Conservation of Angular Momentum

Angular momentum is L = I omega, the rotational cousin of linear momentum p = mv. Just as momentum stays fixed with no external force, angular momentum stays fixed with no external torque. That is the whole law: no net external torque means L_initial equals L_final. A skater pulling her arms in exerts forces on her own arms, but those are internal to her body, so they create no external torque on the system and cannot change L. What they can change is I, her rotational inertia, which drops as her mass moves closer to the spin axis. Since L = I omega must stay the same number, a smaller I forces a bigger omega. She spins faster automatically, not because she tries to, but because the product has to balance.

This is why kinetic energy is a trap here. Rotational kinetic energy is KE = (1/2) I omega², and omega² grows faster than I shrinks when L is held fixed, so KE actually increases. That extra energy is not free: the skater does real muscular work pulling her arms inward against the outward pull of their own circular motion, and that work is exactly where the added kinetic energy comes from. Angular momentum conservation never implies energy conservation; they are separate bookkeeping systems that respond differently to the same change in I.

This connects directly to orbits and planetary spin later in the unit: a planet or comet speeds up at closest approach for the same I-omega tradeoff, and shifting mass on Earth, like seasonal ice melt or atmospheric currents, measurably nudges its rotation rate since Earth's own L must stay constant.