L = Iω — Angular Momentum

Angular momentum for a rotating body — the spin equivalent of linear momentum — with a worked example.

L = I\omega

Angular momentum equals moment of inertia times angular velocity.

What it means

Angular momentum is rotational inertia times spin rate — the quantity that stays conserved when no external torque acts. It explains figure-skater spins, planetary orbits and gyroscope stability: shrink $I$ and $\omega$ must grow to keep $L$ fixed.

Variables

Symbol	Name	Unit	Meaning
$L$	Angular momentum	kg·m²/s	Rotational momentum of the body
$I$	Moment of inertia	kg·m²	Resistance to angular acceleration
$ω$	Angular velocity	rad/s	Spin rate

Assumptions

Rigid body rotating about a fixed axis
I is the moment of inertia about that axis
No external torque (for conservation)

Worked example

Problem

A wheel with moment of inertia 2 kg·m² spins at 3 rad/s. What is its angular momentum?

Given:

I = 2 kg·m²
ω = 3 rad/s

Write the formula: $L = I\omega$
Substitute: $L = 2 \times 3$
Calculate: $L = 6\ \mathrm{kg·m^2/s}$

6 kg·m²/s

The wheel carries 6 units of angular momentum. Because L is conserved without external torque, a spinning skater pulls in their arms (reducing I) and spins faster (increasing ω) — the product stays constant.