Where does a spinning figure skater's energy go when she slows down? Today in physics class we were talking about angular momentum and rotational kinetic energy. My teacher used the classic example of a figure skater spinning on ice - when she pulls her arms in, her angular momentum is conserved and her angular velocity increases, meaning that her rotational kinetic energy also increases. Of course, this increase in energy must come from somewhere - in this case, it comes from the figure skater doing work on her arms and pulling them in toward her body. Then I started wondering - if the figure skater slows her rotation by extending her arms, she decreases her rotational KE. Where is her energy going? Or to put it another way, what force is doing work on the figure skater in order to decrease her energy?
 A: I think there are 2 main sources of confusion:
First, because of gravity, extending your arms feels like work. We're only interested in the radial movement, though, and in this direction, the skater's arms are pulled by the centrifugal force (in the long tradition of spherical cows in vacuum, we could replace the figure skater with two beads on a spinning rod).
Second, the idea of rotational energy as kinetic energy. The relevant work variable is (as already mentioned) the radial extension of the skater's arms, and as far as that's concerned, rotational energy plays the part of potential energy.
Think of the skater pulling in her arms as compressing a spring, and extending the arms as its release.
Going by either the bead or spring model, the rotational energy gets converted into kinetic energy of the arms, accelerated by the centrifugal force in direction of the radial work variable and ultimately dissipating via vibrations when the arms abruptly reach maximal extension.
Of course, if the skater doesn't let her arms be accelerated and slowly extends them instead, the energy dissipates right away, which might be the more realistic approach.
A: 
if the figure skater slows her rotation by extending her arms, she decreases her rotational KE. Where is her energy going? Or to put it another way, what force is doing work on the figure skater in order to decrease her energy?

In the rotating frame, as she extends arms, she gains work from the centrifugal force but does not use it in any useful way, so it transforms into internal energy of her body (her muscles will heat up). In the frame of spectators, rotational energy is transformed into internal energy as well.
