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No kinematic study would be complete without some mention of conservation laws, right? It's a crazy, constantly shifting and mutable universe out there—but we can always count on energy, momentum, and the cheesy gordita crunch to remain unchanged for untold millennia.
The first and foremost conservation law in rotation is the Law of Conservation of Angular Momentum. The Law of Conservation of Momentum*, recall, states that the momentum (mass times velocity) of an object will never change unless an outside force is applied to it. Similarly, the Law of Conservation of Angular Momentum states that the angular momentum of an object will never change unless an outside torque is applied to it.
Angular momentum is the momentum a rotating object picks up as it spins. Just like translational momentum is a measure of both an object's mass and velocity—how hard an object is to move and the speed it's moving at—angular momentum captures how hard an object is to spin (its moment of inertia) and how fast its actually spinning (its angular velocity).
Remember our formula for translational momentum?
p = mv
Well, for angular momentum (L, which stands for "this is a totally Logical symbol for angular momentum"), we have something very similar:
L = Iω
Angular momentum is measured in kg · m2/s, unlike translational momentum, which is measured in kg · m/s. Because of this, there is no law of conservation of every kind of momentum—just the two separate but similar laws of conservation of translational momentum and conservation of angular momentum.
For an example of the Law of Conservation of Angular Momentum in action, just check out the local ice rink (or wait a few years for the Winter Olympics). Figure skaters start spins with their arms and legs extended, moving more of their body's mass farther away from their axis of rotation. This increases their moment of inertia, and they spin slowly. To spin faster, figure skaters will draw their arms and legs into their body, decreasing I—and since L must remain constant, this can dramatically increase their angular velocity.
Having nightmares about the Iron Lotus and don't want to head out on the ice? Try the same experiment in a swiveling desk chair.
Angular momentum and translational momentum are completely analogous—but that does not make them identical. They have different units, and describe different motion.
*We can call this the "Law of Conservation of Translational Momentum" now if you prefer.