Questions tagged [homework-and-exercises]

Applies to questions of primarily educational value - not only questions that arise from actual homework assignments, but any question where it is preferable to guide the asker to the answer rather than giving it away outright. Please READ THE GUIDANCE IN META before asking homework-like questions.

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How do I construct the $SU(2)$ representation of the Lorentz Group using $SU(2)\times SU(2)\sim SO(3,1)$ ?

This question is based on problem II.3.1 in Anthony Zee's book Quantum Field Theory in a Nutshell Show, by explicit calculation, that $(1/2,1/2)$ is the Lorentz Vector. I see that the ...
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54 votes
1 answer
24k views

How long would it take me to travel to a distant star?

Suppose I wanted to travel to one of the recently discovered potentially Earth-like planets such as Kepler 186f that is 490 light years away. Assuming I had a powerful rocket and enough fuel, how long ...
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19 votes
1 answer
6k views

How do I derive the Lorentz contraction from the invariant interval?

While reviewing some basic special relativity, I stumbled upon this problem: From the definition of the proper time: $$c^2d\tau^2=c^2dt^2-dx^2$$ I was able to derive the time dilation formula by using ...
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3 votes
1 answer
3k views

Toppling of a cylinder on a block

A uniform cylinder rests on a cart.The height and diameter is given.coefficient of static friction is given.How can i find the minimum acceleration of block such that the block topples? Morever what ...
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12 votes
2 answers
5k views

Deriving Birkhoff's Theorem

I am trying to derive Birkhoff's theorem in GR as an exercise: a spherically symmetric gravitational field is static in the vacuum area. I managed to prove that $g_{00}$ is independent of $t$ in the ...
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  • 2,706
21 votes
1 answer
12k views

Lorentz Invariant Integration Measure [closed]

When we canonically quantize the scalar field in QFT, we use a Lorentz invariant integration measure given by $$\widetilde{dk} \equiv \frac{d^3k}{(2\pi)^3 2\omega(\textbf{k})}.$$ How can I show that ...
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  • 2,711
6 votes
2 answers
6k views

Force applied off center on an object

Assume there is a rigid body in deep space with mass $m$ and moment of inertia $I$. A force that varies with time, $F(t)$, is applied to the body off-center at a distance $r$ from its center of mass. ...
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  • 175
6 votes
2 answers
2k views

Relation between Wave equation of light and photon wave function?

Suppose in our double slit experimental setup with the usual notations $d,D$, we have a beam of light of known frequency $(\nu)$ and wavelength $(\lambda)$ - so we can describe it as $$ξ_0 = A\sin(kx-\...
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24 votes
8 answers
28k views

Why can't a single photon produce an electron-positron pair?

In reading through old course material, I found the assignment (my translation): Show that a single photon cannot produce an electron-positron pair, but needs additional matter or light quanta. My ...
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  • 243
11 votes
9 answers
10k views

Explanation: Simple Harmonic Motion

I am a Math Grad student with a little bit of interest in physics. Recently I looked into the Wikipedia page for Simple Harmonic Motion. Guess, I am too bad at physics to understand it. Considering ...
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23 votes
2 answers
4k views

Quantization of a particle on a spherical surface

Suppose we have a particle of mass $m$ confined to the surface of a sphere of radius $R$. The classical Lagrangian of the system is $$L = \frac{1}{2}mR^2 \dot{\theta}^2 + \frac{1}{2}m R^2 \sin^2 \...
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5 votes
2 answers
3k views

Shape of a string/chain/cable/rope/wire? [closed]

The height of a string in a gravitational field in 2-dimensions is bounded by $h(x_0)=h(x_l)=0$ (nails in the wall) and also $\int_0^l ds= l$. ($h(0)=h(l)=0$, if you take $h$ as a function of arc ...
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  • 1,502
57 votes
9 answers
20k views

Are we slightly lighter during the day and slightly heavier at night, owing to the force of the Sun's gravity?

Using $g = \frac{Gm}{r^2}$, the force on a point mass located at 1 AU from the Sun ($m = 2 \cdot 10^{30} \text{ kg}$) is about ~0.006 N/kg. Does that mean that, e.g., a 70 kg person is ~42g lighter ...
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8 votes
3 answers
3k views

Can linear momentum be conserved before and after collision in the presence of an external force?

I'm a physics tutor for 11th and 12th grade students. I taught my students that momentum is conserved only when the net external force on the system is zero. But in the books we are following, ...
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  • 6,685
4 votes
2 answers
14k views

What is the relative speed of two near-light speed particles headed towards each other? [duplicate]

I understand that nothing can move faster than light due to time dilation. I want to build upon my understanding of Einstein's theory of Special Relativity, so I came up with this hypothetical problem ...
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  • 1,178
21 votes
2 answers
45k views

What is the force between two charged objects when the space between them is partially filled by a dielectric medium?

I am given two charged particles of same charge at a distance of $r$. They initially apply force $F$. Now an infinite dielectric (of dielectric constant $4$) of width $\frac{r}{2}$ is introduced ...
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  • 564
11 votes
2 answers
24k views

Angular Velocity expressed via Euler Angles

On the top of the fourth page from here, the author trivially derives the components of angular velocity, expressed via Euler angles of the system. I fail to understand how the components of angular ...
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  • 1,104
15 votes
2 answers
6k views

Geodesics equations via variational principle

I would like to recover the (timelike) geodesics equations via the variational principle of the following action: $$ \mathcal{S}[x] = -m \int d\tau = -m \int \sqrt{-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}} $$ ...
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  • 499
10 votes
3 answers
14k views

When is tension constant in a rope?

Consider a massless rope with pulling forces applied at each end. How do we decide if tension is constant or not in a rope? Consider a few example scenarios: For example, if there is a knot in the ...
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11 votes
2 answers
14k views

What Exactly is a Shock Wave?

The Wikipedia defintion of a shock wave pretty much sums up all I've found online about what a shock wave is: A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries ...
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  • 5,204
3 votes
2 answers
364 views

Dancing twins, one travelling in a rocket ship, looking through telescopes at each other [duplicate]

Suppose two twins are put asleep. One is put in a rocket ship and accelerated to 90 percent of the speed of light. They are then both awakened, after which they both train their telescope on each ...
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4 votes
3 answers
15k views

Prove $[A,B^n] = nB^{n-1}[A,B]$

I am trying to show that $[A,B^n] = nB^{n-1}[A,B]$ where A and B are two Hermitian operators that commute with their commutator. However, I am running into a little problem and would like a hint of ...
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  • 940
3 votes
1 answer
5k views

Harmonic oscillator coherent state expectation values

I'm looking to calculate the expected values of a coherent state (of a harmonic oscillator) evolving in time. I know that the $x$ and $p$ expectation values are as in classical motion, but I'm ...
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  • 2,595
3 votes
3 answers
4k views

If string is stretched just by weight, where does the gravitational potential energy goes if only half is converted to elastic potential energy?

If a spring is stretched by a weight of mass m, (so the extension is $\Delta x$) then $ k = \frac W{\Delta x} = \frac {mg}{\Delta x}$. So $ k\Delta x= mg $. When the Spring is stretched by distance $\...
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  • 41
12 votes
2 answers
4k views

Position operator in QFT

My Professor in QFT did a move which I cannot follow: Given the state $$\hat\phi|0\rangle = \int \frac{d^3p}{(2\pi)^3 2 E_p} a^\dagger_p e^{- i p_\mu x^\mu}|0\rangle,$$ he wanted to show that this ...
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  • 243
10 votes
6 answers
8k views

What is the electric field flux through the base of a cube from a point charge infinitesimally close to a vertex?

I'm having some trouble with the following problem: A charge $q$ is placed on the body diagonal of the cube very close to one of the corners (distance $\delta$ from the corner, $\delta$ tending to ...
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  • 317
16 votes
5 answers
25k views

Normalizing the solution to free particle Schrödinger equation

I have the one dimensional free particle Schrödinger equation $$i\hbar \frac{\partial}{\partial t} \Psi (x,t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \Psi (x,t), \tag{1}$$ with ...
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  • 913
5 votes
2 answers
5k views

Does the weight of fluid in a conical container act entirely on the base?

Scenario description Lets assume both containers have a capacity of 300 litres. One is a vertical tube as shown in pic 1 Other one is more or less a V shaped containers as shown in pic 2 Both are ...
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  • 105
3 votes
2 answers
3k views

Proof of Archimedes Principle

Say I have an arbitrary body $\Omega$ and surface $\partial \Omega$ submerged in a hydrostatic fluid of density $\rho$ under the influence of gravity. How does one show Archimedes Principle? i.e. $$\...
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  • 3,394
25 votes
3 answers
9k views

Deriving the speed of the propagation of a change in the Electromagnetic Field from Maxwell's Equations

I've been told that, from Maxwell's equations, one can find that the propagation of change in the Electromagnetic Field travels at a speed $\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ (the values of which can ...
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  • 5,860
8 votes
5 answers
9k views

Commutator algebra in exponents

Considering $X$ and $Y$ such that $[X,Y]=\lambda$, which is complex, and $\mu$ is another complex number, prove: $$e^{\mu(X+Y)}=e^{\mu X} e^{\mu Y} e^{-\mu^2\lambda/2}$$ My attempt (so far) is: ...
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  • 738
15 votes
5 answers
14k views

Derivation of Kepler's laws

How can analytically be derived the Kepler's laws? I found some extremely synthetic equations which from the Newton's laws (in particular $\mathbf{F} = m \mathbf{a}$) tried to obtain the Kepler's ...
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  • 703
13 votes
4 answers
12k views

Capacitor with different charges on each plate

I am confused as to whether/how capacitance changes when each plate has a different charge. For example, consider a coaxial cable and put $20Q$ on the outer cable, and $-Q$ on the inner. Or how about ...
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7 votes
2 answers
3k views

$SU(3)$ irreducible representations with tensor method

I am dealing with the tensor product representation of $SU(3)$ and I have some problems in understanding some decomposition. 1) Let's find the irreducible representation of $3\otimes\bar{3}$ we have ...
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  • 815
7 votes
1 answer
2k views

Conversion of the Polyakov action into the Nambo-Goto action?

I've read that the Polyakov action using an intrinsic metric $h_{\alpha\beta}$ $$\tag{1} S_P ~=~ -\frac{T}{2}\int d^2 \sigma \sqrt{-h}h^{\alpha\beta} \partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu} \...
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  • 9,213
4 votes
1 answer
16k views

Commutator $[\hat{p},F(\hat{x})]$ of Momentum $\hat{p}$ with a Position dependent function $F(\hat{x})$?

I heard from my GSI that the commutator of momentum with a position dependent quantity is always $-i\hbar$ times the derivative of the position dependent quantity. Can someone point me towards a ...
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4 votes
3 answers
3k views

What is the position as a function of time for a mass falling down a cycloid curve?

In the brachistochrone problem and in the tautochrone problem it is easy to see that a cycloid is the curve that satisfies both problems. If we consider $x$ the horizontal axis and $y$ the vertical ...
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  • 173
7 votes
2 answers
767 views

How are the Lagrange points determined?

According to Hyper Physics, there are 5 equilibrium, or Lagrange points of the Earth-Moon system and only 2 of them are said to represent stable equilibrium points. This made me think if there is an ...
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2 votes
3 answers
2k views

Why is $R\cos{a} = mg$ in circular motion compared and not $R = mg\cos{a}$?

Normally, if an object of mass $m$ is inclined to the horizontal at an angle $b$, we set the reaction force of the object on the inclined plane as $R = mg\cos{b}$ (if we resolve the force of gravity ...
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  • 165
12 votes
3 answers
2k views

Is it possible to produce gravitational waves artificially?

Why don't they make a ball with irregularities, say the size of a tennis ball, then spin it very rapidly, so it would produce gravitational waves like a spinning star with irregularities on it? Is ...
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  • 1,097
3 votes
1 answer
2k views

Deriving relativistic Doppler shift in terms of wavelength? [closed]

Consider a star moving with velocity $v$ at an angle $\theta$ with respect to its line of sight to Earth. Show that the relativistic Doppler shift is $$\lambda_{obs} = \frac{1 - \frac{v}{c} cos(\theta)...
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16 votes
2 answers
18k views

Is magnetic force non-conservative? [duplicate]

If magnetic field is conservative, then why not the magnetic force? My professor thinks it is non conservative but he couldn't explain to me why?
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  • 161
11 votes
1 answer
10k views

Is there a simple way of finding the eigenstates of the creation and annihilation operator in QM?

How can I find the eigenstates of creation and annihilation operator in QM? My attempt: Such eigenstate will obey: $$ a^{\dagger} |\psi \rangle = \alpha |\psi \rangle. $$ We can expand $|\psi \...
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  • 22.9k
11 votes
1 answer
2k views

How do I Derive the Green's Function for $-\nabla^2 + m^2$ in $d$ dimensions?

What is the solution to this equation in $d$ dimensions: $$(-\nabla_d^2 + m^2)G(\mathbf{x}, \mathbf{x}') = A \delta(\mathbf{x} - \mathbf{x}'),$$ with the boundary condition that $\lim_{|\mathbf{x} - \...
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  • 19.5k
0 votes
2 answers
5k views

Tricky conceptual question: ball sliding and rolling down incline [closed]

We all are familiar with the classic ball rolling down the incline exercise in rotational dynamics. Here is quite a tricky conceptual problem: You have an incline of fixed height, but the angle of ...
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8 votes
3 answers
6k views

Integrating radial free fall in Newtonian gravity [duplicate]

I thought this would be a simple question, but I'm having trouble figuring it out. Not a homework assignment btw. I am a physics student and am just genuinely interested in physics problems involving ...
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  • 93
7 votes
3 answers
42k views

Flux through side of a cube

I am looking at Griffiths introduction to Electrodynamics 3rd ED. Problem 2.10 asks for the flux of $E$ through the right face of the cube, when a charge $q$ is in the back left corner of the cube. ...
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  • 1,718
8 votes
1 answer
3k views

Find the Hamiltonian given $\dot p$ and $\dot q$

I have these equations: $$\dot p=ap+bq,$$ $$\dot q=cp+dq,$$ and I have to find the conditions such as the equations are canonical. Then, I have to find the Hamiltonian $H$. To answer to the first ...
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  • 1,143
-2 votes
3 answers
2k views

Can you lift a basket up while standing inside it? [closed]

Suppose you stand in a basket and you try to pull the handles of the basket. Will you be able to lift the basket from ground? Similarly, Baron Münchhausen allegedly pulled himself and the horse on ...
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  • 2,577
2 votes
1 answer
919 views

Heisenberg relation

Given that $A(k)=\frac{N}{k^2+\alpha^2}$, show that $\Delta k \Delta x >1$. Considering the above example, according to my textbook, it is written that I must square the above function and ...
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