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This may be a very basic question but I am not seeing how it works. Consider the standard example of an ice skate rotating about his/her center of mass and pulling in his/her arms. The torque is zero so we have conservation of angular momentum. This implies that $\omega$ increases to keep $I\omega$ constant, but then $K_{rot}=\frac{1}{2}I\omega^2$ doesn't stay constant, it increases. This implies that there is work done, but what force is doing this work?

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up vote 3 down vote accepted

The energy comes from the ice-skater's muscles; they have to work to pull their arms in.

There is no external work done on the skater - the energy is converted from the chemical potential energy stored in the skater's body to kinetic energy.

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This is best seen if the system is defined to be just the skater. Then we see a change in internal energy. – user11266 Dec 3 '12 at 2:02
@JoeH "Best" is subjective, but I agree that this is a valuable point of view. I wrote about it extensively on this question: – Mark Eichenlaub Dec 3 '12 at 3:13

Work is being done by the force that rearranges distribution of matter. If you want to pull in your arms, you have to fight centrifugal force.You can see it as being in some kind of a force field, much like gravitational, for example.Work done by your muscles while contracting your arms is just integral of a force over some path.

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