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?

- the kinetic energy decreases
- the kinetic energy stays the same
- the kinetic energy increases

The kinetic energy increases!

K_{i} = ½
I_{i} ω_{i}^{2} = ½ (I_{i}
ω_{i}) ω_{i} = ½ L ω_{i}

K_{f} = ½
I_{f} ω_{f}^{2} = 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.