Final answer:
When a spinning skater pulls in her arms and reduces her rotational inertia by half, her angular momentum remains constant but her rate of spin doubles due to the conservation of angular momentum, resulting in increased rotational kinetic energy.
Step-by-step explanation:
If a skater who is spinning pulls her arms in so as to reduce her rotational inertia by half, her angular momentum will not change because it is conserved when no external torques act on the system. Angular momentum (L) is the product of the moment of inertia (I) and angular velocity (ω), expressed as L = Iω. When the skater pulls her arms in, she reduces her moment of inertia (I); however, since L is conserved, her angular velocity (ω) must increase. Specifically, if I is halved, ω must double to maintain the equality, because L = L' (initial angular momentum equals final angular momentum).
Work is done by the skater when pulling her arms in. She exerts an inward force on each arm, and each arm moves over a certain distance toward the body. Although work is done in this process, which translates into increased rotational kinetic energy, angular momentum remains constant due to the absence of external torques.