A figure skater spins, with her arms outstretched, with angular velocity of ωi. When she moves her arms close to her body, she spins faster. Her moment of inertia decreases, so her angular velocity must increase to keep the angular momentum constant.
Conservation of angular momentum:
Li = Lf → Ii ωi = If ωf
In this process, what happens to the skater's kinetic energy?
The kinetic energy increases!
Ki = ½ Ii ωi2 = ½ (Ii ωi) ωi = ½ L ωi
Kf = ½ If ωf2 = ½ (If ωf) ωf = ½ L ωf
The figure skater does work on her arms and hands as she brings them
closer to her body - that's where the extra energy comes from.