(II) A figure skater can increase her spin rotation rate...

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

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

Key Concepts

Conservation of Angular Momentum

This principle states that in an isolated system with no external torques, the total angular momentum remains constant. In the context of a spinning skater, when the skater pulls in her arms and reduces her moment of inertia, her angular velocity increases in order to conserve the overall angular momentum.

Moment of Inertia

Moment of inertia quantifies an object's resistance to changes in its rotational motion. It depends on both the mass of the object and the distribution of that mass relative to the axis of rotation. For the skater, decreasing the distance of her mass from the spin axis results in a lower moment of inertia, thereby causing an increase in spin rate under conservation of angular momentum.

Relationship Between Angular Velocity and Moment of Inertia

The product of the moment of inertia and the angular velocity of a system remains constant if no external torques are acting. This means that when a figure skater changes her body configuration to reduce her moment of inertia, her spin speed must correspondingly increase to maintain the same overall angular momentum.

Physical Technique in Figure Skating

In practice, figure skaters accelerate their spins by tucking their arms and sometimes legs closer to their body. This action directly reduces their moment of inertia, which by conservation of angular momentum, leads to a faster spin. This is a real-world demonstration of the interplay between body configuration and rotational dynamics.

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