In my humble opinion, this is the best demonstration the world has of conservation of angular momentum. It is also the closest thing to an "experiment" that the old paradigm has to offer . It does not demonstrate what it claims to.
If we take measurements of the esteemed professor's turntable demonstration, one rotation at the extended arms position and another of the contracted arms position and compare the actual results to his predicted results, we find a discrepancy.
I measured the extended rotation between 24:35 and 24:39. Timing it from the point where the weight crosses his shoulder. I took three measurements using a stopwatch. The results are as follows:
I also measured the contracted rotation between 24:52 and 24:54 using the point where the weights line up. Taking more measurements because of the variation in results.
This gives us the following result:
Extended position: 3.6+-0.2 seconds per rotation.
Contracted position: 1.7+-0.2 seconds per rotation.
I use 0.2 seconds for the margin of error because it is pretty much standard practise for a manual stopwatch measurement but I believe that it is excessive for this particular situation. It does however produce a significant discrepancy.
According to his own calculations which can be found at around 23:05 in the video, the expected result from his contracted position based on the measurements for the extended position should be 1.2+-0.1
This is a discrepancy of 0.5+-0.3
About thirty percent slower than predicted by professor Lewin.
The extremely accurate prediction when we conserve rotational energy for his experiment is a two fold increase in angular velocity which is exactly what he is very shocked to discover. Please see my comments on the video.
Angular Momentum Conservation: Astronauts At Play
If angular momentum were truly conserved then the graphs entitled "Figure 3" and "Figure 6" should reflect a flat line.
The demonstration can be found just after 26:00. It is very clear that his result is nowhere near the predicted 12 000 rpm. Please see the section titled "Thought Experiment" in my mathematical physics paper.