A spinning ice skater pulls her arms inward — and immediately spins faster. A planet orbits faster when closer to the Sun. A gyroscope resists being tilted. A spinning top maintains its orientation. All of these phenomena are governed by a single conservation law: the conservation of angular momentum. Angular momentum is to rotation what linear momentum is to translation — a conserved vector quantity that can only change if a torque acts. It is one of the most fundamental conserved quantities in the universe, from subatomic particles to galaxies.
For a point mass moving in a circle of radius r at speed v:
L = mvr = pr
For a rotating rigid body:
L = Iω
where I is the moment of inertia (kg·m²) and ω is the angular velocity (rad/s). Angular momentum is a vector — its direction is given by the right-hand rule (thumb points along rotation axis when fingers curl in direction of rotation). Unit: kg·m²/s.
Angular Momentum of a Point Mass
For a point mass m moving with velocity v at perpendicular distance r from a reference axis:
More generally, L = m v r sinθ, where θ is the angle between the position vector r and velocity v. The direction of L is perpendicular to both r and v.
This is the relevant form for orbital mechanics. The angular momentum of a planet orbiting the Sun:
where v_orb is the orbital speed and r_orb is the orbital radius. Since L is conserved (gravity acts through the centre — zero torque about the Sun), when r decreases (closer to Sun), v must increase. This is Kepler's second law — a planet sweeps out equal areas in equal times — expressed as angular momentum conservation.
Angular Momentum of a Rotating Body: L = Iω
For an extended rotating body, the angular momentum is:
where I is the moment of inertia (the rotational analogue of mass — a measure of how mass is distributed about the rotation axis) and ω is the angular velocity (rad/s).
The moment of inertia depends on both the mass and its distribution:
| Object | Moment of inertia I | Axis |
|---|---|---|
| Point mass at radius r | mr² | Through pivot |
| Solid cylinder / disc | ½mr² | Central axis |
| Hollow cylinder / ring | mr² | Central axis |
| Solid sphere | ⅖mr² | Through centre |
| Thin rod | 1/12 mL² | Through centre, perpendicular |
Mass concentrated further from the axis gives larger I. A hollow cylinder (I = mr²) has twice the moment of inertia of a solid cylinder (I = ½mr²) of the same mass and radius, because all its mass is at the maximum radius.
Newton's Second Law for Rotation
Just as F_net = dp/dt (net force equals rate of change of linear momentum), the rotational equivalent is:
Net torque equals the rate of change of angular momentum. For constant I: τ = I × dω/dt = Iα (Newton's second law for rotation). This is the equation of motion for all rotating systems.
Conservation of Angular Momentum
If the net external torque on a system is zero, its total angular momentum is conserved:
This is one of the most fundamental conservation laws in physics, holding from quantum spin states of electrons to the rotation of galaxies.
The spinning skater
An ice skater spins with arms extended: I₁ = 4.0 kg·m², ω₁ = 2.0 rad/s. She pulls her arms in: I₂ = 1.0 kg·m². Find ω₂.
She spins four times faster. Her kinetic energy also increases: KE = ½Iω² = ½(4.0)(4) = 8 J → ½(1.0)(64) = 32 J. The extra energy comes from the work she does pulling her arms in against the centrifugal tendency to fly outward.
Angular momentum is quantised in quantum mechanics — electrons in atoms can only have angular momentum values that are integer multiples of ħ = h/(2π) = 1.055 × 10⁻³⁴ J·s. Electrons also possess intrinsic "spin" angular momentum of ±½ħ — a purely quantum property with no classical analogue. The conservation of angular momentum governs atomic transitions, selection rules for photon emission, and nuclear decay modes.
The Gyroscope: Angular Momentum and Precession
A spinning gyroscope resists changes to its orientation — a direct consequence of angular momentum conservation. When a torque τ is applied (e.g. gravity on a tilted gyroscope), instead of toppling, the gyroscope precesses — the rotation axis itself slowly rotates. The precession angular velocity:
where M is the gyroscope mass, g = 9.8 m/s², r is the distance from pivot to centre of mass, and L = Iω is the spin angular momentum. Faster spin → larger L → slower precession. Gyroscopes are used in aircraft attitude indicators, ship stabilisers, and the Hubble Space Telescope pointing system.
Frequently Asked Questions
What is angular momentum?
Angular momentum is the rotational equivalent of linear momentum. For a point mass: L = mvr. For a rotating body: L = Iω (moment of inertia × angular velocity). It is a vector quantity measured in kg·m²/s. Angular momentum is conserved when no external torque acts — one of the fundamental conservation laws of physics.
What is conservation of angular momentum?
If the net external torque on a system is zero, total angular momentum is constant: L = Iω = constant. When a spinning skater pulls her arms in (decreasing I), her angular speed ω increases to keep L constant. The same principle governs planetary orbits (Kepler's second law), neutron star formation, and atomic electron transitions.
What is the moment of inertia?
Moment of inertia I is the rotational equivalent of mass — it measures resistance to angular acceleration. I depends on both total mass and how it is distributed: I = Σmr². Mass concentrated far from the axis gives larger I. A hollow cylinder (I = mr²) has twice the moment of inertia of a solid cylinder (I = ½mr²) of the same mass and radius.
Why does a spinning skater spin faster when she pulls her arms in?
Pulling her arms in decreases her moment of inertia I. Since angular momentum L = Iω is conserved (no external torque on the ice, which is nearly frictionless), angular velocity ω must increase to compensate: ω = L/I. Halving I doubles ω. Kinetic energy increases because she does work pulling her arms against the tendency to fly outward.
How is angular momentum related to Kepler's second law?
Kepler's second law (a planet sweeps equal areas in equal times) is a direct consequence of angular momentum conservation. Gravity acts through the Sun, exerting zero torque about the Sun. So L = mvr = constant. Closer to the Sun (smaller r), orbital speed v increases proportionally. Greater distance (larger r), slower speed. This traces equal areas in equal times.
Share this article
Written by
Dr. Marcus WebbTheoretical physicist and science communicator with a PhD from Caltech. Research background in classical mechanics and gravitational physics. Passionate about making advanced physics accessible to all learners.
View all articles by this author →
