A figure skater spins on the ice, holding two weights a distance d from the axis about which he is rotating.  At a certain time, he pulls these weights toward his body as shown, causing his rotational inertia to decrease by a factor of 2.  How does this affect the maximum angular velocity with which any part of his body moves?

To understand how pulling the weights towards his body affects the maximum angular velocity, we need to consider the principle of conservation of angular momentum. According to this principle, the total angular momentum of a system remains constant unless acted upon by an external torque.

The rotational inertia of an object is a measure of its resistance to changes in its rotational motion. When the figure skater pulls the weights towards his body, he effectively reduces the rotational inertia of his body.

Now, let's assume that the angular momentum of the skater before pulling the weights is L1, and after pulling the weights, it becomes L2. According to the conservation of angular momentum, L1 should be equal to L2.

When the skater pulls the weights towards his body, his rotational inertia decreases by a factor of 2. Since the angular momentum is conserved, this means that the angular velocity must increase to compensate for the decrease in rotational inertia.

So, when the skater pulls the weights towards his body, the maximum angular velocity with which any part of his body moves increases. This is because a decrease in rotational inertia leads to an increase in angular velocity to conserve angular momentum.

In summary, pulling the weights towards his body decreases the rotational inertia and increases the maximum angular velocity with which any part of the skater's body moves.