4.19 The Law of Conservation of Angular Momentum

Introduction

The Law of Conservation of Angular Momentum is a fundamental principle in physics that governs the rotational motion of objects. It states that the total angular momentum of an isolated system remains constant over time. This principle is analogous to the conservation of linear momentum and is essential for understanding the behaviour of rotating systems, from spinning ice skaters to orbiting planets.

System and Isolated System

A system is any collection of objects that we choose to analyse.

A system is considered isolated with respect to rotation if there is no net external torque acting on it. Internal torques (forces between objects within the system) always occur in equal and opposite pairs and do not change the total angular momentum of the system.

Statement of the Law

Mathematical Basis

The relationship between net external torque (${\vec{\tau }}_{net}$) and the rate of change of angular momentum ($\vec{L}$) is the rotational analogue of Newton's Second Law:

${\vec{\tau }}_{net}=\frac{\Delta \vec{L}}{\Delta t}$

This equation states that the net external torque on a system equals the rate at which its angular momentum changes.

If the system is isolated, the net external torque is zero:

${\vec{\tau }}_{net}=0⟹\frac{\Delta \vec{L}}{\Delta t}=0$

Therefore the angular momentum is constant:

$\vec{L}=\text{constant}$

Both the magnitude and direction of the angular momentum vector are conserved:

$L_i=L_f$

The Implication: $L=I\omega =\text{constant}$

The angular momentum of a rigid body is given by:

$L=I\omega$

where $I$ is the moment of inertia and $\omega$ is the angular velocity.

If angular momentum is conserved, the product $I\omega$ must remain constant:

$I_i{\omega }_i=I_f{\omega }_f$

This has a profound consequence:

• If $I$ decreases (mass moves closer to the axis), $\omega$ must increase.
• If $I$ increases (mass moves away from the axis), $\omega$ must decrease.

Applications and Examples

1. Spinning Ice Skater

An ice skater begins spinning with arms and legs extended — large $I$, slow $\omega$. When the skater pulls arms and legs in close to the body, $I$ decreases. To conserve $L=I\omega$, the angular velocity $\omega$ increases and the skater spins much faster.

2. Divers and Gymnasts

A diver leaves the board with a certain angular momentum. Pulling the body into a tight tuck position minimises $I$, causing $\omega$ to increase dramatically (fast rotation). Extending the body again before entry increases $I$ and slows the rotation for a controlled entry.

3. Formation of Stars and Planets

A large, slowly rotating cloud of interstellar gas (nebula) collapses under gravity. As the radius decreases, $I$ decreases significantly. To conserve $L$, $\omega$ increases — the cloud spins faster and faster, forming a flattened protoplanetary disk from which stars and planets form.

Key Comparison: Linear vs Angular Momentum Conservation

A force can be applied through the centre of mass (zero torque) — in that case angular momentum is conserved but linear momentum changes.

Effect of Internal Forces

Internal forces (and the torques they create) within a system always occur in equal and opposite pairs (Newton's Third Law) and cancel out. They cannot change the total angular momentum of the system.