TITLE: One-way speed of light experiment
QUESTION [0 upvotes]: Ok, this is (probably) my last question about this topic. My other two (here and here) had some misguided assumptions, but I think this one is better. (There is probably something else I am neglecting, but I'll give it a shot anyway.)
Set up a rotating apparatus with two toothed wheels, one attached to each end of a rod.  (You would know the distance between the wheels because it is just the length of the rod.) Then, you rotate the apparatus, let's say, counterclockwise.  You would then offset the rotation of the wheel farthest from the light source by some known angle (clockwise) so that the light has a possibility of traveling through both wheels (as the light travels from the closer to the farther wheel, the farther wheel would "catch up" and the light would pass through it.)  Since they are both attached to the same rod, I don't think you would need to worry about clock synchronization because both wheels would have to spin at the same rate.  Then, you could adjust the rotation rate of the apparatus until the light passes through both wheels, at which point you would be able to tell how long it took for the light to pass between the two wheels, which would allow you to calculate the one-way speed of light.  (I'm not sure if the location at which the torque is applied to the rod matters, but, if you were to generate the rotation of the apparatus [apply the torque to it] at the center of the rod, I don't think clock synchronization would matter here.)
As a little diagram:
SOURCE --- WHEEL #1 --- ROD --- WHEEL #2 --- DETECTOR
As with my other questions, I'm guessing there is something I haven't thought of.  Please let me know what that thing is (since there most likely is one).  Thank you!
EDIT:
Also, I think that applying the torque to the rod at the center of the rod might be super important here.  Since it takes time for the "signal" that the rod is rotating to travel to each end of the rod, having the torque in the middle would mean that the "signal" would reach both ends at the same time.
EDIT #2:
The math of it all (or at least what I believe is the correct math):
$$v_{\text{one-way}}=\frac{\Delta x}{\Delta t}$$
$$\omega_{\text{rod}}=\omega_{\text{w1}}=\omega_{\text{w2}}=\frac{\Delta \theta}{\Delta t}\text{ so }\Delta t=\frac{\Delta \theta}{\omega_{\text{rod}}}$$
$$v_{\text{one-way}}=\frac{\Delta x}{\frac{\Delta \theta}{\omega_{\text{rod}}}}=\frac{\omega_{\text{rod}}\Delta x}{\Delta \theta}$$
$$\omega_{\text{rod}}=\frac{2\pi}{T}$$
$$v_{\text{one-way}}=\frac{2\pi\Delta x}{T\Delta \theta}$$
where $T$ is the period of rotation for the rod/wheels, $\Delta x$ is the length of the rod (which is also the distance between the wheels), and $\Delta \theta$ is the angle that you offset the farther rod (this is the amount that the farther wheel would then have to rotate during the time $
\Delta t$, which is the amount of time it would take for the light to pass between the two wheels).
EDIT #3:
I wasn't clear when I was talking about applying the torque to the rod.  I meant to apply the torque before the experiment, not during.

REPLY [1 votes]: This question doesn't mention the Veritasium video, but I'll assume it's about that video since the others were. As I said in another answer, Veritasium's variable-speed-of-light model is just standard special relativity in different coordinates. The variant in which the speed of light ranges from $c/2$ to $\infty$ is special relativity in coordinates like $(t',x,y,z)$ where $t'=t-x/c$ and $(t,x,y,z)$ are standard inertial coordinates.
The easiest way to analyze any experiment in the variable-speed coordinates is to analyze it in the standard coordinates and then transform. I'll assume your rod is parallel to the $x$ axis. With respect to standard coordinates, a symmetry argument shows that the rod should have no net twist at any $t$ in its equilibrium rotating configuration (if the torque is applied at the center, or if it's rotating without torque and without friction). Therefore, it is twisted at every $t'$. The amount of the twist is $ω_\text{rod}Δx/c$. This is just enough such that the light moving at $c/2$ or $\infty$ will make it through.
Note that the rod is also twisted with respect to an isotropic inertial frame that is moving in the $x$ direction with respect to the lab frame, due to the relativity of simultaneity, and this "explains" why the light makes it through in both directions in that frame even though the speed of the light relative to the apparatus is $c\pm v$ in that frame. Whether the primed coordinates are inertial or not doesn't matter.
Explaining the twist more directly in either of these primed coordinate systems would be rather difficult. It comes down to the fact that the rigidity of the rod is enforced by electromagnetic interactions between atoms, and those interactions are effectively anisotropic in the primed frames. But what I want to stress is that you don't need to do this analysis. The fact that you'll get the same answer in any coordinates is not a physical assumption; it's purely a statement about the internal consistency of the mathematics.
(In the case of inertial frames, it is a nontrivial physical assumption that the laws of physics are the same with respect to any of them, but that assumption is already built into the physical laws that you're using to do this calculation. You aren't performing an independent test of the assumptions of the theory by calculating the shape of the rod in the primed frame. It's constructed such that the answer must be consistent.)


TITLE: Vectors and Loci of points
QUESTION [1 upvotes]: For some constant vector $\vec{A}$, and for $\vec{r}$ the position vector with respect to the origin,
describe the loci of points defined by the following equations:
a) $(\vec{r} - \vec{A}) \cdot \vec{r} = 0$
b) $(\vec{r} - \vec{A}) \cdot \vec{A} = 0$
I'm not sure how to interpret these. I should be able to look at these and decide that they are circles or whatever have you, but I can't seem to come up with anything that makes sense. I have tried writing out the terms, but nothing enlightening has come to me.

REPLY [2 votes]: If geometrical intuition fails, you can always go with plain old algebra. If $\vec{A}=(a,b)$ and $\vec{r}=(x,y)$, then your two equations write out as:
$$(x-a)x+(y-b)y=0$$
$$(x-a)a+(y-b)b=0$$
A little tampering with those equations should lead you to recognizable forms of the equations of the loci you are after.


TITLE: Given starting number in sequence, find position of second number
QUESTION [0 upvotes]: Given a sequence of numbers $1 - 12$, with starting number $X$ at position $1$, how would you go about finding the position $P$ of number $Y$?
For example, given $X$ = 5 and $Y$ = 4:
The sequence of numbers would be: $[5 \ 6 \ 7 \ 8 \ 9 \ 10 \ 11 \ 12 \ 1 \ 2 \ 3 \ 4]$
So $P = 12$
Or if $X =12$ and $Y = 3$
$[12 \ 1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9 \ 10 \ 11]$
$P = 4$
Hopefully this is clear. This is my first SO post, so feedback on formatting etc is welcome.

REPLY [0 votes]: If $Y \ge X$ then $Y$ is in the $Y-X$ position past $1$.  In other words $P = (Y-X) + 1$.
If $Y < X$ then we must go up to $12$ That is $12-X +1=13-X$ and then $Y$ more places $P= 13 - X +Y$
Or, $P = (Y-X) + 1$ if $Y \ge X$ or $P = [(Y-X) + 1] + 12$ if $Y < X$.


TITLE: Noninteracting electrons in 2D in presence of a magnetic field and a periodic potential?
QUESTION [6 upvotes]: Equally spaced discrete harmonic oscillator levels, called Landau levels, are obtained for noninteracting electrons in 2D in presence of a magnetic field applied perpendicular to the plane. The Landau levels are highly degenerate and the energy eigenfunctions can also be obtained exactly.
In practice, however, the electrons are also moving in an underlying periodic lattice. When we put in a periodic potential, we usually get an electronic bandstructure.
What will happen to the energy spectrum of the theory, when both the effects are present i.e, for noninteracting electrons moving in 2D in presence of a magnetic field and a periodic potential? I was interested in this question because I want to know whether one needs to take into account the energy bandstructure due to a periodic potential to understand quantum Hall effect in addition to Landau levels.
If anyone knows a good reference that solves this problem, please let me know.

REPLY [2 votes]: Statistical Physics, Part 2 by Landau & Livshits contains a chapter on "The symmetry of electron states in a lattice in a magnetic field" (Translation might be unprecise, as I am using my Russian edition.) Basically, the presence of magnetic field breaks the translational symmetry and leads to some interesting quirks, but for weak magnetic fields on still can use Peierls substitution in the effective mass Hamiltonian.
This is an admittedly old resource, but it could serve as a good starting point for serious analysis.


TITLE: Topic for teaching assessment
QUESTION [3 upvotes]: I'm in the position to have a teaching assessment with a tutoring agency next week. This assessment will include me teaching the assessor a topic of my choice in 15 minutes, demonstrating the Socratic method that the agency has chosen as its teaching philosophy.
The problem is that I know very little about the math background of the assessor (I expect GCSE or equivalent), which makes it difficult to find a topic that they will not already be familiar with or have little chance of understanding in 15 minutes.
I am therefore looking for suggestions of math topics satisfying the following criteria;

self-contained enough to be taught in 15 minutes
largely independent of math taught beyond GCSE or equivalent
not already familiar to someone with math GCSE or equivalent
lending itself to the Socratic method, i.e. requiring little factual knowledge, and more about logical thinking and applying concepts. If a few basic concepts need to be introduced at the beginnings, that's fine.
ideally, involving a model (of some sort) that can be played around with, or a graphical representation that can be developed with the student.

So far, I've been mostly thinking in the direction of discrete math, and applications like mechanics, biology, computer science, but have not really found anything satisfying my criteria.
Thanks.

REPLY [2 votes]: I am guessing that they do not want you to teach them something novel, but instead something similar to what you will be teaching your students.  I would recommend transformations of functions:  people have a lot of trouble teaching that topic, so if you do a good job you will make a good impression.