Angular momentum

Angular momentum measures rotational motion the way linear momentum measures straight-line motion. For a rigid body rotating about a fixed axis, L = Iω, where I is the moment of inertia and ω is the angular velocity. For a single particle it is defined as the cross product L = r × p, so its magnitude is L = mvr sin θ, where r is the position vector from the chosen axis and θ is the angle between r and the momentum p. Angular momentum is a vector quantity whose direction follows the right-hand rule, and its SI units are kg·m²/s.

The rotational analogue of Newton's second law connects torque to angular momentum: net torque equals the rate of change of angular momentum, τ = dL/dt. Multiplying torque by the time over which it acts gives angular impulse, which equals the change in angular momentum — exactly parallel to the way linear impulse equals the change in linear momentum.

The most important consequence is conservation. When the net external torque on a system is zero, its total angular momentum stays constant. Because L = Iω, an object that reduces its moment of inertia must speed up to compensate. A figure skater pulling their arms inward spins faster, and a diver tucking into a ball rotates more quickly, without either applying any additional torque. Note that a system can conserve angular momentum while its linear momentum changes, and vice versa — the two conservation laws are independent, each tied to its own kind of external influence.

Angular momentum is tested at several levels. AP Physics 1 covers it within torque and rotational mechanics, including angular impulse and conservation problems. The FE Civil exam includes it in dynamics, and the MCAT touches on angular momentum concepts in the context of atomic and electronic structure, where electrons carry both orbital and spin angular momentum.