A figure skater can increase her spin rotation rate from an initial rate of 1.0 rev every 1.5 s to a final rate of 2.5 rev/s. If her initial moment of inertia was $4.6 \textrm{ kg}\cdot\textrm{m}^2$, what is her final moment of inertia? How does she physically accomplish this change?

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. The angular momentum of the figure skater is assumed to be constant because there's no net external torque acting on the system. The ice is really slippery so we'll just assume that there's no torque there between the skates and the ice. So that means the final moment of inertia times the final angular velocity equals the initial moment of inertia times the initial angular velocity and we can solve for the final moment of inertia by dividing both sides by ω f and we have final moment of inertia is I initial times ω initial divided by ω f. So initially her moment of inertia is 4.6 kilogram meter squared and we'll times that by 1 revolution for every 1.5 seconds divided by the final angular velocity of 2.5 revolutions per second and the moment of inertia must be 1.2 kilogram meter squared. Notice that we didn't need to turn this into radians per second because since we are dividing these revolutions per second the units cancel anyway so the only thing we are concerned with is making sure they are the same, 'revolution per second' in this case and that's fine. And figure skater accomplishes this decrease in moment of inertia from 4.6 to 1.2 by pulling her arms close to her body which is the same as pulling more mass closer to the axis of rotation and that decreases moment of inertia.