Twirling Skater

Consider an ice skater performing a spin. The ice is very nearly a frictionless surface

A skater spinning around has constant angular momentum, but can change his or her Moment of Interia by changing body position. What happens to their rate of rotation when they do so?

Part A

Assume that the skater originally is standing straight up while spinning, with arms extended. He or she then draws his or her arms inwards tightly to the sides. How does the angular velocity change?

Solution

System: The skater is treated as a rigid body in two configurations – one with the arms extended, the other with arms held close.

Interactions: External influences – none. The skater is moving on a frictionless surface.

Model: Single-Axis Rotation of a Rigid Body

Approach:

Diagrammatic Representation

Consider the skater intially as a massless vertical pole with the arms modeled as massless rods of length L_i with a point mass m at each end.

After contracting the skater's arms, the two masses are each a distance L_f from the body

Mathematical Representation

The definition of the Moment of Inertia is:

So that if we calculate the initial Moment of Inertia about the vertical pole that is the skater's torso, we get:

For the "final" configuration the Moment of Inertia becomes:

The Angular Momentum L has a magnitude given by

so the initial angular momentum is

and the final angular momentum is

Since the Angular Momentum is unchanged, the initial and final expressions should be equal. This means that

or

After drawing in his or her arms, the skater is spinning much more rapidly, without the application of any external forces or torques.

Part B

Solution

What if we address this from the standpoint of Energy? If the Energy is conserved (since there is no external work done on the system), then we ought to be able to derive the angular velocity from Conservation of Energy.

From our expressions for Rotational Energy, we know that the energy is given by