Conservation of Angular Momentum

students, imagine a spinning chair at a skate rink 🛼. If you pull your arms in, you start spinning faster without anyone pushing you. That surprising change is one of the best examples of conservation of angular momentum. In this lesson, you will learn what angular momentum means, why it is conserved, and how to use it in AP Physics 1 problems.

What You Will Learn

By the end of this lesson, you should be able to:

explain the meaning of angular momentum and conservation of angular momentum
identify when angular momentum is conserved in a system
use AP Physics 1 reasoning to solve rotation problems
connect angular momentum to torque, rotation, and energy in real-world situations
explain evidence for conservation of angular momentum using examples like spinning skaters, collapsing stars, and rotating chairs 🌟

Angular Momentum: The Rotational Version of Momentum

Linear momentum describes how hard it is to stop an object moving in a straight line. It is given by $p = mv$, where $m$ is mass and $v$ is velocity.

Angular momentum is the rotational version of momentum. For a rigid object rotating about a fixed axis, angular momentum is

$$L = I\omega$$

where $L$ is angular momentum, $I$ is moment of inertia, and $\omega$ is angular speed.

This equation is very important. It tells us that angular momentum depends on both how much mass the object has and how that mass is spread out from the axis of rotation. If mass is farther from the axis, the object has a larger moment of inertia and is harder to spin up or slow down.

For a point mass, angular momentum can also be written as

$$L = r p \sin\theta$$

where $r$ is the distance from the axis, $p$ is momentum, and $\theta$ is the angle between $\vec{r}$ and $\vec{p}$. In AP Physics 1, most problems use $L = I\omega$ for rotating objects.

A useful idea: angular momentum is to rotation what linear momentum is to straight-line motion. Both are linked to how motion changes and what causes that change.

Why Angular Momentum Is Conserved

Conservation laws are powerful because they help us analyze motion without tracking every force in detail. Angular momentum is conserved when the net external torque on a system is zero.

The relationship is

$$\tau_{\text{net}} = \frac{dL}{dt}$$

This means that if the net external torque is zero, then

$$\frac{dL}{dt} = 0$$

so angular momentum stays constant:

$$L_i = L_f$$

This is the conservation of angular momentum.

The key idea is that internal forces inside the system cannot change the system’s total angular momentum. Only outside torques can change it. That is why the choice of system matters so much in physics.

For example, if a skater pulls in their arms, the forces involved are internal to the skater’s body. If external torque from friction is small, the skater’s total angular momentum remains nearly constant. Because $I$ decreases, $\omega$ must increase so that $L = I\omega$ stays the same.

The Big Equation and What It Means

When angular momentum is conserved for a rotating object,

$$I_i\omega_i = I_f\omega_f$$

This equation is one of the most useful tools in this topic. It shows that if the moment of inertia decreases, angular speed increases, and if the moment of inertia increases, angular speed decreases.

This is not because energy appears from nowhere. Instead, the system can change how its motion is distributed. In many cases, work is done by muscles, motors, or internal forces, so rotational kinetic energy may change even while angular momentum stays constant.

Rotational kinetic energy is

$$K_{\text{rot}} = \frac{1}{2}I\omega^2$$

Notice that angular momentum and rotational kinetic energy are not the same thing. A system can conserve $L$ while changing $K_{\text{rot}}$.

That difference is important on the AP exam. A common mistake is thinking that if angular momentum is conserved, energy must also be conserved in the same way. That is not always true.

Real-World Example: The Spinning Skater

A skater begins spinning with arms extended. Then the skater pulls the arms close to the body.

Suppose the initial moment of inertia is $I_i$ and the initial angular speed is $\omega_i$. After pulling the arms in, the moment of inertia becomes smaller, $I_f < I_i$.

If external torque is negligible,

$$I_i\omega_i = I_f\omega_f$$

Solving for the final angular speed gives

$$\omega_f = \frac{I_i}{I_f}\omega_i$$

Because $I_i > I_f$, the final angular speed is greater than the initial angular speed. The skater spins faster. 🌀

Why does this happen? The skater does internal work to move the arms inward. That work can increase rotational kinetic energy. So even though angular momentum stays constant, energy can change.

This example shows a major AP Physics 1 theme: identify the system, check for external torque, and then choose the correct conservation idea.

Example Problem: A Rotating Platform

A student stands on a low-friction rotating platform holding small masses in each hand. The student is spinning with the masses stretched far from the body. Then the student pulls the masses inward.

What happens to the angular speed?

Because friction is very small, the external torque on the system is approximately zero. Therefore angular momentum is conserved:

$$I_i\omega_i = I_f\omega_f$$

When the masses move inward, the moment of inertia decreases. Since $I_f < I_i$, the angular speed must increase.

If the platform initially has angular speed $\omega_i = 2.0\ \text{rad/s}$ and the moment of inertia changes from $I_i = 6.0\ \text{kg}\cdot\text{m}^2$ to $I_f = 3.0\ \text{kg}\cdot\text{m}^2$, then

$$\omega_f = \frac{I_i}{I_f}\omega_i = \frac{6.0}{3.0}(2.0\ \text{rad/s}) = 4.0\ \text{rad/s}$$

So the spin rate doubles.

This kind of calculation is common in AP Physics 1. The important steps are:

identify the system
decide whether external torque is negligible
write $I_i\omega_i = I_f\omega_f$
solve for the unknown

When Angular Momentum Is Not Conserved

Angular momentum is not conserved if the net external torque is not zero.

Examples include:

a spinning wheel slowed by friction
a door rotating because someone pushes on it
a top precessing because gravity creates a torque
a bicycle wheel being twisted by an external force

If there is significant external torque, then angular momentum changes according to

$$\tau_{\text{net}} = \frac{dL}{dt}$$

This equation tells us that torque is the rotational cause of change in angular momentum, just like force is related to the change in linear momentum.

In many AP problems, the challenge is deciding whether a torque is important enough to ignore. For example, a falling object that lands on a spinning turntable may have angular momentum about the turntable axis that becomes part of the system after the collision.

Collisions and Rebounding Motion in Rotation

Conservation of angular momentum is especially useful in rotational collisions. If two objects stick together or interact while external torque is negligible, you can use angular momentum before and after the interaction.

For instance, suppose a small clay ball moving horizontally strikes and sticks to a rotating disk. If the system includes both the ball and the disk, then angular momentum about the disk’s axis is conserved during the collision if external torque is negligible.

The idea is similar to linear momentum collisions. In both cases, you compare the total before and after. The difference is that rotation uses $L = I\omega$ instead of $p = mv$.

A common AP Physics 1 strategy is to choose an axis that makes the analysis simpler. If the external torque about that axis is zero or very small, conservation of angular momentum works well.

Connecting Angular Momentum to Energy and Momentum of Rotating Systems

This topic sits inside the larger unit on energy and momentum of rotating systems. That means you need to connect several ideas:

rotational analogs of linear motion
rotational kinetic energy
angular momentum
torque and rotational change

These ideas are related, but not identical.

Energy helps describe how much work a system can do. Momentum helps describe motion and how it changes when forces or torques act. For rotation, angular momentum is the key quantity that stays constant when net external torque is zero.

In real systems, both energy and angular momentum can matter at the same time. For example, a figure skater pulling in their arms may conserve angular momentum while changing rotational kinetic energy. A satellite orbiting a planet may also show angular momentum conservation when external torque is negligible.

Understanding the difference between these quantities helps you avoid mixing them up on multiple-choice and free-response questions.

Evidence and Everyday Observations

There is strong evidence for conservation of angular momentum in everyday life and in astronomy.

Some examples include:

a spinning skater speeding up when arms come in 🛼
a diver rotating faster after pulling in legs and arms
a collapsing cloud of gas in space spinning faster as it shrinks
a neutron star rotating extremely fast because its radius becomes very small

These examples all follow the same pattern: if the system’s moment of inertia decreases and external torque is negligible, the angular speed increases.

Astronomy gives especially dramatic examples because huge objects can shrink a lot. As a star collapses, its $I$ becomes much smaller, so $\omega$ can become very large. This is a powerful real-world sign that conservation of angular momentum is a universal principle.

Conclusion

students, conservation of angular momentum is one of the most important ideas in rotational motion. The central rule is simple:

$$L_i = L_f$$

when the net external torque is zero.

Use $L = I\omega$ to connect the motion of a system to how mass is distributed around the axis of rotation. Remember that angular momentum can stay constant even when rotational kinetic energy changes. That is why it is important to identify the system, check for external torque, and choose the correct conservation law.

This topic is a major part of Energy and Momentum of Rotating Systems, and it shows up in real life all around you—from skaters and divers to planets and stars 🌌

Study Notes

Angular momentum is the rotational version of linear momentum.
For a rigid object, $L = I\omega$.
For a point mass, $L = rp\sin\theta$.
Angular momentum is conserved when net external torque is zero.
The conservation equation is $I_i\omega_i = I_f\omega_f$.
If $I$ decreases, $\omega$ increases; if $I$ increases, $\omega$ decreases.
Rotational kinetic energy is $K_{\text{rot}} = \frac{1}{2}I\omega^2$.
Angular momentum can be conserved even when rotational kinetic energy changes.
External torque changes angular momentum according to $\tau_{\text{net}} = \frac{dL}{dt}$.
Always choose the system carefully and check whether external torque is negligible.
Common examples include skaters, divers, rotating platforms, and collapsing stars.