11 votes
Accepted

What does the exponentiated generator of scale transformation do when it acts on a function?

Hint: Rewrite the dilation operator $$x\frac{d}{dx}~=~\frac{d}{d\ln |x|}$$ as a translation operator, and use OP's Taylor formula/translation eq. (1) to deduce that $$e^{\lambda x \frac{d}{dx}}f(x)~=~...
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  • 167k
9 votes

How much does the sky weigh?

How much does the sky weigh ? The only meaningful interpretation I can think of is that the question is meant to be What is the weight of the Earth’s atmosphere ? You can estimate this once you ...
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7 votes
Accepted

Problem understanding expectation value of operators defined with density operator in quantum mechanics

For two matrices $M$ and $N$, the definition of matrix multiplication is $$[MN]_{ab}= \sum_c M_{ac} N_{cb}$$ In this case, $$\sum_n \underbrace{\left(\sum_m \rho_{nm}Q_{mn}\right)}_{= [\rho Q]_{nn}}= \...
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  • 50.6k
4 votes
Accepted

What law of thermodynamics is broken here?

This is not a good question. Yes, as stated, the engine appears to violate the first law. It can also be claimed that the second law is violated, but this is not that straightforward because some of ...
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  • 7,257
4 votes
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Dependence (or lack thereof) of forces on frames of reference

When you say "blocks A and B move with a relative acceleration of -3 m/s2" you are considering the motion of one block in the reference frame attached to the other block (block C). But the ...
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  • 6,196
3 votes
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A question regarding equations of motion

The equations of motion only work for constant acceleration. The setup mentioned in your question doesn’t satisfy that criteria. You can use the equations of motion to model the motion of the body ...
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3 votes
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Electric field inside a dielectric sphere placed in a uniform electric field

The solution presented in Griffiths follows a fully systematic method. At the very end, Griffiths makes the comment that "the field inside the sphere is (surprisingly) uniform". Now, the ...
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  • 4,585
2 votes
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Why $\omega=vR$ can not be applied in this question?

It's $v = R\omega$, not $\omega = Rv$.
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  • 7,257
2 votes

A question regarding equations of motion

v=0 after it hits the ground,SUVAT only applies when acceleration is uniform, when it hits the ground acceleration must change from 9.81 to something else, because there is now a new force acting on ...
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  • 21
2 votes

Particle moving around the inside of a semicone - how to model its position up incline?

In polar coordinates, the constraint fixing the particle on the cylinder reads : $$z = r\tan(\alpha)$$ Therefore, the kynetic energy is (setting $m = 1$ for simplicity) : $$T = \frac{1}{2}(\dot r^2 + \...
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  • 3,379
2 votes
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Find the equation for the angle $\theta$ in which the particle leaves the semicircle. No Friction

I don't think this is wrong. $k=0$ represents the case where the large hemisphere has zero mass, so no inertia. I think you are looking for the case where the large hemisphere is "fixed", so ...
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  • 391
2 votes

Change of Metric Under Coordinate Transformation

I think that you are trying to compute the Lie derivative of the metric. If so, there should be no factor of 1/2. Under an infinitesimal shift $x^\mu\to x^\mu +\eta^\mu$ we have $g \to g+ {\mathcal ...
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  • 41.2k
2 votes

Flux through faces of cube if charge is placed at an edge-center

Hint : In general the flux through an oriented open or closed surface $\:\mathrm{S}\:$ due to a point charge $\:Q\:$ is \begin{equation} \Phi_{\mathrm{S}}=\dfrac{\Theta}{4\pi}\dfrac{Q}{\epsilon_{0}} \...
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1 vote

Gauss' Law in differential form applied to charged sphere

Since you said sphere, and not shell, you'll have a total charge $Q$ uniformly distributed over the sphere of radius $R$: $$\rho=\frac Q{\frac 4 3 \pi R^3}$$ Based on symmetry: $$ \vec E(r,\theta, \...
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  • 23k
1 vote

Gauss' Law in differential form applied to charged sphere

I think this problem is ill posed. The divergence operator is not generally invertible. There are many functions with the same divergence. We need further information of electric field on a region of ...
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1 vote

An ideal gas expands into vacuum in an insulated rigid vessel. Which of the followings happens?

I think this could be their point of view, consider a rigid vessel with two chambers seperated by a partition. In this vessel, one of the chambers is filled with the ideal gas and the other chamber is ...
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  • 252
1 vote

How to calculate the moment of inertia of convex polygon? (two-dimensions)

Any polygon can be divided into triangles, as shown. With the common vertex at the origin. Then you need to find the centroid of the area (equal to C of G in this case). To do this, firstly calculate ...
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1 vote

Problem 6 of Sheet 1 - Quantum field theory David Tong - Variation of Lagrangian density

I will try to complement the answer by Thomas, since some things were not that clear to me. We start by varying the Lagrangian $\mathcal{L}(\phi(x),\partial_{\mu}\phi(x),x^{\mu}):$ $$ \begin{split}\...
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1 vote
Accepted

Phase Difference of A Mass-Spring System For Different Starting Points

I put the different phase values in a table to make this easier to talk about. The position can $L,0,R$ meaning left, center and right and the velocity can be $L,0,R$ meaning moving to the left, ...
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1 vote

How do I find the constraint relation in this question?

The speed of the connecting belt must be twice the downward speed of the center of the lower pulley. This means that $α_1 = 2 α_2$. Then: $mg – T – t = ma = mr α_2$ where, $T$, is the connecting ...
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1 vote

Do prescribed glasses transmit light?

Without light transmission, you will not be able to see anything since light is the way of transmitting the picture to your eye.
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1 vote

Different values for the Normal ordering

Hint: In Wick's theorem what remains in the fully contracted term (= the double contration) is the unit-operator $\hat{\bf 1}$ not the zero-operator $\hat{\bf 0}$, so OP's 1st calculation (v3) is ...
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1 vote

Regarding Griffith quantum mechanics problem 2.47: Square double well

Consider the limits $ b \to \infty $ and $ b\to 0 $ separately. In the limit $ b \to \infty $ the two wells are isolated from each other and the result is each particle does not interact with each ...
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  • 3,606
1 vote

Problem with possibly infinite possible solutions

I don't see any obvious problems with your analysis. This problem appears to be statically indeterminate. If so, the "real" value of the tension would be determined by knowing the material ...
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1 vote
Accepted

Kinetic energy of system consisting of rod and rolling cylinder

When setting up the kinetic energy in terms of generalized coordinates, it is in many instances easiest to write $$ T = T_\text{tr,cm} + T_\text{rot,cm} $$ where $T_\text{tr,cm}$ is the translational ...
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1 vote
Accepted

For virtual displacement in the Lagrangian, why is $\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0$?

Generally speaking, you must consider a displacement with $\delta x_i$ to be $t$-dependent, when applying the principle of least action. The section where $\delta\dot{x}_i = 0$ is about symmetries of ...
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  • 391
1 vote
Accepted

Contravariant Vector Component Transformation from Polar to Cartesian

It's because basis are defined with satisfying normalization condition. Basis are covariant, because they are partial derivatives(Consider them as gradient). $$\hat{r} = \frac{\partial}{\partial r} = ...
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