A figure skater spins, with her arms outstretched, with angular velocity of ω_i. 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:

L_i = L_f  →  I_i ω_i = I_f ω_f

In this process, what happens to the skater's kinetic energy?

1. the kinetic energy decreases
2. the kinetic energy stays the same
3. the kinetic energy increases

The kinetic energy increases!

K_i = ½ I_i ω_i² = ½ (I_i ω_i) ω_i = ½ L ω_i

K_f = ½ I_f ω_f² = ½ (I_f ω_f) ω_f = ½ L ω_f

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.