Angular momentum is a physical quantity describing rotational motion and rotational inertia relative to a specified origin or axis. For a particle, angular momentum is defined as the vector cross product of position and linear momentum. Its magnitude depends on the particles distance from the origin, its linear momentum and the angle between those vectors. For a rigid body rotating about a fixed principal axis, angular momentum can often be expressed as moment of inertia multiplied by angular velocity. Angular momentum is measured in kilogram metres squared per second and is a conserved quantity under appropriate conditions.
The conservation of angular momentum is one of the central principles of rotational mechanics. If the net external torque on a system about a chosen point is zero, its total angular momentum about that point remains constant. This does not require every component of the system to retain the same angular velocity or individual angular momentum. Internal interactions can redistribute angular momentum between components while leaving the total unchanged. Conservation therefore explains changes in rotational speed caused by changes in mass distribution, provided relevant external torques are negligible.
A familiar example involves a rotating person changing their rotational speed by moving mass closer to or farther from the rotation axis. Bringing mass inward decreases moment of inertia, so angular velocity increases when angular momentum is conserved. Extending mass outward has the opposite effect. Similar principles apply to rotating astronomical systems, spinning apparatus and mechanical devices. The underlying relationship is that angular momentum depends jointly on the distribution of mass and rotational motion rather than simply on how fast an object spins.
Angular momentum is closely connected to torque through the rotational form of the fundamental dynamical equation. The net external torque equals the rate of change of angular momentum when evaluated consistently about the relevant point or axis. This relationship is more general than the simplified equation involving moment of inertia and angular acceleration because moment of inertia can change and the rotation axis can have more complicated behaviour. Angular momentum can therefore provide a more robust framework for analysing systems involving changing configurations, multiple rotating components and orbital motion.
Angular momentum is important in orbital mechanics. An object moving under a central force can conserve angular momentum about the centre when the force produces no torque about that point. This conservation contributes to the changing orbital speed of bodies as their distance from the central object changes. Keplers area law is closely related to angular momentum conservation for motion under a central force. The same mathematical structure applies to planetary orbits, satellites, comets and other systems where central interactions dominate.
Gyroscopic behaviour is another important application. A spinning object with substantial angular momentum resists changes to the orientation of its rotational axis. When an external torque acts perpendicular to the angular momentum vector, the resulting change in angular momentum can cause precession rather than simply tipping the object directly in the direction of the applied torque. Gyroscopes, stabilisation systems and inertial navigation technologies exploit these principles. The detailed behaviour depends on geometry, angular velocity, mass distribution and applied torques.
Angular momentum has quantum mechanical and classical meanings. In quantum mechanics, angular momentum is associated with operators and quantised measurement values, with orbital and intrinsic spin angular momentum treated as distinct forms. Classical angular momentum varies continuously within the relevant mechanical model. The term spin in particle physics does not mean that a fundamental particle is necessarily a tiny rigid rotating sphere. Quantum spin is an intrinsic angular momentum property governed by quantum mechanics rather than classical rigid body rotation.
The quantity is sometimes confused with rotational energy or angular velocity. These are different concepts. Rotational kinetic energy depends on the square of angular velocity and moment of inertia, whereas angular momentum depends linearly on angular velocity for a rigid body about a fixed principal axis. An object can have angular momentum with relatively low rotational energy depending on its mass distribution and speed, and different systems can have equal angular momentum but different rotational energies. Accurate analysis therefore requires identifying the reference point, axis, mass distribution, external torques and physical model being used.
