Conservation of Angular Momentum

When no external torque acts on a rotating rigid body (or system of particles), its net angular momentum stays exactly constant over time — this is the law of conservation of angular momentum, already derived in §5.2.6 from $\tau=dL/dt$: if $\tau=0$, then $L=\text{constant}$. Since $L=I\omega$, this can be written for an initial and a final state as

The practical consequence is immediate: if $I$ increases, $\omega$ must correspondingly decrease to keep their product fixed, and vice versa.

Ice dancer / figure skater. A dancer spins comparatively slowly with the arms stretched outward (large moment of inertia), and spins visibly faster once the arms are pulled in close to the body. Stretching the arms out increases the dancer's moment of inertia, so — with angular momentum conserved once the spin is underway — the angular velocity must correspondingly decrease, giving a slower spin; bringing the arms in decreases the moment of inertia, so the angular velocity increases, giving a faster spin.

A diver in the air. A diver curls the body into a tight tuck while airborne, deliberately decreasing the body's moment of inertia; since gravity produces no torque about the diver's own center of mass while in free fall, angular momentum is conserved throughout the dive, so the resulting increase in angular velocity lets the diver complete more somersaults before straightening out again just before entering the water. …