Last updated: 2026-06-15

Angular Momentum

Angular momentum is rotation's version of momentum — a measure of how much "turning" a system carries, and one of the most useful conserved quantities in physics. A spinning top, an orbiting planet and a figure skater mid-spin all obey the same bookkeeping. This calculator finds the angular momentum either of an extended spinning body or of a mass moving in a circle.

Two equivalent formulas

For a rigid body spinning about an axis,

the moment of inertia times the angular velocity. For a compact mass moving along a circular path of radius $r$,

the mass times its tangential speed times its distance from the axis. The two agree: a point mass has $I = mr^2$, so $I\omega = mr^2\omega = m(\omega r)r = mvr$.

Worked example

A rigid body with $I = 0.5\ \text{kg·m}^2$ spinning at $\omega = 10\ \text{rad/s}$:

A mass of $2\ \text{kg}$ circling at $v = 5\ \text{m/s}$ on a radius of $0.5\ \text{m}$ gives the same $L = 2 \times 5 \times 0.5 = 5\ \text{kg·m}^2\text{/s}$.

Conservation

When no external torque acts on a system, its total angular momentum stays constant. This single rule explains a surprising range of effects. A figure skater who pulls their arms in lowers their moment of inertia, so their spin rate must climb to keep $L = I\omega$ fixed. A collapsing gas cloud spins faster as it shrinks, seeding the rotation of stars and planets. A diver tucks to flip quickly, then opens out to slow the rotation before entering the water.

Units

The SI unit of angular momentum is the kilogram·metre² per second, $\text{kg·m}^2\text{/s}$, which is identical to the joule·second, $\text{J·s}$. Both formulas — $L = I\omega$ with $I$ in $\text{kg·m}^2$ and $\omega$ in $\text{rad/s}$, and $L = mvr$ in base SI units — land on this same unit, a useful consistency check when setting up a problem.