Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
The direction of frictional force in smooth rolling motion I want to ask about the direction of frictional force in smooth rolling motion which means the rolling object doesn't slide on the surface.
Here is the first case, the wheel rolls to the right so the rotation is clockwise. At the point P, velocity is zero. To m... | Consider the sum of all external torques; calculated about the Center of Mass of the rolling body; then the angular acceleration tells you the direction of relative slipping w.r.t. contact surface. Now, the frictional force will act opposite to the relative slipping tendency.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "11",
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What is the maximum time dilation factor when orbiting a rotating black hole? Suppose one spaceship is stably orbiting a rotating black hole and another is far away from the black hole. What is the maximum time dilation factor between the two ships? Can it be made arbitrarily large, and if so does that require the blac... | You can get the time dilation factor by computing the redshift of a radial photon emitted by someone on a circular orbit, compared to the frequency measured by someone at rest at infinity. The derivation of this formula is a bit involved, but the answer is not too complicated:
$$
\frac{\omega_{emit}}{\omega_\infty}=\f... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/146105",
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Complex Conjugate of Wave Function I've been reading through Griffiths QM book, and the only thing bugging me is they never fully described what $\Psi^* $ should be for any given function. I know it's the complex conjugate at the same time I think I just need concrete examples to solidify it in my head.
What is the cor... | For every $x$ and $t$, $\Psi(x,t)$ is a complex number. $\Psi^*$ is the conjugate of that number, no more, no less. The reason it seems like sometimes it's only the $t$ part that gets conjugated is simply that often it is the only part of the wavefunction that is complex. Let's use your examples:
$\Psi = \sqrt{\frac{2}... | {
"language": "en",
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Does a Michelson-Morley experiment uphold mass-energy equivalence and special relativity? If there is an experiment that best supports $E=mc^2$, is it the Michelson-Morley experiment?
| Nuclear plants (and theoretically fusion plants) work with $e=mc^2$.
For example:
The mass of 2 Protons and 2 Neutrons is bigger than the mass of Helium, which consists of 2 Protons and 2 Neutrons. The difference is emitted as energy when Helium is made by the fusion of 2 Protons and 2 Neutrons (the actual reaction in ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/146408",
"timestamp": "2023-03-29T00:00:00",
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Why do the storms of Jupiter have long life unlike that of Earth? Recently I saw How the Universe Works. In one of the episodes, concerning Jupiter, they told that the storms on Jupiter can survive many, many, times longer than those on Earth.
What is the reason behind it? They said that it is due to its big mass. But... | “Lacking any… surface” (see Autolatry’s answer) in Jupiter is, first, dubious (a metallic hydrogen mantle is conjectured) and second, not very important per se. For example, Uranus and Neptune almost certainly have a relatively dense mantle with a sharp upper boundary, that doesn’t preclude these planets to have very l... | {
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"url": "https://physics.stackexchange.com/questions/146488",
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How is the scattering length in 2d defined? Scattering length is 3d is well-defined. In the literature, one can also see scattering length in 2d. How is it defined? Can we even generalize it to 1d?
| We want to find what with length the center of a incoming wave packet appreciate variations. The scattering approach prefer to talk about time and momentum in detriment of space, but we can talk about space after.
The scatering state always can be written as
$$
\psi^{(+)}_g(t)=\int d\alpha\, e^{-iE_\alpha t} g(\alpha)... | {
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Does zero free current entail zero $\vec H$? There are two kinds of magnetic fields (different authors give them different names), $\vec B $ and $\vec H$ which are related by the equation $$ \vec B = \mu_o (\vec H + \vec M)$$ where $\vec M$ is the magnetization.
Ampere's law for free currents states $$\oint_C \vec H \c... | Yes, I agree with your argument.
Don’t forget $\vec H$ and $\mathrm d\vec l$ are vectors!
$\vec H\cdot\mathrm d\vec l$ is a "dot product",
so the $\oint$ sum the same direction of $\vec H\cdot\mathrm d\vec l$,
Which means if $\vec H$ is a constant
you can take it out the $\int$
and it becomes $\vec H \oint\mathrm d l$ ... | {
"language": "en",
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If angular velocity & angular acceleration are vectors, why not angular displacement?
Are angular quantities vector?
... It is not easy to get used to representing angular quantities as vectors. We instinctively expect that something should be moving along the direction of a vector. That is not the case here. Instea... | Angular displacement is define by change in angle i.e. ${\Delta \theta}=({\theta}_1 -{\theta}_2)$, where $\theta$ is taken as positive in anti-clock wise.
Hence angular displacement ${\Delta \theta}$ has both magnitude and direction. But let when a particle rotate from a point A in $\theta$ angle in anti-clock wise to... | {
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What's the escape velocity of Naked Singularities? Penrose's Cosmic Censorship Hypothesis doesn't hold for naked singularities which means that at least light can escape the singularity. But, if we calculate escape velocity with the given mass only, light shouldn't escape.
How to calculate real escape velocity of naked... | A naked singularity is created the same way as an ordinary singularity, the only difference is that if the angular momentum $J$ is greater than the mass squared, $M^{2}$, then the event horizons (which would be at
\begin{equation}
r = M(1 \pm \sqrt{(1 - (J/M^{2}})^{2}
\end{equation}
cannot exist. Also, There's no suc... | {
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Velocity of car down an Inclined Plane I was learning to make a car game and came across this situation where my car is on an inclined plane. It's initial velocity is 0. Now, the problem is that my memory serves me this formula to calculate the final velocity
$$v_f^2 - v_i^2 = 2as$$
Since, $v_i = 0$, the equation woul... | If i understood your question correctly, you seem to not know the displacement of the car but you still need to increase the velocity of the car.
Velocity of the car has to be expressed in terms of a quantity. I suggest you use velocity as a function of time,
$$v_f = v_i + at$$
$a = g \sin \theta $ as you told correct... | {
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How does interpreting negative energy electrons as positrons solve the negative energy problem? How does interpreting negative energy electrons as positive energy positrons solve the negative energy problem? How does change of “interpretation” without fixing the mathematics have such a profound impact on the underlying... | It basically boils down to the term $e^{-\frac i\hbar E t}$, where the minus can either be included in the energy, making it negative, or into the time. But a negative charge moving backwards in time is exactly the same as a positive charge moving forwards in time, and that is much more sensible than negative energy.
| {
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Where on Earth does the mass of 1 kg actually produce a 1 kg reading on a digital scale? Gravity on Earth varies by about 0.1% between poles and equator. If someone was buying/selling something mass critical e.g. gold, where is the standard place on Earth where a 1 kg mass produces a 1 kg reading as measured by a devic... | These digital scales basically measure the force $F$ required to counteract the weight force when used properly. From a measurement of the force, the scale then converts this to a mass measurement using some conversion akin to $m=F/g$. On different places on Earth, you'll get different "mass" measurements since these d... | {
"language": "en",
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About Right-Hand-Rule and Cross Projuct In "Physics for Scientists and Engineers with Modern Physics - 8th Edition", p. 842, it gives the total magnetic force on the segment of current-carrying wire of length $L$ in a uniform magnetic filed:
$$
\vec{F_{B}}= I\vec{L} \times \vec{B}
$$
According to Cross Product, it sho... | The first example determines the force created by a current carrying conductor in a magnetic field. The second example determines the current created in a conductor moving through a magnetic field. They are both correct.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What does the exponential decay constant depend on? We know the law of radioactivity:
$$N=N_0e^{-\lambda t}$$
where $\lambda$ is the exponential decay constant. My question is: This constant depends of what?
| Here is a table of isotopes versus lifetimes the color code of the lifetimes on the right hand column:
Isotope half-lives. Note that the darker more stable isotope region departs from the line of protons (Z) = neutrons (N), as the element number Z becomes larger
Modeling a nucleus is a many body problem and also a ... | {
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Is there a scientific term for star formation? It might be my stupidity to think that many laymen terms that most people use to describe some physics phenomena usually have a scientifically accepted term or name?
The process of star formation, does it have a specific name generally used in the scientific word? Or is S... | An ADS search for "star formation" turns up about 142,000 articles with "star formation" in the title or abstract. The first article is a 43 page review paper of Star Formation in Galaxies in the Hubble Sequence, written by Robert Kennicutt, Jr, one of the leaders of the field. He never defines anything else to mean st... | {
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Will an object rotate when we apply a force to it? What would happen if the axis of rotation passes through the centre of mass of an object? Will the object rotate when we will apply a force to the object?
Edit: The object is free, is not fixed to an axis of rotation and force is perpendicular to the object.
| The question (even after the edit) is not very clear. So I will make some general statements about forces, objects and rotation.
In order to cause a change in the angular momentum of an object (which is one interpretation of "rotate", although it can mean "stop rotating" too), you need to apply a torque.
A torque resul... | {
"language": "en",
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What does it mean to say "a paramagnetic material is attracted to an external magnetic field?" I'm just having a hard time wrapping my head around what actually goes on when a paramagnetic material is exposed to an external magnetic field. I understand that the individual dipoles line up so that they point in the direc... | Magnetic forces are not easy to apprehend. Personally, I dislike magnets, so in a first step I will use coils. Consider two coils $S_1$ and $S_2$ along the $\vec{z}$ axis at a distance $d$ one from each other. They are fed by a current $\vec{i}=I\vec{e_\theta}$.
As you know, the magnetic field induced by each coil is l... | {
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1-particle non-interacting Green function At $T=0$ in the non-interacting case the $1$-particle Green function for an electron in the excited state $\lambda$ (empty band) is of the form
\begin{eqnarray}
G^{(0)}(\lambda,t-t') = -i \theta(t-t') e^{-i\epsilon_\lambda(t-t')}
\end{eqnarray}
where $H_0 C_\lambda^\dagger|0\ra... | $E$ is not the energy per se of the particle, it is a Fourier parameter. What gives you the possible energy accessible to the particle is the spectral function
$$\rho(E)=-{\rm Im}(G(E))\propto\delta(E-\epsilon_\lambda),$$
which is peaked at the accessible energy of the free particle $\epsilon_\lambda$.
| {
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What do we see while watching light? Waves or particles? I'm trying to understand quantum physics. I'm pretty familiar with it but I can't decide what counts as observing to cause particle behave (at least when it's about lights). So the question is what do we see with our eye-balls?
*
*We point a laser (or any kind... | This is misconception that light is some kind of 'mix' of waves and particles. Instead, It actually IS both waves and particles at the same time, you can't separate them from each other.
So probably, the answer could be: you see particles as well as waves.
| {
"language": "en",
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Capacitance of two non parallel plates What is the formula for capacitance of two non parallel plates at an angle with each other?If the plates were parallel then the value can be calculated as
(PermittivityX area of one plate)/distance between them.But what happens in case the plates are tilted at an angle?The questio... | Assuming that the charge distribution is constant, using the knowledge that capacitance is added in parallel, you could treat your angled plate as being comprised of infinitely many parallel plates, approximating the angle of the plate you would like. You would then be able to integrate across this infinity of plates t... | {
"language": "en",
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Thermodynamics - ideal gas Question:
1 mol of a monoatomic gas at 298 Kelvin acquires a volume of 3 litres. It is expanded adiabatically and reversibly to a pressure of of 1 atm. It is then compressed isothermally and reversibly until its volume becomes 3 litres. Calculate the change in entropy.
What I tried:
As both ... | The gas is expanded adiabatically and then isothermally. Thus the temperature it has at the end of adiabatic expansion stays the same even after the isothermal process.
Ideal Gas equation after adiabatic expansion: $p_aV_a=nRT_a$, where index "a" shows after.
You do not have $V_a, T_a$ in this equation.
However, anothe... | {
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Is it possible to make laser beams visible midair without smoke? My question is: Is it possible to make laser beams viable midair without smoke? I thought it would be a great idea to have a (second) smartphone or pc screen without having a solid screen. The reason why it has to be without smoke is that smoke would be t... | I recently saw a video of a demonstration by a Japanese researcher who came up with a method that used a pair of high-powered (presumably) infrared laser beams that, where they intersected, heated the air enough to turn it into plasma, creating a pulse of white light. It works, but it's slow, low-resolution, & requires... | {
"language": "en",
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Time for a particle undergoing brownian motion to reach a point in a volume I was wondering how one could calculate the average time a particle needs to reach a random point in a small sphere (filled by water) with a radius of maybe $10 \mu m$. I thought of using the Stokes-Einstein-Equation, but then I just get a diff... | First remark, the average hitting time is finite because the volume is finite. None of what I write would make sense in an infinite system.
Let us consider that the target is a ball of radius $a$ at the center of the sphere and let us call $T(\vec r)$ the average hitting time for a Brownian particle starting at positi... | {
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Separability of a Hilbert space and its implications for the formalism of QM In the text I'm using for QM, one of the properties listed for Hilbert space that is a mystery to me is the property that it is separable. Quoted from text (N. Zettili: Quantum Mechanics: Concepts and Applications, p. 81):
There exists a Cauc... | I usually see it in the reverse way, but it is a matter of taste. Hilbert spaces, in general, can have bases of arbitrarily high cardinality. The specific one used on QM is, by construction, isomorphic to the space L2, the space of square-integrable functions. From there you can show that this particular Hilbert space... | {
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Why do the high frequency waves have the most number of modes? While reading the Wikipedia page of Ultraviolet Catastrophe, I came across how Rayleigh and Jeans applied the equipartition theorem. They told that each mode must have same energy. Now as the number of modes are greatest in small wavelengths or large freque... | You need to look at the idea of Separation of Variables for Partial Differential Equations.
You consider a toy universe comprising oscillators in a box: let's think of a cuboid microwave cavity with electromagnetic fields losslessly confined within perfectly reflecting walls.
The Cartesian components of the electric fi... | {
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What is the largest number of bosons placed in a BEC? What is the record for the largest number of bosons placed in a Bose-Einstein condensate?
What are the prospects for how high this might get in the future?
EDIT: These guys reported 20 million atoms in 2008. "Large atom number Bose-Einstein Condensate machines" (pd... | I would suspect the What is the largest number of bosons placed in a BEC would be the largest vat of liquid below boiling point 4He. I think 4He must be in the ground state to be a non boiling liquid and this is effectively a BEC.
| {
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Fourier and inverse fourier transform in QFT According to my lecture notes, the inverse Fourier transform of an operator $\phi(p)$ is given by
$$\phi(x)=\int \frac {d^4p}{(2\pi)^4}\phi(p)e^{-ip\cdot x}.$$
As @WenChern pointed out below, Peskin has a somewhat different formula on p.20:
$$\phi(\mathbf{x},t) = \int \frac ... | Hint:
1. $\phi(x,t)$ at different times are not independent.
2. $\int{d^4p\delta(p^2-m^2)}=\int{d^4p\frac{\delta(p^0-E_p)}{2p^0}}$. The left side of this equation is Lorentz invariant.
This time your question is much clearer.
If $\phi(x)$ is an arbitrary function of $x$, there's nothing confusing. If $\phi(x)$ is... | {
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Does "normal torque" exist? Is there any force called normal torque? If a ruler is spinning, and it hits the floor, obviously it will stop. The floor must be exerting a "normal torque" on the ruler. If this exists, please tell me what it is really called.
| Torque is not a force. You can say there is a torque caused by normal forces, but there is no special name for that.
A normal force comes from acting with a force on an object resting next to a surface. The surface prevents the object from moving through it by producing a reaction force that is necessarily normal (perp... | {
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Where do Newton's Laws not work? I'm working on high school level project about Newton's Laws and I picked topic that describes situations, where they dont work. Can you name any practical cases where they do not work? Why do they not work in special and general relativity or quantum mechanics? Why do they not work on ... | *
*Cartoon physics. For example, after running off a cliff, Wile E. Coyote does not begin to fall until he notices.
*The thing about many physical "laws" is that they are very good generalities, often with exceptions. For example.
*
*You can prove that the angles of a triangle always add up to 180-deg, except for... | {
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What would Earth have been without the Moon? Would Earth rotation have been more slowed down because of the tidal effect from the Sun, as seems to be the case with Mercury and Venus? Due to the giant impact hypothesis the angular momentum from the impact was increased and split.
If Earth not would have been a two part ... | As shown in a previous answer,
Assuming they have the same density (the Sun's average density is not much smaller than that of the moon) , if they had the same apparent size in the sky, then the mass M of the object will grow as r3 (because M=4/3ρπR3 and R=θr), so the force actually grows linearly with r.
this impli... | {
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Gibbs free energy intuition What is Gibbs free energy? As my book explains:
Gibbs energy is the energy of a system available for work.
So, what does it want to tell? Why is it free? Energy means ability to do work. What is so special about this energy? Can anyone simply explain?
I just want a math-free intuition.
| Short answer: Gibbs free energy $G = U + PV - TS$ combines internal energy $U$, pressure $P$, volume $V$, temperature $T$, and entropy $S$ into a single quantity that measures spontaneity. With that I mean that processes lowering the Gibbs free energy of your system will spontaneously occur, and equilibrium is reached ... | {
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Using tracking detector in a double slit experiment, what would we see? Let's say we put tracking detector (eg. a cloud chamber or a more advanced device) behind the double slits.
What would we see?
I think the interference pattern is three dimensional. So there are hyperbolic regions where waves cancel. So in these r... | That's an excellent suggestion for an experiment since Claus Jönsson has performed his experiment with electrons at a double slit. Since the electrons must be in a vacuum, it is questionable whether the diffraction pattern shows in a cloud chamber.
However, if it succeeds it may be, that in a single-photon experiment o... | {
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Can the relative position of two masses connected by a spring become negative?
Consider the following diagram. Two masses of 1 kilogram each are attached by a spring of 1 N/m. The $x$-axis is chosen such that $x_1(0)=0$ and $x_2(0)=L$ where $L$ is the length of the spring in relaxation (no restoring force acting on th... | I don't think this is the correct answer to the problem. The correct answer is pretty straightforward and simple (it's a sine function times a constant). Check out Kleppner and Kolenkow (Intro to Mechanics, page 128) for the correct solution. Also, I've done this problem using the Lagrangian, but I've never seen Laplac... | {
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Cooling a satellite Satellites are isolated systems, the only way for it to transfer body heat to outer space is thermal radiation. There are solar panels, so there is continuous energy flow to inner system. No airflow to transfer the accumulated heat outer space easily. What kind of cooling system are being used in sa... | As an example, the International Space Station (ISS) has external thermal radiators. They looks similar to solar panels, but instead of pointing the flat side towards the sun, they point towards empty space. An ammonia loop carries heat from various parts of the space station to the radiators.
This is a picture of a ra... | {
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Kirchhoff's current law with a nonlinear resistor It is said that by Kirchhoff's current law
$$
\frac{e - v_c}{R_1} = c\frac{dv}{dt} + f(v_c) + i\tag{1}
$$
and from Kirchhoff's voltage law
$$
v_c(t) = iR_2 + L\frac{di}{dt}\tag{2}
$$
from the following diagram:
It is easy to see equation (2) but I dont see how equatio... | Kirchhoff's Voltage Law states that the voltage around a closed mesh or loop is zero. In this case, taking an 'imaginary loop current', $i$ around the 'central' mesh in your circuit, you get:
$$v_c(t)-v_{R2}(t)-v_L(t)=0$$
where $v_{R2}(t)$ and $v_L(t)$ are the voltage drop across the variable resistor $R_2(t)$ and indu... | {
"language": "en",
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How was it proven that a quantum entanglement measurement of particle A, affects properties of particle B If I understood the wikipedia article correctly, quantum entanglement claims that information travels instantly between entangled particles. An act of measurement on one of the entangled particles changes (I think ... | This video well explains Bell's Stern–Gerlach experiment that is cited as proving that quantum entanglement is not explained by predetermined "hidden variables".
https://www.youtube.com/watch?v=ZuvK-od647c
Basically, the experiment involves detecting the spin of two entangled particles via the use of 3 randomly determi... | {
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How do microwaves heat moisture-free items? Today I learnt that microwaves heat food by blasting electromagnetic waves through the water molecules found in the food.
Does that mean food with 0% moisture (if such a thing exists - dried spices?) will never receive heat from a microwave oven? And how in that case is a mic... | I don't think it's dielectric heating .. it's more akin to inductive heating .. the establishment of eddy currents (largely surficial) which heat through resistance heating. Best material would be moderately electrically conductive
substance .. hence ceramics containing metal atoms heat - where ceramics containing e... | {
"language": "en",
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What does $m^*>m_e$ imply? (the effective mass of electron is larger than its rest mass) From what I understand, the concept of effective mass is just something people come up with to make electrons and holes obey the equation of motion
$$
\vec{F}=m^* \vec{a}
$$
without dealing with the charge carrier and the crystal a... | Second derivative of kinetic energy with respect to momentum equals inverse mass of a particle. In a metal, you have a band structure defined through the dispersion relation of the form E(k) where k is wave vector of electron. Second derivative of this expression can be also taken to be some sort of inertia of a partic... | {
"language": "en",
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Newton's third law and punching a glass or a feather According to Newton's third law, action force equals reaction force in terms of magnitude.
When I punch a glass, the glass punches me back. If I exert a greater force on the glass, it will break.
Suppose a glass could sustain 100N force and that my muscles can exert... | The question as it stands is not very clear. What you describe as a "200N force" is what you would measure if your fist hit a force probe. What is in fact physical is the momentum of your fist. If you imparted some momentum to your fist and set it in motion on a collision course with the feather, then Newton's laws wo... | {
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How do you define a "universe" (in the context of multiverse)? How do you define a universe (in the context of multiverse)?
The traditional definition of universe is something like
"The Universe is all of spacetime and everything that exists therein".
But in multiverse theory one talks about "universes" and this raises... | It is a pseudoproblem of definition. Some people define universe as everything that could ever possible exist, to them the word multiverse is an oxymoron. But those who like to use the idea of a multiverse, use it encompassing different things depending on context. For instance, Max tegmark defines 4 different levels o... | {
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How do I make a Gaussian process generate a stream of events? I would like to generate a stream of "random" gaussian events (on for event - off for no-event) that obeys a set standard deviation and mean. what algorithm could I use?
thank you
| First method:
The principle
You can always transform one distribution to another through a transformation of the independent variable, so we get if $p:\mathbb{R}^N\to(\mathbb{R}^+\bigcup\{0\})^N$ is a probability density function of $N$ variables and we transform the independent variables $\mathbb{R}^N\to\mathbb{R}^N$ ... | {
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What is Optimal Control Theory and Controllability Theory? I was exploring the methods to analyze the evolution of a system from one quantum state to another using a suitable Hamiltonian. Some searching led me to the keywords Controllability Theory and Optimal Control Theory. Although both these topics are concepts in ... | The quantum equivalent to the optimal control of densities is a topic of interest in the areas of large population control, optimal mass transport and mean field games (try https://arxiv.org/abs/1810.06064) in the control theory. The idea in the related literature is to transport a density given the individual micro-dy... | {
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Why is Graphene Transparent? Graphene is always in the news now a days and its key features are that it is; very strong, conductive and transparent. It is so transparent that each layer of graphene will only absorb 2% of Light passing through it.
But what is it about the structure of Graphene which makes it (almost) tr... | Graphene is only transparent because it is very thin (one atom thick). If it absorbs 2% per layer then just a few hundred layers would absorb almost all light and that would still be a very thin sheet of graphite.
The question should be why does graphene absorb so much light compared to diamond which really is transpa... | {
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What is the procedure (matrix) for change of basis to go from Cartesian to polar coordinates and vice versa? I'm following along with these notes, and at a certain point it talks about change of basis to go from polar to Cartesian coordinates and vice versa. It gives the following relations:
$$\begin{pmatrix}
A_r \\
A_... | What you need to do to derive the transformation matrix is first wrtie out all the relation between both coordinations.
For example, the relation between cartesian coordinates and polar coordinates are:
$$x=r\cos \theta \hspace{1cm} y=r\sin \theta $$
$$r=\sqrt{x^2+y^2} \hspace{1cm} \theta = \tan^{-1} \frac{y}{x}$$
From... | {
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First law of thermodynamics In the first law of thermodynamics, we learned that $W$ and $Q$ are path-dependent quantities, but how are $Q$ and $W$ defined?
I mean $W = \int_{\gamma} p(s) ds$ would be one possibility, where $\gamma$ is a path from volume 1 to volume 2, but how can I calculate $p$ as a function of $V$ in... | As you wrote $W = \int_{\gamma} p(s) ds$ you are implying that the pressure, I assume you denoted pressure by $p$, is function of only the volume $V$ and the dummy integration variable is in fact the volume. That is not the case, the thermal equation of state even for the simplest system involves the (absolute or empir... | {
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Does constant velocity of center of mass imply linear momentum is conserved? I know that if momentum is conserved for a system, you can derive that the velocity of the center of mass of that system is constant. I was wondering if the second condition also implies the first: if I can demonstrate that the velocity of the... | yes, you assumption is correct, for an isolated system, conervation of linear momentum is equivalent to the velocity of the center of mass being constant. The term with variable mass from another answer is incorrect. You can only have variable mass in a non-isolated sysrtem.
| {
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Complex comjugate of Schrodinger equation: paradox in matrix form? We can take the complex conjugate of schrodinger equation, and obtain
$$
-\frac{\hbar^2 }{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi = i \hbar \frac{\partial \psi}{\partial t}
$$
$$
-\frac{\hbar^2 }{2m}\frac{\partial^2\psi^*}{\partial x^2} + V(x... | The origin of the eigenvalue equation $H\phi=E\phi$ is the separation ansatz $$\psi(x,t)=\exp{\left(-i\frac{E}{\hbar}t\right)}\phi(x)$$
If you conjugate this, this will obviously change the sign of the exponent and therefore you will the same eigenvalue.
What you are trying to state would be something like "if $\lambda... | {
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What will happen to a permanent magnet if we keep the same magnetic poles of two magnets close together for a long time? What will happen to permanent magnet's magnetic field or magnetic ability if we keep same magnetic poles of two permanent magnet for long time?
Will any magnetic loss happen over the long period of ... | I'm trying this now. I have 2 magnets
(round with hole in middle). One is suspended on top of the other with a 1/2"plastic pipe.This site won't accept my pic as is. I believe energy is being used to keep the one magnete suspended. As a result, these magnets will lose their strength and eventual... | {
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Why does room temperature water and metal feel almost as cool as each other? From what I've read about heat, temperature and conductivity, I understand that the reason water at room temperature feels colder than most other things at the same temperature (like wood, air, cotton) is because of its higher thermal conducti... | The specific heat of air is more than the specific heat of oxygen in water. Therefore, the water is colder than air. neglect the specific heat of hydrogen in water as its density is negligible comparing to the oxygen. Specific heat of oxygen is 0.9 J/g/cCelcius, specific heat of air about 1 J/g/c
| {
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Charged dielectric/conductor in capacitor It is a standard problem to consider a dielectric or a conductor between the parallel plates of a capacitor. But what happens to capacity, voltage, charge, inserting between the plates of an ideal capacitor a charged dielectric or a charged conductor (without contact with the p... | There are two case it depends on the battery is connected with constant voltage then There are two main cases:-
1)When Battery is connected $V$=constant
a)Capacity:-
$$C_{0}=\frac{Q}{V_{0}}$$.For a medium permitivity constant will be written as In general,
$$\epsilon_{m}=\epsilon_{0}K$$($\epsilon_{m}$ the permitiv... | {
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Is there a confirmation of dark matter signal? Dark matter, as we know does not emit light, so confirmation of its presence is indirect. Are there any recent indirect confirmations of dark matter. A place one would look for in detecting dark matter would be to detect gravitational effects of this matter on other. In t... | To date there is nothing published (and serious) that makes a confirmed detection of dark matter particles. Thus, the only evidence in favor of its existence remains from indirect methods: calculate the mass that should be there based on visible sources (stars, galactic powder, etc), and use this mass to calculate the... | {
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Formula for the magnetic field due to a current loop I need expressions for the $\mathbf B$ field generated by a circular current loop at a point off-axis from the ring's axis of symmetry.
The ones I came across on the internet aren't very convincing. I verified them with Mathematica, and none seems to be correct ─ I'm... | Remember, $\vec{\nabla} \times \vec{B} = \mu_0 \vec{J}$, and $\vec{J}$ will be zero anywhere except on the loop itself, where it will be singular. Do you mean, perhaps, that the line integral around the loop equals the current, a la Ampere's law?
| {
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Physical examples where changing the order of limits yields wrong result In mathematics it is generally not allowed to change order of limits. For example it is not always true for a sequence of functions $f_n$, that $\int_a^b \left(\sum_{n=0}^\infty f_n(x) \right) dx = \sum_{n=0}^\infty \left(\int_a^b f_n(x) dx\right)... | For example, in statistical mechanics, you get different results for systems with spontaneous symmetry breaking, say, for a ferromagnetic, depending on whether you first take the limit $N\rightarrow\infty$ or $H\rightarrow 0$ when calculating average magnetization (http://www.encyclopediaofmath.org/index.php/Quasi-ave... | {
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Half-integer eigenvalues of orbital angular momentum Why do we exclude half-integer values of the orbital angular momentum?
It's clear for me that an angular momentum operator can only have integer values or half-integer values. However, it's not clear why the orbital angular momentum only has integer eigenvalues.
Of ... | I had already answered this question, but my answer had a fatal flaw. This should be correct.
First, we have to make a physical demand. We demand that a rotation by $2\pi$ of spatial configuration (distinct from an internal configuration, i.e. spin) leaves physics invariant. From a study of the first homotopy group $Z... | {
"language": "en",
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Why is the gravitational force $10^{38}$ times smaller than the strong nuclear force? Also, why is the weak interaction force $10^7$ times smaller than the strong nuclear force?
| The Anthropic answer to this question is that if gravity were a lot stronger, then the evolution of the universe would have proceeded in a different way, it would have collapsed just after the Big Bang. One can speculate that all possiblities really exists, but we can obviously only find ourselves in those universes wi... | {
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Electric field inside a diode When a voltage is applied to a diode (forward or reversed bias) the depletion zone is changed due to charges change in this region. My question is in both case (forward or reversed bias), how the electric field that is responsible of moving the charges in the P and N region is established ... | You are talking about a p-n junction which forms a diode. Now a diode junction is very small compared to the rest of semiconductor. The diode junction can in a first approximation be taken as an insulator since it does not have any mobile charges compared to the rest of the region. When a voltage is applied at the ends... | {
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Temperature of electroweak phase transition How does one estimate the temperature at which electroweak phase transition (EWPT) occurred? Somewhere I have read it is around 100GeV but the reason was not explained.
| I don't think that an unambiguous justification can be given because the dynamic of the electroweak symmetry breaking (EWSB) is still unknown. We don't have a well established theory describing how the Higgs scalar potential evolves with the temperature. When people talk about the scale of the EWSB, they usually refer ... | {
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Why do floating water drops form spheres? Consider a drop of water floating in an inertial frame in STP air (e.g., the ISS). Intuitively, the equilibrium shape of the drop is a sphere.
How would one prove that? Is it equivalent to showing that the minimal surface area for a simply connected volume in $\mathbb{R}^3$ wit... | Another way to look at it is the following.
The main force on the molecules will come from other water molecules and be due to cohesion. The system will try to minimize it's energy and bond the molecules together as much as possible. This means minimizing the surface which results in a sphere.
| {
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What does amplitude in wavelength of light physically mean?What oscilates with time in photon? Like amplitude in wavelength of water waves signify the displacement of water particles about their mean position.
| Although you can (as you obviously know) think of electromagnetic radiation as either a particle or a wave, it's easier in this case to think of it as a wave.
As a thought experiment, if you wave a magnet near a piece of wire, an electric potential will be induced in the wire. Likewise, if you pass current through a w... | {
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Can velocity be an undefined quantity? We have the image below displaying the uniform velocity by time-distance graph. At every point velocity is constant but what if distance and time both become zero as at origin in the graph is? The velocity must become undefined as $\frac{0}{0}$ is undefined in mathematics or shall... | Actually you have plotted the graph of displacement $x(t)$,
And here the $x(t)=vt$ ,
$v$ is some constant.
now lets take the slope of the graph i.e
$ dx/dt =v $
The slope is constant (equal to $v$ and we physically call it velocity)
now what is the slope at (0,0)?
since the slope is constant it will be still $v$,righ... | {
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How is energy transferred in Joules law of heating? Joule's law of heating states that an accelerated electron loses its energy, which is then converted into heat energy, by colliding with vibrating atom i.e ions in their lattice site. but we know atom consist of electrons and a nucleus. Where does it collide? How does... | That's a very hard question to answer with the appropriate level of detail! Very broadly speaking in an ideal metal all atoms are forming a perfectly regular crystal lattice. Conduction band electrons can move freely around these atoms, which makes it easy to pass a current trough the metal.
In a (theoretical) metal w... | {
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How can electric field representation be obtained from Enge representation using Maxwell's equations? Suppose we have a long electric capacitor. Let $L$ be its length ($z$ coordinate), $W$ its width ($y$ coordinate), and $D$ its full height (full aperture; $x$ coordinate). Let $L\gg W\gg D$.
The shape of it is similar ... | Let $E_{0}$ be the $x$ component of the electric field at $z\rightarrow-\infty$. Then the $x$ component of the electric field $E\left(0,0,z\right)
$ on the reference orbit can be approximated by $E_{x}\left(0,0,z\right)
=E_{0}F\left(z\right)$.
By Gauss's law $\nabla E\left(x,y,z\right)=0$ near the reference orbit. As... | {
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Why are position and velocity enough for prediction and acceleration is unnecessary? In classical mechanics, if you take a snapshot and get the momentary positions and velocities of all particles in a system, you can derive all past and future paths of the particles. It doesn't seem obvious why the position and its fir... |
Why doesn't anything need the second derivative (acceleration)?
Only Newton's gravity law does not use acceleration in the expression for force. In electromagnetic theory with retarded fields, forces are functions of past positions, velocities and accelerations of the charged particles.
| {
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A precise definition of macroscopic and microscopic objects What are the formal definitions of "macroscopic object" and "microscopic object". How can one differentiate between them? I mean, is there any fixed condition by which we can distinguish between them?
| I suspect this will be closed as "opinion based". I don't believe there is a canonical answer. Usually microscopic scale relates to phenomena that occur on a level much smaller than the system under consideration (atoms in a crystal when you are thinking about the crystal, for example).
There is an analogy with micro- ... | {
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Different frequencies working together How do the different waves of EM spectrum present in the environment not interfere with each other? If they do, how does everything work properly?
The radio waves of mobile phones and wi-fi work together. Why don't they collide with each other, since they are physically present?
| It's a good question, but the answer isn't particularly simple. The wikipedia page on the Superposition Principle is a decent starting point. If you feel like you understand the basics, there's a nice simulation of superimposing different frequency components that might illustrate the idea for you.
| {
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What is the energy of a Gaussian wave packet? Suppose we have a potential barrier situation, that is $V(x)$ is zero everywhere except on the interval $[-a,a]$, where it is equal to some $V_0 > 0$. Introduce some Gaussian shaped wave packet to the left of the barrier, moving right.
What is the energy of the packet (i.e... | As Nemis L. pointed out, the expectation value $\langle H\rangle$ is constant, because of Ehrenfest's theorem:
$$\frac{d}{dt} \langle H \rangle = \frac{1}{i \hbar} \langle [ H,H ] \rangle = 0.$$
The other way of seeing this is that the state can be written as a superposition of orthogonal energy eigenstates.
Obligatory... | {
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What's the true reason behind thermal expansion? Thermal expansion is a normal concept everyday. There are 2 explanations:
1, thermal expansion result in stress, then result in deformation
2, thermal expansion result in deformation, then result in stress
I am confused about it. Could you explain the thermal expansion?
| Roughly speaking solid matter is on a lattice form,
A three-dimensional lattice filled with two molecules A and B, here shown as black and white spheres.
The molecules fit like LEGO , the forces tying them together are mainly the spill over electric field forces , attractive and repulsive forming the patterns of th... | {
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Resource cost and noise effects in quantum teleportation of multible (entangled) qbits Suppose you have n qubits that are in an unknown state (may be entangled, etc).
Can you teleport this state by teleporting each qubit individually (using a Bell state and a classical channel)?
If not, how many classical bits and wha... | Yes, a many-qubit state (even if it entangled) can be teleported by teleporting each qubit separately using one (perfect) Bell pair and two bits of classical communication. (This is the idea of quantum teleportation: The qubit, including all of its quantum correlations, is teleported.)
This can be seen by observing th... | {
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London equation and Maxwell equation I am reading Tinkham's Introduction to Superconductivity (2nd ed) (Amazon link).
On pages 4-5, they state that:"
The second London equation 1.4, when combined with the Maxwell equation ${\rm curl}\ h = \frac{4\pi J}{c}$ leads to:
$$\nabla^2 h = \frac{h}{\lambda^2}$$
where equati... | Hints:
*
*Given the Maxwell equation:
$$
\nabla\times\mathbf h=\frac{4\pi}{c}\mathbf J
$$
Take the curl of both sides, what do you get? How can (1.4) be used here?
*Vector calculus tells you
$$
\nabla\times\nabla\times\mathbf a=\nabla\left(\nabla\cdot\mathbf a\right)-\nabla^2\mathbf a
$$
What assumptions must you m... | {
"language": "en",
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planewave Ansatz for modelling phonon dispersion in crystals From Ashcroft's "Solid State Physics", for one-dimensional monatomic Bravais lattice, the equations of motion of ions are:
\begin{equation}
M\ddot u(na)=-K[2u(na)-u([n-1]a)-u([n+1]a)]
\end{equation}
And we seek solutions of the form:
\begin{equation}
u(na,t)\... | This is the classical treatment to model vibrations in solids, using the analogy with vibrations of a one-or-two dimensional monatomic or diatomic chains. Which basically boils down to writing Newton's equation of motion to find out the force on each mass when the whole system constitutes of masses attached by Hookean ... | {
"language": "en",
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Is Boltzmann distribution contradicting with the fundamental assumption of statistical thermodynamics? In equilibrium statistical physics the fundamental assumption of statistical thermodynamics states that the occupation of any microstate is equally probable (i.e. $p_i=1/\Omega, S=-k_B\sum p_i\,{\rm ln}\,p_i=k_B{\rm l... | To clarify, the full assumption is that all states are equally probable, in the absence of any knowledge about the state of the system.
With the Boltzmann distribution (AKA canonical ensemble) this assumption doesn't apply since we have knowledge about the system. In particular we know that if the system is put into co... | {
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Cosmic Background Radiation and redshift vs. temperature I get that the CMB has gone from high energy photons, to low microwave photons today. And that is probably due to the redshift from the expanding Universe. But, since CMB is a black body it is temperature dependent, i.e. it has a high peak at high temperatures, b... | The blackbody radiation due to the cosmic microwave background was in thermodynamic equilibrium with the rest of the Universe at a temperature of about 3000K when it "decoupled".
The decoupling event (I never understand why it is often call recombination, since the protons and electrons were never combined previously),... | {
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General question about the potential barrier problem: Why does $\exp( kx)$ diverge when $x>0$ in the case when $E < V(x)$? For the two images below, the first potential barrier has particles approaching it where $E > V_o$ & the second has a particle that has $E < V_o$, where $E$ is the energy of the particles and $V_o$... | Just try to solve the Schroedinger's equation. You'll find that for $x>0$, if you take the solution as $e^{ikx}$, then
$$k=\frac{\sqrt{2m(E-V_0)}}{\hbar}.$$
Now if $E<V_0$, then $k$ is imaginary, i.e. $k_2=ik$ is real, and thus $e^{ikx}=e^{k_2x}$ is real. Thus in case $E<V_0$ one of the solutions vanishes at infinity, ... | {
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Why does a pitcher with lemon juice have foam, while one with pure water does not? Whenever I pour water into lemon juice (pouring directly from the tap into the pitcher, not quietly along its edge) I get a foam on top:
The same pitcher with water (same water tap, pitcher, time between the water poured and the picture... | Manufactured 'lemon juices' such as cordials typically contain emulsifiers which can act as surfactants to lower the surface tension of the liquid. This would result in a foam forming more easily when air becomes 'trapped' during pouring water into the jug.
Even homemade lemon squash which often has honey added as a sw... | {
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Does the diffracted electron radiate photons? When electron is diffracted after the slit it might follow different direction, than before the slit. That means, that going through the slit it gains some acceleration. And accelerated charge emits photons.
Thus - does the diffracted electron radiate photons?
| I believe the answers given so far are off the mark and do not directly answer the OP's question: do diffracted electrons emit bremsstrahlung radiation/soft photons?
So I give the answer here: The answer is a resounding yes!
However since the diffracted electrons are scattered in the far forward direction (tiny scat... | {
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Tensor Product vs. Direct Product for three spin-1/2 particles Let us consider three spin-1/2 particles and only focusing on their intrinsic spin $S$. The Hilbert space has then to be $\mathcal H = ℂ^2 ⊗ ℂ^2 ⊗ ℂ^2$. The spin can be described by $V ∈ \text{SU(2)}$ and the fundamental representation $\mathcal D_{1/2}$ wi... | This is mainly about your question 2.
A decomposition of the Hilbert space into a direct sum $\mathcal H = \mathcal H_1 \oplus \mathcal H_2$ represents that the system can be in a state in $\mathcal H_1$, or a state in $\mathcal H_2$ (or a superposition, of course). In your example, $\mathcal H = \mathcal D_{3/2} \opl... | {
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How confident can we be that the speed of light in a medium is constant? I have recently found this article http://news.harvard.edu/gazette/1999/02.18/light.html it tells that physicists have been able to slow the speed of light. Is this hokum? If not how is it possible to use light as a measure of distance? Surely doi... | That article is about slowing the speed of light in a physical medium (note the mention of the 'exotic medium' in the third paragraph), a common and well-known phenomenon (even ordinary transparent mediums like water or glass will slow light waves traveling through them to some extent), the newsworthy aspect was just t... | {
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How to calculate U-spin for baryons I am trying to calculate U-spin for sigma baryons. I don't know why U-spin for $\Sigma^{+}$ and $\Sigma^{-}$ and $\Sigma^{*+}$ is 1/2, but for $\Sigma^{*-}$ is 3/2?
I know that $\Sigma^{+}$ and $\Sigma^{-}$ are octet, and $\Sigma^{*+}$ and $\Sigma^{*-}$ are decuplet, but I don't und... | U-spin is very similar to isospin which interchanges u and d quarks, except, now, U-spin interchanges d and s quarks, so, in the Eightfold-way weight multiplets, it transitions to lower right from upper left---unlike for left to right for isospin.
So, behold!, in the baryon octet, the $\Sigma^-$ is part of a U-doublet... | {
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Electron scattering to measure the nuclear radius I have been taught that you can find out the size of a nucleus of an atom by firing electrons at high velocities at the atom. This causing scattering due the positive charge of the nucleus and diffraction due to the small de Broglie wavelength.
Part of this relies on th... | If we evaluate the DeBroglie wavelength of electrons (e.g. with this online calculator), then we find that an electron energy of 1 eV leads us to $1.2\times10^{-9}\,\mathrm m$ or about 1 nm (that's the size of small organic molecules with 2-3 benzene rings), 15 eV for the size of a hydrogen atom. If we use 1 GeV electr... | {
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Given a wave function $\psi(x)$, is there always a potential $V(x)$ such that $\psi(x)$ is an eigenstate? Given any unit norm wave function $\psi(x)$ which is in the Hilbert space, can we always find a $V(x)$ such that the $\psi(x)e^{-i\omega t}$ is a solution of the corresponding Schrödinger equation? (I mean the Hami... | As the question is about physics and not mathematics the point is that the Schrödinger equation is only defined if a real valued potential exists.
So any wave function - defined as a solution of the given equation - has necessarily at least one corresponding potential. If someone wants to solve your intended problem ma... | {
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Raising and Lowering indices of tensor Why we use metric tensors $g$ to raise or lower indices of tensors, why not using other (invertible) order-2 tensors to do the job?
| Given a finite-dimensional vector space $V$ over some field, there is no natural isomorphism with $V^*$, its dual vector space. If $V$ is equipped with an inner product, this can be used to define an isomorphism between $V$ and $V^*$ by sending any linear form $\omega\in V^*$ to the unique vector $v_\omega \in V$ that ... | {
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How do electromagnetic waves travel in a vacuum? This is perhaps a total newbie question, and I will try to formulate it the best I can, so here it goes. How does an electromagnetic wave travel through for example, the vacuum of space?
I usually see that waves are explained using analogies with water, pieces of rope, ... | Photons are released in packets. So,when the source is generating the electromagnetic wave it's actually being released in packet forms. These packets are self entities that travel on it's own and doesn't need any medium to sustain it.
| {
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Why aren't (domestic) kettles insulated? In my experience of buying and using kettles, I have come across none which are insulated.
The obvious reasons as to why it would be beneficial being that heating time would be reduced, similarly, less power hence money would be required to heat an arbitrary volume of water. Som... | Most kettles are silver to minimize heat loss through radiation. (They also have small exit holes at the top to minimize heat loss of steam because conversion of liquid water to steam requires latent heat)
I expect the reason that there is usually no thermal insulation is that kettles heat water very quickly and becaus... | {
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Do particles always flow from high to low pressure? In a recent question, it was stated that
particles in high pressure air always flow to lower pressure.
In a pipe with a constriction, fluid flows from from low to high pressure after the constriction. (From here.)
How are these concepts related?
| It doesn't necessarily flow from higher pressure to lower, but from higher energy to lower energy, as per Bernoulli's theorem. That's the answer you want here when you are pertaining to the venturimeter.
Bernoulli's theorem states that total energy remains same between any two points. Total energy includes pressure ene... | {
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Was everything in the Universe "created" from light? Is the following true?
The only matter existing directly after the big bang was electromagnetic radiation.
| No.
We don't know what happened in the very early stages of the Big Bang because we have no experimentally tested theory that takes us back that far. However, courtesy of the LHC we have an experimentally tested theory that takes us back to a time called the electroweak epoch, and we can use this theory to answer your ... | {
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What's the closed-form of the sum relating to the DOS of simple harmonic motion? In order to calculate the density of states of single particle in the simple harmonic potential, we would calculate that
$$
D(\epsilon)=\sum_{n}\delta(\epsilon-\epsilon_n)
$$
where $\epsilon_n=(n+1/2)\hbar\omega$. In the limit $\hbar\omega... | There is a formal manipulation that can answer your question using well-known formulas. So, let me write
$$
\epsilon_n=(2n+1)\epsilon_0
$$
being $\epsilon_0=\hbar\omega/2$ and
$$
\delta(\epsilon-\epsilon_n)=\int_{-\infty}^{+\infty}\frac{dt}{2\pi}e^{-i(\epsilon-\epsilon_n)t}.
$$
Then,
$$
D(\epsilon)=\int_{-\in... | {
"language": "en",
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Pressure equation working to get units kg m/s$^2$, but stuck? Why does $kg/m^3 \cdot m/s^2 \cdot m$ become $kg\,m/s^2$ (i.e. one newton)
When I do the working, I get $kg/(s^2 m)$ (i.e. metres is on the bottom rather than the top)
My working is:
\begin{align}
\frac{kg}{m^3} \cdot \frac{m}{s^2} \cdot m&= \frac{kg}{m\... | By multiplying the units you gave in the question, it looks like you're referring to $P=\rho g h$, where $P$ is pressure, $\rho$ is mass density, $g$ is gravitational acceleration, and $h$ is the depth in some fluid (of density $\rho$) at which the pressure is being measured/calculated. Recall that the definition of pr... | {
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Is differential geometry used in solid state? I'm an undergraduate in physics interested in a career in solid state. While I know that any additional math is helpful--I am on time constraints, and can only take a few supplemental classes.
That said, is differential geometry used much in solid state physics? I'm aware o... | There is theory of dispersion in crystals. One can say that the differential geometry is used there. I think it is Group theory + differential geometry.
| {
"language": "en",
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Dumbbell rotation
What if apply impulse to dumbbell consisting of two masses and massless rod? It is clear for me that left dumbbell will move straight line. But it is not clear for me what kind of motion will have right dumbbell. It will certainly have rotational component, but will its center of mass move?
Let's def... | It looks like you've already done all the work. It is correct. Yes the center of mass will move.
Any unbalanced force on the object will cause an acceleration of the center of mass. $F=ma$
Any unbalanced force on the object that does not go through the center of mass will provide a torque and a rotational acceleratio... | {
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Centripetal force in this example? I have a general question regarding the centripetal force. In the example of a ferris wheel where there is a normal force pushing up against the person and the gravitational force pulling the person down, which force is centripetal? I know that the centripetal force counters the linea... | The component of the net force pointing to the center is the centripetal force. In your Ferris wheel example, we can choose three points: top (12 o'clock), bottom (6 o'clock) and halfway (3 o'clock). At 12, centripetal is normal force plus gravitational force; at bottom they are on opposite direction ($F_c=F_n-F_g$) an... | {
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Is there a physical interpretation of a tensor as a vector with additional qualities? What is a tensor? has been asked before, with the most highly up-voted answer defining a tensor of rank $k$ as a vector of a tensor of rank $k-1$.
But if a scalar is defined as a physical quantity with a magnitude, a vector as a physi... | If you want antisymmetric tensors, there are well known geometric building blocks. For instance if you take two orthogonal vectors you can multiply them to get the oriented plane they span (with the orientation determined by the order you multiplied them). Similarly for three mutually orthogonal vectors, note there a... | {
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What position of the center of gravity can make the front wheels of the car lift off the ground? I have a question regarding the position of center of gravity required to just lift off the front wheel of a vehicle
Consider a vehicle of mass $m$ having a center of gravity at height $h = 0.5m$ from the ground. The coeffi... | It's just a simple sign-error.
You wrote
$$ mgl_2+~ma0.5=0~.$$
But when you accelerate the car to the right then you get a reaction force $ma$ which points in the other direction, so your $m \vec a$-arrow should point to the left. The equation then reads
$$ mgl_2-~ma0.5=0~.$$
Then you have your torque balance about th... | {
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How much additional light does Earth receive from the Sun due to Earth's gravitational field? I was reading about how gravity affects light, and that got me wondering how much additional light is collected by the Sun due to the Earth's gravitational field.
Is it a significant amount of light (>1% of total light)? Is it... | The amount of deflection of light (the bending of the null geodesic) passing a star is a deflection angle of 4m/R where m is the mass of the star and R is the radius of the star. The mass of the sun is $2 \times 10^{30}\rm\,kg$, where as the mass of the Earth is $6 \times 10^{24}\rm\,kg$. The radius of the earth is $6... | {
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If two photons collide, does the resulting particle have zero velocity? If two photons traveling in opposite directions along the same line collide, will the resulting particle have a velocity of zero relative to the rest of time space in the instant of the collision?
| Generally no, because velocity is not a conserved quantity. It is momentum that is conserved in all interactions. For photons, the magnitude of momentum is simply
$$ p = \frac{E}{c} = \frac{h\nu}{c} = \frac{h}{\lambda}, $$
so photons with different energies/frequencies/wavelengths will have different momenta. If the to... | {
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What is the effect of torque steering on a motorcycle with a long wheel base? From experience, it appears motorcycles with a large wheelbase coupled with a long handlebar (say a modified Harley Ape Hangar) shows a noticeable tendency to veer to a side during acceleration vs a motorcycle with a short wheelbase and short... | Although your question is simple, the answer involves some rather complex modeling of dynamic systems to analyze and explain the stability of the bicycle (as well as motorcycles) and what parameters affect it. Here is an on-line article that doesn't go too deeply into the mathematics, but gives you an idea of the thing... | {
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Ampere's law of circular path when "bulging" out
The picture shows a use of Ampere's law. A circular path is chosen.
$$\oint \vec B \bullet \mathrm{d}\vec l=\mu_0 I_{encl}$$
When using Ampere's law we are talking about the current enclosed. That means, the current through the surface enclosed by the path.
The text to ... | The explanation of this is enclosed in Stoke's theorem. In the most general formulation, Stokes theorem asserts that
$$\oint_{\partial\Omega}\alpha = \int_\Omega\text d\alpha$$
where $\partial\Omega$ is the boundary of the cycle $\Omega$ and $\text d\alpha$ is the exterior derivative of the form $\alpha$.
In the case o... | {
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Mass in special relativity? Is the mass of a object at rest defined by $$E=mc^2$$
where $m$ is the rest mass. I.e. does the rest mass include every thing from thermal to gravitational potential energy and every other possible energy that it could have at rest. And thus if we write the following:
$$total\ energy=mc^2+p... | No, $E=mc^2$ covers only the mass energy of the object. Requirement of gravitational potential or thermal energy correctly require additional terms to the energy equation.
It's worth noting they are usually neglected because they are so small compared with the mass energy. Consider a rock of mass 1kg a few metres above... | {
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"answer_id": 1
} |
Why does a plane look like it's slower than a car? Recently, I was in a car and noticed a plane in the sky. What was interesting was that the plane seemed to go slower than my car because I passed the plane. Is there a physics reason to this?
| There can be only two possibilities:1. In case the plane and the car were moving in the same direction the plane was indeed flying with slower speed than the car.2. In case the plane was flying with greater speed than the car the plane was going in some other direction. The component of velocity of the plane in the dir... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/159849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Apparent depth and virtual image position
(source: bbci.co.uk)
Why does the virtual image appear right on top of the object and not a little to the right, for example? Is it explained by some formula or just symmetry of the geometry?
| The answer is that, when looking at an object, in order to determine where it is, you need several rays coming at slightly different angles and from all over the object's surface. Using Snell's Law and tracing back the rays to the point where they converge you get the full image and its apparent position. The next imag... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/159906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Symplectic geometry in thermodynamics There seems to be analogues between Hamiltonian dynamics and thermodynamics given the Legendre transforms between Lagrangian and Hamiltonian functions and all of Maxwell's relations. Poincarè tried to generalise classical mechanics to the realm of statistical thermodynamics with er... | No answers yet? So let's take a shot at a (partial) answer:
Therefore as my main question, does symplectic geometry underpin thermodynamics?
No. In thermodynamics, we're dealing with a Legendrian submanifold of a contact manifold (cf Wikipedia). Thermodynamic variables are canonical coordinates on that manifold.
Mora... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/159980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
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