Spinning nauseatingly fast
The world record for revolutions per minute for an ice skater is 308 RPM, held by Natalia Kanounnikova. You can see a video of the spin here.
When she starts the spin and her leg is extended, she's going somewhere between 1-2 revolutions per second. Let's average this and say her initial angular velocity is 1.5 revolutions/sec. She has to do work to move her leg in and get herself to speed up. To see this, calculate the ratio of her final rotational kinetic energy to her initial rotational kinetic energy (it's >1 so she had to do some work).
Details and assumptions
- Neglect any friction with the ice.