A spinning ice skater pulls her arms inward and immediately spins faster — no external force applied, no extra energy added. A planet moves faster when closer to the Sun. A gyroscope resists being tilted even without anything holding it upright. All of these are the same phenomenon: conservation of angular momentum. Just as linear momentum is conserved when no external force acts, angular momentum is conserved when no external torque acts. The skater spins faster because reducing her moment of inertia forces her angular velocity to increase — the product Iω stays constant.
Angular momentum is the rotational analogue of linear momentum in every meaningful sense: it's a vector, it's conserved, it obeys an equation (τ = dL/dt) that mirrors Newton's second law, and it explains a huge range of phenomena that linear mechanics can't touch — from the stability of bicycle wheels to the formation of spiral galaxies.
- Angular momentum L = Iω — what moment of inertia is and how it differs from mass
- Conservation of angular momentum — when it applies and how to use it
- The relationship τ = ΔL/Δt — the rotational analogue of Newton's second law
- Real examples: figure skaters, gyroscopes, planetary orbits, and neutron stars
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.
Angular Momentum in Quantum Mechanics
In quantum mechanics, angular momentum is quantised. Orbital angular momentum magnitude is L = ℏ√(l(l+1)), where l = 0, 1, 2, … is the orbital quantum number and ℏ = h/(2π) = 1.055 × 10⁻³⁴ J·s. The z-component is L_z = mℏ, where m = −l, …, 0, …, +l. Electron spin is an intrinsic angular momentum with s = ½: S = ℏ√(3/4) = (√3/2)ℏ. This quantum angular momentum has no classical analogue — it exists even for point particles. Spin determines the magnetic properties of atoms and the structure of the periodic table (Pauli exclusion principle).
Frequently Asked Questions
What is angular momentum?
Why does a spinning skater speed up when pulling in their arms?
What is the moment of inertia?
What is the relationship between torque and angular momentum?
Is angular momentum always conserved?
How is angular momentum related to Kepler's second law?
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