Compute the moment of inertia of a point mass, rotational kinetic energy, and angular momentum, and apply conservation of angular momentum.
Moment of inertia (I) is rotation's version of mass: it measures an object's resistance to a CHANGE in its rotational motion. For a single point mass at a distance r from the rotation axis, I = m*r^2 -- notice the SQUARE, which means mass located farther from the axis contributes much more resistance to being spun up or slowed down than the same mass located close in. This is exactly why figure skaters and divers control their spin rate by moving their arms and legs closer to or farther from their body's rotation axis, without changing their total mass at all. Rotational kinetic energy follows the same pattern as ordinary kinetic energy, with I replacing m and angular velocity (omega, in radians per second) replacing v: KE_rot = (1/2)*I*omega^2.
Angular momentum, L = I*omega, is the rotational parallel to ordinary momentum (p=mv), and it obeys the same kind of conservation law: with no external torque acting, a system's total angular momentum stays constant. This is precisely why a spinning ice skater speeds up dramatically when pulling their arms in: reducing I (by bringing mass closer to the axis) forces omega to increase to keep L=I*omega unchanged.
Mastery looks like: They compute moment of inertia for a point mass, rotational kinetic energy, and angular momentum, and correctly apply conservation of angular momentum to find a new angular velocity.
Common stumbles: Forgetting to square the radius in I=m*r^2; treating angular momentum as conserved even when an obvious external torque (like friction slowing a real spinning top) is acting.