Say I am out in space rotating with my arms extended. If I slowly pull my arms in, my angular velocity speeds up to conserve angular momentum, and my rotational kinetic energy has increased. That doesn't bother me, cuz I know I had to expend some chemical energy to bring my arms in. I did work against the centrifugal force. Now suppose that instead of starting out with my arms extended, i start out rotating with them tucked against me. If I slowly extend my arms, say at a small constant speed, my angular velocity slows down and I lose kinetic energy. But i still had to do work against the centrifugal force to keep my arms from accelerating outward. So I expended chemical energy, and now I have less kinetic energy. Where did the energy go?