rotation motion?

Question

two sphere have equal densities and are subject only to their mutual gravitational attraction. Which the following quantities must have the same magnitude for both spheres.
a) acceleration
b) velocity
c) kinetic Energy
d) Displacement from the center of mass
e) Gravitational force
I think it is e

An ice skater is in a spin with his arms outstretched. If he pulls in his arms, what happens to his rotational kinetic energy?
a) increase
b) decrease
c) remains constant but non-zero
d) remains zero
i think it is a

Answer 1

You are right on both counts.

Given that the two spheres may be of different dimensions and hence different masses, all the other quantities can be different. The magnitude of the gravitational force between them is the same (though the direction is different). The comment about their having the same density is a red herring.

For the skater, pulling in the arms causes the spinning rate to rise so as to keep the angular momentum constant. But the kinetic energy goes as the square of the spinning rate, so the result is a net gain in the kinetic energy. This energy comes from the work done by the skater when he pulls his arms in.