Conservation of Angular Momentum

Opening the Idea

The conservation of angular momentum is the rotational twin of the conservation of linear momentum. Just as the linear momentum of an isolated system remains constant when no external force acts, the total angular momentum of a system is constant when no external torque acts. This principle governs everything from a spinning ice skater pulling in her arms to the orbital motion of planets.

For a system of particles, the total angular momentum $\mathbf{L}$ is the vector sum of the angular momenta of all individual particles. When the net external torque $\boldsymbol{\tau}_{\text{ext}}$ on the system is zero, we have:

which immediately tells us that $\mathbf{L}$ is constant in both magnitude and direction. This is the law of conservation of angular momentum.

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The Law in Detail

For a rigid body rotating about a fixed axis, the angular momentum about that axis is $L_z = I\omega$, where $I$ is the moment of inertia about the axis and $\omega$ is the angular speed. If no external torque acts about that axis, then:

This single equation explains a wide range of phenomena. When $I$ changes, $\omega$ must change in the opposite way to keep the product constant.

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Properties and Their Derivations

Property (I): The Spinning Ice Skater Effect

When a rotating body changes its moment of inertia without any external torque, its angular speed changes inversely.

Proof: Consider a person sitting on a rotating stool with arms outstretched, holding heavy weights. The system (person + stool + weights) rotates with angular speed $\omega_i$ and has moment of inertia $I_i$. When the person pulls the arms inward, the moment of inertia decreases to $I_f$. Since no external torque acts about the vertical axis, angular momentum about that axis is conserved:

Therefore:

Since $I_f < I_i$, we get $\omega_f > \omega_i$ — the rotation speeds up.

Property (II): The Dancer's Pirouette

A ballet dancer spinning on one toe demonstrates the same principle. Starting with arms extended, the moment of inertia is large and the spin is slow. Bringing the arms close to the body reduces the moment of inertia dramatically, and the spin becomes very fast.

Quantitative example: If a dancer's initial moment of inertia is $I_i = 4.0\ \text{kg m}^2$ and she spins at $\omega_i = 1.0\ \text{rev/s}$, then after pulling arms in to $I_f = 1.6\ \text{kg m}^2$:

The spin rate increases by a factor of 2.5.

Property (III): The Diver's Tuck

A diver jumping off a springboard has some initial angular momentum about the centre of mass. In mid-air, no external torque acts (ignoring air resistance), so angular momentum is conserved. By tucking the body (reducing $I$), the diver increases angular speed, allowing multiple somersaults before entering the water. Just before entry, the diver extends the body (increasing $I$), which reduces angular speed to nearly zero for a clean entry.

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The Vector Nature of Conservation

Angular momentum is a vector quantity. Conservation means both magnitude and direction remain constant. This is why a spinning gyroscope or a bicycle wheel resists changes to its orientation — the direction of its angular momentum vector tends to stay fixed in space unless a torque acts. …