classical mechanics - Moment of a force about a given axis (Torque) - Scalar or vectorial?

I am studying Statics and saw that:

The moment of a force about a given axis (or Torque) is defined by the equation:

$M_X = (\vec r \times \vec F) \cdot \vec x \ \ \ $ (or $\ \tau_x = (\vec r \times \vec F) \cdot \vec x \ $)

But in my Physics class I saw:

$\vec M = \vec r \times \vec F \ \ \ $ (or $\ \vec \tau = \vec r \times \vec F \ $)

In the first formula, the torque is a triple product vector, that is, a scalar quantity. But in the second, it is a vector. So, torque (or moment of a force) is a scalar or a vector?

Answer

It is obviously a vector, as you can see in the 2nd formula.

What you are doing in the first one is getting the $x$-component of that vector. Rememebr that the scalar product is the projection of one vector over the other one's direction. Actually you should write $\hat{x}$ or $\vec{i}$ or $\hat{i}$ to denote that it is a unit vector. That's because a unit vector satisfies

$\vec{v}\cdot\hat{u}=|v| \cdot |1|\cdot \cos(\alpha)=v \cos(\alpha)$

and so it is the projection of the vector itself.

In conclusion, the moment is a vector, and the first formula is only catching one of its components, as noted by the subindex.

quantum mechanics - Do the canonical commutation relations have any connection to geometry?

I was wondering if the canonical commutation relations have any connection to geometry? If so, could you explain the connection in fairly simple and intuitive terms?

projectile - Could one fire a bullet with sufficient speed to leave the Earth?

Consider a gun or rifle fired directly upwards. My original question was what speed would be required to escape the Earth.

The escape velocity from the surface of the Earth is the classic $$v_e = \sqrt{ 2GM \over r } \approx 11,000 \text{ m/s}$$ and bullets typically (see for example) leave the muzzle with a maximum speed one order of magnitude lower, ~$1,000$ m/s. Terminal velocity of bullets in STP is another magnitude lower, ~$100$ m/s.

Even if a bullet were fired with speed $v_e$ that of course would not be sufficient due to drag which would slow the bullet down. So a theoretical required speed $v_T > v_e$.

• If the bullet were fired with anything close to $v_e$ or $v_T$, would it would burn up very rapidly in STP? I understand that rockets typically don't achieve anything like $v_e$ until they are high in the atmosphere at least in part for this reason


• And hence is there no speed possible in realistic conditions with which a bullet could be fire and escape the Earth?


• If it is possible, what model of drag should one use to calculate $v_T$ and concretely, does anyone have an estimate of its value?

velocity - Light traveling through a medium

Does the frequency of light change when it travels across an interface between two media? What happens to the light wavelength and the light velocity at the interface?

I've gotten different answers some say all three change and some say frequency is constant or some say that the wavelength is constant...

What's the right answer?

Answer

Frequency is constant. If it were otherwise, you would have more (or fewer) wave crests hitting the interface per unit time than leaving it, which would lead to some sort of pileup that gets worse and worse as time progresses. Another way to think about it is the atoms/electrons/electric field lines/whatever at the interface respond immediately to their neighbors (the wave is essentially continuous), even if those neighbors are in a different medium.

Wavelength can change, and it does so in such a way that $f \lambda = c/n$ at all locations. Here $f$ is the frequency, $\lambda$ is the wavelength, $c$ is the speed of light in vacuum, and $n$ is the index of refraction. As you can see, going from air ($n \approx 1$) to glass (say $n = 1.4$) will result in the wavelength of optical light to decrease by a factor of $1.4$. Also note that in general $n$ can depend on $f$, leading to colors dispersing as in rainbows and chromatic aberration.

visible light - What is the trajectory of a photon moving through a vacuum?

Since electromagnetic energy is carried by photons and moves in forms of waves, does it mean that a single photon when propagating through space doesn't follow the straight path but instead always moves up and down, up and down like a wave. If so another question arises the speed of propagation of light in vacuum is fixed meaning that it will always take the same amount of time for it to travel from point A to point B, but if a photon always moves up and down it will also mean that it travels longer distance than the distance between A and B and so it ill travel faster than light propagates, is it even possible, could you please clarify these concepts for me?

electric circuits - On this infinite grid of resistors, what's the equivalent resistance?

I searched and couldn't find it on the site, so here it is (quoted to the letter):

On this infinite grid of ideal one-ohm resistors, what's the equivalent resistance between the two marked nodes?

With a link to the source.

I'm not really sure if there is an answer for this question. However, given my lack of expertise with basic electronics, it could even be an easy one.

Answer

Nerd Sniping!

The answer is $\frac{4}{\pi} - \frac{1}{2}$.

Simple explanation:

Successive Approximation! I'll start with the simplest case (see image below) and add more and more resistors to try and approximate an infinite grid of resistors.

Mathematical derivation:

$$R_{m,m}=\frac 2\pi \left( 1 + \frac 13 + \frac 15 + \frac 17 + \dots + \frac 1 {2m-1} \right)$$

thermodynamics - Slow thermal equilibrium

I have a question which is inspired by considering the light field coming off an incandescent lightbulb. As a blackbody radiation field, the light is in thermal equilibrium at temperature $T$, which implies that each normal mode has a mean energy given by Planck's law, and a random phase. Thus, if I were to look, microscopically, at the electric field, I would see a fairly complicated random function that can only really be considered constant at timescales $\tau\ll \hbar/k_B T$ (at which the corresponding modes have next to no amplitude and therefore do not affect the electric field's time dependence).

This illustrates a general aspect of thermal and thermodynamic equilibrium: they are only relevant concepts when the systems involved are looked at on timescales far longer than their relevant dynamics.

My question, then, is this: are examples of slowly-varying systems (where by "slowly" I mean on the timescales of seconds, or preferably longer) that can be considered to be in thermal equilibrium on timescales longer than that known?

Answer

Globular clusters are an example of such a system. Their velocity distributions are close enough to Maxwellian that, observed instantaneously, they would appear to be in thermal equilibrium unless you paid very close attention to the high velocity tail, which is truncated. Depending on the number of stars in the cluster, you may not even be able to tell this apart from shot noise in the measurement. Over longer timescales, though, the cluster is evaporating. This happens because occasionally a star will gravitationally scatter a little bit and end up on an orbit whose energy is slightly positive and escape the system. This lowers the binding energy of the system, making it easier for another star to scatter onto an orbit that lets it escape, and so on. Eventually all that is left is a pair of stars locked in a Keplerian orbit.

There are some more details on the relevant timescales in this answer, and an excellent reference on the topic is this book (chapter 7, particularly 7.5, 7.5.2). Here is some content available online that explains that GCs are well-modelled by quasi-Maxwellian distribution functions, that they can in some contexts be treated as isothermal, etc. Be warned, the rabbit hole of globular cluster dynamics is deep.