What Is Angular Momentum and Why Is It Conserved?

Angular momentum is the rotational equivalent of linear momentum. It is conserved when the resultant external torque on a system is zero, meaning the system’s total angular momentum remains constant.

This is an HL-only concept in A.4 Rigid body mechanics. For a rigid body rotating about a fixed axis,

$$L=I\omega$$

where $L$ is angular momentum, $I$ is the moment of inertia, and $\omega$ is angular velocity. Its SI unit is $\text{kg m}^2\text{ s}^{-1}$. Angular momentum is a vector directed along the axis of rotation.

Moment of inertia depends on both the mass and its distribution about the axis. Moving mass farther from the axis increases $I$, while moving it closer decreases $I$.

A resultant torque changes angular momentum:

$$\tau=\frac{\Delta L}{\Delta t}$$

Therefore, if $\tau_{\text{external}}=0$, then $\Delta L=0$, giving

$$I_i\omega_i=I_f\omega_f$$

Torque is the rate of change of angular momentum, so an external angular impulse produces $\Delta L=\tau\Delta t$. If external torques cancel, the angular impulse is zero and total $L$ cannot change. Internal torques transfer angular momentum between parts of the system but cannot alter its total.

Conservation of angular momentum does not imply conservation of rotational kinetic energy. As the skater pulls inward, their muscles do work, so $E_k=\frac{1}{2}I\omega^2$ can increase while remains constant. Thus, conserved momentum and conserved kinetic energy are separate conditions.

For example, a skater has $I_i=4.0\ \text{kg m}^2$ and $\omega_i=3.0\ \text{rad s}^{-1}$. After pulling in their arms, :

$$(4.0)(3.0)=(2.0)\omega_f$$

so $\omega_f=6.0\ \text{rad s}^{-1}$. The skater rotates faster because $I$ decreases, not because angular momentum increases.

Exam technique: State the system, establish that the resultant external torque is zero, and then apply $I_i\omega_i=I_f\omega_f$. A common misconception is that angular momentum is always conserved; it is conserved only when no resultant external torque acts on the chosen system.