Angular Momentum of a Rotating Body

Linear momentum is mass times velocity, p = mv. Angular momentum is the rotational twin: L = I omega, where I is rotational inertia (how spread out the mass is from the axis) and omega is angular velocity. Just like p measures how hard it is to stop something moving in a straight line, L measures how hard it is to stop something spinning, or to tilt its spin axis. A spinning bicycle wheel resists being tipped over not because of some mystery force, but because tipping it means changing the direction of a large angular momentum vector, and that takes a real torque applied over time.

The reason L isn't just linear momentum relabeled is that a spinning object can have zero center-of-mass velocity and still have plenty of angular momentum. A disk spinning in place about a fixed axis through its center never translates anywhere, so its linear momentum p = mv is zero. But every bit of mass in that disk is moving in a circle, and summed up, that motion gives I omega, a nonzero angular momentum. Rotational inertia I plays the role mass played before, but I depends on how the mass is arranged relative to the axis, not just how much mass there is.

Angular momentum connects directly to torque the way linear momentum connects to force: torque is the rate of change of L, just as force is the rate of change of p. That link is what lets angular momentum be conserved when no outside torque acts, setting up the skater-pulling-in-her-arms behavior and satellite orbit problems that follow this topic. Direction matters too: L is a vector along the rotation axis, found with the right-hand rule, which is why a spinning top's angular momentum points straight along its spin axis, not off to the side.