*
*Coulomb gauge: $\mathrm{div}\mathbf{A}' = 0 $. Where an arbitrary function can be chosen so that it satisfies the condition $\nabla^2 \lambda= -\nabla \cdot \mathbf{A}$. By solving the equation, one can get the following function.
In practice, they proceed as follows: imagine that we know that a given $\mathbf{A}$ solves the Maxwell's Equation. We can always find a gauge transformation that converts it to a new solution $\mathbf{A}'$ that satisfies the Lorentz Gauge condition $$\mathrm{div}\mathbf{A}' = 0. $$ Since we can always do this, rather than imposing this condition after solving Maxwell's Equations, we can require that we are looking for solutions of this type $$\mathrm{div}\mathbf{A} = 0. $$ before we solve that equation.
*For Lorenz gauge $\mathrm{div}\,\mathbf{A}' + {1 \over c^2}{\partial \mathbf{\varphi}' \over \partial t} = 0$, an arbitrary function can be chosen so that it satisfies the condition $\Box^2 \lambda= - \nabla \mathbf{A} - \frac{1}{c^2}\frac{\partial \varphi}{\partial t}$.
Just as a clever choice of coordinates can make solving a problem easier, so
we will find that a clever choice of gauge can make finding solutions easier .
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Doppler's effect use While i was in high-school i learn't the Doppler's Effect which if i remember correctly is:
*
*The Apparent change in the frequency of sound caused due the relative motion between the sound and the observer.
This phenomenon seems obvious, but what i would like to know is, what use does Doppler Effect have in real life. Why is it useful?
| It can be used to measure speed - that's how police radar guns and speed cameras work. Radar waves from the gun/ camera are reflected off the moving vehicle, and the wavelength is shifted according to the speed of the vehicle relative to the gun/ camera.
In astrophysics, looking at light from distant galaxies, we notice that certain characteristics of the light are shifted in wavelength due to the Doppler effect. This is known as red-shift, as we notice the light is mostly shifted to the longer-wavelength (red) end of the spectrum. This tells us that distant galaxies are moving away from us, which is the primary piece of information that led to the development of the Big Bang theory.
| {
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What sustains the rotation of earth's core (faster than surface)? I recently read that the earth's core rotates faster than the surface.
Well, firstly, it's easier to digest the concept of planetary bodies, stars, galaxies in rotation and/or orbital motion.
But, what makes a planet's core rotate? And in the earth's case, faster than its surface?
Secondly, I am aware that the core's rotation is what lends the earth its magnetic field but.. what keeps it going in the first place?
| The earth's rotation is slowing down, due primarily to tidal interaction with the moon (transfering angular momentum to the moon). This frictional loss occurs near the surface. On this basis, if the rotational coupling of the inner core to the mantle is imperfect, it would be expected to lag with respect to the slowdown. I think precessional changes (cycle length roughly 23000 years) are probably more important for generating rotational mistmatches. Then it all gets mixed up with/by
thermal convection and magnetic fields as well.
| {
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Best example of energy-entropy competition? What are the best examples in practical life
of an energy-entropy competition which favors entropy over energy?
My initial thought is a clogged drain -- too unlikely for the
hair/spaghetti to align itself along the pipe -- but this is probably
far from an optimal example. Curious to see what you got. Thanks.
| One of the nicest examples I know is the Kosterlitz-Thouless phase transition in the XY model. What is cool is that the transition is driven by the condensation of vortices which have an energy that diverges logarithmically with the size of the system. You would think they couldn't contribute at all because of this, but it turns out their entropy also diverges in the same way, so the free energy $F=E-TS \propto (c-k_BT) log(R)$ where $c$ is a parameter and $R$ is the size of the system. At sufficiently large $T$ the entropy terms wins and the system undergoes a transition through formation of vortices.
p.s. after rereading the question I realize my answer does not involve "practical life" but I'll leave it anyway.
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Searching books and papers with equations Sometimes I may come up with an equation in mind, so I want to search for the related material. It may be the case that I learn it before but forget the name, or, there is no name for the equation yet. In this case, I may be able to recall a reference book. Searching in Internet can be a fast way though.
However, there are cases that I get some idea and equations (maybe a modified one) myself. So I want to know whether there are any other people working on it before. Because of the vagueness, it is difficult to search it by keywords because I do not know the 'name' of this idea: it is either too board or no results.
I can try to search google scholar, arxiv and maybe the prola, but there is no support of equation pattern matching. For example, entering ∇⋅V in google give you no useful results (it is better if you input the 'divergence of potential').
Is there any good way to search papers in this case? It is even better that I can use the combination of equations and keywords.
Edit: Another reason is that Mathematician should have done some deep analysis on the related mathematical topics. Yet it is difficult to know their results because there is a gap between the terms used in physics and mathematics. It would be really useful if we can find and learn their results.
| You can perform $\LaTeX$ search - that is, write formula in LaTeX in an appropriate search engine:
*
*http://www.latexsearch.com/
However, as one can type the same expression in different ways and with different symbols, I never used it it practice. (Anyone did?)
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Is there a name for the derivative of current with respect to time, or the second derivative of charge with respect to time? This measurement comes up a lot in my E&M class, in regards to inductance and inductors.
Is there really no conventional term for this?
If not, is there some historical reason for this omission?
| It is change of current in unit time. If there is a current, there will be a magnetic field. If there is a change in current, the possibility is an acceleration of charge which leads to the production of electromagnetic waves. So it could be an electric field and/or magnetic field.
| {
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Can I parameterize the state of a quantum system given reduced density matrices describing its subparts? As the simplest example, consider a set of two qubits where the reduced density matrix of each qubit is known. If the two qubits are not entangled, the overall state would be given by the tensor product of the one qubit states. More generally, I could write a set of contraints on the elements of a two-qubit density matrix to guarantee the appropriate reduced description.
Is there is a way to do this more elegantly and systematically for arbitrary bi-partite quantum systems? I'm particularly interested in systems where one of the Hilbert spaces is infinite dimensional, such as a spin 1/2 particle in a harmonic oscillator.
| Density matrices often admit an interesting geometric interpretations when you map them to the space of generalized Bloch vectors, see for example the book I. Bengtsson, K. Życzkowski, Geometry of quantum states, 2006. I won't be surprised if it turns out that the result has something to do with the coset space $SU(2N)/[SU(N)\times SU(N)]$, where N is the dimensionality of the Hilbert space of a single qubit.
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Is it possible to obtain gold through nuclear decay? Is there a series of transmutations through nuclear decay that will result in the stable gold isotope ${}^{197}\mathrm{Au}$ ? How long will the process take?
| the alchemists have dreamed about the production of gold (Z=82) from some cheap material and lead (Z=79) was their favorite choice. They were just using a wrong science - namely primitive chemistry instead of nuclear physics. But otherwise their choice of lead was OK. And indeed, lead became the element that was transmuted into gold for the first time sometime in 1980 (and maybe even in 1972). See
http://chemistry.about.com/cs/generalchemistry/a/aa050601a.htm
One has to remove three protons which costs a lot of energy. Needless to say, the transmutation remains economically unacceptable. That's true for other strategies, too.
Best wishes
Lubos
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Which experiments prove atomic theory? Which experiments prove atomic theory?
Sub-atomic theories:
*
*atoms have: nuclei; electrons; protons; and neutrons.
*That the number of electrons atoms have determines their relationship with other atoms.
*That the atom is the smallest elemental unit of matter - that we can't continue to divide atoms into anything smaller and have them retain the characteristics of the parent element.
*That everything is made of atoms.
These sub-theories might spur more thoughts of individual experiments that prove individual sub-atomic theories (my guess is more was able to be proven after more experiments followed).
| The history of atoms is definitely intertwined with quantum mechanics. There are many features of the quantum theory that make atomic nature of our world apparent. But here I'd like to state an earlier result.
Thomson's 1897 discovery of the electron not only showed that atoms exist but also that they have substructure.
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What's the difference between running up a hill and running up an inclined treadmill? Clearly there will be differences like air resistance; I'm not interested in that. It seems like you're working against gravity when you're actually running in a way that you're not if you're on a treadmill, but on the other hand it seems like one should be able to take a piece of the treadmill's belt as an inertial reference point. What's going on here?
| (Running up a treadmill) = (expend energy to keep feet moving at a constant speed) + (other effects)
(Running up a hill) = (expend energy to keep feet moving at a constant speed) + (energy to lift center of gravity by hill height) + (other effects)
| {
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Why doesn't air freeze? I am in no way experienced in the Physics field so this question may seem a bit silly but i'd appreciate an answer :)
Why doesn't air freeze?
| Air does freeze just at temperatures and pressures we don't often experience.
There is an entire industry around producing and distilling liquid air. You take air and compress it. As a result the air increases in temperature; the air is allowed to cool. Then the air is expanded by venting it into a new chamber. The result is a much colder gas.
Through cycles of compression, cooling and venting we can get air to condense into a liquid.
If liquid nitrogen is then placed in a vacuum chamber, it will freeze.
| {
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Why are physicists interested in graph theory? Can you tell me how graph theory comes into physics, and the concept of small world graphs?
(inspired to ask from comment from sean tilson in):
Which areas in physics overlap with those of social network theory for the analysis of the graphs?
| Richard Feynman reformulated quantum mechanics (and quantum field theory) in terms of a path integral, meaning that in order to find the likelihood of some process occurring, you take a kind of weighted average over all potential trajectories. The weighting function is the exponentiated "action," $\exp(iS/\hbar).$ and the dominant contribution comes from paths which extremize this function, i.e. classical trajectories.
Typically -- almost always -- this integral is too hard for anyone to do (let alone define rigorously), so Feynman developed a perturbation theory, an expansion in terms of graphs. The nature of the graphs depends on the interactions and coupling constants of your model -- that is, on the action.
An (oversimplified) example: Suppose you only had one degree of freedom, x, and the action is $S_0(x) = i\hbar x^2/2$, so that $\exp(iS_0/\hbar) = \exp(-x^2/2).$ (You can ignore $\hbar$ in this example.) Then the path integral is $\int \exp(-x^2/2) dx$ and equals $\sqrt{\pi}$. However, if we add a cubic "interaction" term, so $S = S_0 - i\hbar a x^3$ then we can expand $\int \exp(iS/\hbar)$ in powers of $a,$ the first nonzero contribution being $a^2 \int (x^3)^2 \exp(-x^2/2) dx,$ which you can do exactly. (The term linear in $a$ is zero because the three powers of $x$ can't be paired up ["contracted"].) The graph for this term has two vertices (the two powers of $a$), each with three edges attached (the three powers of $x$ in the interaction term).
So graphs are ubiquitous in QFT!
| {
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Buckyballs in vacuum I've read about the idea that buckyballs and other nanostructures could be used to hold drugs and things until they reach certain places in the body and then get released.
So I was wondering, if you created a buckyball in air, so that some molecules that are in air (such as oxygen and nitrogen) were inside the buckyball. if you then put the buckyball in a vacuum, would the air be able to escape, or would it be trapped in the buckyball.
(And vice versa, a buckyball created in a vacuum put into the atmosphere, would any molecules fit through into the buckyball?)
| It is just that the buckyball's faces are not holes as in popular view; there is pretty much electron probability density there forming quite a strong barrier. Also the size of the faces is comparable to the size of a, for instance, oxygen molecule, so I'm pretty sure that it is impossible for a buckyball to release a molecule just because of one atmosphere pressure difference; yet I don't really think that anything can be put inside without a targeted, careful procedure.
EDIT: A picture of C60, with, I hope, van der Waals surface. This shows that this structure is indeed pretty dense.
| {
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Why does holding something up cost energy while no work is being done? I read the definition of work as
$$W ~=~ \vec{F} \cdot \vec{d}$$
$$\text{ Work = (Force) $\cdot$ (Distance)}.$$
If a book is there on the table, no work is done as no distance is covered. If I hold up a book in my hand and my arm is stretched, if no work is being done, where is my energy going?
| In my humble opinion, I don't really think this is much of a problem that needs so much clarification. You must understand that the "energy" you know in Physics has absolutely, I mean absolutely nothing to do with the energy that your body cells expend. You can actually expend energy while doing physical (of Physics) work but who cares! Physics only works with what it has defined to be "work" and if you don't do this kind of work believe me you haven't done any work as far as Physics is concerned. The definition of work in Physics would have told you already it doesn't depend on what you feel or what your cells do. It is just force and distance. You are exhausted because you climbed some stairs even though Physics would say you have increased your potential energy. So isn't that a contradiction! Why are you feeling weak when you have increased your potential energy? The bottom line is: physical work and the one you think is actually real work are just totally different. While the former was invented by physicists, the latter is what happens in your cells of which we don't care about as long as Physics is concerned.
| {
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Can I levitate an object without using an electromagnet? I know that it's possible to make an object levitate using an electromagnet to hold it up.
But is it also possible to do this with regular magnets? Is there a special kind of magnet I need in order to have one powerful enough to hold an object up?
I'm asking because I have this idea in mind where I want to make a decorative item that is levitating an inch or so above its container.
A is the object I want to levitate.
B is the container.
C shows the magnetic field that is repelling A and B from each other achieving the levitation.
The size of this would be as small as a food plate, maybe even smaller. B is barely a pound or two in weight.
Is that possible?
| Answer is NO.
I would summarise the question as below.
'Is it possible to make a system with 2 parts made of permanent magnets so as to have an arrangement of one part floating due to magnetic repulsion of the other part above it ( effectively the gravitational pull gets balanced by the magnetic repulsive force )
As we know the likes are repulsive and unlikes are attractive in magnetism,
Suppose the north pole (upside) of the magnet placed below (base magnet) is repulsing with the north pole of magnet (floating magnet) which is placed directly above the base magnet with its north pole facing down. Then during this time, the upside of the Floating magnet is attracted by upside of the base magnet.
This combination of repulsion and attraction forces creates a couple in the Floating magnet(unless it is hinged some where without any physical contact with the surroundings or base magnet)
Due to this force the Floating magnet will rotate and stick on to base magnet immediately with its southbpole mating with north pole of base magnet having north pole as upside.
Hence this cannot be used as a decorative item.
But yes, it can be used as a decorative item with Floating magnet if the one magnet is a controllable ectromagnet which can take care of the magnetic strength for balancing of the Floating magnet.
Hope this clarifies the query.
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Why do we think there are only three generations of fundamental particles? In the standard model of particle physics, there are three generations of quarks (up/down, strange/charm, and top/bottom), along with three generations of leptons (electron, muon, and tau). All of these particles have been observed experimentally, and we don't seem to have seen anything new along these lines. A priori, this doesn't eliminate the possibility of a fourth generation, but the physicists I've spoken to do not think additional generations are likely.
Question: What sort of theoretical or experimental reasons do we have for this limitation?
One reason I heard from my officemate is that we haven't seen new neutrinos. Neutrinos seem to be light enough that if another generation's neutrino is too heavy to be detected, then the corresponding quarks would be massive enough that new physics might interfere with their existence. This suggests the question: is there a general rule relating neutrino masses to quark masses, or would an exceptionally heavy neutrino just look bizarre but otherwise be okay with our current state of knowledge?
Another reason I've heard involves the Yukawa coupling between quarks and the Higgs field. Apparently, if quark masses get much beyond the top quark mass, the coupling gets strong enough that QCD fails to accurately describe the resulting theory. My wild guess is that this really means perturbative expansions in Feynman diagrams don't even pretend to converge, but that it may not necessarily eliminate alternative techniques like lattice QCD (about which I know nothing).
Additional reasons would be greatly appreciated, and any words or references (the more mathy the better) that would help to illuminate the previous paragraphs would be nice.
| As the Gluons don't have any charge, the position of Gluon 7 and 8 takes the free Spinaxis on the PionLeft and Right answered Oct 20 Hsch31
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How long a straw could Superman use? To suck water through a straw, you create a partial vacuum in your lungs. Water rises through the straw until the pressure in the straw at the water level equals atmospheric pressure. This corresponds to drinking water through a straw about ten meters long at maximum.
By taping several straws together, a friend and I drank through a $3.07m$ straw. I think we may have had some leaking preventing us going higher. Also, we were about to empty the red cup into the straw completely.
My question is about what would happen if Superman were to drink through a straw by creating a complete vacuum in the straw. The water would rise to ten meters in the steady state, but if he created the vacuum suddenly, would the water's inertia carry it higher? What would the motion of water up the straw be? What is the highest height he could drink from?
Ignore thermodynamic effects like evaporation and assume the straw is stationary relative to the water and that there is no friction.
| I think we can most easily consider the problem from the perspective of energy. For a unit area column the external energy put in equals the volume of the column times the air pressure. The gravitational energy is the mass of water raised times the average height of the water. So
as we pull the water up, we the water is gaining kinetic energy until the water reaches the static limit (roughly 10meters), but at this point the average water in the tube has only risen by half that amount, so the rest is kinetic energy of (upward) water motion. So the vacuum energy in dimensionless units is $h$, while the gravitational energy is $\frac{1}{2}h^2$. The solution is $h=2$.
When we reach 2 times the static limit (20 meters) then the gravitational energy in the water matches the "vacuum" energy we put in, so that would represent the high point of the oscillation. So I think we would get 2 times the static limit. The water velocity will be messy to solve for, as the amount of water moving in the column depends upon height, so just look only at net energy as a function of water height... Of course he will only get a sip of water, then the column would start to fall........
| {
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Why did this glass start popping? I remember a while ago my father dropped a glass lid and it smashed. It looked something like this. When that happened, for about 5 minutes afterwards, the glass parts were splitting, kind of like popcorn, and you could hear the sound. I was just wondering why this happened, and the particles didn't just sit quietly in their own original parts?
| Due to internal stress in the material.
This stress might be there due to fabrication technology or due to heat cycling while usage (less likely).
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Is it possible for information to be transmitted faster than light by using a rigid pole? Is it possible for information (like 1 and 0s) to be transmitted faster than light?
For instance, take a rigid pole of several AU in length. Now say you have a person on each end, and one of them starts pulling and pushing on his/her end.
The person on the opposite end should receive the pushes and pulls instantaneously as no particle is making the full journey.
Would this actually work?
|
Is it possible for information (like 1 and 0s) tO be transmitted in anyway faster than light.
No.
Born2Smile said the same thing (which I +1'd) but I figured it's worth repeating for emphasis. It'd be a violation of causality. For some more details on why this is not allowed, in addition to Born2Smile's answer, see What are some scenarios where FTL information transfer would violate causality?.
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Aspherical lenses - perfect analytical shape It's known that single spherical lens cannot focus parallel beam of monochromatic light into single (diffraction-limited) point, so it has to have aspherical shape to achieve that.
Is perfect analytical lens shape is known that is able to focus light into a single point (again, light is monochromatic)? Is there a universal perfect solution with conic factor, or higher-order components are always required?
| The problem you are talking about is called the spherical aberration. Spherical lenses are much easier to make, while from geometrical point of view the ideal focusing surface is a parabola. Since the light on optical instruments goes close to optical axis, one uses the paraxial approximation where sphere and parabola are the same up to the quadratic term.
There is a large variety of aspheric lenses, with parabolic lenses among them. But it is usually simpler to use combinations of lenses to deal with spherical and other aberrations.
| {
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Why are materials that are better at conducting electricity also proportionately better at conducting heat? It seems like among the electrical conductors there's a relationship between the ability to conduct heat as well as electricity. Eg: Copper is better than aluminum at conducting both electricity and heat, and silver is better yet at both. Is the reason for this known? Are there materials that are good at conducting electricity, but lousy at conducting heat?
| This is true only for metals. Diamond, for example, is barely a semiconductor. But it has a better heat conductivity than any metal.
| {
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How cold does it need to be for spit to freeze before hitting the ground? What is the dominant form of heat transfer between warm water and cold air?
If a $100 mg$ drop of water falls through $-40 C$ air, how quickly could it freeze?
Is it credible that in very cold weather spit freezes in the half a second it takes to reach the ground?
| Yes, I just did it. Arced a small spheriod of spit (no phlem) and it hit the ground and rolled instead of splattered. -15 degrees F, light wind in parking garage (not sure if that matters) attempted with numerous quantaties of spit and trajectories but small quantity with upward trajectory is what got the tiny frozen droplet.
| {
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Do high/low pass lenses exist? For an experiment I will hopefully be soon conducting at Johns Hopkins I need two different lenses.
The first needs to allow all wavelengths above 500 nm to pass (thus a high pass filter) and cut off everything else.
The second needs to allow all wavelengths below 370 nm to pass (thus a low pass filter) and cut off everything else.
My knowledge of optics is middling. I know that good old glass cuts of UV light, but I was hoping for something more specific. Does anyone know of the theory necessary to "tune" materials to make such filters?
Truth be told, I'm an experimentalist, so simply giving me a retail source that has such lenses would get me to where I need to go! But learning the theory would be nice as well.
Thanks,
Sam
| You may consider to use a prism to separate the different frequency light (essentially you are performing a Fourier transform). Then physically blocked those light with wavelength higher than 500 nm and lower than 370 nm. Finally, you merge these light ray together. This will be the device you want and you can freely adjust the frequency range.
| {
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Primer on Liquid/Ink Splatter Physics I'm looking for primer material on the modelling the physics of liquids. In particular I want to make a small simulation (I'm a programmer by profession) of throwing ink at a board, much like this.
http://www.pond5.com/stock-footage/670934/ink-splatter.html
I'm researching ways to approach the problem before I start banging on the keyboard. But I'm finding it hard to find an article about procedural generation of splatters and pools of liquid. There are a lot of search results for After Effects plugins, but this is of limited use. It could be possible that an open source AE plugin exists out there that I can analyze, but I would prefer to have a good maths based article to draw from.
My math skills are quite basic so I'm fully prepared for a hard slog on this one. I would like to make some sort of road map for myself to get to a level where I can make a simulation such as this.
Guidance is much appreciated.
| There are two separate physical considerations regarding the video you posted:
*
*The physics of a spherical drop hitting a dry surface
*Multiple spherical drops hitting the surface at different times
Once the first is solved, you should be able to simulate the second easily.
Regarding the first point, a quick arxiv search returns the following results:
*
*Impact of a Viscous Liquid Drop
*Rayleigh-Plateau instability causes the crown splash
*Thin Film Formation During Splashing of Viscous Liquids
*How micropatterns and air pressure affect splashing on surfaces
| {
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Would you be weightless at the center of the Earth? If you could travel to the center of the Earth (or any planet), would you be weightless there?
| Correct. If you split the earth up into spherical shells, then the gravity from the shells "above" you cancels out, and you only feel the shells "below" you. When you are in the middle there is nothing "below" you.
Refrence from Wikipedia Gauss & Shell Theorem.
{I am using some simplistic terms, but I don't want to break out surface integrals and radial flux equations}
Edit: Although the inside of the shell will have zero gravity classically, it will also have non zero gravity relativistically. At the perfect center the forces may balance out, yielding an unstable solution, meaning that a small perturbation in position will result in forces that exaggerate this perturbation.
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Is there a limit to loudness? Is there any reason to believe that any measure of loudness (e.g. sound pressure) might have an upper boundary, similar to upper limit (c) of the speed of mass?
| Well, the short answer is: there is, when the hydrodynamic approximation (that fluid is composed of small "fluid particles" i which real particles move in the reference frame of the "fluid particle" like in stationary fluid) breaks.
The upper bound can be approximated with wave amplitude equal to ambient pressure, so that the pressure is going down to 0 in wave minimas (this plus minus corresponds to cavitation); yet this corresponds to loudness of $\infty$dB.
| {
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How is the classical twin paradox resolved? I read a lot about the classical twin paradox recently. What confuses me is that some authors claim that it can be resolved within SRT, others say that you need GRT. Now, what is true (and why)?
| Its easy to resolve from the person on earth - he took the doppler effect out of the equation during his observations of the space ship and observed that the clock on the rocket ship was running slow, and confirmed this when the ship arrived back in port, by comparing the two clocks.
The observation according to the traveller is somewhat of a mystery. The traveller looks back at earth, removes the doppler effects and apparently views the earth time running slowly.
Without taking his eye of the two clocks, at some point the doppler effect removed time on earth must speed up and overtake that of the travellers clock. When does this happen? - isn't this in contradiction to what we are taught is observed?
| {
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How does mass leave the body when you lose weight? When your body burns calories and you lose weight, obviously mass is leaving your body. In what form does it leave? In other words, what is the physical process by which the body loses weight when it burns its fuel?
Somebody said it leaves the body in the form of heat but I knew this is wrong, since heat is simply the internal kinetic energy of a lump of matter and doesn't have anything do with mass. Obviously the chemical reactions going on in the body cause it to produce heat, but this alone won't reduce its mass.
| Weight loss occurs when a large amount of calories or fat is turned into energy through excercise or activity. That waste is then released from the body in one's urine, stool, & sweat. When your body converts fat into accessible energy, the process generates heat that is used to regulate body temperature, according to MayoClinic.com. Once stored fat is converted to energy, your body uses it to fuel activity and metabolic functions in much the same way it uses immediate energy from food. MayoClinic.com notes that waste material produced during the conversion of body fat into energy, specifically water and carbon dioxide, leaves your body through urine, sweat and exhaling.
| {
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What temperature can you attain with a solar furnace? A solar furnace is a device that concentrates the sun's light on a small point to heat it up to high temperature. One can imagine that in the limit of being completely surrounded by mirrors, your entire $4\pi$ solid angle will look like the surface of the sun, at about 6000K. The target will then heat up to 6000K and start to radiate as a blackbody, reaching thermal equilibrium with the sun.
The question is: is there any way to surpass this temperature, perhaps by filtering the light to make it look like a BB spectrum at higher temp, then concentrating it back on the target?
| There is no limit to the degree of concentration. In theory, the entire output of the Sun could be concentrated into a small point.
| {
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How fast a (relatively) small black hole will consume the Earth? This question appeared quite a time ago and was inspired, of course, by all the fuss around "LHC will destroy the Earth".
Consider a small black hole, that is somehow got inside the Earth. Under "small" I mean small enough to not to destroy Earth instantaneously, but large enough to not to evaporate due to the Hawking radiation. I need this because I want the black hole to "consume" the Earth. I think reasonable values for the mass would be $10^{15} - 10^{20}$ kilograms.
Also let us suppose that the black hole is at rest relative to the Earth.
The question is:
How can one estimate the speed at which the matter would be consumed by the black hole in this circumstances?
| If the black hole simply swalled matter, and didn't lose any energy, it probably isn't too hard a calculation, just assume the earth is unsupported mass that falls into the BH, which grows in mass as it adds more stuff. The problem, is we know this isn't how it would happen, and some significant fraction of swalled mass will be released as energy, maybe one to a few percent of mC**2. So the energy liberated from swallowing mass, is orders of magnitude greater per unit mass than an H bomb. Clearly most of the planets mass would be blown away, and only a small amount would end up incorporated into the BH. I'd bet this would happen extremely rapidly, and the shock wave that rips the planet apart would probably only take a few seconds. Note freefall time to the center of the earth is probably more like a half hour (order of magnitude), so most of the planet wouldn't even begin to fall before the released energy blasted it apart.
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Why does my wife's skin buzz when she's using her laptop? When my wife uses her laptop, if I touch her skin, I can feel a buzz. She doesn't feel the buzz, but she can hear it if I touch her ear.
So I'm guessing it's a faulty laptop, and she's conducting an electrical current.
But why would she not feel anything, and what would it be that she would be hearing when I touch her ear?
More info:
The effect is only intermitent - it's pretty reliable in a single session on the laptop, but some sessions it won't happen and others it will.
I had the same sensation with a desk lamp that I had several years ago, with no moving parts (as far as I could tell)
The effect only occurs when I move my finger - if I'm stationary, I don't notice anything.
I was playing with my son, and noticed the same buzz. First I thought he was touching the laptop. Then I realised he had skin-to-skin contact with my wife who was using the laptop.
| Its due to residual transductance of the live AC current into the shielding of the device. Get an earthed plug and it will disappear. I've experienced the issue in dozens of shielded but unearthed electrical appliances, not just laptops. It feels exactly like a vibration (when you move the finger over the surface) but it's not, my guess is that the mechanoreceptive nerve-endings in the skin react to the small current.
| {
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"url": "https://physics.stackexchange.com/questions/2824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 5,
"answer_id": 1
} |
Impedance formula for an edge coupled differential stripline in an asymmetrical stackup? I've been searching and have not been able to find a formula for calculating the impedance of differential lines on inner layers where the dielectric above and below it is not symmetrical. I've seen plenty of examples of symmetrical stackups, but that would not help in my case.
Does anyone know what the formula would be for this?
Thanks!
Edited for more information:
For those that don't know, a stackup consists of all the different layers of a PCB (Printed Circuit BOard). You can have, for example, a 4 layber PCB which has individual copper material & glass epoxy stacked on top of each other. This allows the board to have more layers for routing the nets of the circuits.
| Of course it doesn't matter what voltage the voltage planes are held at so assume both are at ground. Now, take advantage of symmetry: assume that the voltages on the two strips are opposite. Use the same logic as with the presence of a charge near a conducting plane implies the presence (for calculation purposes) of an image charge. Therefore the problem reduces to one of determining the impedance of a single strip in a trough. I'll draw something up in paint:
You may be able to look up the impedance of the above single asymmetric stripline. If not, you can calculate it. A reference I quickly found that seems useful is:
http://lss.fnal.gov/archive/tm/TM-1270.pdf
but I bet you can find better in your electrical engineering library.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/2891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Accuracy of the Boltzmann equation I have had this question for some time now. Hopefully someone can answer it.
I know that the Boltzmann equation is widely regarded as a cornerstone of statistical mechanics and many applications have been explored with a linearized version.
I also know that it's extremely hard to obtain exact solutions, which has started a considerable amount of investigation looking for an equally good (or acceptable) formalism to analyse systems that otherwise would be impossible or would take a great deal of computational resources to obtain a solution using the Boltzmann equation.
In spite of this, I never heard a precise description about the degree of accuracy (in comparison with experiments) that can be drawn from the Boltzmann equation. Obviously, I expect that accuracy depends on the system at hand, however, it would be great to hear about some specific examples.
Recommended readings would also be appreciated.
Thanks in advance.
| I provided a response to question in the same context here:
Experiments that measure the time a gas takes to reach equilibrium
However, specifically to your question, this is a good quick overview:
http://homepage.univie.ac.at/franz.vesely/sp_english/sp/node7.html
Where you find out that with certain simplifying assumptions, the Boltzmann distribution is a solution to the Boltzmann equation.
Again, it is interesting to understand how boltzmann distribution are related to the Arrhenius equation:
http://en.wikipedia.org/wiki/Arrhenius_equation
The fact that these theories underpin most of 20th century technology I think is testament enough to accuracy within relevant regimes.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/2933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
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