Angular Momentum L=I·ω

Angular momentum: L = r × p = I·ω

Think of angular momentum as the spinning counterpart to linear momentum. For a single particle you write it as L = r × p, and for a rigid body it becomes L = I·ω, with I the moment of inertia and ω the angular velocity. The units are kg·m²/s. What makes it useful is the conservation law: when no external torque acts, the total L doesn't change. Take a figure skater with I = 0.5 kg·m² turning at ω = 10 rad/s, so L = 5 kg·m²/s. Pull the arms in and I drops to 0.25, which forces ω up to 20 rad/s. Quantum mechanics adds a twist, since there L is quantized: L = √(l(l+1))·ℏ.

Applications

Gyroscopes hold their orientation thanks to conserved L, which is why they show up in inertial navigation, smartphones and spacecraft. Satellites steer their attitude using reaction wheels and control moment gyros. Kepler's 2nd law, the bit about sweeping equal areas in equal times, is really just orbital L staying put. Pulsars spin once every few milliseconds because a collapsing stellar core shrinks I so sharply that ω has to skyrocket. Even atomic electrons carry angular momentum, both orbital and spin.

FAQ

Why does the skater spin faster? Bringing the arms in lowers I. With no external torque, L = I·ω stays fixed, so ω has to climb to compensate.

Is L a vector? It is. The direction points along the rotation axis, set by the right-hand rule, and each component is conserved on its own.

Why are pulsars so fast? When a massive star collapses, its radius falls by a factor of about 10⁵. Since I scales with radius squared, that means I drops by roughly 10¹⁰, and ω rises by the same factor. The result is a spin period of milliseconds.