Angular Momentum
A figure skater spins slowly with arms outstretched, then pulls them in and whirls faster — without any push. A bicycle stays upright only while its wheels turn. A planet sweeps around the Sun forever. All three are governed by one of the most elegant conservation laws in physics: the conservation of angular momentum.
The rotational cousin of momentum
Ordinary (linear) momentum measures how hard it is to stop something moving in a straight line. Angular momentum, written L, is its rotational counterpart: it measures how hard it is to stop something spinning. Just as a heavy, fast truck has lots of linear momentum, a heavy, fast-spinning flywheel has lots of angular momentum.
For a single particle moving at distance r from a pivot, angular momentum combines that distance with the particle’s mass and velocity. For an extended spinning body, it is captured more compactly by two quantities.
Here I is the moment of inertia — the rotational equivalent of mass — and ω is the angular velocity, how fast it spins in radians per second.
Moment of inertia: where the mass sits matters
Moment of inertia is not just about how much mass an object has, but about how far that mass is from the axis of rotation. Mass placed far from the axis contributes far more, because the contribution grows with the square of the distance.
This r² is the secret behind the spinning skater. With arms outstretched, her mass sits far from the axis and her moment of inertia is large. Pull the arms in and that mass moves close to the axis, shrinking I dramatically.
The conservation law
When no external twist (torque) acts on a system, its total angular momentum stays exactly constant. This is the conservation of angular momentum, and it is as fundamental as the conservation of energy.
If a spinning object’s moment of inertia shrinks, its angular velocity must rise to keep L constant. The skater pulls her arms in, I drops, and ω shoots up — she spins faster purely by rearranging her own mass, with no outside push at all.
Torque: how angular momentum changes
To change an object’s angular momentum you must apply a torque τ — a twisting force, related to torque. The relationship mirrors Newton’s second law: torque equals the rate of change of angular momentum.
No torque, no change in angular momentum — that is the conservation law restated. A door swings open faster when you push at its outer edge precisely because that gives more torque for the same force, changing the door’s angular momentum more quickly.
Angular momentum is a vector
Angular momentum has a direction as well as a magnitude — it points along the spin axis, given by a right-hand rule. This directional stubbornness explains some of the most counterintuitive phenomena in mechanics.
- A spinning gyroscope resists being tilted, because changing its angular momentum direction requires a torque.
- A moving bicycle stays upright thanks largely to the angular momentum of its wheels.
- A spinning top precesses, slowly tracing a cone, instead of simply falling over.
From skaters to galaxies
The same law operates across every scale. A collapsing cloud of gas spins faster as it shrinks, which is why young stars and their planetary disks rotate rapidly. Neutron stars — collapsed stellar cores just kilometres across — spin many times a second because they inherited the angular momentum of a vast, slowly turning star and concentrated it into a tiny volume. Planets orbiting the Sun sweep out equal areas in equal times, Kepler’s second law, which is nothing but angular momentum conservation in disguise. From a pirouette to a pulsar, it is the same physics.
Frequently asked questions
Why does a skater speed up when pulling in her arms?
Pulling her arms in moves mass closer to the spin axis, lowering her moment of inertia I. Since angular momentum L = Iω is conserved with no external torque, a smaller I forces a larger ω — so she spins faster. Extending the arms again slows her back down.
Is angular momentum always conserved?
It is conserved only when no net external torque acts on the system. Internal rearrangements — like a skater moving her arms — don’t count, so they leave L unchanged. An outside twist, such as friction at the pivot, can change a system’s angular momentum.
How is angular momentum different from linear momentum?
Linear momentum describes motion in a straight line and equals mass times velocity. Angular momentum describes spinning or orbiting motion and equals moment of inertia times angular velocity. They are conserved independently, which is why a system can change its rotation without changing its straight-line motion.
