Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Meaning of wave function squared, notational confusion For a non-degenerate ground state in a system with $N$ electrons, we may write the wave function as,
\begin{equation}
\psi(r_1, r_2,..., r_N)
\end{equation}
Where the $r_i$ represent the position of the $i^{th}$ electron. When we say the square of this wave functio... | This is known as the electron density and it is obtained by integrating over the coordinates of all electrons except one.
For simplicity, let us take a 2-electorn case with wavefunction $\psi(r_1, r_2)$ and ignore the spin. Then the electron density is
$$\rho(r) = \int dr_1 |\psi(r_1, r)|^2 + \int dr_2 |\psi(r,r_2)|^2 ... | {
"language": "en",
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Bell State Measurement Algorithm I'm relatively new to quantum computation and am taking a course in it. I was wondering if it is possible to code an algorithm which would be able to take an input of a 2 qubit state and perform a bell state measurement in order to which of the four bell states the two qubits are in. Or... | A Bell state is prepared with Hadamard gate applied on first qubit and CNOT gate with control on first qubit and target on second qubit.
As a quantum computation is reversible, putting transpose conjugate of above mentioned gate in reverse order returns the original input.
So, assume that you have some unknown Bell sta... | {
"language": "en",
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How can $1/V$ be equal to $0$ in Boyle's Law? In relation to ideal gases, Boyle's Law states that pressure is inversely proportional to volume under constant temperature. In other words,
$$P \propto 1/V$$
Below is a graph that plots pressure, $P$, against inverse volume, $1/V$.
How can $1/V$ ever equal zero? How is th... | Formally we have
$$
\lim_{V\rightarrow \infty} \frac{1}{V}=0
$$
I.e. the statement is true in the limit of infinite volume. For an ideal gas, this can be interpreted as saying that as the confining volume for the gas becomes infinite, the gas no longer exerts a normal pressure on the walls of the volume.
| {
"language": "en",
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Rolling down a slope I am having some trouble understanding rotational dynamics. If we have a cylinder that we give an initial velocity and rotation such that it satisfies the non slip condition ($ v_{cm} = \omega R$) and let it roll down a slope where its weight component down the slope equals the static friction up t... | There is a condition for object to be rolling without slipping. As show above by BioPhysicst, the static firctional force $f$ equals to (adopts the inertial moment for a cylinder of mass $M$ and radius $R$ : $ I = \frac{1}{2} M R^2$,
$$
f = \frac{1}{3} M g \sin \theta \tag{1}
$$
and this frictional force is arised f... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/605226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Submarine buoyancy Maybe a silly question but..
A submarine has ballast tanks to control it's buoyancy. To submerge, water is allowed to fill the ballast tanks which displaces the air inside.
When a submarine is submerged it contains a fixed amount of air compressed and uncompressed. By filling the ballast tanks with c... | The buoyant force is equal to the weight of the water that is displaced by the submarine. It comes down to the average density of the total submarine volume versus the density of water. By replacing water in the ballast tanks with air you are effectively shifting this balance because the air is about 1000x less dense... | {
"language": "en",
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Free Body Diagram for a Body on a Smooth Inclined Plane Consider a body of mass $m$ on a smooth inclined plane with the normal force and weight labelled as shown:
Now, usually when the forces are broken into its components, we get an FBD like this:
From this we can obtain the equation:
$$ \Sigma F_\perp = 0\quad \imp... | If instead of decomposing into perpendicular and parallel force components, you use the 'regular' $x$ and $y$ axis as you did to get Eq. (2), then you will have a non-zero acceleration in both $x$ and $y$ directions. So, what you really should end up with for Eq. (2) would be $$ma_y=mg-N\cos\theta\tag2$$
because the fo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Where does the law of conservation of momentum apply? Take the scenario of a snowball hitting a tree and stopping. Initially, the snowball had momentum but now neither the snowball nor tree have momentum, so momentum is lost (thus the law of conservation of momentum is violated?). Or since the tree has such a large mas... | Momentum of the body will be transferred to earth, let's exaggerate the numbers in your favor and assume your snowball is $10kg$ and you managed to throw it with $v_s = 200\frac{m}{s}$ where subscript "s" stands for snowball. Mass of the earth is approximately $5.97\cdot10^{24}kg$. For simplicity assume it was an elast... | {
"language": "en",
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Mass sliding down a plane Say we have a mass $m$ sliding down a plane with friction. Will the plane move too if it's mass is finite? Because the force of friction acting on the mass $m$ must react on the plane too but isn't it supposed to dissipate and be unusable.
| Yes, assuming the the surface on which the plane rests is frictionless (or has a small enough coefficient of friction such that the static friction between plane and earth is 'small enough' it will move in the opposite direction).
Consider the following sketch:
I drew the FBD for the block and the incline (or the plan... | {
"language": "en",
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What makes a standing wave a wave? Well, this is my physics professor's question and it really made me think a lot about standing waves, realising I don't actually understand it. What makes a standing wave a wave? How could I explain it? What actually makes wave a wave?
| As John Rennie had already written a nice answer, I would just like to add something more. Standing waves can form under a variety of conditions in a medium which is finite or bounded. But they don't usually form under any circumstances. They require that energy be fed into a system at an appropriate frequency, i.e., w... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/606189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can 1 kilogram of radioactive material with half life of 5 years just decay in the next minute? I wondered this since my teacher told us about half life of radioactive materials back in school. It seems intuitive to me to think this way, but I wonder if there's a deeper explanation which proves me wrong.
When many atom... | In order for that to happen in the real world you need to start with about 3.8 million kilograms of that material.
Here is how you come up with that number. You start from the the formula connecting the half-life to the number of particles over time
$$ N(t) = N_0 \left(\frac{1}{2}\right)^\frac{t}{t_{1/2}} $$
Now you re... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why have Majorana fermions not been detected? There are two types of fermions - Dirac's and Majorana's. Majorana's fermions are their own antiparticles and they have not been detected yet. Sometimes, it is conjectured that e.g. neutrinos could be Majorana fermions.
However, it seems to me that there should be no Majora... |
Since there is no distinction between particles and antiparticle in case of Majorana fermions, a bunch of Majorana fermions should completely annihilate
It is not that simple, to model a baryon dominated universe. The same thoughts for Dirac fermions lead to the problem of why there is baryon asymmetry in the observe... | {
"language": "en",
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Cancellation of vacuum bubbles in the correlation function $$
G_K = \langle\Omega|T\prod_{i=1}^{2K}\phi(x_i)|\Omega\rangle.
$$
A problem sheet question asks you to 'Show that the contribution to $G_K$ of $O(λ_L)$, $L$ integer, in which all external points are connected to a single cluster contains no bubble diagrams.'
... | Yep, you are right and the problem sheet is wrong. See Timo Weigand's notes section 2.8, and specifically equation (2.145) which reads
$$\left\langle \Omega \right| T \left\{\prod_i \phi(x_i) \right\}\left| \Omega \right\rangle = \sum (\text{all partially connected diagrams with } n \text{ external points}). $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How would one estimate the age of a black hole? The age of the universe is a question that has long been asked and supposedly been answered.
My question however is a lot more specific rather than a general age of the entire universe. How would one estimate the age of an object that is without information? As for instan... | It can't be done. As you write, black holes are without information (more commonly people say black holes have "no hair"). Therefore you can't directly estimate the age of a black hole.
What can be done (and it's not always possible) is to use auxiliary information to guess at the age of the black hole. For example, if... | {
"language": "en",
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Force analysis in a compound pulley system At 4:56 of this video on compound pulley systems with the following schematic set up:
The professor does a force analysis of the forces on the pulley A , and strangely (for me) , he finds that there is tension acting vertically at two diametrically opposite ends of the pulley... | When analyzing pulley systems we do not normally take into account the force of the loop on top of the pulley. We would use the tension of the rope on each side of the pulley. Discounting small frictions in the pulley and rope, the tension is the same throughout the rope. So pulley A has rope tension on each side for a... | {
"language": "en",
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Do different kind of waves in a medium interact with each other? Say for example, air and sound can both propagate to air. Now, suppose we have a sound and light wave travelling close by, how would the two waves interact with each other? Suppose I made a very large sound in the path of the ray, would that make any devi... | Carl is right, and if you are flying at max cruise in a newer Boeing airliner and have a window seat on the wing you can actually see the index of refraction "kink" in the air above the wing's skin at about 2/3rds of the way back along the chord, where the air flowing over the wing goes locally supersonic for a bit.
| {
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Why does falling water become less and less focused as it falls down? So, I was observing my water purifier fill up my water bottle and I noticed that the beam of water coming through it became less and less focused as it falls down more.
Now, I do understand the concept of surface tension but I don't get it why this h... | I guess by "less and less focused" you mean water turning from streamline to turbulent flow. Initially the velocity of flowing water is less than some threshold velocity, so its flow is ordered( streamline), but when it moves down, velocity increases and flow becomes turbulent. The threshold velocity above which flow b... | {
"language": "en",
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Will the speed of object be the same for observers with different speeds if the speed of the object is closer to the speed of light? If we assume an object X traveling at a speed $V$ that is a bit slower than the speed of light (let's say 1000 miles/sec.), what will be the difference in the observations of the speed o... | $v_{XA}=0$. The value of $v_{XB}$ will be the result of the relativistic velocity addition of velocities $V$ and $c-V$, which is greater than $c-V$ and less than $c$.
| {
"language": "en",
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Partial derivative in Newtons Second law Newton's second law states Force is the time derivative of momentum. But is it a total derivative or partial derivative? What is the reason behind it?
| As a rule of thumb, physically meaningful quantities use total derivatives. Partial derivatives are mostly mathematical tools that help us express total derivatives, since the implicit functional dependencies that even allow us to speak of a difference between partial derivatives and total ones aren't inherent features... | {
"language": "en",
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Problem regarding components of forces in conical pendulum
For the motion of conical pendulum we can write equations as
$$T_{F}\cos\theta =mg$$
$$T_{F}\sin\theta=\frac{mv^2}{R}$$
$T_{F}$ represents tension in the string.
$v$ is the velocity of bob at this instant.
$R$ is the radius of circle.
But if we split $mg$ into... |
you have three forces that must be in equilibrium, these is independent of the coordinate system that you chosen
x-y system
$$\sum F_x={\frac {m{v}^{2}}{R}}-T_{{F}}\sin \left( \theta \right)= 0$$
$$\sum F_y=-mg+T_{{F}}\cos \left( \theta \right)=0 $$
x'-y' system
$$\sum F_x'={\frac {m{v}^{2}\cos \left( \theta \right) }... | {
"language": "en",
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If a jet engine is bolted to the equator, does the Earth speed up? If a jet engine is bolted to the equator near ground level and run with the exhaust pointing west, does the earth speed up, albeit imperceptibly? Or does the Earth's atmosphere absorb the energy of the exhaust, and transfer it back to the ground, cancel... | A jet engine requires fuel.
Let us build it over a huge oilfield, next to a refinery. Over the years (centuries?) that it runs, the evacuation of the oil deposit will cause the Earth to lose mass and also to shrink slightly in diameter through subsidence. This will tend to speed up the rotation of the Earth.
There is n... | {
"language": "en",
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Are multiple images an exclusive propertiy of gravitational lenses? Strong gravitational lensing produces multiple images if the object lens and observer are in a specific alignment. This is due to there being multiple stationary points in the time arrival surface for the paths the light can take. So far I understand t... |
[W]hat prevents a classical lens, for lack of better term, from displaying the same behaviour?
Classical lenses are designed to produce a single image under normal circumstances.
For a gravitational lens corresponding to a point mass the angle of ray deflection increases the closer it passes to the center, while for ... | {
"language": "en",
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Meeting point of magnets Suppose, we leave two magnets in space close to each other with no other force acting on them apart from the attractive force between them. One is stronger than the other but, they have the same mass. Would they meet at the middle or closer to the stronger?
It’s important to clarify that althou... | Newton's Third Law says that the force exerted on Magnet #1 by Magnet #2 is equal in magnitude to the force exerted on Magnet #2 by Magnet #1. Since they have the same mass, they will therefore accelerate at the same rate, and they will therefore meet exactly in the middle.
The relative strength of the magnets is a re... | {
"language": "en",
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How much of the momentum is being transfer in case of collision relative to elastically and momentum difference? if we take a look at two objects that has been collision ,we may use the the third law of newton to calculate the velocity of each one if we know the force they use on each other. for example if we take newt... | We can say that momentum is conserved in the collision. So, in your example: $m_1 = 1000$, $v_1 = 100 \implies p_1 = 10000$. The total momentum after the collision is also $10000$. But there are infinite combinations of velocities for object 1 and 2 that fulfill that requirement. There is no formula to find the final v... | {
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What does applying a potential difference mean? I have been studying the physics of accelerating electrons through a vacuum using an electron gun. It says that a potential difference (p.d.) is applied between the filament lamp and an anode. The electrical energy of the electrons is converted to the K.E as the electrons... | Applying a potential difference means modifying a system such that the potential energy at point A is higher than at point B. In the case of an electric potential, this potential difference is caused by the electromagnetic force.
$$F_{el} = -\frac{1}{4\pi \epsilon_0}\frac{qQ}{r^2}$$
The magnitude of a potential differe... | {
"language": "en",
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A statistical definition of temperature In order to define temperature in terms of statistics, "Concepts of Thermal Physics" by Blundell states:
Let us assume
that the first system can be in any one of $Ω_1(E_1)$ microstates and the
second system can be in any one of $Ω_2(E_2)$ microstates. Thus the whole
system can ... | Your argument is correct. What is not told explicitly here is that before the systems come in contact you can consider $E_1,E_2$ fixed. After you bring them in contact they are no longer fixed but the quantity $E=E_1+E_2$ is fixed. This leaves only $E_1$ as the free parameter (or $E_2$, just pick one). This means that ... | {
"language": "en",
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What is the contribution of sea quarks to proton mass? I've read in this article that the 3 valence quarks make up only 0.2% of the mass of a proton (or neutron) and the remaining 99.8% comes from gluons. From other sources we know that there are an infinite number of sea quarks popping in and out of existence, being... | The answer is 98.5% is the contribution of the sea of virtual gluons, quarks and antiquarks (that is, the interaction between the virtual gluons, quarks and antiquarks in the sea) to the total rest mass of the proton.
The masses of the quarks entering the proton when summed have mass less than 15 MeV, and the proton ... | {
"language": "en",
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Why are infrared images not great? Why are infrared images not great?
Simply googling infrared images, and looking at pictures of things like streets, people, animals, etc. . . I can't quite pin down why infrared images aren't great. I can think of a few ideas but I can't convince myself of any of them. I'd appreciate... | It is often the number of pixels in the sensor that is limiting the quality.
The sensors are often made of a thermoelectric material. When it absorbs radiation, the temperature of the pixel goes up and this induces a voltage (like in piezoelectrics), which is then read out. It is probably useless to make small pixels b... | {
"language": "en",
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Is Joule's law of heat generated by electricity only applicable for ideal resistors? I have learned about Joule's law of heat generated by electricity. It states that $H=VIt=I^2Rt$. But here's where I'm confused: if the whole electrical energy does not get converted into thermal energy then how can we write $H=W=VIt=I^... | In the case of a DC fan, while the contacts supply a DC voltage, a commutator changes several times per second its polarity in the electromagnets of the motor, in order to keep a torque between stator and rotor. So the voltage (and current) in the device is far from continuous, and its relation is not given simply by O... | {
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In QFT why does the degree of the interaction terms in Lagrangian start from 3? I'm new to QFT so it's not obvious to me why there is no quadratic interaction terms in Lagrangians.
For example, the Lagrangian for a real scalar field is
$$L=\frac{1}{2}\partial_\mu \phi \partial^\mu \phi-\frac{1}{2}m^2\phi^2-\sum_{n\geq ... |
...why does the degree of the interaction terms in Lagrangian start from 3?
...What's the reason that we can't add terms like $g\phi^2$ to the free field Lagrangian?
You can add such a $g\phi^2$ term if you want, but it's not usually called an "interaction" term, since if $g$ is constant you can just combine it wit... | {
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Time spent by a comet inside the Earth's orbital (Kepler problem re-visit) I come across this interesting problem comet-x-earth. It was an exam problem asking the time that a comet will be spent inside the Earth's orbital. I make an illustration for the problem:
The comet is in a parabolic orbit (i.e. its total energy... | I try to work it out following the clues from G.Smith for a complete record.
in 2-d polar coordinate, the velocity $\vec{v}$
$$
\vec{v} = \dot{r} \hat{r} + r \dot{\theta} \hat{\theta} = v_r \hat{r} + v_{\theta} \hat{\theta}
$$
The radial component $ v_r = \dot{r} = \frac{dr}{dt}$. The time $T$ from point $A$ to poi... | {
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RMOND theory and Schwarzschild metric In a recent paper https://arxiv.org/abs/2007.00082, two authors propose a new model (RMOND) which seems succesful to explain many cosmological data. Obviously, the GR theory must have as classical limit the modified equations of gravity of RMOND. Is the Schwarzschild metric modifie... | Of course the Schwarzschild metric is modified by this new RMOND theory.
Newton's gravity is the weak field limit of Schwarzschild metric. And MOND is a modification of Newton's gravity. Therefore any relativistic extension of MOND theory (including RMOND) has to modify the Schwarzschild metric at least in the weak fie... | {
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Rotational Dynamics Problem Involving Multiple Rods
In the figure shown, the rod $OA$ of length $1 \ \rm m$ is rotating
about the fixed point $O$ and the rod $AB$ of length $1\ \rm m$ is
rotating about $A$, in a vertical plane. At a certain moment, when
$OA$ is making an angle $60^\circ$ with the horizontal, the rod ... | This is a standard kinematics problem.
The steps to solve these is to first designate the degrees of freedom of the system. Use the two orientation angles $\theta_1$ and $\theta_2$ is conventionally what is used for pin joints.
Then track the points of interest (since dynamics are not involved, each center of mass isn... | {
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What is quasineutral in plasma physics? What is quasineutral gas? In plasma physics, what exactly quasineutral means? In definition of plasma, it is written that plasma is a quasineutral gas of charged and neutral particles which exhibit collective behaviour.
| Quasi-neutrality in a plasma frequency is just saying that the following holds on distances larger than the the Debye length:
$$
n_{e} = \sum_{s} \ Z_{s} \ n_{s} \tag{0}
$$
where $n_{s}$ is the number density of electrons, $Z_{s}$ is the charge state of ion species $s$, and $n_{s}$ is the number density of ion species... | {
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Calculation of time for electronic transitions How do we determine the time for electronic transitions in atoms or in semiconductor devices, from one energy level to another?
| The lifetime of an atom in an excited state can be estimated from the Fermi Golden rule:
$$
\frac{1}{\tau_{i\rightarrow f}} = \Gamma_{i\rightarrow f} =\frac{2\pi}{\hbar}|\langle i|\hat{V}|f\rangle|^2\delta(E_i-E_f)
$$
What seems to pose the problem in this expression is the delta-function.
In a semiconductor it is actu... | {
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What 'sets' the curvature in the FRW metric? In the FRW metric, the way I understand it, there are 4 key quantities, $P$ (pressure), $\rho$ (matter density), $\Lambda$ (the cosmological constant) and $K$ (the curvature constant).
$P$ and $\rho$ are fairly self explanatory, while $\Lambda$ we attribute to 'dark energy',... | The curvature is set my the densities of the fluids comprimising the Universe.
Consider the first FL-equation
\begin{equation}
\frac{3}{R^2} \left( k + \frac{\dot{R}^2}{c^2} \right) = \frac{8 \pi G}{c^2}\rho_{tot}
\end{equation}
with
\begin{equation}
\rho_{tot}(t) = \rho_{\gamma}(t) + \rho_{M}(t) + \rho_{\Lambda}(t)
\e... | {
"language": "en",
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Why don't you feel gravity the same way you feel a car's acceleration? If you are in an accelerating car or spaceship you feel an apparent force pushing you backwards, which as I understand is due to your own body's inertia (it wants to keep its current velocity but the vehicle's velocity keeps changing as it accelerat... | gravity and acceleration feel the same to humans.
so you do in fact feel it.
| {
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Translational and Rotational Kinetic Energy Does an object that only spins have translational kinetic energy?
Since $v=\omega r$ ($\omega$ is angular velocity), if substituting $v$ with $\omega r$, won't that allow an object that only spins to have translational kinetic energy?
Thinking about the other way, won't an ob... | You are confusing different types of motion. If a body only spins but stays at the same point, say $r=0$, then it has no translational kinetic energy for $v = w r = 0$.
If a point mass moves on an orbit, say circular, then one can speak of both, linear and angular energies, which however must be identical $E = mv^2/2 =... | {
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Why is the tension equal to the spring force here?
Here the block is oscillating and to solve this question I took the tention in the string to be equal to the spring force
But if that's the case a particle in the junction of the spring and the rope will expirence a net force of zero yet it still it goes up and down e... | You are right, but usually we consider an ideal, massless string and same for the spring. So that particle in the junction has zero mass. In reality the tension changes both along the spring as well as along the string so any small portion experience an acceleration due to the gradient in tension.
| {
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Deriving equation 9.5 relating to the path integral in Peskin and Schroeder I have not encountered a great deal about the path integral but I am trying to learn it, I would like to derive by hand each step in Peskin and Schroeder however I am hitting a wall at equation $9.5$ on page $279$:
$$U(x_a,x_b;T) = \int_{-\inft... | I'm not sure why you think that the propagator can't be a function of $(x'-x_b)$, it lies within the domain for that parameter and has the effect of shifting the endpoint of the propagator by a fixed distance. Nevertheless, this is actually immaterial since we don't encounter the object $U(x_a, \ x'{-}x_b, \ T{-}\epsil... | {
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Validity of Ampère's law in terms of $H$ We know that the auxiliary magnetic field $\bf{H}$ is
$$\mathbf{H}=\frac{1}{\mu_{0}} \mathbf{B}-\mathbf{M}$$
and
$$\mathbf{\nabla} \times \mathbf{H}=\mathbf{J}_{f}$$
but this differential equation is generally not valid at the boundary of a magnetized body due to an abrupt chan... | The contour integral across the boundary should not be considered as the direct application of Stoke's theorem in this area, but in a spirit similar to the principle of analytical continuation.
The Stokes' theorem
$$
\oint_C \mathbf{H}\times d\bf{l} = \iint_A \boldsymbol\nabla \times \mathbf{H}\cdot d\mathbf{a} = \math... | {
"language": "en",
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If displacement is 0, does that mean initial velocity equals final velocity? For instance, one of the kinematic equations is :
$$v_f^2 = v_i^2 + 2ad$$
where $v_f$ is final velocity, $v_i$ is initial velocity, $a$ is acceleration, and $d$ is displacement.
Say for instance a guy rides a bike in circles for hours with ini... | As @AccidentalTaylorExpansion and @David White discuss in their answers, your relationship is only valid for constant linear acceleration. Your situation is rotational, not linear motion. Also, acceleration is a vector and is not constant for circular motion.
For circular rotational motion of a particle about a fixed... | {
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Pure Rolling Motion A small doubt in rolling motion. As we know, for pure rolling, the velocity of the lowermost point has to be zero wrt the ground or whatever platform it may be on.
Suppose, we have a body (let's say a ring) and it has velocity $v_0$. Assuming the floor has friction, which serves as the torque to cau... | You can use either method, but only your top equation is correct. If you define the rotation axis at the ground, then you must get rid of the $Mvr$ term (or, equivalently, $v=0$ in that term). At any given instant, the motion of the rolling ring is exactly the same as it would be if it were rotating about the instant... | {
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Will liquid nitrogen evaporate if left in an unopened container? SOS! I left work today and got a horrible feeling that I forgot to put the lid back on a large container of liquid nitrogen which contains many racks of frozen cells in it. If this did happen, how long would it take liquid nitrogen to evaporate? Does it s... | Anecdotally, a well-insulated cylinder of LN$_2$ which is 3 feet tall and 6 inches in diameter will take several (maybe 3 or 4) days to completely evaporate if open to the air. The rate at which your nitrogen level falls will be determined by the exposed surface area-to-volume ratio as well as the quality of the conta... | {
"language": "en",
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Finding the centre of mass in polar coords with double integrals The centre of mass of a body can be found using the general formula:
$$
\bar{\boldsymbol{r}} = \frac{1}{M} \int \boldsymbol{r} \ \mathrm{d}M
$$
(RHB, p. 195)*. When I try to use this method with polar coordinates however, it fails.
Consider a circular dis... | The integral you worte down is wrong. The right one would be
$$
\vec r _{CM} = \frac \mu M\int_0^{2\pi}\int_0^a\begin{pmatrix} r\cos(\phi)\\r\sin(\phi) \end{pmatrix} r \text d r \text d \phi = \begin{pmatrix}0\\0\end{pmatrix},
$$
since
$$
\int_0 ^{2\pi}\sin\phi\,\text d\phi = \int_0 ^{2\pi}\cos\phi\,\text d\phi = 0
$$
| {
"language": "en",
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Calculating RMS Radius of a Globular Cluster I am coding a simulation of a globular cluster in python, starting with particles of the same mass and
$0$ initial velocity in a random spherical distribution. I am trying to investigate the root-mean-square radius of the system over time and trying to see how long it takes ... | Isn't a better metric just the radius that includes half of your original particles (or half the mass if the particles are of different masses) ?
If the evaporation rate is still so high that you feel even this is compromised, then you could instead work out the half-mass radius for particles that are within some arbit... | {
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Induced electric field inside a perfect conductor If I have a magnetic field and I place a coil that is a perfect conductor in this field that I'm rotating.
This would mean that there is a change in the flux through the area of the coil which would induce an emf in the coil.
But shouldn't the electric field inside a pe... | Yes. The electric field in a conductor is zero but because the coil is being rotated in a magnetic field, there is a change in magnetic flux and the emf that’s generated is given by
$$\epsilon = - \frac{d \phi}{dt}$$
where $\phi$ is the magnetic flux.
Since the magnetic field is assumed to be constant (and obviously th... | {
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Professors gave two different solutions for the same problem. What is the right one?
"A ball $m$ moves horizontally with velocity $v_0$, colliding with a spring $k$ fixed to a block mass $M$ such as shown in the diagram. The block $M$ has no friction with the ground. What is the maximum compression of the spring?"
Th... | The comments made by @Ben51 and @David White are both correct that the second answer is incorrect unless the mass $M$ is fixed, which it is not since it's on a frictionless surface, or that $M$ is infinitely large (or at least $M>>>>>m$) so that its movement after the collision is immeasurably small. But the problem do... | {
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Why are interacting field theories called nonlinear? Explanation for interacting EM field, in particular The classical equation of motion for the electromagnetic field interacting with a charged fermion field $\psi$ of charge $eq$ is given by $$\Box A^\mu(x)=j^\mu(x)$$ where $j^\mu(x)=eq\bar{\psi}(x)\gamma^\mu\psi(x)$.... | I think the following is a clean and correct way to see why the equation for $\psi$ is nonlinear. The solution of the first equation is $$A^\mu(x)=\int d^4y ~G(x-y)j^\mu(y)=eq\int d^4y ~G(x-y)\bar{\psi}(y)\gamma_\mu\psi(y)$$ where $$G(x-y)=\int\frac{d^4p}{(2\pi)^4}\frac{e^{ip\cdot(x-y)}}{p^2+i\epsilon}.$$ Substituting ... | {
"language": "en",
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Why is Minkowski metric diagonal? Why is the Minkowski metric a diagonal in a 4x4 matrix? What does the diagonal do?
| Two facts are sufficient to make Minkowski tensor globally diagonal:
*
*Symmetry of the metric tensor $g_{ij} = g_{ji}$, the simple reason for this can be found in the answer to this question Why is the metric tensor symmetric?. For every symmetric matrix one can switch to the basis of eigenvectors, where it is diago... | {
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What does the "true" visible light spectrum look like? When I google "visible light spectrum", I get essentially the same image. However, in each of them the "width" of any given color is different.
What does the "true" visible light spectrum look like, then? It can't be that each and every image search result is corre... |
...the "width" of any given color is different
This is to be expected, since the unit of the horizontal axis might be different.
Two separate bands that are equally wide on a frequency (energy) scale will always have different width on a wavelength based chart.
| {
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Is a wave function a ket? I just started with Dirac notation, and I am a bit clueless to say the least. I can see Schrödinger's equation is given in terms of kets. Would I be correct to assume if I were given a wavefunction, say $\Psi(x)=A\exp(-ikx)$, would I be able to just use the notation $\lvert \Psi\rangle =A\exp(... | A ket is an abstract vector which describes the state of your system. A wavefunction is a representation of that vector in a particular basis, usually the position basis (although you will sometimes also hear about, e.x. "momentum-space wavefunctions"). The way the position-space wavefunction $\psi$ is defined is by le... | {
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How does Inertia opposes change in state? The first line of Wikipedia states that Inertia opposes change in velocity. My teachers also told that Inertia opposes change in state of rest or motion.
But I would like to know how?
Suppose a body is at rest as seen from an inertial frame and a net external force is applied t... | Inertia is an intrinsic property of matter. The term inertia is used in two different senses. In the directional sense, it means "momentum". Think of it as suddenly stopping while traveling by car. You would move forward. Is there any physical force that makes you move forward, no, there isn't. This is why we don't hav... | {
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Are there fields (of any kind) inside a black hole? It is said that nothing escapes from black holes, not even light. All particles are now thought to be excitation of different fields (electric field, electromagnetic field, photon field, etc).
Does it follow that there are no fields (of whatever kind) inside the event... | Sure, the reisner-Nordström solution is given by:
$$ds^{2} = -f(r)dt^{2} + \frac{1}{f(r)}dr^{2} + r^{2}d\theta^{2} + r^{2}\sin^{2}\phi^{2}$$
where
$$f = 1 -\frac{2M}{r} + \frac{q^{2}}{r^{2}}$$
This is a solution to the Einstein-Maxwell equations with 4-vector potential:
$$A_{a}ds^{a} = \frac{q}{r}dt$$
So, this solution... | {
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What are the necessary and sufficient conditions for a wavefunction to be physically possible? Often times it is stated in books that a quantum state is physically realizable only if it is square integrable. For example, in Griffiths (2018 edition) page 14 he stated
Physically realizable states correspond to the
squar... | This answer is meant as a supplement to Valter Moretti's one.
Quantum systems are defined by their Hamiltonian, however, this is normally not their single observable. We have the basic observables stemming from the space-time context (coordinate, momentum, spin), plus the Hamiltonian, plus other observables stemming fr... | {
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Why is intensity required here?
Q)A point source of light is placed at the centre of curvature of a hemispherical surface. The radius of curvature is r and the inner surface is completely reflecting. Find the force on the hemisphere due to the light falling on it if the source emits a power W.
I tried attempting this... | (I don't have enough rep to comment)
The equation that you have used is derived for normal incidence (specifically for elastic collision assuming light as a particle).
| {
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Why does the bullet have greater KE than the rifle? A rifle is fired, and the bullet moves much faster than the rifle, and momentum is conserved.
My question is whether the kinetic energy of the bullet is greater than the kinetic energy of the rifle because the mass of it is smaller, therefore the force acting on it (f... | I think the easiest way to work this out is to create an arbitrary model and simplify away some of the complexities that don't directly related to the question. So Let's say we have a bullet that is 10g, a 'gun' that is 10kg. Furthermore, our very simple gun creates a constant acceleration on the bullet while it is f... | {
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Can Galilean transformation be derived from length invariance? Given length invariance in Euclidean 3D space between two inertial frames:$$ds^2=ds'^2$$
Can Galilean transformation be derived like Lorentz transformation derived from space-time interval invariance?
| I think what we should start from is not invariance of a scalar $ds^2$, but co(ntra)variance of a $3$-vector. Note that in terms of split-complex numbers the $2$-dimensional Lorentz transformation$$dt^\prime=\gamma(dt-\beta dx),\,dx^\prime=\gamma(-\beta dt+dx)$$satisfies$$dt^\prime+jdx^\prime=\gamma(1-j\beta)(dt+jdx).$... | {
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Speed of light in the Schwarzschild metric or Gullstrand metric In the Schwarzschild metric, the speed of light will be given by:
$$\frac{\mathrm{d}r}{\mathrm{d}t}= \Big(1-\frac{2M}{r}\Big)$$
and in the Gullstrand metric, it will be given by:
$$\frac{\mathrm{d}r}{\mathrm{d}t}= 1- \sqrt{\frac{2M}{r}}$$
As is evident the... | I will focus on the part of your question regarding the Schwarzschild metric.
I assume that what you have done to find that expression is to set $\mathrm{d}s^2 = 0 $, considered radial motion only, and simplified the expression for the metric to get the ratio
$$
\left(\frac{\mathrm{d}r}{\mathrm{d}t}\right)^2 = \left(1 ... | {
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Are Newton's laws always an approximation because of the non-existence of inertial frames? It's said that perfect inertial frames don't exist.
Here
Is there any true inertial reference frame in the universe?
Newton’s laws hold in inertial frames, now due to their non-existence then is it that Newton’s laws are always a... | To specify, Newton's law are in any case an approximation for $| \vec{v} | << c$ of Einstein's special relativity laws. Since in special relativity the idea of inertial frame is crucial, too, the question remains.
The point is that this laws are mathematically valid without approximation in the precise conditions by wh... | {
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Will a muon decay in an empty universe? Imagine, a deep empty universe consisting of only one particle muon. Will it decay?
As there isn't any change in its surroundings and thus time will lose its meaning. But if the muon will decay in that situation, doesn't it prove that time is something deeper than our experience ... | This question can be generalized to whether or not anything can "happen" in empty Minkowski space where there are no external indications of time progressing. If you were to look at the muon's world line in an empty universe or a non-empty universe, the result at a certain point along that line will always be the same ... | {
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Differential form of Massieu’s function Massieu’s function is given by:
$$F_{M}=-\frac{U}{T}+S$$
And its differential form is given by:
$$dF_{M}=\frac{U}{T^{2}}dT+\frac{P}{T}dV$$
Well, it seems that:
$$\frac{\partial S}{\partial T}=0$$
How can that be?
| Note that $\mathrm{d}(-U/T)= -\frac{\mathrm{d}U}{T} + \frac{U}{T^2}\,\mathrm{d}T$ and hence $$ \mathrm{d}F_{\mathrm{M}} =-\frac{\mathrm{d}U}{T} + \frac{U}{T^2}\,\mathrm{d}T + \mathrm{d}S \quad .$$
Using $ \mathrm{d}U=T\, \mathrm{d}S - p\,\mathrm{d}V$ in the above equation gives the desired result.
| {
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Under what circumstances is the equation for magnetic energy in terms of magnetic vector potential correct? In Jackson section 5.16 he discusses different ways of calculating the energy in a magnetic field. One way (eq. 5.149) expresses it in terms of the magnetic vector potential and the current density:
$$
W = \frac1... | In the magnetostatic case, we know that $\nabla\cdot \vec{J} = 0$. Further, we assume that the currents are localized, so we may say that $\vec{J}\to 0$ at infinity. With these in mind, we can examine the effect of a gauge transformation.
$A\to A + \nabla f$.
\begin{align}
\int d^3 x\ \left(\vec{A} + \nabla f\right)\c... | {
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Is this sensible index placement for a Christoffel symbol of the second kind? I am trying to wrap my head around index notation and I have found a question in a textbook I am unsure of the answer of. It gives the expression:
$$
t_{jk} = Γ^{i}_{jk} u_{i}
$$
and asks for an explanation as to whether this has sensible in... | If by a vector you mean contravariant vector $u^i$ then this wouldn't be correct since you can not contract it with the Christoffel symbol $\Gamma^{i}_{jk}$ and obtain $t_{jk}$. You can perform the contraction between upper and lower indices.
| {
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Are neutrons an actual particle? I'm a senior student in high school and we are learning about Particle Physics in Physics. I wanted to ask a question about neutrons. Is there a possibility that neutrons may not even be a particle, just a bond, relationship or pairing between electrons and protons? Can neutrons just be... |
or is the theory we have have no correct and there is no need for further debate.
You are a hundred years too late to be able to play with the models of nuclear physics. Physics reasearch at present has progressed to the level that has shown that protons and neutrons, not only are the two versions of the"same" partic... | {
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If $g_{ij}$ is a tensor of type $(0,2)$, what is kind of tensor is $\partial_{i}g_{jk}$? Suppose $g_{ij}$ is a tensor of type $(0,2)$, then what type of object is $\partial_{i}g_{jk}$? Is it even a tensor, and if so, of what type? Is the $\partial_{i}$ still a differential with respect to the $i$-th variable in this co... | As the comments mention, $\partial_{i} g_{jk}$ isn't a tensor in general. You can confirm this by calculating how the object transforms.
If you take the covariant derivative of a general type $(0,2)$ tensor $t_{ij}$ then you'll get a type $(0,3)$ tensor $\nabla_{k} t_{ij}$. Although if by $g$ you mean the metric, then ... | {
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What causes the mass of a black hole? I heard in a lecture about the Higgs mechanism that the mass of a black hole has nothing to do with the Higgs mechanism. The point was made in relation to the proton mass being largely due to the energy content within the proton, not the mass of the constituent quarks.
I don't know... | General relativity does not distinguish between matter and energy. As far as gravitation goes they are the same and are related by Einstein's famous equation $E = mc^2$. The gravitational field of a proton is the same whether we treat it as a mass of $1.673 \times 10^{-27}$ kg or an energy of $938.2$ MeV.
So whether or... | {
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Why does general homogenous solution of differential equations modelling circuits die off after a long time? I was reading this answer in Elecronic engineering stack exchange which said that when solving the linear second order differential equation modelling circuits having ac source. We only need to account for parti... | Physically, the homogeneous solution is a motion that can occur without any driving term. If it were to continue forever, that would be perpetual motion. Every real system has some dissipation, so such motions must decay away when there is no power source. Ultimately it is a consequence of the 2nd law of thermodynamics... | {
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Prove $\Lambda^T \eta \Lambda = \eta$ How can one prove that $\mathbf{\Lambda^T \eta \Lambda} = \mathbf{\eta}$ in special relativity, where $\mathbf{\Lambda}$ is the Lorentz transformation and $\eta$ is the Lorentz metric? Also, how does this correspond to the intuitive definition of the invariance of the Lorentz metri... | $x^u x_u = n_{u\beta} x^{\beta} x^u = n_{u \beta} \Lambda^u _\gamma x^\gamma \Lambda^\beta _\omega x^\omega = n_{u \beta} \Lambda^\beta _\omega \Lambda^u _\gamma x^\gamma x^\omega$
$[m] \equiv$ matrix of m.
By the definition of matrix multiplication,
$n_{u \beta} \Lambda^\beta _\omega = n \Lambda =[n \Lambda]^u _\omeg... | {
"language": "en",
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Confused about linear displacement in circular motion We know that velocity is the derivative of displacement w.r.t. time. Keeping this mind, suppose that we are traveling across a circle where we go from point A to point B. In textbooks, it is said that the linear displacement is the arc length $AB$ given as, $s=r\the... | The definition of velocity is: $$\mathbf v = \frac{d\mathbf r}{dt} = \lim_{\Delta t\to0}\frac{\mathbf {\Delta r}}{\Delta t}$$ In the case of circular motion, when $\mathbf {\Delta r}$ (whose modulus is the lenght of straight line between 2 points in the arc) is too small), that modulus can be approximated by the lengt... | {
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Variation of Einstein Tensor In my working, I found it hard to solve this action
$$S = \int d^4x \sqrt{-g} \partial_\mu\phi \partial_\nu\phi G^{\mu\nu} $$
My goal is to find variation to metric:
$$ \delta S = \int d^4x (\delta{\sqrt{-g}) \partial_\mu\phi \partial_\nu\phi G^{\mu\nu}} + \int d^4x {\sqrt{-g}(\delta[{ \... | This calculation is tedious.
You will need:
$$\delta R_{ab} = \nabla_{b}(\delta \Gamma^{c}_{ac}) - \nabla_{c}(\delta \Gamma^{c}_{ab})$$
$$\delta \Gamma^{c}_{ab} = \cfrac{1}{2}g^{cd}\Big(\nabla_{b}\delta g_{da} + \nabla_{a}\delta g_{db} - \nabla_{c} \delta g_{ab}\Big)$$
$$G_{ab} = R_{ab} - \cfrac{1}{2}g_{ab}R$$
First of... | {
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Does Zeno's Paradox hold the simplest key to explain the widely-held continuous spacetime belief in physics? Richard Feynman in this 60's public lecture claimed it's easy to prove "physical space cannot be discrete automata", otherwise it will soon violate existing physical observations towards the end of this lecture.... | Zeno's paradox is based on incorrect logical reasoning, as he did not take into account that shorter length steps are associated with shorter time steps as well. And incorrect logical reasoning can not hold the key to anything.
| {
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Dimensional regularization: order of integration This is a two-loop calculation in dim reg where I seem to be getting different results by integrating it in different orders. I am expanding it about $D=1$. What rule am I breaking?
$$\int \frac{d^{D} p}{(2\pi)^D}\frac{d^{D}q}{(2\pi)^D}\frac{p^2+4m^2}{(q^2+m^2)((q-p)^2+m... | I'd say, when you set
$$
\int \frac{d^{D}q}{(2\pi)^D}\frac{1}{(q^2+m^2)((q-p)^2+m^2)}=\frac{1}{m}\frac{1}{p^2+4m^2}
$$
the correct answer actually has a $+\mathcal O(d-1)$ piece. The $p$ integral has $1/(d-1)$ divergences which, when multiplied by the missing subleading piece, leaves a finite contribution.
We can do th... | {
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What is the spectrum of the Laplacian operator in a domain? I heard in a lecture that if you know the spectrum of the Laplacian operator in a domain, you actually know everything about the domain. What does this mean?
| This problem is known as "hearing the shape of a drum". Is it possible to know the shape of a drum (a 2d membrane with a fixed boundary) only by hearing the sound it produces (its spectrum) ?
As nice as the idea sounds, the answer is negative. Gordon, Webb and Wolpert gave a counterexample, with two domains ("drums") t... | {
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Conceptual difficulty with friction and inertia Consider this -
A box lies on the floor of a bus which starts accelerating. Two cases arise
$1.$ If the static friction is great enough, the box accelerates with the bus with acceleration equal to that of the bus in the direction of force
$2.$ If the static friction is no... | The flaw in your reasoning is that you have confused two different situations in case 2.
You have described one situation in which the bus has constant acceleration, and the force of friction is not enough to match the acceleration of the bus. In this case, DanDan0101's answer is completely correct: the box will slide ... | {
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Why does low magnitude of force imply that all components of a system have same acceleration?
The surface between block M and ground (green) is frictionless, the surface between the both block M and m have coefficient of static friction $\mu$. An external force $\vec{F}$ is applied on the right block of mass $M$
It ... | If you apply a large enough $F$, you will be able to pull out the right block of mass $M$ from under the right block of mass $m$. The assumption of "small magnitude of F" is to prevent that from happening (and avoid pulling off the left block of mass $m$ from the top of left block of mass $M$, and also prevent the stri... | {
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What happens to the velocity if you decrease the radius in uniform circular motion? PROBLEM
Suppose we have a frictionless table and we have a mass attached to a string rotating in a circular motion. If we decrease the radius by pulling string downwards (from a hole in the table), what will be the effect on the speed o... | Conservation of angular momentum dictates that it will speed up as the area swept out per unit time is a constant.
as for your first solution i have no idea where u got that the velocity square is directly proportional to rT as in this case r is a variable not a constant
| {
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Conjugate momentum notation I was reading Peter Mann's Lagrangian & Hamiltonian Dynamics, and I found this equation (page 115):
$$p_i := \frac{\partial L}{\partial \dot{q}^i}$$
where L is the Lagrangian. I understand this is the definition of conjugate momentum, but I wanted to know if there is a particular reason for ... | *
*The index of generalized coordinates $q^1, \ldots, q^n$, is conventionally$^1$ a superscript/upper index in physics.
*The Lagrangian momenta $p_i:=\frac{\partial L}{\partial \dot{q}^i}$ have a subscript/lower index since they transform under general coordinate transformations $q^i\to q^{\prime j}=f^j(q,t)$ as com... | {
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Can the Hamiltonian be interpreted as the "speed" of unitary evolution? The Schrodinger equation
$$i\hslash \frac{d}{dt} \psi = H \psi$$
means a quantum state $\psi(t)$ evolves unitarily, that is,
$$\psi(t) = \exp(-\frac{i}{\hslash} H t) \psi(0)$$
where $\psi(0)$ is the initial state at time $t = 0$.
Suppose if we sca... | Sure, the Hamiltonian operator is the generator of 1-parameter unitary time flow in the sense of Stone's theorem.
| {
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What is our time-size in spacetime? Are we spaghetti or flat blobs? In special and general relativity time is treated as a dimension ($ict$, being $t$ a real number). Computations usually revolve around describing world lines, and events (crossings of world lines).
But objects can't physically be time-lines. Future unf... | We are very much not flat blobs. The spaghetti is much closer. We are on average about 70 light-years long in the time direction.
So even spaghetti does not really convey the concept of how anisotropic we are. A spaghetti noodle is roughly 1 mm diameter and 300 mm long. For a 6’ human a spaghetti-proportional length is... | {
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Could the observable universe be bigger than the universe? First of all, I'm a layman to cosmology. So please excuse the possibly oversimplified picture I have in mind.
I was wondering how we could know that the observable universe is only a fraction of the overall universe. If we imagine the universe like the surface ... | Yes, it's possible in principle that we see the same galaxy more than once due to light circling the universe. It wouldn't necessarily be easy to tell because each image would be from a different time in the galaxy's evolution.
There is a way to test for this. The cosmic microwave background that we see is a 2D spheric... | {
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Singularity in Robertson Walker metric with flat spatial slices In Sean Carroll's GR book, pg. 76, a special case of the Robertson-Walker metric, where the spatial slices are flat is given by
$$ds^2=-dt^2+a^2(t)[dx^2+dy^2+dz^2].$$
It was said that $t=0 $ represents a true singularity of the geometry (the 'Big Bang') an... | If you're looking for some physical quantities that blow up as $t \rightarrow 0$ in FRW cosmology, you can note that the Hubble factor $H = \frac{\dot{a}(t)}{a(t)}$ diverges at this point. Similarly, to pin home the fact this is a proper singularity, if you calculate the curvature invariants, e.g. $R$, $R^{\mu \nu}R_{\... | {
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What is a simple example illustrating the concept of "commensurate" and "incommensurate" order in condensed matter physics? In a wide range of contexts in condensed matter physics, e.g this paper, the concepts of commensurate and incommensurate orders are invoked to describe particular ordered phases. I think I have so... | Commensurate means that the ration of two parameters is an integer or, more generally, a ratio of integers.
For example, if we consider a one-dimensional lattice with period $a$, and we impose on top of it a perturbation potential $$V(x)=V_0\cos(kx)$$ with spatial period $$\lambda=\frac{2\pi}{k}.$$
The sutuations where... | {
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Newton's Laws of Motion, Pulleys, Rope and tension I was solving some questions to apply my concepts, and I came across the atwood machine and pulley block problems.
Consider the following for example:
The pulley is massless and frictionless, string, too, is ideal.
Why does the book say that the tension in the green s... | The pulley is at rest, so the net force must be zero. Since there are two tensions $T$ trying to push it down, the tension of the upper string must be equal to twice this value, so that the vector sum is zero:
\begin{equation}
F_{\text{net}}=F'-T-T=0\to F'=2T
\end{equation}
If the pulley has a mass $M_p$, then
\begin{e... | {
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What does $\rm kg m^{-2}$ mean in simple terms? I'm reading a research paper on fish stocking density and they measure the density in terms of $\rm kg m^{-2}$ (i.e. stocking density is 1.27 $\rm kg m^{-2}$). I've googled it but still don't have a good grasp on what a kg m is. Is kg m just an alternate way of writing ... | You can think of negative exponents as fractions, for example
$$
x^{-3}=1/x^3
$$
Then, in your case $kgm^{-2}$ express a ratio $kg/m^{2}$ , that would mean amount of kilograms of fish per meter squared, which is the common unit for denoting area.
| {
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Homogeneous (projective) coordinates and spinors When a complex number is considered as the stereographic projection from a sphere to the Argand plane, and then is represented by two “homogeneous coordinates” (in order to allow for a “point at infinity” corresponding to the point of projection) such that ζ=ξ/η, what ar... | To appreciate the discussion a bit more I would recommend comparing their discussion with the discussion (and main reference) here.
You can check that re-interpreting
$$\zeta' = \frac{a \zeta + b}{c \zeta + d}$$
as a matrix equation
$$\begin{bmatrix} \zeta' \\ 1 \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatr... | {
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How to identify non-hermitian Hamiltonians? Can anyone explain how can I prove that hamiltonians like $H=p^2+ix^3$ or $H=p^2-x^4$ are non-hermitian.
I think in first case, as the conjugate of $H$ is not equal to $H$, we can say it is non-hermitian. Is it a valid argument?
And what about the 2nd case?
Please help.
| Technically you'd need to specify the domain on which these operators act and the inner product... Assuming the domain is the real line and the usual inner product
$$
\langle \phi\vert\psi\rangle = \int_{-\infty}^\infty dx\,\phi^*(x)\psi(x) <\infty
$$
then the first clearly isn't because $\hat x$ and $\hat p$ are Herm... | {
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Is the many-worlds interpretation really just an interpretation? Is the many-worlds interpretation just a different interpretation to quantum mechanics or does it contain some different predictions?
In other words, is it possible theoretically to conduct an experiment that checks the many-worlds interpretation?
| An interpretation is a mapping from the formalism of QM to the real world. It does not, by definition, disagree with the predictions of QM at all.
Many world believes that the various possible outcomes of experiments actually exist, Bohm claims that particles are "real" and move along complex paths, Copenhagen says tha... | {
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What are the the actual similarities between Electric Circuits and magnetic circuits I have surfed google to find an answer to that but all I keep seeing is "differences" and not actual similarities.
So what exactly do electric circuits and magnetic circuits have in common.I wanna know as many as possible.
| The concept of a "magnetic circuit" exploits a one-to-one correspondence between the equations of the magnetic field in an unsaturated ferromagnetic material to that of an electrical circuit. Using this concept the magnetic fields of complex devices such as transformers can be quickly solved using the methods and techn... | {
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If electrons can be created and destroyed, then why can't charges be created or destroyed? I read on Wikipedia that electrons can be created through beta decay of radioactive isotopes and in high-energy collisions, for instance when cosmic rays enter the atmosphere. Also, that they can be destroyed using pair annihilat... | It’s important to note that electrons can only be created and destroyed in interactions which respect the fundamental symmetries of nature and their associated conservation laws.
So, charge conservation is one such conservation law; “number-of-electrons conservation” is not. (Though before radioactivity was discovered,... | {
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What is the interpretation of zero probability in physics? Impossibility of an event implies vanishing of it's probability. But the reverse is not true.
This post in math stack exchange posts says why zero probability doesn't necessarily mean impossible events.
Then why do we act like it is, in physics ,i.e., how is va... | The square of the wavefunction $\vert\psi(x)\vert^2$ is a probability density, not a probability. The probability of finding the system in a small bin of
width $dx$ centred at $x_0$ is very nearly $\vert\psi(x_0)\vert^2 dx$ and thus very nearly $0$ if $\vert\psi(x_0)\vert^2= 0$, but the exact calculation yields
$$
P=\... | {
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Is no acceleration a cause or consequence of no net force? If a body is moving with constant velocity, or is at rest, then the net force on it must be $0$. If the net force on a body is $0$, then it must be moving with constant velocity or must be at rest.
Is $0$ net force a consequence of being at rest or moving with ... | Many answers say the answer is “both” or that the question is really philosophical. Perhaps that is true, but it does not seem like a reasonable view to take in this context.
Beginning physics students are usually taught that a force is a “push or pull”. Later, when they learn physics in a slightly more rigorous manner... | {
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Does bottle water rise a little bit on full moon days?
High tides and low tides are caused by the Moon. The Moon's
gravitational pull generates something called the tidal force. The
tidal force causes Earth—and its water—to bulge out on the side
closest to the Moon and the side farthest from the Moon. ... When
you're ... | Since water is compressible it will expand when there is less gravitational pull towards the center of the Earth, which is the case on the Moon-facing side of the Earth. It's just not very much because water is not particularly well compressible.
The water surface in the bottle will also, like the oceans in the simplif... | {
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Entropy of a gas of $N$ distinguishable Particles Suppose I have a container contain $N$ particles, all of which are distinguishable. How would I calculate the entropy of this gaseous system. The Hamiltonian of the system is simply the sum $\sum_{i=1}^N \frac{p_i^2}{2m_i}$ and I believe (not 100% sure) that the partiti... | Yes, exactly, this will be the partition function. However, there is no need to go to spherical coordinates. Just take the integral in Cartesian coorindates:
$$
\int e^{-\beta p_i^2/(2 m_i)} = \sqrt{\frac{2 \pi m_i}{\beta}}
$$
You will have the product $3 N$ $1D$ gaussian integrals:
$$
\prod_{i=1}^{N} \sqrt{\frac{2 \pi... | {
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Why do we set the number of degrees of freedom of an electron in a gas to 2 rather than 3? I was reading up on basic properties of plasmas and came across this description in the Wikipedia on how to convert between eV and Kelvin units to describe the amount of energy of the electrons:
The SI unit of temperature is the... |
Why do we set the number of degrees of freedom of an electron in a gas to 2 rather than 3?
In a magnetized plasma, most electrons are said to be gyrotropic -- their velocities can be decomposed into parallel and perpendicular (with respect to the background, quasi-static magnetic field), where the latter is assumed t... | {
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} |
Which is the centripetal term here? In spherical coordinates the acceleration can be written as
$$\textbf{a} = \dot{\textbf{v}} = \ddot{r} \hat{\textbf{r}} + \dot{r} ( \dot{θ} \boldsymbol{\hat{\theta}} + \sin θ \dot{\phi} \boldsymbol{\hat{\phi}}) + \dot{r} \dot{θ} \boldsymbol{\hat{\theta}} + r \ddot{\theta} \boldsymbo... | By definition, if the acceleration points towards the origin it is centripetal. So if $\ddot{r} - r \dot{θ}^2 - r \sin^2 θ \dot{\phi}^2<0$ then the radial component is centripetal.
You are most likely getting confused with introductory systems of uniform circular motion where we say "centripetal acceleration is $r\dot\... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/625355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Does physics explain why the laws and behaviors observed in biology are as they are? Does physics explain why the laws and behaviors observed in biology are as they are? I feel like biology and physics are completely separate and although physics determine what's possible in biology, we have no idea how physics determi... | Every biological system is comprised out of zillions of elementary particles. Each of these particles interacts with a fraction of the others. Every interaction can be accounted for with the help of elementary particle physics. So yes, every biological system can be accounted for in a physical way. You and I are huge c... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/625503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
"answer_count": 12,
"answer_id": 9
} |
Where, in the standard model, do resonance widths come from? For example, the Z boson decays and hence the propagator is:
$$
\frac{1}{p^2-m^2-i \Gamma}
$$
Where does this arise in QFT, is it that the lagrangian mass is complex or is it that when we compute the self-energy and get the renormalised mass we find that the ... | The bubbles in the expansion of the$Z$ propagator become imaginary when the internal particles can go on-shell. The imaginary part is just the inverse lifetime of the $Z$.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/625629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What exerts force, the electric field of a charged body or the body itself? When another charged body is placed in the electric field of a charged body, the field exerts a force of attraction or repulsion or the body itself?
| Simplistically, we can say that it is the field that exerts the force. Say the Sun disappears in an instant, it is known popularly that the earth should revolve around the position where the Sun existed for 8 minutes, the time taken for the information of Sun's disappearance to reach earth. So it is not the body itself... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/625757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Does the $\rm s$ squared in acceleration $\rm m/s^2$ have a geometrical interpretation? Is there a geometric interpretation of $\rm s^2$ in $\rm m/s^2$ as an actual square? Is it right to see the time forming a square like there is an actual square of side 2 in 2 $\rm m^2$?
If that's wrong, Why is that an invalid inter... | The unit $\mathrm{m/s^2}$ represents metres-per-second added/removed per second, so $$\mathrm{\left[\frac{m}{s^2}\right]=\left[\frac{m}{s}/s\right]}.$$
Now, in classical non-relativistic mechanics space is three-dimensional while time is one-dimensional. With space being three-dimensional, we can see 1D, 2D og 3D space... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/625835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why does light from a source like sodium lamp undergo abrupt changes in phase?
"If we use two sodium
lamps illuminating two pinholes (Fig. 10.11) we will not observe any
interference fringes. This is because of the fact that the light wave emitted
from an ordinary source (like a sodium lamp) undergoes abrupt phase cha... | The light from these two sources forms an incoherent not a coherent superposition. Perhaps this is meant by "abrupt phase changes".
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/626296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
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