Describe how work is done by a skater pulling in her arms during a spin. In particular, identify the force she exerts on each arm to pull it in and the distance each moves, noting that a component of the force is in the direction moved. Why is angular momentum not increased by this action?

Work is done by a skater pulling in her arms during a spin,

The work done in a rotational motion is the change of kinetic energy of the body per unit time and derived to be

W = Tw

Where,

T = torque acting on the body

w = angular velocity

Consider the center line of the body of the skater as the axis of rotation, when the skater pulls in her arm during the spin, the angular velocity of the body increases leading to the rise in angular kinetic energy of the body.

The force exerted on each arm to pull the arm in is in the plane which is perpendicular to the plane of the axis and the distance each arm move is radian, where r is the radius of arm from the body axis.

The angular momentum remains conserved during all this process as no external torque is acting on the body

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 ∑ τ =0,  L₀ =

Explanation:

In a circular turning movement, when the arms are extended and then contracted in two possibilities:

- They are lowered the force of gravity is what pulls them, the tension of the muscle becomes zero to allow this movement.

    In this movement the force is vertical(gravity) and the movement of the center of mass of each arm is vertical, so that the work is the weight value of the arm by the distance traveled by the center of mass.

- Another possibility is that the arms have stuck to the body, in this case the person's muscles perform the force, this force is horizontal and the displacement is the horizontal of the center of mass of the arms from the extended position to the contracted

In these movements the torque of the external force is equal for each arm, but in the opposite direction, so they are canceled where a net torque of zero, this causes the angular momentum to be preserved, which changes is the moment of inertia of the system and therefore you must also change the angular velocity to keep your product constant

          ∑ τ =0

                L₀ =

              I₀ w₀ = I w