Birefringe Benefits

This week, I set a plastic container on our kitchen counter in the sun, and noticed an interesting effect:

The reflection of the sunlight off the counter gave a rainbow pattern. I recalled seeing a similar effect in demonstrations of polarized light. The stresses frozen into plastic cause a change in the light's polarization, which you can observe by putting it between two polarizing filters:

Wikipedia

I've mentioned polarizing filters before, in the context of reflection, but as I thought about this situation more, a problem occurred to me: The plastic is birefringent, which means it rotates the polarization of light passing through it, but the light from the sun is unpolarized, so rotating it should have no effect. The light only gets polarized afterward when it reflects off the countertop. To understand why, I have to get into the weeds a bit, so I suggest reading that earlier post I linked to before continuing.

Oscillibations

A common question that arises for over-caffeinated physicists is, why does carrying a mug of coffee make it slosh over the rim? Ever since Marika got us our Apple Watches, I've been wanting to use the accelerometers in the watch, which allow it to measure your arm's orientation and motion, to record how my hand moves when walking with a mug. Sadly, I couldn't find an app that would allow me to download the data... until now! I recently decided to have another look, and found HemiPhysioData, designed to help people recovering from injuries track their progress regaining movement.

Accelerometers are a type of sensor that measure acceleration in a particular direction. A simple example is a weight on a scale: This measures a force, which is a mass multiplied by an acceleration. Typically these are used with just the acceleration due to gravity, but if you lift or lower a scale suddenly, you can increase or decrease the acceleration it reads. Inside the watch is a 3-axis accelerometer, which measures the acceleration through the face, top, and side of the watch. Using these measurements, it estimates a direction for gravity, since 9.8 m/s^2 is (hopefully) much more than an average person experiences otherwise. Subtracting that from the total acceleration gives just the wearer's contribution from moving around. We can also use it to find the orientation of the watch. All these measurements are spit out by the app as columns of a file:

1. ID columns, giving info about the run
2. Timestamp, measured at 100 Hz
3. Roll/Pitch/Yaw Euler rotations
4. Rotation vector x/y/z
5. Estimated gravity x/y/z
6. User acceleration x/y/z (total accel. minus gravity)
7. Quaternion rotation w/x/y/z
8. Raw acceleration x/y/z

The Euler, vector, and quaternion rotations are all methods for expressing the orientation of the watch. We can use these to rotate the user acceleration into the wearer's reference frame, rather than the watch's frame.

I decided to try comparing two runs: carrying an empty mug with a normal walking pace, and carrying a full mug being careful not to spill. Here's the output of the sensors for those two runs:

There's a clear periodicity to both datasets, but the differences between the two aren't clear. Instead of looking at the time-domain, we can look at the frequency spectra:

Now we can see that the empty mug has a few spikes between 4-7 Hz. If you look at Figure 5b in the paper I linked at the top, you can see this is the upper end of the frequencies that most excited the liquid in their mug. The paper points out that changing the radius of the mug will shift the resonance frequency, so the difference could be explained by the size of the mug.

The paper suggests a few methods for decreasing the risk of spilling, including dividing the cup into many small tubes, adding foam, or using a "claw grip", but I'll leave you with their comments on the suggestion of walking backwards to prevent spillage:

Of course, walking backwards may be less of a practical method to prevent coffee spilling than a mere physical speculation. A few trials will soon reveal that walking backwards, much more than suppressing resonance, drastically increases the chances of tripping on a stone or crashing into a passing by colleague who may also be walking backwards (this would most definitely lead to spillage).

Bottle Throttle

Last week I got a question from my nephew Ezra: How does the bottle-flipping trick work? What's the best amount of water to use?

In case you're unfamiliar with the phenomenon, Ezra sent along a demo of one of his flips:

As a first approximation, I figured the water should stay fairly stable in the bottom of the bottle, and the main factor that dictates whether it lands upright is how much water is in it, and the range of impact angles that cause it to tip onto its base. I pictured the landing like this:

What matters here is the height of the center of mass, which we can calculate with

where mb and mw are the masses of the bottle and the water. The tipping point will be when the center of mass is over the contact point: Farther left, and it will fall on its side, farther right and it will stay upright. The maximum value of θ then is

We can plot this for different water levels to find the best height of water (assuming a 500 ml bottle, per this page):

This model gives the optimum water level as 12%, but I wasn't entirely confident in my simplified model. I wondered whether anyone had looked at this problem in detail, and lo and behold, an arXiv paper called Water Bottle Flipping Physics!

The paper looked at 3 cases: a rigid bottle, similar to the model I came up with; a can with a pair of tennis balls, which is a simpler model with mass moving around inside; and finally the water bottle:

Figure 3

The key finding in the paper was that the bottle's rotational momentum gets absorbed by the water. This happens in such a way that the bottle stops rotating with its base pointing down. In my model above, I didn't consider an existing rotational velocity on landing, which could easily tip the bottle. According to the paper, for the water bottle the ideal filling fraction is

where M is the ratio of the mass of water in a full bottle to the mass of the bottle itself. For the numbers I used, I get M = 56 and f = 12% again! It seems at least for the 500 ml bottle, the approximation works great, but for other sizes they'll diverge.

Pulsar-Teacher Association

On Thursday, the NANOGrav project, along with international partners, made the announcement that they had detected a stochastic gravitational-wave background! This week, I thought I'd talk a bit about the news, and how the discovery was made.

First though, we should talk about what a stochastic gravitational-wave background is. Gravitational waves are produced whenever large amounts of mass move around in an asymmetric way. In the case of (still undetected) continuous waves, a bump on the neutron star, or for CBCs a pair of black holes or neutron stars. In the case of stochastic waves, we're talking about galaxies colliding, which is a much slower process. Since the movement is slower, that means the frequency is lower, on the order of nanohertz, or about 1/(32 years). That range of frequencies is far below what LIGO, or even LISA can detect:

Wikipedia

The orange region on the left is the background signals we're talking about, and the type of detectors used are called Pulsar Timing Arrays (PTAs). Pulsars are rapidly-spinning neutron stars, which produce pulses of radio-frequency signals at extremely regular intervals. They were initially referred to (jokingly) as LGMs, or "little green men", since it seemed like regular radio bursts would be a hallmark of an intelligent species.

The strength of a gravitational wave depends in part on the size of the masses that are moving. Since this background signal is due to entire galaxies moving, the gravitational waves are a million times stronger than those detected by LIGO! You might wonder then, why they were not detected before the CBCs that LIGO found. While I was thinking about this myself, an analogy occurred to me: Shifts in the Earth's tectonic plates are responsible for both earthquakes and continental drift. Even though the drift is on a significantly higher scale than the earthquakes, it's much harder to detect, due to the long periods (low frequency) involved, while earthquakes are picked up every day.

Since the first detection by Jocelyn Bell in 1968, many more pulsars have been found. The regular signals from these pulsars can be thought of as distant clocks ticking, from which the idea of pulsar timing arrays was conceived. A passing gravitational wave will cause a change in the signal's arrival time on Earth, but that change will depend on the direction of the pulsar, and the direction and polarization of the gravitational wave.

An isotropic signal means it should be the same in all directions. In 1983, Hellings and Downs suggested a method to detect such a signal: If two pulsars are affected by the same gravitational wave background, then the measurement of those pulse deviations on Earth should depend on the strength of that background, the noise in our measurements, and the orientation of the pulsar relative to Earth. By averaging the correlation between two pulsars over a long period, we can reduce the noise (which should be uncorrelated) and increase the background signal. Hellings and Downs derived a specific curve that that correlation should follow, according to the angle between the pairs of pulsars. After 15 years of collecting data from 67 pulsars, the collaboration presented this comparison to the expected curve:

Figure 1c

The points clearly deviate from the straight line that would result from no stochastic background signal, and instead follow the predicted curve, indicating a background signal is present. It's exciting to have another part of the gravitational wave spectrum filled in, and I look forward to more results from PTA groups!