Problem — Why the skater spins faster

Problem

Why does a figure skater spin faster when they draw their arms in towards their body? What is conserved and what is not?

Solution

The angular momentum $L = I\omega$ is conserved, because no appreciable external torques act about the vertical axis. By drawing the arms in, the skater reduces their own moment of inertia $I$; to keep $L$ constant the angular velocity $\omega$ must increase — which is why they spin faster.

The rotational kinetic energy is not conserved: $E_\text{cin,rot} = \dfrac{L^2}{2I}$ increases as $I$ decreases. The extra energy is supplied by the muscular work done to pull the arms towards the axis. $\ev{L=I\omega \text{ is conserved};\quad I\downarrow \Rightarrow \omega\uparrow;\quad E_\text{cin,rot}\uparrow \text{ (muscular work)}}$

Links

Topics: Rotational dynamics Concepts: Conservation of angular momentum · Rotational kinetic energy