# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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