2.5.5.2. Conservation of Angular Momentum

Table of Contents

Why angular momentum can stay constant

Conservation of angular momentum is one of the most important ideas in rotational motion. It says that if no external torque acts on a system, the total angular momentum of that system does not change.

Angular momentum may look complicated at first, but the conservation rule is simple. Nature allows angular momentum to be transferred between parts of a system, but if there is no external twisting influence, the total amount remains constant.

If the net external torque on a system is zero, then the total angular momentum is constant.
$$\tau_{\text{ext, net}} = 0 \quad \Rightarrow \quad \mathbf{L}_{\text{total}} = \text{constant}$$

This is the rotational partner of conservation of linear momentum.

From torque to conservation

For rotation, torque plays the role that force plays in straight line motion. The connection is

$$\boldsymbol{\tau}_{\text{ext, net}} = \frac{d\mathbf{L}}{dt}$$

This equation means that external torque changes angular momentum. If there is no net external torque, then

$$\frac{d\mathbf{L}}{dt} = 0$$

so angular momentum stays the same over time.

External torque changes total angular momentum. Internal torques do not change the total angular momentum of the whole system.

This point is very important. Forces between parts of the same system can change how the angular momentum is shared inside the system, but they cannot change the total if no external torque acts.

What “system” means

To use conservation correctly, you must choose a system. The system may be one object, two colliding objects, or something more complicated like a planet and a moon.

After choosing the system, ask whether any external torque acts on it. If the answer is no, angular momentum is conserved.

For example, if two skaters push off from each other on smooth ice, each skater may gain angular momentum, but the total for both skaters together remains constant if external torque is negligible.

Conservation for a rigid body

For a rigid body rotating about a fixed axis, angular momentum is often written as

$$L = I\omega$$

If no external torque acts, then

$$I_i \omega_i = I_f \omega_f$$

where the subscripts $i$ and $f$ mean initial and final.

This is especially useful when the moment of inertia changes.

For rotation about a fixed axis with no external torque,
$$I_i \omega_i = I_f \omega_f$$

If $I$ decreases, then $\omega$ must increase. If $I$ increases, then $\omega$ must decrease.

A classic example, the spinning skater

A figure skater spinning with arms extended has a larger moment of inertia. When the skater pulls the arms inward, the moment of inertia becomes smaller. Since external torque is approximately zero, angular momentum is conserved.

So,

$$I_{\text{out}} \omega_{\text{out}} = I_{\text{in}} \omega_{\text{in}}$$

Because $I_{\text{in}} < I_{\text{out}}$, it follows that

$$\omega_{\text{in}} > \omega_{\text{out}}$$

The skater spins faster, not because angular momentum increases, but because the same angular momentum is now carried with a smaller moment of inertia.

Rendering scientific sketch…

Skater pulling in arms

Orbiting motion

Conservation of angular momentum also appears in orbital motion. When a planet moves around the Sun, the gravitational force points toward the Sun. This means the torque about the Sun is zero, so angular momentum is conserved.

As the planet moves closer to the Sun, it moves faster. As it moves farther away, it moves slower. This happens because the angular momentum remains constant.

This idea helps explain why planets sweep out equal areas in equal times, which is closely related to Kepler's second law.

Collisions and angular momentum

In collisions involving rotation, angular momentum is often conserved even when kinetic energy is not. This is very useful because many real collisions are inelastic.

For example, suppose a lump of clay strikes a rotating disk and sticks to it. During the short collision time, external torque may be negligible, so

$$L_{\text{before}} = L_{\text{after}}$$

But the kinetic energy usually decreases because some of it becomes heat, sound, or deformation.

Angular momentum can be conserved even when mechanical energy is not.

This is similar to linear collisions, where momentum may be conserved while kinetic energy changes.

Example of rotational sticking collision

Imagine a disk initially at rest. A small mass $m$ moves with speed $v$ and hits the edge of the disk of radius $R$, then sticks.

Before the collision, the disk has no angular momentum. The mass has angular momentum about the disk center equal to

$$L_i = m v R$$

if it strikes tangentially.

After the collision, the mass and disk rotate together with angular speed $\omega$, so

$$L_f = I_{\text{total}}\omega$$

Conservation gives

$$m v R = I_{\text{total}}\omega$$

and therefore

$$\omega = \frac{m v R}{I_{\text{total}}}$$

This kind of problem is a standard use of angular momentum conservation.

Rendering scientific sketch…

Mass sticking to edge of disk

When angular momentum is not conserved

Angular momentum is not always conserved. If there is a net external torque, then angular momentum changes.

A door rotating on hinges is an easy example. If you push the door, your push creates an external torque, so the door's angular momentum changes.

Another important detail is the choice of axis or origin. Torque and angular momentum are defined relative to a chosen point. Conservation works when the net external torque about that same point is zero.

Common situations

The table below shows some typical cases.

Situation	External torque approximately zero?	Angular momentum conserved?
Skater pulling in arms	Yes	Yes
Planet orbiting the Sun, about the Sun	Yes	Yes
Clay sticking to rotating disk during short collision	Often yes	Yes
Spinning wheel slowed by friction at axle	No	No
Door being pushed open	No	No

Problem solving strategy

When solving problems about conservation of angular momentum, the main steps are to identify the system, choose the axis, and decide whether the net external torque about that axis is zero or negligible. Then write

$$L_i = L_f$$

and express each side in a useful form, such as $L = I\omega$ for rigid rotation or $L = mvr_\perp$ for a moving particle.

Be careful to use the same axis before and after the event.

To apply conservation of angular momentum correctly, use the same system and the same reference axis throughout the calculation.

Final idea

Conservation of angular momentum explains many striking physical effects. A skater spins faster by pulling inward, a diver tucks to rotate faster, planets change speed along their orbits, and rotational collisions can be analyzed even when energy is not conserved.

The central message is simple. Without external torque, angular momentum does not disappear and does not appear from nowhere. It remains constant.

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