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How did we get the formula $d U = nCvdT$? Our teacher taught us that for any thermodynamic process, dU=nCvdT where Cv is molar specific heat capacity at constant volume and dU is change in internal energy. How did we get this formula and why is it valid for all processes
| $dU=nC_{v}dT$ is only valid for all processes in the case of an ideal gas where the internal energy is considered to be purely kinetic depending on temperature only. We get this formula for ideal gases by combining the first law, the ideal gas law. and the general definition of the molar specific heat at constant volume.
You can find a derivation here:How can internal energy be $\Delta{U} = nC_{v}\Delta{T}$?
Hope this helps.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can red and blue light interfere to make fringes in young’s double split experiment? Supposing in young’s double split experiment, I cover one slit with red filter and the other slit with blue filter. The light coming out from the first slit would be red and the second slit would be blue. Would there be any interference fringes? I tried googling this question but all the answers say that two different monochromatic lights cannot interfere and hence no interference pattern. But, we do know that if we did the experiment using white light, there is a pattern(for a few fringes). So what should be the right answer?
| You would have two single slit diffraction patterns overlapping on the screen.
The two pattern have different spacings between their fringes. For example the blue spacings are smaller than the red spacings. Like any beat pattern you will find points where bright fringes from blue coincide with bright fringes from red. These points will form a double slit pattern with a color mix of magenta.
| {
"language": "en",
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WKB approximation derivation for $EI understand that we can write any complex wavefunction on polar form $A\exp(iθ)$ with both $A,θ$ real. Following the logic of Griffiths on WKB (here, page 291):
*
*We write the energy wavefunction in the previous form.
*For $E>V$, we insert the previous form in S.E and demand $A^"=0$. The reason we can do this approximation is because we are lead to equation (8.6), which can give indeed real $A$ for $A^"=0$.
But what happens when $E<V $? In this case if we write again the wavefunction in the form $A\exp(iθ)$ with $A,θ$ real, then we cannot apply $A^"=0$, because equation (8.6) would not be able to give real $A$ (since $p^2$ will now be negative).
So what do we do to overcome this problem for $E<V $?
| OP has a point: The polar form of the complex wavefunction $\psi$ is not useful in the classically forbidden region$E<V$ because the complex TISE then doesn't separate into 2 real equations.
Alternatively, consider the approach of Ref. 1 (which happens to be problem 8.2 in Ref. 2). Here the wavefunction is assumed to be on the semiclassical form
$$ \psi~=~\exp\left(\frac{i}{\hbar}\sigma\right), \tag{46.1} $$
where
$$ \sigma~=~\sum_{n=0}^{\infty}\left(\frac{\hbar}{i}\right)^n\sigma_n \tag{46.3}$$
is a complex power series in Planck's constant. The leading coefficient satisfies
$$ \sigma_0~=~\pm \int p \mathrm{d}x, \qquad p=\sqrt{2m(E-V)},\tag{46.5}$$
In eq. (46.5) the momentum $p$ is imaginary in the classically forbidden region $E<V$. The next-to-leading coefficient
$$ \sigma_1 ~=~-\frac{1}{2}{\rm Ln}(p) \tag{46.8}$$
is given in terms of a complex logarithm.
References:
*
*L.D. Landau & E.M. Lifshitz, QM, Vol. 3, 2nd & 3rd ed, 1981; $\S46$.
*D. Griffiths, Intro to QM, 1995; problem 8.2.
| {
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How does this derivation of the proper time derivative of a covariant vector work? Define the operator $\frac{D}{D\tau}$ by its action on an arbitrary contravariant vector $A^\lambda$:
$$\frac{DA^\lambda}{D\tau} = \frac{dA^\lambda}{d\tau} + \Gamma^\lambda_{\mu\nu} \frac{dx^\mu}{d\tau} A^\nu$$
(with the motivation that this allows us to express the geodesic equation in a nice form).
Now, in an attempt to deduce the corresponding action on a covariant vector $\frac{DA_\lambda}{D\tau}$, contract with an arbitrary covariant vector $B_\lambda$:
$$\begin{align}
B_\lambda\frac{DA^\lambda}{D\tau} &= B_\lambda\frac{dA^\lambda}{d\tau} + B_\lambda\Gamma^\lambda_{\mu\nu} \frac{dx^\mu}{d\tau} A^\nu \\
&= \frac{d(A^\lambda B_\lambda)}{d\tau} - A^\lambda\frac{dB_\lambda}{d\tau} + B_\nu\Gamma^\nu_{\mu\lambda} \frac{dx^\mu}{d\tau} A^\lambda \\
&= \frac{d(A^\lambda B_\lambda)}{d\tau} - A^\lambda\left(\frac{dB_\lambda}{d\tau} - B_\nu\Gamma^\nu_{\mu\lambda} \frac{dx^\mu}{d\tau}\right) \\
\end{align}$$
The following is the bit which I don't understand - we claim that the term in brackets must be the derivative of this covariant vector, i.e.
$$
\frac{DB_\lambda}{D\tau} = \frac{dB_\lambda}{d\tau} - B_\nu\Gamma^\nu_{\mu\lambda} \frac{dx^\mu}{d\tau}
$$
I agree that it is definitely a vector, but I don't see how we can make the leap to saying that it is certainly the form that this operator takes when acting on a covariant vector. Is there something I'm missing?
I considered that it might have something to do with the fact that if you substitute this definition and rearrange you obtain
$$
B_\lambda\frac{DA^\lambda}{D\tau} + A^\lambda\frac{DB_\lambda}{D\tau} = \frac{d(A^\lambda B_\lambda)}{d\tau}
$$
but I can't see exactly where this leads.
| This essentially follows from that $\frac{D}{D\tau}=\nabla_{\dot{x}}$ and from how a connection $\nabla$ acts on tensors.
| {
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In an optical system, does the Point Spread Function apply to all light? Or only Point light sources? So in optics, the Point Spread Function (PSF) describes how an optical system responds to a point source of light. My understanding is that this is due to diffraction and the wave-like nature of Light.
This would lead me to believe that this should apply to any light entering the optical system (indeed, we know there is a diffraction limit for the resolving power of an optical system). After all, if I drew a ray from my camera to any object reflecting/emitting light into my camera, that light should be diffracted the same as if it were a point source, no?
Where I'm getting tripped up is that point sources seem to be spread far more than I'd expect from looking at an image. A star for example, can be spread across dozens of pixels despite being as close to a point source as is practically possible. And this effect occurs even on diffraction limited systems where the resolution of the sensor is at or lower than the diffraction limit of the optics and so, I would assume that diffraction could not be observed. If the same kind of blurring observed in stars was applied to the rest of the image, all fine detail would be lost.
So what is different about a star/true point source of light, versus everything else? Does the PSF apply to all light, but is extremely narrow and so is only noticeable for extremely intense light sources like a star? Or is there something "special" about a truly point source of light that causes it to be "blurred" more than other "broader" features in an image?
| For a point source the spread varies as inverse square of the distance. However for a linear source it varies as receprocal of the distance near the source. So the variation of intensity depends on the geometry of the source specially at close distance.
| {
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How do non-periodically varying currents produce electromagnetic waves? Electromagnetic radiation is created by the varying/accelerating of a system of charges and currents. Suppose that the time dependence of the charges and currents are $\rho(x,t)$ and $J(x,t)$. Then the subsequent radiation will have the same time dependence. In Jackson it is stated that we can assume $\rho$ and $J$ have harmonic time dependence because we can build up any "arbitrary" function as a superposition of sinusoidal functions via Fourier analysis. My understanding of Fourier series is that we can only do this for periodic functions. We always refer to electromagnetic radiation as a wave because of the harmonic time dependence but the radiation has the same time dependence as $\rho(x,t)$ and $J(x,t)$. So what do we do, if the charges and currents are accelerating but not periodically? Then the electromagnetic radiation would not be a wave I think. So why does Jackson state that we can assume harmonic time dependence without losing any generality?
| The confusion is between Fourier series, which is expansion for periodic functions, and Fourier transform - which is an expansion for arbitrary functions (satisfying certain mathematical conditions.)
Fourier transform can be though of as a generalization of Fourier series. Some would probably even say that Fourier series simply a particular case of the Fourier transform, although certain care is required when switching from one to another.
Finally, it is mentioning a useful trick: a function defined in an interval, can be extended periodically beyond this interval and expanded in Fourier series.
Since Maxwell equations are linear equations, we can Fourier transform the sources (currents and charges) and the fields, and solve the equation in Fourier space. Thus, even non-periodic fields can be represented in terms of periodic waves.
A related topic is the electromagnetic field radiated by an accelerated point charge: see Liénard-Wiechert potential and Larmor formula.
| {
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How can a grain of sand be "spaghettified" when nearing a black hole? I have a hard time wrapping my head around this "spaghettification" process that apparently takes places when getting close to a black hole.
Gravity is proportional to the distance of the object exerting gravity. Everything in space is unimaginably HUGE. This applies to, e.g., both distances and size of different bodies. Earth is big, but our sun could contain more than one million Earths in it. And a grain of sand is negligible compared to Earth.
The difference in gravity between the "front" and the "rear" of a grain of sand sent towards a black hole should/must be negligible, considering how weak gravity as a force is, compared to e.g., magnetism. So how can this weak force and miniscule gravity gradient cause spaghettification on something as small as a grain of sand?
(I believe I can understand why something as big as a star would become spaghetti when closing in towards a black hole.)
Second question, if grain of sand would spaghettify in these circumstances, would something much smaller like a hydrogen atom also spaghettify? A water molecule? A free electron? Neutrinos?
| Spaghettification does not always happen outside the event horizon. You could fall through the event horizon of a supermassive black hole, like the one at the center of our galaxy, without suffering any immediate harm. But soon enough (quite soon, in fact) you would get close enough to the singularity to be spaghettified. And so would a grain of sand or a molecule.
| {
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Does work done by a non-conservative force involve distance rather than displacement? I am a new physics teacher and struggling to piece out the nuance of work calculations for my Advanced Placement (AP) students.
I feel like after a fruitful year of distinguishing between vector and scalar quantities for the use of kinematics and Newton's laws, all distinctions have been blurred in the work and energy unit. In the textbook and all resources I've found, work done by a force (no distinction between non-conservative and conservative) is found by the dot product of the force and displacement, but displacement is represented by $d$ (distance) rather than $\Delta x$. Then, we get to the topic of conservative and non-conservative forces, and it becomes clear that only conservative forces are path independent (so when displacement is zero, work done by the force is zero). So it seems that work done by non-conservative forces might be the dot product of force and distance.
Honestly, I hate this unit, as it feels like there is a lot of hand-waving and ambiguity in the way the topics are presented... and I want to make it clearer for my students but unfortunately am struggling myself to define the terms and assumptions with precision.
| Both conservative and nonconservative forces do work as the path integral $\int _L \vec F \cdot d\vec s$.
If force and path are antiparallel (as for friction*) and force is constant in magnitude along the path, since the dot product of antiparallel vectors is negative the product of their magnitudes, we can replace the dot product with the magnitude product and pull -F out of the integral, leaving $-F \int_L ds =-FL$
If the force vector is constant in magnitude and direction along the path (as in gravity*), we can pull the whole vector out of the integral and simplify to obtain the high school version
$\vec F \cdot \int_L d\vec s = \vec F \cdot \vec s$
*...as typically framed in high-school appropriate problems.
| {
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Husband and Wife with Metal and Fabric Parachutes Let's suppose: an open parachute made of fabric with a lady hanging below, starting from stand still, falls from 5000 meters. Another parachute made of thick metal, with the same size and shape as the fabric one and with the lady's husband hanging below, is also falling from 5000 meters. Which one would reach the ground first?
According to Newton's physics, if there is no air to consider, the two would reach the ground at the same time, because they would have the same rate of acceleration.
Now, with air. According to what I've learned so far, air resistance is affected by the shape and velocity of the falling object, given the same air density. The mass of the objects is not a factor directly, unless it contributes to the velocity of the objects. If the two parachutes have the same acceleration from the gravitational pull, then their velocity would remain the same because the air resistance is also the same.
This seems to conclude that the two parachutes would reach the ground at the same time. The lady would survive. That we know. Would the husband also survive, given the dome of the metal parachute is tall enough for him not to be crushed upon the "gentle" landing? What am I missing?
|
If the two parachutes have the same acceleration from the gravitational pull, then their velocity would remain the same because the air resistance is also the same.
You are mixing force and acceleration in this statement. They have the same gravitational acceleration (different gravitational forces) and they have the same air resistance force (different air resistance accelerations). You can add accelerations to accelerations or you can add forces to forces, but you cannot add forces to accelerations.
Either way when you add forces (and divide by mass) or when you add accelerations you will find that the husband will hit the ground going much faster than the wife. For example, adding accelerations, the husband’s gravitational acceleration down is the same as the wife, but the air resistance acceleration up is much smaller (same force divided by larger mass), so the net downward acceleration is larger and he is faster on impact.
| {
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"timestamp": "2023-03-29T00:00:00",
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How does Bolztmann Brain explain experience of time? If a Boltzmann Brain existence is fleeting due to the absence of supporting organs and environment, how does the thought experiment account for the passage of time it experience in its brief existence? Even if the brain pops into existence complete with a lifetime worth of memory, how does it explain going through those memory as if one is living through them one day at a time?
| Well, if you think about it carefully, you will realize you only ever have the experience of the passage of time in the current moment. You never have any experience of passage of time in a past moment, you only have a memory of said experience, even if the experience seems to have taken place just a split second ago. This memory of the past does not need to be real (formed by an actual experience), it can just be an artefact of the Boltzmann brain, a file in your memory that appeared randomly. So the only thing the Boltzmann brain has to do is to provide an experience of a single moment of time. This can be done, in principle, without organs and so on.
Let me reiterate, the Boltzmann brain concept is hard to wrap one's head around because the moment it needs to create is not the one where you are first examining the concept, checking it for logical consistency, thinking about your past, checking whether you experience the passage of time etc. It only ever needs to create is an experience of now, the now that is happening as you finish reading the last word of this sentence.
| {
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How can QFT perturbation be used for electron-positron scattering? I'm studying scattering and perturbation theory in QFT from Peskin and Schroeder book. After all the calculations and theory developing, they made a calculation for $$e^-e^+\to\mu^-\mu^+$$ scattering. Even though I understood the calculations, one thing I don't agree on is using perturbation approach for this kind of problem. How can the interacting Hamiltonian be considered small when the force is attractive between them, and gets bigger and bigger the closer they get? for $e^-e^-\to\mu^-\mu^-$ scattering I can somewhat understand that they don't get close enough for the interacting energy to be big enough, but what about the case I mentioned above?
| According to the path integral approach (qualitatively), they take all possible paths, which presumably includes arbitrary closeness...which may mean up to some UV limit in the renormalization scheme. How those infinities cancel is a question for experts.
Moreover: Feynman diagrams are an expansion in momentum space, so spatial information isn't available. From the point-of-view of a former Deep Inelastic Scattering experimentalist, elastic electron scattering from a charged object is factored into a point-point particle scattering cross-section (Rutherford scattering) times a structure function $G_E(Q^2)$, something like:
$$ \frac{d\sigma}{d\Omega} = \Big(\frac{m^2\alpha^2}{4p^2\sin^4(\theta/2)} \Big)^2 \times G_E(Q^2) $$
(which only includes the charge scattering, not magnetic moments). $Q^2$ is the squared 4-momentum transfer of the virtual photon.
The structure function $G_E(Q^2)$ can be related to the Fourier transform of the radial charge distribution at wavelength $\lambda =\hbar c/\sqrt{Q^2}$.
In say electron-muon point-on-point scattering (which I choose to avoid $s$-channel annihilation or $u$-channel exchanges), the structure function is:
$$ G_E(Q^2) = 1 $$
which means all length scales contribute equally,
including impact parameter $b \rightarrow 0$. No divergence, and this only at tree-level.
| {
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Potential due to line charge: Incorrect result using spherical coordinates Context
This is not a homework problem. Then answer to this problem is well known and can be found in [1]. The potential of a line of charge situated between $x=-a$ to $x=+a$ ``can be found by superposing the point charge potentials of infinitesmal charge elements. [1]'' Adjusting from [1] ($b\to a$), the answer to the problem below is
$$
\boxed{
\Phi{\left(r, \frac{\pi}{2},\pi\pm \frac{\pi}{2}\right)}
=
\frac{1}{4\,\pi\,\epsilon_o}
\frac{Q}{2\, a}
\left[
\ln{\left(\frac{ a+ \sqrt{a ^2 + r^2}}{ -a+ \sqrt{a ^2 + r^2}}\right)}
\right] \,
\,.}
$$
Yet, because I am practicing using the curvilinear spherical coordinate system, I attempted to work this problem in that system. I know that
$$\Phi( \mathbf{r} ) = \frac{1}{4\,\pi\,\epsilon_o}
\int
\frac
{
1}
{
\left\| \mathbf{r}-\mathbf{r}^\prime\right\|
}\rho(\mathbf{r}^\prime) \,d\tau^\prime \,$$
I also know that
$$
\rho(r,\theta,\varphi) = \frac{Q}{2\,a} \,\frac{H{\left(r-0\right)}- H{\left(a-r \right)}}{1}\,\frac{\delta{\left(\theta-\frac{\pi}{2}\right)}}{r}\, \frac{\delta(\varphi-0) + \delta(\varphi-\pi) }{r\,\sin\theta} \,.$$
Further, since
\begin{equation}
\begin{aligned}
x
&=
r \sin\theta \cos\varphi ,
\\
y
&=
r \sin\theta \sin\varphi ,
\\
z
&=
r \cos\theta ,
\end{aligned}
\end{equation}
I know that
the expression of the distance between two vectors in spherical coordinates is given by the equation
\begin{align}
\|\mathbf{r}-\mathbf{r}^\prime\|
=
\sqrt{r^2+r'^2-2rr'\left[ \sin(\theta)\sin(\theta')\,\cos(\phi-\phi') +\cos(\theta)\cos(\theta')\right]}.
\end{align}
Finally, we are given that the obervation points, $\mathbf{r}$, are restricted as given by the equation
$$\mathbf{r} = \left(r, \frac{\pi}{2},\pi\pm \frac{\pi}{2}\right) .$$
Putting these togehter, we have that
$$
\Phi{\left(r, \frac{\pi}{2},\pi\pm \frac{\pi}{2}\right)} = \frac{1}{4\,\pi\,\epsilon_o}
\int
\frac
{
\frac{Q}{2\,a} \,\frac{H{\left(r^\prime-0\right)}- H{\left(a-r^\prime \right)}}{1}\,\frac{\delta{\left(\theta^\prime-\frac{\pi}{2}\right)}}{r^\prime}\, \frac{\delta(\varphi^\prime-0) + \delta(\varphi^\prime-\pi) }{r^\prime\,\sin\theta^\prime}
}
{
\sqrt{r^2+r'^2-2rr'\left[ \sin(\theta)\sin(\theta')\,\cos(\phi-\phi') +\cos(\theta)\cos(\theta')\right]}
}
\, {r^\prime}^2\,\sin\theta^\prime\,dr^\prime\,d\theta^\prime\,d\phi^\prime
\,.$$
Based on the point of observation, we rewrite the potential according to the equation
$$
\Phi{\left(r, \frac{\pi}{2},\pi\pm \frac{\pi}{2}\right)} = \frac{1}{4\,\pi\,\epsilon_o}
\frac{Q}{2\,a} \, \int
\frac
{
\left[H{\left(r^\prime-0\right)}- H{\left(a-r^\prime \right) } \right] \, \delta{\left(\theta^\prime-\frac{\pi}{2}\right)} \, \left[\delta(\varphi^\prime-0) + \delta(\varphi^\prime-\pi) \right]
}
{
\sqrt{r^2+{r^\prime}^2-2\,r\,r^\prime\, \sin(\theta')\,\cos(\pi\pm \frac{\pi}{2}-\phi') }
}
\,dr^\prime\,d\theta^\prime\,d\phi^\prime
\,.$$
Upon taking the angular integrals I rewrite the potential according to equation
$$
\Phi{\left(r, \frac{\pi}{2},\pi\pm \frac{\pi}{2}\right)} = \frac{1}{4\,\pi\,\epsilon_o}
\frac{Q}{2\,a} \, \int_0^a
\frac
{
2 }
{
\sqrt{r^2+{r^\prime}^2 }
}
\,dr^\prime
\,.$$
I know that
$$
\int \frac{dx}{\sqrt{x^2 \pm a^2}} = \ln{\left(x+ \sqrt{x^2 \pm a^2}\right)}
\,.
$$
Therefore,
$$
\Phi{\left(r, \frac{\pi}{2},\pi\pm \frac{\pi}{2}\right)}
=
\frac{1}{4\,\pi\,\epsilon_o}
\frac{Q}{ a}
\left[
\ln{\left(r^\prime+ \sqrt{{r^\prime}^2 + r^2}\right)}
\right]_0^a \,.
$$
Upon evaluation of the limits of integration, I have the incorrect result that
$$
\boxed{
\Phi{\left(r, \frac{\pi}{2},\pi\pm \frac{\pi}{2}\right)}
=
\frac{1}{4\,\pi\,\epsilon_o}
\frac{Q}{ a}
\left[
\ln{\left(\frac{ a+ \sqrt{a ^2 + r^2}}{r}\right)}
\right] \,
\,.}
$$
Question
The result should be identical no matter what coordinate system that I choose. I have a gap in my understanding. Please help by identifying and stating the error in my analysis?
Bibliography
[1] http://hyperphysics.phy-astr.gsu.edu/hbase/electric/potlin.html
|
Adjusting from [1] $(b→a)$, the answer to the problem below is
$$\Phi{\left(r, \frac{\pi}{2},\pi\pm \frac{\pi}{2}\right)}
=
\frac{1}{4\,\pi\,\epsilon_o}
\frac{Q}{ a}
\left[
\ln{\left(\frac{ a+ \sqrt{a ^2 + r^2}}{ -a+ \sqrt{a ^2 + r^2}}\right)}
\right]$$
This is wrong by a factor of 2, in the original answer they use the linear density $\lambda$, which you substituted by $Q/a$, while it should be $Q/2a$.
$$\Phi{\left(r, \frac{\pi}{2},\pi\pm \frac{\pi}{2}\right)} = \frac{1}{4\,\pi\,\epsilon_o}
\frac{Q}{2\,a} \, \int_0^a
\frac
{
2 }
{
\sqrt{r^2+{r^\prime}^2 }
}
\,dr^\prime$$
From here you can use that the integrand is symmetric under the change of variables $r' \to -r'$, so you can write $2\int_0^a = \int_{-a}^a$ and the desired answer follows trivially.
Another option is to start with your final expresion $$\Phi{\left(r, \frac{\pi}{2},\pi\pm \frac{\pi}{2}\right)}
=
\frac{1}{4\,\pi\,\epsilon_o}
\frac{Q}{ a}
\left[
\ln{\left(\frac{ a+ \sqrt{a ^2 + r^2}}{r}\right)}
\right]$$
And realise that $f(a):=\ln{\left(\frac{ a+ \sqrt{a ^2 + r^2}}{r}\right)}$ is odd under the change $a\to-a$. So you can rewrite $f(a)=\frac{1}{2}(f(a)-f(-a))$. This also gives you the desired result.
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How to test resonance frequency of spring using sound? I've done experiment with spring and mass to determine the natural frequencies of 4 springs. The first experiment went well but I had some problem when I want to test the resonant frequency. I'll explain the context first.
After the spring test, the spring was tested for its resonance frequency using frequency generator, amplifier, and speaker with tube to focus on the wave. I set up the spring stretched vertically and blasting the sound perpendicular to the spring. The string were stretched about 1.7x of its length (it was clamped both side).
In theory the resonance is at 26 Hz but after trying for 4 hours blasting the sound, I can't make the spring in resonance, the same goes for the other 3 springs
What did I do wrong?
I still unfamiliar with the correct methods to find the resonant frequency
| The resonant frequency depends just as much on mass density as on stretch and spring constant. You also have to consider whether you are looking at a transverse wave or a longitudinal wave. Blasting from the side is a transverse wave, similar to a wave on a string. A mass hanging from the spring is a longitudinal effect. I don't know what kind of spring you have, but many springs experience a significant change in loop density, and therefore mass density, when oscillating.
Unlike a string, stretching a spring vertically can result in a non-uniform tension and mass density. Loops at the top must support the force due to the stretch and the weight of the spring. If they are further apart than loops at the bottom, tension at the top is greater and mass density is smaller. Both of these would contribute to a greater wave speed at the top than at the bottom. A greater tension would make the spring force a greater effect at the top and make loop distances more uniform.
You don't say enough about your spring to know what does or does not matter most, but these are the first things thatcome to mind.
| {
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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.
| Mass is not converted to energy. Mass is energy. Sometimes mass energy can be converted into heat energy or light energy, or vise versa (as in chemical or nuclear reactions). Thermodynamics always applies regardless to these transfers or conversion of energy. In most thermodynamics and chemistry contexts, the amount of heat or mechanical (pressure) or chemical energy released or absorbed is incredibly small compared to the mass energy of the reactants and products - so when making computations, it is typically assumed that mass is constant, and only the other forms of energy are accounted for.
For example, in a reaction of 1 kg of hydrogen and oxygen gas:
$$ H_2+\frac 1 2 O_2 \rightarrow H_2O$$
the resulting water vapor product weighs about 0.1 micro grams ($1\times 10^{-10}$ th of a kg) less than the original mixture. This minute difference accounts for the mass energy released as heat and light energy. But in the context of chemistry, this is usually accounted for by saying the mass is constant (which is nearly true, and certainly to the limits of our measurements), while the heat and light energy came from the change in "chemical potential energy" associated with the reaction.
Bottom line is: $E=mc^2$ and thermodynamics are both always true, but the former is typically not relevant in chemistry or most everyday thermodynamic problems.
| {
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How does the magnetic field strength (in Teslas) change when two cylindrical magnets are pulled appart? I have two cylindrical magnets aligned such that the opposite poles are facing each other (N-S N-S).
I am trying to find a mathematical relationship that models the change in the magnetic field strength (B - measured in Tesla) at the midpoint of the two magnets when they are pulled apart (with a distance between them denoted r).
I have found many seemingly conflicting resources that say the relation is one of the following:
*
*B=1/r.
*B=1/r^2.
*B=1/r^3.
I am very unsure of which is applicable to my scenario. I should note that I have a high school level understanding of magnetism so I struggle to understand some of the more complex explanations. I would appreciate it very much if someone could provide some insight into this.
| A simple model of your scenario would be to replace both your magnets with ideal magnetic dipoles, which is a decent approximation and quite useful because the fields of finitely sized permanent magnets are difficult to write down. Electromagnetism has an important and useful property known as superposition. The total electric field in a volume of space is the sum of all electric fields, and likewise for magnetic fields.
Therefore, ignoring any slight effects that each magnet's individual field might have on the domains and field of the other, you can apply superposition and know that the total field in space is the sum of the fields of each magnet. This gives us a good starting point. Now we just approximate each magnet as an ideal dipole, which has a neat closed form expression for magnetic field, and we can write down an exact expression for the total field everywhere in space.
The field of a dipole scales as $1/r^3$, so we can assume that the sum of their fields will also scale as $1/r^3$ far away from the magnets. The field between the two is a bit trickier, but you should be able to compute it given your expression for the total field. If the magnets are separated by a distance $a$, then the field at the midpoint should scale as $1/(a/2)^3$.
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Confusion regarding Relation between Frequency and Loudness The mathematical relation between intensity and loudness is:
$$ I_{L} = 10 \log_{10}\left( \frac{I}{I_{0}} \right) $$
where $I$ is the sound intensity and $I_{0}$ the reference intensity. The unit of sound intensity is expressed in decibels (dB).
The mathematical relation between intensity and frequency is:
$$ I = 2 \pi^{2} \nu^{2} \delta^{2} \rho c$$
with $I$ the sound intensity, $\nu$ the frequency of sound, $\delta$ the amplitude of the sound wave, $\rho$ the density of the medium in which sound is travelling and $c$ the speed of sound.
This indicates that as frequency increases, loudness must also increase.
Then, why most physics textbooks mention that Loudness does not depend on Frequency?
| Personally, I am not sure where exactly you found that loudness is not frequency dependent. Loudness is a psychoacoustic quantity (measure) and according to the well known Fletcher-Munson curves (Equal Loudness Contours) there is strong frequency dependence on loudness.
You may be referring to the Sound Intensity Level (SIL), which is calculated with the first equation you provided. There seems to be a rough rule of thumb being used in the acoustics community relating Sound Intensity Level with loudness, which states that there is "roughly" doubling of loudness for a $10 ~ dB_{SIL}$ increase.
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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.
| As a simple(ish) special case of the principle set out by @Cleonis, imagine two initially stationary charges, starting a short distance apart. Under the Coulomb (electrostatic) force, they will accelerate apart (if they're like charges) or together (if they're opposite charges).
Now think about this same pair of charges from the point of view of an observer who is moving perpendicular to the displacement between the two charges, so that, to this observer, the charges appear to be initially moving parallel to each other and perpendicular to the spacing between them.
Because the displacement between the two charges is perpendicular to the frame-of-reference motion, it will not appear Fitzgerald-contracted. However, there will be time dilation, so that to this observer, it will appear to take longer for the two charges to reach any given distance apart than it did to the stationary observer; i.e. the acceleration of the charges will appear to be smaller to the moving observer than to the stationary observer; i.e. (again) the force between the charges will appear to be smaller to the moving observer than to the stationary observer. That is, an observer who encounters moving (in the same direction, perpendicular to the spacing between them) charges will think there is an extra force between them, acting in the opposite direction to the Coulomb force, compared with an observer who encounters stationary charges. That extra force is what we call the magnetic force.
| {
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On Bohr's response to EPR If I understand correctly, the EPR paper (1935) points out that quantum mechanics is incomplete theory if it describes individual particles and measurements. This is true by the mathematical formalism. But already in 1926 quantum mechanics had its statistical interpretation, and in 1930 Heisenberg in his Chicago Lectures admits that position and momentum can be known exactly. So why didn't Bohr just give a short reply: $$\text{"It's a statistical theory."}$$
| Because that would be conceding to Einstein's view. Einstein believed quantum theory is statistical and not fundamental, and the EPR provides an argument for that view. Bohr believed quantum theory is fundamental, and thus did not agree with Einstein's characterization of the theory as incomplete.
| {
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Is the temperature of the hottest star's core known? WR 102 is believed to be the hottest star in the observable universe, whose surface temperature is $210,000 ^\circ K$. But the related wikipedia entry does not say anything about the temperature of its core. So,
*
*Is the core temperature of this star known?
*Is the core temperature of this star the highest core temperature, as well?
| The core temperature of this star isn't known, but can be estimated from theoretical models of stellar evolution, using the observed luminosity and photospheric temperature as constraints.
These suggest that WR102 is currently (it will have lost a lot of mass during its earlier life) about 16 times the mass of the Sun and is probably burning carbon or oxygen in its core (Sander et al. 2019). To accomplish this requires core temperatures of $\sim 10^9$ K. This is a short-lived phase, possibly lasting a few thousand years prior to core collapse and is presumably why WR-O stars are rare (WR102 is one of only a few known in our Galaxy).
It is possible that the star is even closer to core collapse, burning neon or even silicon in its core. This would mean temperatures about 4 times higher still. That is unlikely, because such an evolutionary phase might only last a few days.
So the answer to your question is, probably. Or it could be another of the WR-O stars, such as WR144, which is 3 times as luminous and probably about twice as massive. The hottest core temperature will be in the massive star that is closest in time to ending its life with a core collapse and it isn't possible to securely tell which that is based solely on external appearance.
In particular, surface temperature is not a good guide to core temperature. For example, the red supergiant Betelgeuse is also near the end of its life, probably not quite as close as the WR-O stars, but will have a core temperature that might only be slightly lower. Massive stars of initial masses of 15-30 solar masses will go through red supergiant phases and may be red supergiants (with low surface temperatures) even just prior to core collapse.
| {
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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?
| Strictly speaking, you are right. However, in magnetohydrodynamics, it happens that people refer to the force per unit charge $ \frac{{\bf J}}{e}\times {\bf B}$ as emf, with the implicit assumption that its line integral over a path provides the real emf.
| {
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Motivation behind introducing creation/annihilation operators into the Dirac equation When studying the Klein-Gordon equation, the introduction of creation/annihilation operators was justified by recognizing a harmonic-oscillator-like equation which we know how to quantize. Is there a similar justification when introducing these operators for the Dirac equation? Most of the resources I have looked at simply state them and move on.
| In general the reason is that it was desired to find an operator solution to the equation, so that the fields could be operators in analogy to the observable operators that we have in nonrelativistic quantum mechanics.
I don't have an explicit proof of it, but a professor of mathematical physics who I spoke to told me that "putting operators in place of $a$ and $a^\dagger$ gives the most general operator solution to the equation of motion". At that point it can be shown that these operators must be annihilation and creation operators.
| {
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Is there anything truly "stationary" in the universe? Ok, so I read this question and it got me thinking about something. Is there anything genuinely stationary in our universe? What does it mean to be stationary or devoid of any motion? If there isn't anything stationary, can there be a time when a thing is stationary and devoid of any motion in the future? Is a reference point always needed to classify a particular thing as stationary? I may be sitting right now, not making the slightest movement, but that does not mean I am not in motion. I am in motion, in reference to the earth, the solar system and the milky way galaxy
Also, what would happen if, say, a "stationary" object was present in our universe? What would be the conditions required for this anomaly?
P.S. I have taken a look at this question too, but it doesn't completely answer the particular question I am asking, hence this question
| The laws of physics do not allow us to distinguish who in a pair is moving and who is stationary. As such, the question of whether or not there is anything in our world that is truly stationary is meaningless, because we have no way at all of determining whether anything in our world is stationary to begin with.
| {
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Question on the relation between the Hubble constant and the absolute magnitude of Type Ia supernovae I would like to ask a question about the relation between the Hubble constant and the absolute magnitude of supernovae
I have read that supernovae alone cannot fix $H_0$ and that there is a degeneracy between $H_0$ and absolute magnitude. Why? How do Riess et al. find the absolute magnitude then?
| Observation of Type Ia supernovae yield a redshift and an apparent magnitude at peak brightness. If you don't know the absolute magnitude at peak brightness then the apparent magnitude does not give you its distance. A measurement of distance, as well as redshift, is required to determine the Hubble parameter.
The absolute magnitude of Type Ia supernovae is determined from those that occur in (comparatively) local galaxies, where the distances are known by other means (e.g. Cepheid variables). See for example, Sandage et al. (1996); Altavilla et al. (2004).
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How do you account for all the photons and plethora of quantum particles in the box between the double slits and the back wall? Does the particle being shot not interact with the all the particles that must be consuming the space in the box before the back wall? How do we know it’s the same photon traveling the distance to register on the detector wall? Don’t photons ( for example) pass its energy on to the very next one in its path? Maybe it’s just the energy pushing the last photon in line against the wall..?
| You take the "particle" characteristic of Photons too far, they are not like atoms or molecules in your box. T do not bump in each other. For the double slit experiment you have to think of light as a wave.And it has nothing to do with the double slit experiment. If you look at some object, the light comes in your ey without "bumping" to other light particles, otherwise you would not be able to see the object.
| {
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What experiment would disprove string theory? I know that there's big controversy between two groups of physicists:
*
*those who support string theory (most of them, I think)
*and those who oppose it.
One of the arguments of the second group is that there's no way to disprove the correctness of the string theory.
So my question is if there's any defined experiment that would disprove string theory?
| String excitations, rspt lack thereof. Problem is however that unless you believe in a theory with a lowered Planck scale, you'll have to go to energies we'll never be able to reach in the lab to test this regime. But at least in principle it's falsifiable.
| {
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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?
| Magnetic field in this case (a set of magnets in space, no relativity involved) is conservative, which means it has a potential -- each positional configuration of charges (or dipoles in this case) has its fixed energy which does not depend on history or momenta of charges. So, the work you put or get from displacing them is just exchanged with the potential energy of the field, which means no energy is created or destroyed, just stored.
| {
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Is it possible that all neutron stars are actually pulsars? I'm assuming that what I've been told is true:
We can only detect pulsars if their beams of electromagnetic radiation is directed towards Earth.
That pulsars are the same as neutron stars, only that they emit beams of EM radiation out of their magnetic poles.
So, isn't it possible that neutron stars emit EM radiation in the same fashion as pulsars, just not in the right direction for us to detect it?
| Pulsars are a label we apply to neutron stars that have been observed to "pulse" radio and x-ray emissions. Although all pulsars are neutron stars, not all pulsars are the same. There are three distinct classes of pulsars are currently known: rotation-powered, where the loss of rotational energy of the star provides the power; accretion-powered pulsars, where the gravitational potential energy of accreted matter is the power source; and magnetars, where the decay of an extremely strong magnetic field provides the electromagnetic power. Recent observations with the Fermi Space Telescope has discovered a subclass of rotationally-powered pulsars that emit only gamma rays rather than in X-rays. Only 18 examples of this new class of pulsar are known.
While each of these classes of pulsar and the physics underlying them are quite different, the behaviour as seen from Earth is quite similar.
Since pulsars appear to pulse because they rotate, and it's impossible for the the initial stellar collapse which forms a neutron star not to add angular momentum on a core element during its gravitational collapse phase, it's a given that all neutron stars rotate.
However, neutron star rotation does slow down over time. So non-rotating neutron stars are at least possible. Hence not all neutron stars will necessarily be pulsars, but most will.
However practically, the definition of a pulsar is a "neutron star where we observe pulsations" rather than a distinct type of behaviour. So the answer is of necessity somewhat ambiguous.
| {
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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.
| Although not specifically answering the needs of Cedric H. (symplectic geometry) I find Introduction to Dynamics by Percival & Richards one of the best (and simplest) introductions to Lagrangian and Hamiltonian dynamics, particularly canonical transformations and so on. I point out this book because it is probably not so well known.
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Is acceleration an absolute quantity? I would like to know if acceleration is an absolute quantity, and if so why?
| In standard Newtonian mechanics, acceleration is indeed considered to be an absolute quantity, in that it is not determined relative to any inertial frame of reference (constant velocity). This fact follows directly from the principle that forces are the same everywhere, independent of observer.
Of course, if you're doing classical mechanics in an accelerating reference frame, then you introduce a fictitious force, and accelerations are not absolute with respect to an "inertial frame" or other accelerating reference frames -- though this is less often considered, perhaps.
Note also that the same statement applies to Einstein's Special Relativity. (I don't really understand enough General Relativity to comment, but I suspect it says no, and instead considers other more fundamental things, such as space-time geodesics.)
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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?
| Since I'm not entirely content with the answers to date, here's my take - everyone seems to agree on the basics of forces generated by pressure differentials. If you took a rigid uncooked spaghetto with a cross section of $A$, the case is quite clear - on the cylinder's base in your mouth, a force of $p_{in} A$ is trying to push the spaghetto out, and on the other end, a force of $p_{out} A$ pushes it inwards. The forces generated by the pressures on the sides even out.
It becomes more confusing when referring to cooked spaghetti since a) the cooked spaghetto won't stick straight out from your mouth and b) we intuitively don't want it to transmit forces because "you can't push on a string".
However, these complications do not change the underlying principle. Imagine cutting the spaghetto right outside your mouth while you keep sucking on the inner part. Clearly, the free spaghetto wouldn't feel any net force from air pressure after the cut and just float away/fall down. By cutting, you have changed the total picture by adding $p_{out} A$ orthogonal to the cut surface, but removed the push of $p_{in} A$, so before there must have been a net force on this surface before if there is none afterwards.
I know that it's not the qusestion, but in this case it is much simpler to look at the basic thermodynamics - the system is basically a microcanonical ensemble, with $E = p_{out} V_{out} + p_{in} V_{in}$ under the constraint that $V_{in} + V_{out} = \mathrm{const}$, and if $p_{in} < p_{out}$, the minimum energy (and thus equilibrium) state is clearly that with $V_{out}=0$.
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What property of objects allow them to float? I used to think that the shape of an object determines its ability to float (boat-shaped objects are more likely to float, and spheres tend to sink). But my friend, who is fond of making me look stupid, took me to the local lake showed me a sphere that floated and a boat-shaped object made out of iron that sunk.
Is it based on the mass? I'm not sure that is possible, because I've seen really heavy things (like airplane carriers) float, and really light things (like my friend's iron boat) sink.
What property of certain objects allow them to float, if any?
| DENSITY
It is because of densities of the object that is floating and the liquid in which it is floating.
If an object have density lower than a fluid it will float otherwise it will sink.
Density of entire object [mass / volume] should be
taken into account and not merely the
density of material it is made up of.
*
*A ship made up of iron floats in sea because density of ship i.e. mass of ship/ volume of ship is less than that of water. Here though density of iron is more than water hollowness of ship makes its volume large hence density of "ship" is lower than that of water.
*In dead sea you and I can float.
*In mercury iron nails float.
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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.
| This is a nice example which shows understanding does not come automatically after completing a calculation. But calculation still serves the (perhaps the most) important guide. Nobody in the above has mentioned the discussions given in \ittext{Landau & Lifshitz, Mechanics (BH, 3rd ed.), page 112}. I think these discussions have already elucidated the issue. Unfortunately, they proceeded using Euler angles. I have reformulated their discussions here. Hope that helps.
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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?
| As your question is very general, I can suggest a general answer: when a water wave is hitting a wall for example, you can "trap" an air bubble between the wave and the walls. This bubble can be compressed, the pressure will be higher and when the water moves, this bubble "explode" emitting a sound (which is nothing else than a pressure wave).
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Angular Momentum and Average Torque
Refer to number 6. This is the one I'm stuck on. So angular momentum is conserved right, so initial angular momentum is equal to final angular momentum. Initial is 7.87 so final must be 7.87, right? And so average torque is just change in angular momentum / change in time, so 0/7=0. What am I doing wrong?
| The angular momentum of the rod is 0 at the beginning because it is not rotating.
I would proceed like that:
*
*by conservation of angular momentum, calculate the final rotational speed of the rod
*with that given, calculate the final angular momentum of the rod
*You have that the torque gives the variation (with time) of the angular momentum. So if the torque is constant you just have "torque = angular momentum / $\Delta t$".
I can be more specific if you want. Tell me where you find a problem.
Edit: Apparently the steps are done in the previous questions, so this should just be a "put everything together question".
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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?
| Those laser tape-measures operate in an interesting way, that relies on the speed of light to determine distance. So conversely, if you have a known distance, then with the same equipment you should be able to estimate c.
What the tape measures do is modulate the intensity of the outgoing laser according to the intensity of the reflected light. It's basically an oscillator whose frequency depends on the optical propagation delay. The commercial products use the resulting frequency to determine a distance to display.
If you can get at the oscillator output, and set up to measure a known distance, you should be able to estimate c as the frequency in Hz times the round-trip distance in meters.
| {
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Books for general relativity What are some good books for learning general relativity?
| Spacetime Physics by Taylor/Wheeler
(a red book, with white text on it)
and this book as well
General Relativity, Black Holes, and Cosmology by Andrew J. S. Hamilton 4 December 2021 Available here (free)
*
*https://jila.colorado.edu/%7Eajsh/courses/astr5770_21/text.html
footnotes
my view on it | Really a great introductory book, has references, examples, analogies, and is funny especially in the beginning. It has exercises, and further readings(I advice checking out as well)
spacetimephysics - At eftaylor.com
The 1st and 2nd edition is also good; (my view that is)
| {
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What software programs are used to draw physics diagrams, and what are their relative merits? Undoubtedly, people use a variety of programs to draw diagrams for physics, but I am not familiar with many of them. I usually hand-draw things in GIMP which is powerful in some regards, but it is time consuming to do things like draw circles or arrows because I make them from more primitive tools. It is also difficult to be precise.
I know some people use LaTeX, but I am not quite sure how versatile or easy it is. The only other tools I know are Microsoft Paint and the tools built into Microsoft Office.
So, which tools are commonly used by physicists? What are their good and bad points (features, ease of use, portability, etc.)?
I am looking for a tool with high flexibility and minimal learning curve/development time. While I would like to hand-draw and drag-and-drop pre-made shapes, I also want to specify the exact locations of curves and shapes with equations when I need better precision. Moreover, minimal programming functionality would be nice additional feature (i.e. the ability to run through a loop that draws a series of lines with a varying parameter).
Please recommend few pieces of softwares if they are good for different situations.
| There is an add-in for Microsoft Word called Science Teacher's Helper.
http://www.helpscience.com
SmartDraw is also an excellent program for creating diagrams.
http://www.smartdraw.com
| {
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Can extra-solar gamma rays reach the Earth's surface? Can gamma rays of high enough energy entering our planet's atmosphere reach the surface (50% probability)?
Or, in other words, is there a window for extremely high-energy gamma rays like for the visible spectrum and radio?
This figure, from Electromagnetic Spectrum, shows that the penetration of gamma rays increases with increasing energy, but it seems to level out at about 25 km altitude:
There are no units on the X-axis, and thus it does not show the energy for the highest energy gamma rays for this figure. This does not rule out a window at even higher energies.
| I found a reference through Google Books, Very high energy gamma-ray astronomy by Trevor Weekes, which says that the atmosphere is essentially opaque to high-energy gamma rays, equivalent to a meter-thick wall of lead. We are able to do gamma-ray astronomy with ground-based telescopes by detecting the decay products of the gamma rays' interactions with atmospheric particles, but the photons themselves never (well, essentially never) reach the ground.
From page 13:
The earth's atmosphere effectively blocks all electromagnetic radiation of energies greater than $10\text{ eV}$. The total vertical thickness of atmosphere above sea level is $10^{30}\ \mathrm{g\ cm^{-2}}$, and since the radiation length is $37.1\ \mathrm{g\ cm^{-2}}$, this amounts to more than 28 radiation lengths. This is true up to the energy of the highest known cosmic rays (some of which may be gamma rays).
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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 example, for starters, outside a Schwarzschild black hole horizon, particles can move in any direction, but time only goes one way. i.e. forward. Inside the horizon, particles can only move inward toward the central singularity, i.e. one way, but time can go either forward or backwards. This is a result of the "light cones tipping over". That's different.
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Quantum Field Theory cross sections integrals Where can I find some examples of cross sections calculations in QFT done step-by-step? Those integrals are a little horror. For example - a simple scalar+scalar -> scalar+scalar at the tree level in a theory scalar $\phi^4$ ?
| Peskin and Schroeder tends to be the book used in most introductory QFT courses, so you'll definitely find all things there done in a pretty detailed way.
Warren Siegel has an online book which is also pretty good, Fields.
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Unambiguous distinguishing of quantum states by local measurement Let's have two orthogonal n-particle quantum states: $|\psi \rangle$ and $|\phi \rangle$. In theory it is always possible to make an unambiguous measurement.
However, things get complicated when one restricts oneself to a certain class of measurements. With so called LOCC (Local Operations and Classical Communication, that is, we have to measure particles separately, but we are allowed to communicate and to have dependence for measurements on the outcomes of previous measurement) still it is possible to unambiguously distinguish any two states (see: Walgate et al., Local Distinguishability of Multipartite Orthogonal Quantum States, Phys. Rev. Lett. 85, 23: 4972-4975 (2000) arXiv:quant-ph/0007098).
With fixed local operations (and thus classical communications only after all measurements are done) sometimes we can't unambiguously distinguish between $|\psi \rangle$ and $|\phi \rangle$.
*
*Is there any simple argument why?
*Are there any simple criteria which says which orthogonal states can be unambiguously distinguished with local measurements and communication only after them?
| (Edited to correct TeX typo)
I complete Tim Goodman's answer in answer to get something more systematic. A local measurement has to be written as a tensorial product of two observables A⊗B. And it can only distinguish (with probability 1) its eigenstates. The states |ϕ⟩ and |ψ⟩ of Tim's example cannot be written as eigenstates of a tensorial product.
Note that this does not correspond exactly to states which cannot be written as tensorial product of states. For example, if let's use the following 4 states :
$|\psi_0\rangle=|00\rangle$ , $|\psi_1\rangle=|1+\rangle$,
$|\phi_0\rangle=|01\rangle$ , $|\phi_1\rangle=|1-\rangle$ and let's try to distinguish the ψs from the ϕs.Tim's arguments are still valid, even if each the 4 states is a product state and is orthogonal to all the 3 others. Furthermore, each pair of state is locally distinguishable.
I think locally distinguishable subspaces has something to do with to the direct sum of locally orthogonal subspaces, but I don't exactly know how to write it.
| {
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Can the coefficient of static friction be less than that of kinetic friction? I was recently wondering what would happen if the force sliding two surfaces against each other were somehow weaker than kinetic friction but stronger than static friction. Since the sliding force is greater than the maximum force of static friction, $F > f_s = \mu_s F_N$, it seems that the surfaces should slide. But on the other hand, if the force of kinetic friction is greater than the applied force, there'll be a net force $\mu_k F_N - F$ acting against the motion, suggesting that the surfaces should move opposite to the direction they're being pushed! That doesn't make sense.
The only logical resolution I can think of is that the coefficient of static friction can never be less than the coefficient of kinetic friction. Am I missing something?
| This answer is speculative - not based on my experience with friction.
Logically, there's no reason kinetic friction has to be velocity-independent. You could have kinetic friction that increases with velocity. That way, if you push on something with more force than static friction, the thing would accelerate up to some certain velocity at which kinetic friction equaled the applied force, and then accelerate no more.
If that speed were very slow, you could say that kinetic friction is greater than static friction for all normally-encountered speeds without a paradox.
However, as you pointed out, kinetic friction would have to be less than or equal to static friction for speeds right next to zero.
| {
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How do neutron stars burn? Is it decay or fusion or something else?
*
*What makes a neutron star burn, and what kind of fusion/decay is happening there?
*What is supposed to happen with a neutron star in the long run? What if it cools, then what do the degenerated matter looks like after it cools? Will
the gravitational equilibrium be ruined after some burn time? How does it explode if it can explode at all?
| Neutron stars can actually "burn" and leave quark matter as an ash, according to some hypotheses.
The Bodmer-Witten hypothesis states that quark matter made of up, down, and strange quarks might be the most stable form of matter. If one could find a way to melt neutrons and protons into stable, strange quark matter, this could unleash a very exothermic event, which can be described as a combustion process. Some argue that the core of a neutron star, due to its high density, might be an ideal place for this phase transition to happen. So in short, the neutron star could "burn" into a quark star and produce an explosion. The Bodmer-Witten hypothesis is a bit controversial but has very interesting astrophysical imiclations.
This is the theoretical basis of the quark nova model:
https://en.wikipedia.org/wiki/Quark-nova
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Two-point correlation function for planar Potts model Fastest known method for computing Potts model partition function (Bedini and Jacobsen's "A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings") uses a "tree decomposition" of the square lattice. How can this method be extended to computing two-point correlation function? An obvious approach is to re-run it for every pair of vertices fixed to a particular value, but that ends up with a lot of redundant computation, is there a more efficient way?
Example splitting scheme for grid
| *
*First, what we clam is that our method is the best for arbitrary planar graphs, due to the fact that the treewidth for a planar graph of size N scales a O(N^1/2). For the particular case of the square lattice, our method is not different from the traditional TM, since a strip of width L has a tree decomposition of width L+1 which is a path decomposition.
*Second, in the Fortuin Kasteleyn (FK) representation you can't really fix vertices to any particular state since the degrees of freedom sit on the bonds. Nevertheless it's known that the two point function in the state representation equals the probability of the two vertices to be connected in a FK cluster. So what one should do, perhaps, is to give a special mark to connectivity states touching the two vertices.
Andrea Bedini
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Why does one experience a short pull in the wrong direction when a vehicle stops? When you're in a train and it slows down, you experience the push forward from the deceleration which is no surprise since the force one experiences results from good old $F=m a$. However, the moment the train stops one is apparently pulled backwards. But is that a real physical effect or just the result from leaning backwards to compensate the deceleration and that force suddenly stopping?
So far the answers basically agree that there are two spring forces involved, for one thing oneself as already guessed by me and for the other the vehicle itself as first suggested in Robert's answer. Also, as Gerard suggested the release of the brakes and some other friction effects might play a role. So let's be more precise with the question:
Which effect dominates the wrong-pull-effect? And thus, who can reduce it most:
*
*the traveler
*the driver
*the vehicle designer?
edit Let's make this more interesting: I'm setting up a bounty of 50 100 (see edit below) for devising an experiment to explain this effect or at least prove my explanation right/wrong, and by the end of this month I'll award a second bounty of 200 150 for what I subjectively judge to be the best answer describing either:
*
*an accomplished experiment (some video or reproducibility should be included)
*a numerical simulation
*a rigorous theoretical description
update since I like both the suggestions of QH7 and Georg, I decided to put up a second bounty of 50 (thus reducing the second bounty to 150 however)
| You are pulled wrong way by your own hands and legs, which were stressed to keep you decelerating together with the vehicle.
When the vehicle suddenly stopped accelerating (this is how friction works: it is opposite to speed, and then speed suddenly reaches zero) then this tension keeps pulling you until you react and reconfigure.
| {
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logarithmic wind speed profile Under some atmospheric stability condition, over flat terrain, it has been observed for a while that the ratio between wind speed at height $h_1$ above the earth and the wind speed at height $h_0$ is $\log\frac{h_1}{h^*}/\log\frac{h_0}{h^*}$ where $h^*$ is related to the terrain (called roughness length). (see for example http://en.wikipedia.org/wiki/Log_wind_profile)
What are the theories (with some details or references please) that explain this rule. Please put only your prefered theory (and hence one per post).
Thanks in advance
| My guess is this is a property of turbulent boundary layers over rough surfaces. You may be able to associate the roughness and windspeed to a Strouhal number, and together with the Raynolds number give you at what situation you have this $\log()$ law be dominant.
As for a derivation of this behavior, I am hoping someone with fluids experience can actually answer your question.
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Learning physics online? I'm thinking of following some kind of education in physics online. I have a master degree in Computer Science and have reasonable good knowledge in physics. I would like a program of 1-2 years and I'm more interested in particle physics.
Is there any good online program that offer something similar?
| Here are a few that don't appear to have been mentioned:
[EDIT: I will list one per answer, as the spam filter won't allow me to post more than one link at a time.]
*
*I second MIT's OCW as the very best resource for self-learners.
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Mathematics of AdS/CFT To date, what is the most mathematically precise formulation of the AdS/CFT correspondence, and what are the most robust tests of the conjecture?
| when one talks about "truly rigorous" mathematical physics, there is really no good treatment of ordinary quantum field theory yet. So of course, there is no "truly rigorous" framework to discuss AdS/CFT whose one side is a quantum field theory and another side is something even more complicated - a description of a quantum gravity theory in terms of string/M-theory.
At the physics level, nothing has changed since 1997 or early 1998: the claim was known in the right form from the very beginning. The string theory dynamics in the AdS space is exactly equivalent to the dynamics of the field theory on the boundary. While the field theory may be defined by the lattice - up to some issues with adjusting supersymmetry at long distances - the gravitating string theory side is only known from various limiting descriptions, including perturbative string theory, Matrix theory, and other dual descriptions, besides various terms calculable from SUSY etc. But there is no counterpart of the "lattice" that would allow us to define string theory "completely exactly" in any background.
Nathan Berkovits has made the longest steps to prove AdS/CFT in a Gopakumar-Vafa way, using his pure spinors etc: the world sheet of string theory directly degenerates into Feynman diagrams in the right limit. However, no physicist is too curious about such things because the map clearly works. I believe that the tests derived from the BMN-pp-wave duality - which show that even all strings and their interactions on the AdS side exactly match to the boundary CFT side - are among the most stringent tests of the AdS/CFT correspondence - especially for the canonical background AdS5 x S5 of type IIB related to N=4 SYM in d=4.
This industry has transformed to the integrability business in recent years and increasingly began to overlap with the research of the people who study N=4 SYM scattering amplitudes via twistors although the role of the AdS gravity in the latter remains largely invisible.
Best wishes
Lubos
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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.
| *
*Nonlinear Optics - Robert Boyd
*Photons-Atom Interactions - Cohen Tannoudji
*Photons - Cohen Tannoudji
*Classical Field Theory - A.O Barout (Dover book)
*Problems in General Physics -I.E Irodov (Undergrad/Highschool problem sets)
*Classical Field Thory - Jan Rezwuzki (extremely rare!, Polish academy of sciences)
*Abstract Algebra -Charles Pinter (Dover Book)
*Topology - Mendelson (Dover Book)
*Foundations of Mechanics- Marsden (Very Advanced)
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Books that develop interest & critical thinking among high school students I heard about Yakov Perelman and his books. I just finished reading his two volumes of Physics for Entertainment. What a delightful read! What a splendid author. This is the exact book I've been searching for. I can use it to develop interest for science (math & physics) in my students.
His math books:
*
*Mathematics Can Be Fun
*Figures for Fun
*Arithmetic for entertainment
*Geometry for Entertainment
*Lively Mathematics
*Fun with Maths & Physics
His physics books:
*
*Physics for Entertainment (1913)
*Physics Everywhere
*Mechanics for entertainment
*Astronomy for entertainment
*Tricks and Amusements
I want to get all the above books. Because books from author like this cannot be disappointing. But unfortunately not all of them are available. :(
I also read another amazing book How to Solve It: A New Aspect of Mathematical Method by G.Polya. This books actually teaches you how to think.
In the similar lines if you have any book suggestions (with very practical problems & case studies) for physics & Math (Please don't differentiate between math & physics here. If someone can develop interest in one of the subject they will gain interest in other.) please contribute.
Cross Post: https://math.stackexchange.com/questions/10543/books-that-develop-interest-critical-thinking-among-high-school-students
| try
The Cartoon Guide to Physics by Larry Gonick
funny and smart!
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PSF Measurements In Fluorescence Imaging Quite a technical question!
I have measured the Point Spread Function of 100nm fluorescent breads with my Olympus scanning head. I'm two-photon exciting the beads with a wavelength of 800nm and focused in the sample with a 100x with N.A.: 1.4
The theory suggests me the resolution of such a system to be:
$$d=\frac{0.7\cdot\lambda}{N.A.}\approx 400nm$$
Now that value should be equal (with at most a 18% correction) to the FWHM of the 2D gaussian I obtain on the image.
But from my analysis of the images I obtain a FWHM close to 2um. Now surely the formula is for an ideal optical setup but a factor five seems to me quite strange! Is that possible to obtain such a result, in the case of very not ideal optical elements, or should I look for some sort of problem with the acquisition sw that tells me the pixel dimension of the images?
| I've found the solution to my problem. First of all, I had a factor 2 discrepancy due to the pixel dimension and, more important from an optical point of view, the laser beam was underfilling the entrance pupil of the objective. This means the focusing was worse!
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Explanation: Simple Harmonic Motion I am a Math Grad student with a little bit of interest in physics. Recently I looked into the Wikipedia page for Simple Harmonic Motion.
Guess, I am too bad at physics to understand it. Considering me as a layman how would one explain:
*
*What a Simple Harmonic motion is? And why does this motion have such an equation $$x(t)= A \cos(\omega{t} + \varphi)$$
associated with it? Can anyone give examples of where S.H.M. is tangible in Nature?
| I guess this thread shows everyone has their own tastes when it comes to this topic!
First of all, it's called harmonic motion because sine and cosine are the elementary harmonic functions. Recall that in general a harmonic function is a solution of Laplace's equation (which shows up everywhere in physics), and in we initially study $sin(x+vt)=sin(kx+\omega t)$ since these are the building blocks of solutions of the wave equation. In QM, the principle of superposition (from ODE's) takes on an entirely different, physical meaning.
It's also obvious that the equation of harmonic motion is the projection of $e^{i\omega t}$ onto the real axis, which is a standard trick to derive this equation.
Now an important example of harmonic oscillators in nature are the atoms in a solid. This is basic 1905 Einstein, in 1905 Einstein showed that atoms in a solid can be treated as quantum harmonic oscillators, and that this explains the behavior of specific heat at low temperatures.
Another important example is that the EM field can be considered as three space where each point is a quantum harmonic oscillator. In fact, the EM field is quantized by considering the Fourier components of the field to be creation and inhalation operators for photons of a given frequency $\omega$, which are exactly alike to the raising and lowering operators of the QM harmonic oscillator.
The final standard remark is that almost all small oscillations in nature can be approximated by $F=-kx$, by a Taylor expansion of the force $F(x)$ to first order in $x$.
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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
| Notion of simultaneity. Because of speed of light is so big, it looks true in our day to day affairs.
But it really is a non existent thing [due to special relativity].
2 people in 2 different places can't say "at the same instant".
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How do contact lenses work? I understand how telescope, microscope and glasses work.
But how do contact lenses work?
| Contact lenses work in the same way as glasses, by adding or subtracting wavefront curvature.
This basically adjusts where the focal point of light entering the eye is, glasses and contact lenses are designed to adjust it so that the focal point of the light lies on the retina.
The main differences I can see between glasses and contact lenses, is that the contact lens faces are both curving outwards like this (( rather than this () or this )( like traditional glasses.
This is so that the lens can rest comfortably on the outside of the eye.
If we remember the laws of optics, we remember that the light is only affected by the changes in refractive indices, in the boundaries between air and glass for example.
In normal glasses the two surfaces are air to glass and glass to air.
In contact lenses we have again two surfaces, but it's air to contact lens then contact lens to eye.
Note I say contact lens not glass as contact lenses are made from a variety of materials.
The focal point adjustment in a contact lens happens mostly at the outer boundary - the one between air and the contact lens.
The fact that the other surface is curved the 'wrong' way (for some types of lens requirements) means that it's effect must be calculated and compensated for by the outer surface.
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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 ?
| To get a rough estimate of the Avogadro number, one can also use a method similar to that used by Loschmidt ([1] http://iweb.tntech.edu/tfurtsch/Loschmidt/LOSCHMID.HTML). Gas viscosity can be measured (see, e.g., http://www.phywe.com/index.php/fuseaction/download/lrn_file/versuchsanleitungen/P3010201/e/LEC01_02_LV.pdf - gas flow through a capillary is measured there). Gas viscosity is equal (up to a coefficient) to a product of gas density, average molecular speed, and the mean free path [1]. As the average molecular speed is of the same order of magnitude as the sound velocity in gas, one can estimate the mean free path (I assume that it is not too difficult to measure sound speed). The size of a molecule equals (up to a coefficient) the mean free path times (liquid volume / gas volume) (http://en.wikipedia.org/wiki/Loschmidt_constant ).
EDIT (9/28/2013): It is not easy to measure volume of liquid air at home, as its temperature is very low, but the above measurement can be performed with CO2, as solid CO2 (dry ice) is easily available (it costs about a dollar per pound, as far as I know). It does not matter for our purpose that dry ice is solid, rather than liquid.
| {
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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?
| Helicity and chirality are not the same thing in the massless limit. They are unrelated. Helicity is an extrinsic physical property related to the alignment of spin and momentum; chirality is related to weak interactions. Chirality is more akin to electric charge or strong color charge than it is to momentum.
| {
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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?
| In mechanical systems, I would say the difference is whether the forces involved are due to static or quasi-static situations in which the forces are due to weight/gravity, springs, etc. If the forces result from accelerations then we have a dynamic system, whereas the former would be a kinematic system.
In the transmission of light example of the original questioner, I don't understand the distinction that is being made. All of the phenomena are related to the interaction of particles and light which to my way of thinking is a dynamic system. But that's at a lower level.
| {
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Why is it hopeless to view differential geometry as the limit of a discrete geometry? This is a follow-up question to Introductions to discrete space-time:
Why is this line of thought hopeless?
Classical mechanics can be understood
as the limit of relativistic mechanics
$RM_c$ for $c \rightarrow \infty$.
Classical mechanics can be understood
as the limit of quantum mechanics
$QM_h$ for $h \rightarrow 0$.
As a limit of which discrete geometry
$\Gamma_\lambda$ can classical
mechanics be understood
for $\lambda \rightarrow 0$?
| Not exactly sure what you are asking. For a cubic lattice, the limit as the lattice spacing goes to 0 recovers all of classical physics, and, if you want to discretize time, then as $\Delta{}t$ approaches 0, then finite difference equations become differentials.
For example, Low energy sound waves in a crystal don't see the discreteness, and it can be treated as a continuous medium.
| {
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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?
| On a planetary scale the earths surface is a good insulator and we do not lose significant net heat to space or gain net heat from the sun. So heat generated may build up over time. Radioactive decay heat and gravitationally produced friction would tend to melt the interior of a sufficiently large rocky planet. Heavy elements would sink toward the center and thus conservation of angular momentum would cause the resulting core to spin faster than the surface. It might take the heavy elements a long time to sink to the core. This relative motion core vs magma might help generate the earths magnetic field. (Lots of ideas but no one seems to have a verifiable theory on what causes the earths magnetic field.) Friction between the core and the magma would tend to reduce the difference in rotation rate over time. Only thing we know for sure is that earths magnetic field has continued to decline over the last few centuries we have been making measurements.
| {
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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.
| If you are willing to go down to microscopic scales, a nice example of "entropy winning" is the phenomenon of depletion forces. Large particles in a suspension of smaller ones feel an effective attractive force, even if the interaction between all particles is just hard-wall. The attractive force arises because the volume available to the smaller particles increases when the larger ones get sufficiently close, and hence their entropy increases. See e.g.
Sho Asakura and Fumio Oosawa,
"Interaction between particles suspended in solutions of macromolecules",
J. Pol. Sci. 33 (1958) 183-192.
Depletion forces can be measured directly and are quite important for biological systems.
| {
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Is it possible to separate the poles of a magnet? It might seem common sense that when we split a magnet we get 2 magnets with their own N-S poles. But somehow, I find it hard to accept this fact. (Which I now know is stated by the magnetic Gauss's Law $\vec{\nabla}\cdot \vec{B} =0.$)
I have had this doubt ever since reading about the quantum-field-theory and I know I might sound crazy but is it really impossible to separate the poles of a magnet?
Is there some proof/explanation for an independently existing magnetic monopole?
| I suspect your problem is you may want to think about it rhetorically. Magnetic poles are really just a mental shortcut useful to provide a bit of intuition to something that is inherently just math. We don't have physical entities called mag poles, we have a magnetic field, and it works as if it were generated by currents (and maybe spin, which may or may not work like moving charge (a current)). So cut your magnet, and you have two similar, but shorter pieces, and a local concentration of field lines is usually called a "pole", and polarity refers to the signed value of the magnetic field normal to the surface.
| {
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How efficient is an electric heater? How efficient is an electric heater?
My guess: greater than 95%. Possibly even 99%. I say this because most energy is converted into heat; some is converted into light and kinetic energy, and possibly other forms of energy.
Anyone other opinions? (This is not homework. I am just curious and I'm having a discussion with a friend who says an electric heater is horribly inefficient, less than 5%.)
| It was a good answer Mark. Of course by drawing a lot of current some Joule heating will happen outside the house as well, in the transmission lines and transformers especially. So the efficiency will get lower depending upon where you draw the (electrical box). Some of the energy from the TV and refrigerator will also escape from the house before being degraded to heat (harmonics in
the electric lines, noise, and light from the picture tube (I'd bet these loses are well under one percent)).
In any case air sourced heat pumps are far less efficient than the theoretical limit. The figure I've seen is that a quality heat pump might give you about 3.5 times as much heat as the electrical power input. Not bad compared to the alternatives, but a very long way from Carnot efficiency. Using real (im)perfect working materials is really a drag.
| {
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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?
| I guess you are really looking for this wikipedia page : http://en.wikipedia.org/wiki/Synthesis_of_noble_metals#Gold .
In short, there are gold synthesis technique, but they apparently all need some external energy (either $\gamma$-ray or neutron capture) and are not restricted to nuclear decay. One of them has for intermediate step the nuclear decay${}^{197}Hg\rightarrow{}^{197}Au+e^+$ with a 2 days half life. The unstable ${}^{197}Hg$ is obtained from a stable $Hg$-isotope by $\gamma$-ray irradiation (${}^{198}Hg+\gamma\rightarrow {}^{197}Hg +n$.)
| {
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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).
| I think that the points made about Einstein's theoretical explanation for the observed Brownian motion and the observed Perrin experiments on it are quite valid. But perhaps one could quibble that actually the forces on the pollen were produced by molecules...not by atoms... and perhaps one could resist the point by what is more than a quibble: it proved the reality of things that were too small to be seen, on the scale of atoms, but atomic theory is a little more than that.
Rutherford's alpha particle scattering experiments played a major role, too, besides giving the idea of atomic structure (even though it is called in the O.P. sub-atomic, which is true). The combination of Rutherford and Rayleigh and Einstein--Perrin and the Millikan oil-drop experiment might be the best experimental verification of atomic theory. After all, an entire theory needs several reinforcing experiments about quite a variety of phenomena to really support it, a point which was also made by Einstein as quoted in the answer by Mr. Goldberg.
| {
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Spectral Line Width and Uncertainty principle so I've been at this for about 3 - 4 hours now. It is an homework assignment (well part of a question which i've already completed). We did not learn this in class. All work is shown below.
An atom in an excited state of $4.9 eV$ emits a photon and ends up in the ground state. The lifetime of the excited state is $1.2 \times 10^{-13} s$.
(b) What is the spectral line width (in wavelength) of the photon?
So lets look at what I have done so far. I have done the following:
$$\Delta E \Delta t = \frac{\hbar}{2} $$ but
$$E = h f$$ so
$$\Delta f = \frac{1}{4\pi \Delta t}$$
but if I take $\Delta f$ and convert it into wavelength using $\lambda f = c $ then it gives me the wrong answer. I've tried MANY variations of the above formulas.
The correct answer is $0.142 nm $
Can anyone give me a hint?
| Okay so a buddy helped me out.
You had to use the following formula:
$$ \Delta \lambda = hc \( \frac{1}{E_1} - \frac{1}{E_2} \) $$
| {
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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?
| For sake of argument I will compare a climber maintaining constant speed up a hill and a treadmill runner. If the climber suddenly stopped spending any amount of energy climbing the hill, gravity will tug on him as $mg sin \theta$ while additionally, friction acts in the same direction to slow him down. If a person on the treadmill does the same thing, not only is $mg sin \theta$ pulling him down, so will the friction in the same direction. In both cases they are countering the work done by these two forces so that there is no net change in kinetic energy. Assuming climber maintained the same amount of velocity at which the treadmill runs over, the friction should be perfectly identical with the only difference being what is moving relative to what. Hence there would be no difference in the energy spent neglecting the air drag, of course.
| {
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Ice skating, how does it really work? Some textbooks I came across, and a homework assignment I had to do several years ago, suggested that the reason we can skate on ice is the peculiar $p(T)$-curve of the ice-water boundary. The reasoning is that due to the high pressure the skates put on the ice, it will melt at temperaturs below $273 K$ and thus provide a thin film of liquid on which we can skate. It was then mentioned as fun fact that you could ice-skate on a planet with lakes of frozen dioxide because that gas has the $p(T)$-curve the other way round.
My calculations at that time told me that this was, pardon my french, bollocks. The pressure wasn't nearly high enough to lower the melting point to even something like $-0.5$ degrees Celsius.
I suppose it is some other mechanism, probably related to the crystal structure of ice, but I'd really appreciate if someone more knowledgeable could tell something about it.
| The assertion that the skate does not exert enough pressure to melt ice is wrong. Imagine that the skate is lowered vertically until it touches a perfectly flat surface of ice. The initial contact area (before the blade starts to sink into the ice) would be incalculably small and the initial pressure incalculably large because of curvatures. A typical freestyle blade’s “rocker” has a radius of 6 feet; its “hollow” of 7/16 to 10/16 inch. The blade is typically 0.15 inch thick, so its two edges have “bite” angles of 7 to 10 degrees. The rate at which an edge could melt ice and sink in would be limited by heat conduction. In a dynamic situation, with the skater gliding along at a good speed, viscous dissipation in the thin layer of lubricating water would generate some of the heat. If the skater’s trajectory is curved but the rocker’s curvature multiplied by sin(tilt) is poorly matched to the curvature of the trajectory, then there will be additional friction and sound effects as the edge chews up the ice.
| {
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Where can we find information of International Physics/Astrophysics conferences? Where do you check and put them usually?
Let's make a wiki ~
| There is a number of sites which lists conferences, e.g.
*
*Conference Listing in Mathematics, Physics and Chemistry - Mandl (this one is good and aesthetic; once I attended a conference I had found there)
Also:
*
*Associations/societies list their conferences
*Some research groups homepages have list on conferences in their field.
*Facebook, your faculty homepage, etc
(However, none of the current solutions is perfect. I have one idea what to do... but let's wait - or contact me.)
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Hawking radiation and quark confinement The simple picture of Hawking radiation is that a pair-antiparticle pair is produced near the event horizon, then one falls into the black hole while the other escapes. Suppose the particles are quarks-antiquarks, which experience quark confinement thanks to QCD. If one of them is swallowed by the black hole, its partner is left alone. Eventually the quark gains enough energy and turns into a hadronic jet.
Is my line of thinking correct? If yes, is it (or QCD in general) taken into account when calculating Hawking radiation?
| Most calculations of the Hawking effect assume free quantum fields. This assumption breaks down for strongly coupled quantum chromodynamics. As the Hawking temperature is much lower than the QCD deconfinement temperature, there isn't enough energy to hadronize "virtual" quark-antiquark pairs. Instead, the particle just outside the event horizon gets pulled into the black hole by the confining QCD flux tube. It can't escape.
Actually, for astronomical sized black holes, the Hawking temperature is so low only massless gravitons and photons can be radiated. Even neutrinos are too heavy, never mind quarks.
But for sufficiently light mini black holes, we can have the creation of hadron pairs, with one hadron escaping, and the other falling into the black hole.
| {
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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?
| Graph theory is very useful in design and analysis of electronic circuits.
It is very useful in designing various control systems.
E.g. Signal Flow Graphs and Meson's Rule make your life a lot easier while trying to find transfer functions.
Also, while solving differential equations numerically Graph Theory is used for mesh generation.
| {
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What does it mean that the universe is "infinite"? This question is about cosmology and general relativity.
I understand the difference between the universe and the observable universe. What I am not really clear about is what is meant when I read that the universe is infinite.
*
*Does it have infinite mass or is it
dishomogeneous?
*How can the universe
transition from being finite near the
big bang and infinite 14 billion
years later? Or would an infinite
universe not necessarily have a big
bang at all?
| If the basic question is how we define whether the universe is finite or infinite, then the most straightforward answer is that in a finite universe, there is an upper bound on the proper distance (which is defined as the distance between two points measured by a chain of rulers, each of which is at rest relative to the Hubble flow).
"Does it have infinite mass[...]?" -- GR doesn't have a scalar quantity that plays the role of mass (or mass-energy) and that is conserved in all spacetimes. There is no well-defined way to discuss the total mass of the universe. MTW has a nice discussion of this on p. 457.
"[...]or is it dishomogeneous?" -- I don't understand how this relates to the first part of the sentence. You can have homogeneous or inhmogeneous cosmological solutions.
"How can the universe transition from being finite near the big bang and infinite 14 billion years later? Or would an infinite universe not necessarily have a big bang at all?" -- This was asked again more recently, and a good answer was given: How can something finite become infinite?
| {
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What is the difference between a white object and a mirror? I was taught that something which reflects all the colors of light is white. The function of a mirror is the same, it also reflects all light. What's the difference?
Update:
But what if the white object is purely plain and does not scatter light? Will it be a mirror?
| White color is associated with reflected (not absorbed) light. White paint usually includes a titanium oxide component, whose absorption is in UV.
The difference between a shiny surface and a Lambertian surface is its roughness, if light is reflected in a collective manner it look shiny. Any smooth surface is shiny given a grazing angle.
A mirror is usually coated with a metal surface, silver for example. Silver's metallic behavior makes most of the light reflected back (about %95 percent reflection, which is why it has a greyish tint), however it becomes transparent for UV light.
A mirror can be engineered to have more reflectance than silver by using multi-layer reflection and interference, using dielectric materials instead of metal. They are called dielectric mirrors.
To sum up, white color is due to non absorbed white light, reflection is due to non transmitted light.
| {
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What is terminal velocity? What is terminal velocity? I've heard the term especially when the Discovery Channel is covering something about sky diving. Also, it is commonly known that HALO (Hi-Altitude, Lo-Opening) infantry reaches terminal velocity before their chutes open.
Can the terminal velocity be different for one individual weighing 180 pounds versus an individual weighing 250 pounds?
| You can find a good article here: http://en.wikipedia.org/wiki/Terminal_velocity
In the context you provide, terminal velocity is the maximum speed that an object in free fall reaches in the atmosphere.
When an object is falling, or in free fall, there are two forces that determine whether it will accelerate downwards or not:
*
*gravity (trying to accelerate the body downwards)
*air friction (trying to push the body upwards)
Initially, as the body is not moving, there is no air drag, and the object starts falling due to gravity.
Now, as the object speeds up, the gravity contribution remains constant, whereas the drag increases with the speed of the object.
Finally a point is reached where the drag is so much that the object does not accelerate anymore. Velocity stays constant and it is called terminal velocity.
The value for it is proportional to $\sqrt{m}$ so clearly objects of different weights have, in general different terminal velocities (heavier objects having higher values), but there are also other factors to account for, like how aerodynamic the object is. A sphere has higher terminal velocity than a sheet of metal of the same mass.
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How come gas molecules don't settle down? If the earth's gravity exerts a net downward gravitational force on all air molecules, how come the molecules don't eventually lose their momentum and all settle down? How is the atmosphere is still miles thick after billions of years?
| The other answers are correct but to understand them you have to get an idea of how much thermal energy does an average molecule have.
According to Maxwell-Boltzmann distribution, the most probable speed of an air (say, nitrogen) molecule at room temperature is $v_p = \sqrt { \frac{2kT}{m} } = 422 m/s$. Without collisions with other molecules it can travel upwards $h=\frac{v_{0}^{2}}{2g}$ = 9 kilometers before the gravity stops it and pulls back to Earth.
Basically, potential energy of molecules in gravitational field is too small compared to their kinetic energy to keep them low.
Update: Still, gravity is the reason why we have atmosphere after billions of years.
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Good introductory papers and books on laser physics and pulsed lasers I am looking for good introductory papers and/or books on the principles of lasers. In particular, I am interested in pulsed laser technology.
I understand that
Gould, R. Gordon (1959). "The LASER, Light Amplification by Stimulated Emission of Radiation"
was one of the principal papers published by the disputed inventor himself. What are other good sources?
| This answer contains some additional resources that may be useful. Please note that answers which simply list resources but provide no details are strongly discouraged by the site's policy on resource recommendation questions. This answer is left here to contain additional links that do not yet have commentary.
*
*Rüdiger Paschotta: Encyclopedia of Laser Physics and Technology, online version or paper version.
*You can look for LIA Handbook of Laser Material Processing or Laser Processing of Engineering Materials by John C. Ion.
*Orazio Svelto, Principles of Lasers, 4th edition. Springer, 1998. This is a good all-round book about lasers.
*Femtosecond Laser Pulses: Principles and Experiments by Claude Rulliere (ed.) helped me a lot during the writing on my master thesis on femtosecond pulsed lasers.
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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?
| The information about the pushes will be received on the other end with the speed of sound in the substance of the pole. For any real material it is much slower than the speed of light (for a steel rod it would be about 5000 m/s).
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Anti-gravity in an infinite lattice of point masses Another interesting infinite lattice problem I found while watching a physics documentary.
Imagine an infinite square lattice of point masses, subject to gravity. The masses involved are all $m$ and the length of each square of the lattice is $l$.
Due to the symmetries of the problem the system should be in (unstable) balance.
What happens if a mass is removed to the system? Intuition says that the other masses would be repelled by the hole in a sort of "anti-gravity".
*
*Is my intuition correct?
*Is it possible to derive analytically a formula for this apparent repulsion force?
*If so, is the "anti-gravity" force expressed by $F=-\frac{Gm^2}{r^2}$, where $r$ is the radial distance of a point mass from the hole?
Edit:
as of 2017/02 the Video is here (start at 13min):
https://www.youtube.com/watch?v=mYmANRB7HsI
| This is not correct, but Newton believed this. The infinite system limit of a finite mass density leads to an ill defined problem in Newtonian gravity because $1/r^2$ falloff is balanced by density contributions of size $r^2\rho$, and there is no well defined infinite constant-mass-density system. The reason is that there is no equilibrium of infinite masses in Newtonian gravity--- you need an expanding/contracting Newtonian big-bang.
This is subtle, because symmetry leads you to believe that it is possible. This is not so, because any way you take the limit, the result does not stay put. This was only understood in Newtonian Gravity after the much more intricate General Relativistic cosmology was worked out.
| {
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If photons have no mass, how can they have momentum? As an explanation of why a large gravitational field (such as a black hole) can bend light, I have heard that light has momentum. This is given as a solution to the problem of only massive objects being affected by gravity. However, momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum.
How can photons have momentum?
How is this momentum defined (equations)?
| The reason why the path of photons is bent is that the space in which they travel is distorted. The photons follow the shortest possible path (called a geodesic) in bent space. When the space is not bent, or flat, then the shortest possible path is a straight line. When the space is bent with some spherical curvature, the shortest possible path lies actually on an equatorial circumference.
Note, this is in General Relativity. In Newtonian gravitation, photons travel in straight lines.
We can associate a momentum of a photon with the De Broglie's relation
$$p=\frac{h}{\lambda}$$
where $h$ is Planck's constant and $\lambda$ is the wavelength of the photon.
This also allows us to associate a mass:
$$m=p/c=h/(\lambda c)$$
If we plug in this mass into the Newtonian gravitational formula, however, the result is not compatible with what is actually measured by experimentation.
| {
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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?
| See http://en.wikipedia.org/wiki/Thermal_conductivity In metals, I think it generally has to do with the higher valence band electron mobility, but it gets more interesting elsewhere.
In metals, thermal conductivity approximately tracks electrical conductivity according to the Wiedemann-Franz law, as freely moving valence electrons transfer not only electric current but also heat energy. However, the general correlation between electrical and thermal conductance does not hold for other materials, due to the increased importance of phonon carriers for heat in non-metals. As shown in the table below, highly electrically conductive silver is less thermally conductive than diamond, which is an electrical insulator.
| {
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Polar vs non-polar fluid In the book "Vectors, Tensors, and the Basic Equations of Fluid Mechanics" by Rutherford Aris I read the following:
If the fluid is such that the torques
within it arise only as the moments of
direct forces we shall call it
nonpolar. A polar fluid is one that is
capable of transmitting stress couples
and being subjected to body torques,
as in polyatomic and certain
non-Newtonian fluids.
Can someone help me understand this? In particular, it would be helpful if someone could give me another definition of polar and nonpolar fluids.
| It is about the stress tensor; it is almost always assumed that it is symmetric to satisfy angular momentum conservation. Yet, there are some fluids capable of creating rotation from squeezing (like those spintops with pistons) and thus have some antisymmetric part in their stress tensors.
Aris just calls those fluids polar, what is pretty correct but makes confusion with electromagnetic properties -- I believe that "fluid with non-symmetric stress" or "couple stress" are better keywords.
| {
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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
| If you need extremely sharp filters because your wavelengths in question are either close together or need to be sepearated by a high degree, look at filters from http://www.semrock.com (no affiliation).
Other than that a spectral device (grating or prism based) combined with knife edges as described by hwlau is best, but as he said, you will need a 2nd such device to recombine your desired wavelengths into a single beam and there will be losses (non-brewster reflections on the prism, higher diffraction orders in the grating case, optical surface quality limitations) in the process.
| {
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What is the physical meaning of the connection and the curvature tensor? Regarding general relativity:
*
*What is the physical meaning of the Christoffel symbol ($\Gamma^i_{\ jk}$)?
*What are the (preferably physical) differences between the Riemann curvature tensor ($R^i_{\ jkl}$), Ricci tensor ($R_{ij}$) and Ricci scalar ($R$)? For example why do the Einstein equations include the Ricci tensor and scalar, but not the Riemann tensor?
To be clear, by "physical meaning" I mean something like - what physical effect do these components generate? Or, they make the GR solutions deviate from Newton because of xxx factor... or something similarly physically intuitive.
| As for the 'physical meaning' of Christoffel symbols, there is a sense in which they don't have a physical meaning, because the information they encode is not really information about the curvature of space but about the geometry of the coordinate system you're using to describe the space.
As for an intuition about them, they encode how much the basis vector fields change for infinitesimal changes in the coordinates being used. This is why in a flat space (i.e. locally) it is always possible to make them zero: transform to a coordinate system where the basis vector fields don't change from point to point.
To know how the spacetime curves, you can look at how the metric function changes from point to point. To see this, you can look at how the basis vectors
change from point to point (since the metric is completely determined by the
basis vectors). This is the information the Christoffel symbol encodes.
| {
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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?
| The only practical limit to sound-pressure might be when the medium were to be compressed into a black hole. Although I do not know about the sound-propagating features of black holes. Long before that the medium would come apart, f.e. into plasma.
After all, you compare two things: an upper limit of a speed with a power. How much power can you put into a particle to speed it up with c in mind?
PS: I'd rather not guess what kind of sound-generator would be necessary. One might ask the band "Disaster Area" (https://secure.wikimedia.org/wikipedia/en/wiki/List_of_minor_The_Hitchhiker%27s_Guide_to_the_Galaxy_characters#Hotblack_Desiato)
| {
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The final death of a black hole What are the different death scenarios for a black hole? I know they can evaporate through Hawking radiation - but is there any other way? What if you just kept shoveling more and more mass and energy into the black hole?
| Hawking radiation is a very slow process of the black hole losing energy and shrinking. If you counter this by supplying a little bit of matter or energy falling into the black hole you can easily overcome it and sustain the black hole.
Other than Hawking radiation I don't think there is any known process for black holes to shrink. The area theorem in classical general relativity states that the area of black hole horizon always increases in any physical process. So at least classically there is no way for black holes to die, or even shrink a little. Hawking radiation evades that because it is a quantum process (which is also why it is a slow process).
As for the final stage of the evaporation, I think the honest answer is nobody knows. The logical possibilities are either the black hole shrinks to nothing and disappears, or it leaves behind some long-lived "remnant". Either one of this possibilities has its strong and weak points, but ultimately you'd need to know more about a quantum theory of gravity to know for sure.
| {
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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.
| I encountered this question by accident and also had the same question a while ago. I found a TED talk about this subject with the title The mathematics of weight loss. The author is Ruben Meerman, he describes himself as:
I am [sic!] surfer with a physics degree who fell in love with science
communication.
The video of the talk can be found on the TED website (https://ed.ted.com/on/dgLmO0cP) or on YouTube.
As I understand the talk, the main weight loss is due to CO2 (breathing) and water. CO2 is the main effect though.
I attached three screenshots to show the look and feel.
| {
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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?
| It would take a long time if we do a back of the envelope calculation.
*
*the black hole would exert a force of 1g at around 20 km (assuming 10^20 kg of mass).
*if we can reasonably assume that the mass inside this sphere is going to be absorbed quickly, that would mean the black hole mass increases correspondingly.
*on the other hand this extra mass can be calculated to be around 10^20 kg too. So we can expect the 1g radius not to increase significantly.
*I believe that mass with less than 1g of pull will take a long tine to spiral inside the black hole, as its size (Shwartzchild radius) would be in the micrometer scale and the sizes involved in the kilometre scale.
Hope this helps!
| {
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Rotate a long bar in space and get close to (or even beyond) the speed of light $c$ Imagine a bar
spinning like a helicopter propeller,
At $\omega$ rad/s because the extremes of the bar goes at speed
$$V = \omega * r$$
then we can reach near $c$ (speed of light)
applying some finite amount of energy
just doing
$$\omega = V / r$$
The bar should be long, low density, strong to minimize the
amount of energy needed
For example a $2000\,\mathrm{m}$ bar
$$\omega = 300 000 \frac{\mathrm{rad}}{\mathrm{s}} = 2864789\,\mathrm{rpm}$$
(a dental drill can commonly rotate at $400000\,\mathrm{rpm}$)
$V$ (with dental drill) = 14% of speed of light.
Then I say this experiment can be really made
and bar extremes could approach $c$.
What do you say?
EDIT:
Our planet is orbiting at sun and it's orbiting milky way, and who knows what else, then any Earth point have a speed of 500 km/s or more agains CMB.
I wonder if we are orbiting something at that speed then there would be detectable relativist effect in different direction of measurements, simply extending a long bar or any directional mass in different galactic directions we should measure mass change due to relativity, simply because $V = \omega * r$
What do you think?
| In your calculations you assume that your propeller is a rigid body.
You cannot use that assumption, when your speeds are not much smaller than the speed of light. Because "there are no rigid bodies in relativity".
| {
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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.
| 1 - Use a voltage detector pen in USB connector GND to see if there is a bad isolation between AC ground and DC ground (this is first option not because probability, but because it could be dangerous). It's normal a little fugue current between primary and secondary of transformer but the AC plug have 3 conectors, 2 for voltage and the third is for GND (ground), perhaps your home installation haven't a good grounding, check that! is very important.
2 - Beware of what is on your wife's laptop screen when her skin buzz =P
picture source
| {
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Why GPS is at LEO? Why GPS/GLONASS/Galileo satellites are on low earth orbit?
Why geostationary orbit is so bad? Sattelites might be placed there 'statically' and more precise...
The only problem I can see is navigation close to poles, but they have this problem anyway.
| This question is slightly faulty... Part of the US GPS system is geostationary (the WAAS component). It's used in conjunction with the non-geostationary birds for higher precision fixes. While primarily used for aircraft instrument approaches, there are off the shelf USB GPS computer peripherals that use WAAS.
| {
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What is planetary surface temperature given constant sub-surface temperature? If a planet of radius $R_1$ has a constant sub-surface temperature $T_0$ at $R_0<R_1$, what is the long-term equilibrium surface temperature $T_1$? Say we assume constant thermal diffusivity of the planet material, surface emissivity $1.0$, no atmosphere, and no incoming radiation. I figure the temperature profile is harmonic, i.e. $T=a+b/R$, for constants $a$ and $b$, and that we can use $(R_0,T_0)$ to eliminate one of these constants. Is there enough information to obtain the other constant?
| You want to equate the conductive energy flux at the surface which is a constant times T1-T0 -given your geometry and conductivity you can determine the coupling factor, with the radiative heat flux sigma*T**4. You could also add in the CMB, which is simply the same sigma*T**4 using the CMB temperature. Then you simply have a nonlinear algebraic equation in T0 to solve. It should be solvable by iteration without much effort.
| {
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movement of photons In a typical photon experiment the photon is depicted as moving across the page, say from right to left.
Suppose we were actually able to witness such an experiment, from the side (to position of reader to a page).
If the photon is actually moving from left to right can I, standing at 90 degrees to the motion, see the photon?
| Being very careful, we'll assume you mean a situation in which there exists only a single photon and two detectors placed at right angles to one another, all of which exists in a vacuum and effectively isolated from the rest of the universe: the answer would be 'no'. In practice of course, very few photons and detectors exist in such perfect isolation.
| {
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When one thinks of a field of operators in QFT, is it reasonable to think of a matrix being associated with each point in space time? Is it correct to visualize operators existing as matrices parameterized by spacetime coordinates in the context of QFT?
| I'm not quite sure if you're asking the same question in the title of your post as you are in the body of your post. But the answer to the question in your title is yes, you can think of an operator field as having a matrix associated with each point in space. That's what a field is, after all: a mapping that associates some value (in this case, an operator, which can be represented by a matrix) with each point in space.
However, keep in mind that there's a subtle difference between an operator and a matrix. An operator is an abstract entity that has some defined action on quantum states. There are various ways to describe the action of that operator, and giving the matrix representation in a certain basis is just one possible way. For example, in the basis $\{|1\rangle, |2\rangle, |3\rangle, \cdots\}$, the particle creation operator can be represented as
$$a^\dagger = \begin{pmatrix}0 \\ 1 & 0 \\ & \sqrt{2} & 0 \\ & & \sqrt{3} & \ddots\end{pmatrix}$$
but the same operator can also be represented by its action on a generic basis state,
$$a^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$$
among other ways.
The point is, the operator itself is something more than any of these representations. So don't get stuck thinking that one operator corresponds to just one particular matrix. If you have an operator field, then sure, you can choose a particular basis and use that basis to create a matrix out of the operator at each spacetime point, and then you'll have a matrix at each point, like you were talking about. But you could choose a different basis, and have a different matrix representation at each point, but it'd still be the same field. You could even do certain things without picking a basis at all, and in that case you wouldn't have any matrices at all. You'd be working directly with the operators themselves.
| {
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Why is quantum entanglement considered to be an active link between particles? From everything I've read about quantum mechanics and quantum entanglement phenomena, it's not obvious to me why quantum entanglement is considered to be an active link. That is, it's stated every time that measurement of one particle affects the other.
In my head, there is a less magic explanation: the entangling measurement affects both particles in a way which makes their states identical, though unknown. In this case measuring one particle will reveal information about state of the other, but without a magical instant modification of remote entangled particle.
Obviously, I'm not the only one who had this idea. What are the problems associated with this view, and why is the magic view preferred?
| In fact your view is quite close to the 'official' one; entanglement occurs just because both particles are described with one wave-function; the magic is in our classical habit of thinking that separate objects are described with separate "coordinates".
| {
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Vortex in liquid collects particles in center At xmas, I had a cup of tea with some debris at the bottom from the leaves. With less than an inch of tea left, I'd shake the cup to get a little vortex going, then stop shaking and watch it spin. At first, the particles were dispersed fairly evenly throughout the liquid, but as time went on (and the vortex slowed, although I don't know if it's relevant) the particles would collect in the middle, until, by the time the liquid appeared to almost no longer be turning, all the little bits were collected in this nice neat pile in the center.
What's the physical explanation of the accumulation of particles in the middle?
My guess is that it's something to do with a larger radius costing the particles more work through friction...
| I jokingly call it Vortex Physics, as it doesn't appear to obey the normal rules. At high speed, where centrifugal force prevails, the heavier particles are forced to the outside, like in a Dyson vacuum cleaner, or industrial cyclone dust separator, from which he got the idea. But at very low speed when the leaves start to drag the bottom, it behaves like a hammer, when you throw it. The heavy part (the head) stays near the middle, and the handle around the outside. They orbit around their common center of gravity like the earth and the moon. The earth wobbles a bit, but the moon does most of the orbiting. If you floated a hammer in a bowl and spun it, it would behave the same.
| {
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Where's the best place to add weight to a Pinewood Derby car? A little background: a Pinewood Derby car is a small wooden car that races down an inclined track, powered only by gravity. You are allowed to add weight to the car up to a certain limit.
Here is a recommendation to add the weight to the upper back of the car, to maximize the potential energy. My own gut feel is to add it so that the weight is evenly distributed between all 4 wheels. I'd like some advice that is well grounded in science.
| UPDATE: Having totally misunderstood what a pinewood derby car was, my previous answer probably wasn't very helpful (I thought it was like a go-cart). Here is a proper answer:
First off, let me say the recommendation you read is still wrong, for reasons that will be seen in a minute.
The car starts off inclined at an angle $\theta$, with the additional mass a distance $x$ behind the front axle, and a distance $h$ above the line passing through both axles. The change in height of the additional mass relative to the front axle is then given by $d_x + d_h$, where $d_x$ is the decrease in relative elevation of the point a distance $x$ behind the front axle on the line between the two axles, and $d_h$ is the decrease in relative elevation between the mass relative and this point. Clearly, $d_x = x \sin(\theta)$ and $d_h = h (\cos(\theta) - 1)$. Notice that $d_h$ is negative, since the change in relative elevation actually increases for positive $h$. Thus the total change in height (which is proportional to the additional potential energy) is given by $\Delta H = x\sin(\theta) - h(1-\cos(\theta))$. Thus, the best position for the additional mass is very low down (even below the line of the axles if possible), at the back.
| {
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Is there a maximum possible acceleration? I'm thinking equivalence principle, possibilities of unbounded space-time curvature, quantum gravity.
| I like previous answer but:
1) I believe that in the provided formula the mass of the electron should have a power of one (not two)
2) It is valid for electrons only, because it uses their Compton wavelength.
By the way, there is such a thing as "Caianiello’s maximal acceleration". In his 1985 paper Caianiello demonstrated the existence of a maximal acceleration. It is a consequence of Heisenberg’s uncertainty relations. An example may be found here.
| {
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What would the electromagnetic field of a massless electron look like The Standard Model gives non-zero mass to the electron via the coupling to the Higgs field. Issues of renormalizability aside, this is fundamentally unrelated to the fact that the electron couples to the EM field. However, if the Higgs mechanism did not operate - that is, if there were no vacuum symmetry breaking, the electron field would have no effective mass term. In QFT perturbation theory, this model offers no special difficulty.
My question is, what is the classical limit of this theory, if it has one? Does the electron acquire a purely EM mass? If the low energy renormalized mass is set to zero (is there an obstacle to doing that?) what do the classical field configurations look like? My puzzlement is related to the classical description of the EM field of a relativistic charged particle - namely, that it becomes "flattened" in the direction of motion, just like a Lorentz-contracted sphere. That is, the field is weaker than for a motionless particle in the longitudinal direction, and stronger in the transverse directions. The limiting case is that the field becomes concentrated on a 2-d surface transverse to the particle location, where it has infinite strength - obviously an unphysical situation. So what actually happens?
| With a massless charged fermion, the electromagnetic coupling strength will be totally screened, with $1/e^2$ increasing logarithmically with the scale $R$. The electric field will be Coulomb multiplied by this logarithmic factor in $R$.
Actually, the massless charged fermion will be travelling at the speed of light, but this logarithmic screening will still happen.
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"language": "en",
"url": "https://physics.stackexchange.com/questions/3394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Your favorite Physics/Astrophysics blogs? What are the Physics/Astrophysics blogs you regularly read? I'm looking to beef up my RSS feeds to catch up with during my long commutes. I'd like to discover lesser-known gems (e.g. not well known blogs such as Cosmic Variance), possibly that are updated regularly and recently.
Since this is a list, give one entry per answer please. Feel free to post more than one answer.
| Skulls in the Stars
Exceptional for its extremely clear, basic-level introductions to phenomena in optics I would otherwise never have heard of, and for interesting historical posts resulting from digging around in the archives of old journals and scientists' personal letters.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/3432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 24,
"answer_id": 17
} |
Expansion of the space-time metric If the space-time metric is expanding with the expansion of the universe, if I could travel back in time, would I be less dense than the matter in that previous era?
| The answer is 'no'. The Hubble expansion does not enlarge the distance between atoms in your body. It even does not enlarge distances between stars in our Milky Way. (Nor between the Milky Way and Andromeda for that matter.)
This question has been asked here more often. See for instance the answers in: Why does space expansion not expand matter?
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/3518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why does a ballerina speed up when she pulls in her arms? My friend thinks it's because she has less air resistance but I'm not sure.
| A very simple explanation is the following: the arms of the ballerina are pulled outwards by the centripetal force she experiences by spinning. When she pulls her arms in, she is doing work by more than counteracting this force and this is what makes her spin faster.
This is due to the fact that the spin velocity is related to the effort that the ballerina makes in pulling in her arms. The closer the arms, the more force she has to use to pull them in future or keep them in position, and the faster she spins.
When I was in high school, we did a more hardcore version of this experiment by sitting on a chair that could spin, holding two heavy weights outwards. Then someone would spin the poor test subject and ask him to pull in the weights... The results were quite scary... :-)
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/3611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 1
} |
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