6.4 Conservation of Angular Momentum

Angular momentum (L = Iω) stays constant when the net external torque on your chosen system is zero. That is why a spinning skater speeds up when she pulls her arms in: her moment of inertia drops, so her angular velocity rises to keep L the same.

Why This Matters for the AP Physics 1 Exam

Conservation of angular momentum shows up in both multiple-choice and free-response questions, and it pairs naturally with the rotational ideas from Unit 5 and the rest of Unit 6. You will often need to compare a quantity before and after an interaction, or predict whether angular velocity increases, decreases, or stays the same when something about the system changes.

On free-response, especially the first question (the qualitative or quantitative reasoning question), naming a principle is not enough. You have to walk through the steps that connect "no external torque" to "L is constant" to your final claim. Saying "by conservation of angular momentum" without explaining why the net external torque is zero will not support a stronger score.

Key Takeaways

• Total angular momentum of a system is the sum of each part's angular momentum about the same axis.
• If the net external torque on your chosen system is zero, the total angular momentum stays constant.
• A nonrigid system can change its angular speed by moving mass toward or away from the axis, even when L does not change, because I changes.
• Internal torques between parts of a system cancel by Newton's third law, so they cannot change the system's total angular momentum.
• If the net external torque is nonzero, angular momentum transfers between the system and the environment, and the change equals the angular impulse.
• Your choice of system decides whether a torque counts as external or internal, so pick the system before you decide if L is conserved.

Conservation of Angular Momentum

Adding Up Angular Momentum

The total angular momentum of a system about a chosen axis is the sum of the angular momenta of all its parts about that same axis. Every contribution has to be measured relative to the same axis before you add them.

• All objects, particles, and components contribute to the total.
• Use the same rotational axis for every part of the system.
• In a multi-object system, each object adds to the total angular momentum about the chosen axis.

Example: For a rotating stool holding a student and two handheld masses, the total angular momentum about the stool's axis is

$L_{total} = L_{stool} + L_{student} + L_{masses}$

Each term is calculated about that same axis. If no net external torque acts on the system, this sum stays constant even though the parts can exchange angular momentum with each other.

What Changes Angular Momentum

Any change in a system's angular momentum must come from an interaction with its surroundings.

• When two parts interact, the angular impulse part A exerts on part B is equal and opposite to the angular impulse B exerts on A. This is Newton's third law, and it means angular momentum can move between parts while the system's total stays constant (if there is no net external torque).
• You can choose system boundaries so the total angular momentum is constant.
• Including all objects that exchange angular momentum in one system makes those torques internal rather than external.
• Even with constant angular momentum, angular speed can change if the shape changes.
• Moving mass closer to or farther from the axis changes the moment of inertia.
• As I decreases (mass moves inward), ω increases to keep L constant. The reverse happens when I increases.

When an external torque acts, it delivers angular impulse that changes the total angular momentum.

• Angular impulse equals the change in angular momentum. For a constant net external torque, angular impulse is $\tau_{net}\Delta t$.
• The change in angular momentum equals the net angular impulse applied.

Choosing a System

Conservation in Interactions

Angular momentum is conserved in all interactions. Whether the angular momentum of your selected system stays constant depends on how you draw the system. If the system includes all interacting objects, the torques between them are internal, so the total stays constant. If you pick a smaller system, its angular momentum can change because external torque transfers angular momentum between it and the environment.

• This works for isolated systems and for systems interacting with their surroundings.
• It holds no matter what kind of forces are involved.
• During a rotational collision, total angular momentum before equals total after, as long as there is no external torque about the axis.
• This is how you analyze skaters, stools, and collisions involving rotation.

Zero Net External Torque

When the net external torque on a system is zero, its angular momentum stays constant.

• This applies to single objects and to collections treated together.
• With no external torque to change rotation, angular momentum is preserved.
• Examples: a student on a low-friction stool, a skater pulling in their arms, or a low-friction lab setup.
• For the fixed-axis problems in AP Physics 1, this means the angular momentum about that axis does not change.

Nonzero Net External Torque

A nonzero external torque transfers angular momentum between the system and its environment.

• The change in a system's angular momentum equals the angular impulse from external torque.
• Angular momentum moves into or out of the system depending on torque direction.
• Examples: pushing a door open, braking a spinning wheel, or slowing a rotating disk.

System selection decides whether L changes. Zero net external torque about the chosen axis means constant angular momentum. Nonzero net external torque means L changes by the amount of angular impulse delivered. Choosing a larger system can turn what looked like an external torque into an internal interaction.

How to Use This on the AP Physics 1 Exam

Problem Solving

Most conservation problems follow the same setup:

1. Pick your system and your axis. Confirm the net external torque about that axis is zero (or close enough to ignore during a short collision).
2. Write $L_i = L_f$, which usually becomes $I_1\omega_1 = I_2\omega_2$ for a shape change or $L_i = I_f\omega_f$ for a collision.
3. Solve for the unknown and check that the result makes sense (smaller I should give larger ω).

Free Response

When you justify a claim, do not stop at "angular momentum is conserved." Explain the chain: identify the system, state that the net external torque about the axis is zero, conclude that L is constant, then connect L = Iω to your prediction. In a collision, point out that the pivot force acts at the axis and exerts no torque, so angular momentum about the pivot is conserved.

Common Trap

A frequent mistake is treating kinetic energy like it is conserved in these problems. When a skater pulls her arms in or a ball sticks to a rod, angular momentum is conserved but rotational kinetic energy usually is not, because internal work (muscles) or an inelastic collision changes it.

Practice Problem 1: Figure Skater Spin

Apply conservation of angular momentum:

$L = I\omega = \text{constant}$

Where:

• $L$ is angular momentum
• $I$ is moment of inertia
• $\omega$ is angular velocity

Since angular momentum is conserved when no external torques act on a system:

$I_1\omega_1 = I_2\omega_2$

Rearranging to solve for the final angular velocity:

$\omega_2 = \omega_1 \times \frac{I_1}{I_2}$

Substituting the known values:

$\omega_2 = 2.0 \text{ rad/s} \times \frac{4.0 \text{ kg}\cdot\text{m}²}{1.6 \text{ kg}\cdot\text{m}²} = 5.0 \text{ rad/s}$

The skater's angular velocity increases from 2.0 rad/s to 5.0 rad/s as she pulls in her arms.

Practice Problem 2: Student on a Rotating Stool

Use conservation of angular momentum. Since there is negligible external torque on the student-stool system, the total angular momentum before and after must be equal:

$I_1\omega_1 = I_2\omega_2$

Solving for the final angular speed:

$\omega_2 = \frac{I_1\omega_1}{I_2} = \frac{3.0 \text{ kg}\cdot\text{m}² \times 2.4 \text{ rad/s}}{2.0 \text{ kg}\cdot\text{m}²} = 3.6 \text{ rad/s}$

The student's angular speed increases from 2.4 rad/s to 3.6 rad/s after pulling the masses inward. This is a direct example of how a nonrigid system can change its angular speed without any change in angular momentum, by redistributing mass relative to the rotation axis.

Practice Problem 3: Collision and Angular Momentum

This problem uses conservation of angular momentum during a collision.

Given values:

• Rod length: $\ell = 2.0 \text{ m}$
• Rod mass: $m_r = 0.5 \text{ kg}$
• Ball mass: $m_b = 0.2 \text{ kg}$
• Ball velocity: $v_b = 15 \text{ m/s}$
• Initial angular velocity of rod: $\omega_i = 0$

Choose the system to be the rod + ball, and analyze angular momentum about the pivot. During the short collision, the pivot force acts at the pivot, so it exerts zero torque about that axis. Therefore, angular momentum about the pivot is conserved.

The rod's moment of inertia is: $I_{\text{rod}} = \frac{1}{3}m_r\ell^2 = \frac{1}{3}(0.5)(2.0)^2 = \frac{2}{3}\,\text{kg}\cdot\text{m}^2$

The ball's moment of inertia after sticking is: $I_{\text{ball}} = mr^2 = (0.2)(2.0)^2 = 0.8\,\text{kg}\cdot\text{m}^2$

So the total final moment of inertia is: $I_f = \frac{2}{3} + 0.8 = \frac{2}{3} + \frac{4}{5} = \frac{10}{15} + \frac{12}{15} = \frac{22}{15}\,\text{kg}\cdot\text{m}^2$

The ball's initial angular momentum about the pivot is: $L_i = mvr = (0.2)(15)(2.0) = 6.0\,\text{kg}\cdot\text{m}^2/\text{s}$

Conservation of angular momentum gives: $L_i = I_f\omega_f$

So: $\omega_f = \frac{6.0}{22/15} = \frac{90}{22} = 4.09\,\text{rad/s}$

The rod-ball system rotates at approximately 4.1 rad/s immediately after the collision.

Common Misconceptions

• "Angular momentum is conserved no matter what." It is conserved for the whole universe, but for the system you pick, L only stays constant if the net external torque about your axis is zero. Choose your system to make that true.
• "If L is constant, kinetic energy is too." Not usually. When a skater pulls in her arms, ω goes up and rotational kinetic energy increases because her muscles do work. In a sticking collision, kinetic energy drops.
• "You can add angular momenta about different axes." Every part must be measured about the same axis before you add. Mixing axes gives a meaningless total.
• "Mass moving inward makes the spin slow down." It is the opposite. Smaller moment of inertia means larger angular velocity to keep L the same.
• "A force always changes angular momentum." A force that acts along the axis or whose line passes through the axis produces no torque about that axis, so it does not change L. This is why the pivot force in a rod collision can be ignored.
• "Internal forces can change the system's spin." Internal torques come in equal and opposite pairs and cancel, so they cannot change the total angular momentum of the system.

Related AP Physics 1 Guides

• 6.3 Angular Momentum and Angular Impulse
• 6.5 Rolling
• 6.6 Motion of Orbiting Satellites
• 6.1 Rotational Kinetic Energy
• 6.2 Torque and Work