# Muireall’s Shortform

• Since Raemon’s Thinking Physics exercise I’ve been toying with writing physics puzzles along those lines. (For fun, not because I’m aiming to write better exercise candidates.) If you assume an undergrad-level background and expand to modern physics and engineering there are interesting places you can go. I think a lot about noise and measurement, so that’s where my mind has been. Maybe some baseline questions could look like the below? Curious to hear anyone’s thoughts.

### Pushing a thermal oscillator

You’re standing at one end of a grocery aisle. In your cart, you have a damped oscillator in a thermal bath, initially in equilibrium.

You push the cart, making sure it moves smoothly according to a prescribed velocity profile, and you bring it to a stop at the other end of the aisle. You then wait for the oscillator to reach equilibrium with its bath again.

The final temperature is

1. Cooler than before

2. Exactly the same

3. Hotter than before

4. Not enough information. More than one of the above may be true because of one or more of the following:

1. You can only answer in expectation.

2. It depends on the properties of the oscillator.

3. It depends on the cart’s trajectory.

### Thermal velocity and camera speed

You’re observing a particle undergo thermal motion in a fluid. It’s continuously bombarded by fluid molecules that effectively subject the particle to a white noise force and velocity damping. You estimate that it tends to lose its momentum and change direction on a timescale of 1 millisecond.

You want to get some statistics on the particle’s velocity. You know the average velocity is zero, but there will be some variance that depends on temperature. You recall that in equilibrium that the particle should have velocity with probability proportional to the Boltzmann factor , giving a root mean square thermal velocity .

You calculate velocity by taking pairs of pictures at different times, then dividing the change in position by the time step. Your camera has an effectively instantaneous shutter speed.

In experiment 1, you use a time step of 0.1 milliseconds to measure velocity. In experiment 2, you use a time step of 10 milliseconds.

You collect distributions of measured velocities for each experiment, giving root mean square velocities and , respectively. What do you find?