Figure Skater Question?

Question

From physics, we know that when a figure skater is spinning and pulls her arms in, her rotational velocity increases (to conserve angular momentum) However, the rotational kinetic energy increases. So my question is, where does this extra energy come from? I read that it is from biochemical potential energy (from your muscles), but I am not sure if this is correct. Thanks in advance! :)

Answer 1

Your have the conservation of kinetic energy part correct. It I1 = skater's moment of inertia with arms out, and I2 = moment of inertia with arms in, then

w2 = I1/I2*w1

Now kinetic energy is:

KE1 = 1/2*I1*w1^2
KE2 = 1/2*I2*w2^2 =1/2 I1^2/I2 *w1^2

KE2 > KE1 since I2 < I1 so the extra energy has to come from the work done from the skater pulling her arms in.

Answer 2

This is true. The skater has to do work against the "centrifugal force" that tends to pull her arms outward. It takes energy to do that work. The energy then manifests itself as increased energy of rotational motion.

In a simpler example, imagine a rotating spacecraft, with two weights attached to ropes extending on opposite sides of the craft (perpendicular to the rotation axis). It requires energy (maybe from a battery) to winch the weights in toward the craft. That energy then makes the craft spin faster.