Response to John Mandlbaur

This is a response to John Mandlbaur's analysis of the conservation of angular momentum found here. Since he asked me on Quora to address his work without prejudice, I will do so. I will analyze his work with the same level of rigor I apply to my own experiments. At the end I will propose my own experiment.

Description of the experiment

I will only deal with your analysis of Walter Lewin's spinning platform demonstration, which makes up the first part of his essay, as I believe it will be sufficient to show why I don't accept your conclusion.  You measures Lewin's rate of rotation before and after he brings in his arms by timing the period of Lewin's rotation in the video. You then calculates the ratio of his measured final and initial angular velocities and compare it to the prediction Lewin calculated in the video (beginning around the 20 minute mark)

You find that the initial angular velocity divided by the final angular velocity is about 2.12 ± 0.2 while Lewin predicted a ratio of 3. You then conclude, after similar analyses of similar videos, that the law of conservation of angular momentum is not true.

What You Did Right

You were right to do an experiment. It is always good to test hypotheses, and to not take anything for granted. This experiment is sound in principle: you use the law of conservation of angular momentum to predict what the ratio of the angular velocities should be, then you measure the angular velocities and calculate their ratio and compare. If you find with high confidence that the measured and predicted ratios differ, you may begin to doubt the law of conservation of angular momentum.

You were also right to attempt to estimate the error in his your measurements. No measurement is perfect, and an imperfect measurement puts limits on how confident we can be in our experimental conclusions.

However, having read your analysis I do not share your confidence in your conclusion. I think you ignored several other sources of error that should stop us from concluding with high confidence that the prediction and measurement differ.

What You Did Not Do

There are two major sources of uncertainty in this experiment that we need to account for. The first is the uncertainty in the moments of inertia, the second is friction.

Moment of inertia

You did put error bars on your time measurements, but we also need to put error bars on the moment of inertia values. If we do not have an idea of how precise the prediction is, we can't correctly compare the measured result, and we won't know how confident to be in our conclusions.

Since Lewin was trying to show how to calculate simple moments of inertia, he did not actually measure his moment of inertia, nor did he do the incredibly complicated problem of calculating the moment of inertia of a real human body. Instead, he used some simple estimates, like modeling himself as a 75 kg cylinder with a radius of 20 cm. If we want to use Lewin's demo as a real experiment we need to understand how much error there is in this model. 

One way to estimate how much error there is in the moment of inertia value is to use a slightly more realistic model and see by how much the two models differ. Instead of a 20cm radius cylinder let's model Lewin as a cuboid with a width of 40 cm and a depth of 10 cm. These are roughly my dimensions. If he is 75 kg (which is another rough estimate) then this moment of inertia works out to 1.0625 kg m^2, compared to the 1.5 kg m^2 we get from Lewin's estimate. That's a difference of about 30%. If we want to be cute, we can treat these two estimates as data points and calculate their average and standard deviation to get the first moment of inertia to be  1.28 ± 0.22 kg m^2. This is still an incredibly rough estimate, but at least it gets us some idea of the amount of error on those values, which starts at roughly 17% at the least.

And I have not included estimates of the error of using 75 kg -- which is a fairly standard human mass physicists use in practice calculations and not an actual estimate of Lewin's mass -- or the error in the estimate of the length of his arm, or the fact that the final moment of inertia calculation ignores the weights he is holding, or any of the errors in the calculation of the final moment of inertia (which is even more complicated because some of his mass in his arms is extended).

Suppose, for example, that there is a 20% error in both moment of inertia values. Then the ratio predicted angular velocities also has an error of about 28% (this is rough statistics; the variances add, so multiply the error by square root of two). But then the predicted ratio of angular velocities is 3 ± 0.84, which overlaps significantly with the measured ratio of 2.12 ± 0.2! So we cannot conclude that angular momentum is not conserved.

Of course the 20% is a made up number, but the point is that the conclusion about angular momentum depends on the error on the predicted value. Unless we know how precise our prediction is, we can't draw any meaningful conclusions. If John wants to convince me of his conclusion he must convincingly quantify the error on the prediction.

Friction

You also neglected the fact that Lewin's demonstration has friction in it. The law of conservation of angular momentum is true in the absence of friction. A fully rigorous experiment would not predict that angular momentum is completely constant, but that it decreases at a mostly steady, hopefully small rate. Again, since Lewin is doing a demo and not a full experiment he doesn't bother to include the complication of friction. When I do a similar experiment with my students we do include friction.

Unfortunately, I can think of no great way to account for friction in this experiment. We do not have access to the platform Lewin used so we cannot measure it directly. It may be possible to compare his rotation speed at two different points in time and calculate the rate at which the speed has changed, but it looks tricky because he keeps changing his shape and the camera angle.

Without a way to quantify how much friction there is all we can say is that the angular momentum in the final position is less than the angular momentum in the initial position, and hence that the ratio of angular speeds should be somewhat less than 3.  How much less? We don't know, and until we know we can't actually make much of a conclusion. However, it is telling that your measured rate is in fact smaller than 3, consistent with some angular momentum being lost to friction.

Your original analysis essentially says "since 2.12 ± 0.2 does not overlap with 3 ± 0 so angular momentum is not conserved." The problem is that both the 3 and the 0 are not the right numbers to put there. To get the right numbers we need much better measurements or calculations of the moments of inertia and the friction. As it stands the real values could be 3 ± 0.5 or 2.23 ± 0.5 or 100 ± 99. We just don't know.

Comments on other sections

The criticisms above apply to the other experiments mentioned in your essay. In all of the referenced demonstrations the moment of inertia is either estimated or calculated using simple models with no estimate of the error. Friction is also not taken into account (except for the second video with the ping-pong ball, where the experimenter finds that reducing the time of the experiment --- and hence the amount of time friction can act --- makes the measured ratio more nearly match the friction-less prediction). The paper linked to in the third section is not published in any journal. It does only basic estimates of the error on the angular momentum or moments of inertia. The fact that Figures 3 and 6 do not show error bars on the data points is telling.

If you really want to stick to analyzing these videos you should start by figuring out ways to carefully quantify the error in the moment of inertia measurements and find ways to account for the friction in each case. Without doing that your analysis will not be persuasive.

A Proposed Experiment

I perform an angular momentum experiment with my 12th grade physics students nearly every year. The purpose is slightly different than your, but with slight re-purposing it can be made to work. I don't have any leftover data that I can use right now, but I can propose an experiment.

I have a spinning arm that can carry weights that slide in and out along each side of the arm. The arm has a photogate speedometer that can be connected to a digital device that tracks angle to measure the angular speed of the arm. There is a pulley in the center of the arm. String can be attached to the sliding weights to make their change their radius of rotation, hence changing the moment of inertia of the arm.

Here is my proposed experiment.

1. Measure the amount of friction in the arm. Assume that the frictional torque is constant. Get the arm spinning with the weights in a fixed position and measure the angular speed over time with the digital readout. The slope of the angular speed tells us the ratio of the torque to the angular momentum. Repeat this for several different configurations of the arm to determine if the torque depends on the moment of inertia or not.
2. Measure the moment of inertia of the arm with weights in several different positions. I can do this by attaching a hanging mass to a pulley so that a string turns the arm as the mass falls. Once the mass has fallen a fixed distance, the arm will have gained a fixed amount of rotational kinetic energy. Measuring the rotational speed once the mass has fallen allows us to calculate the moment of inertia of the arm. Repeating this procedure several times for each arm position and accounting for friction gives us error values for all the moment of inertia values we will use.
3. Get the arm spinning with the weights in an extended position. Collect several turns of angle data to measure the initial angular velocity. Pull on the pulley string to pull the arm into a compact position. Collect several turns of angle data to measure the final angular velocity. Record the time along with each angular velocity measurement.
4. Repeat step 3 multiple times and with multiple different weight positions.
5. Using the angular velocities for 3-4 and the moments of inertia from 2, calculate the initial and final angular momenta. Calculate the mean angular momenta and standard deviations.
6. Subtract the torque from step 1 times the time between the two speed measurements from the initial angular momentum to account for friction. Take the difference between the corrected initial angular momentum and final angular momentum. Calculate error bars for this difference using the error bars for all the values that go into it. Do this for each different arrangement of extended and contracted positions.
7. If angular momentum is conserved, the differences calculated in 6 should be consistent with zero within an error bar. The more times I repeat each measurement the closer the average should be to zero.

My proposed experiment accounts for all of my issues with your experiment. It gets actual measurements of the moments of inertia and their errors instead of simplified calculations. It accounts for friction. It also gives well-quantified error bars on the velocity measurements by repeating the experiment multiple times.

My experiment still has some limitations. It does not account for air resistance, which is another source of error. As part of my experiment I would give some back-of-the-envelope estimations of the air resistance torque. If my estimations got me a significant number, I would come up with some ways to measure it experimentally. I expect that the main uncertainty will be either my measurement of the friction or the angular speeds. I don't know the precision limits of the digital reader off-hand.

My question to you is this: if I do this experiment and document my results thoroughly, and I find that the answers are consistent conservation of angular momentum, will it change your mind? If not, are there any changes I could make to this proposed experiment that would make it more convincing to you?