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Conservation of
Angular Momentum

Discover why spinning objects speed up when they pull inward, and master the law that governs every rotating system in the universe.

AP Physics 1

What Is Angular Momentum?

Angular momentum is the rotational analog of linear momentum. Just as a moving object resists changes to its straight-line motion, a spinning object resists changes to its rotation.

Angular momentum is a vector quantity: it has both magnitude and direction (determined by the right-hand rule).

The Key Quantities

\( I \) : Moment of Inertia
How mass is distributed relative to the axis. Units: \( \text{kg} \cdot \text{m}^2 \)
\( \omega \) : Angular Velocity
How fast the object spins. Units: \( \text{rad/s} \)
\( L \) : Angular Momentum
The product \( I\omega \). Units: \( \text{kg} \cdot \text{m}^2/\text{s} \)

Why Does It Matter?

Arms Out Large \( I \), Small \( \omega \) Arms In Small \( I \), Large \( \omega \)

An ice skater pulling in their arms decreases \( I \), so \( \omega \) must increase to keep \( L \) constant. This is conservation of angular momentum in action.

A separate arms-in comparison

For this illustration the moment of inertia decreases from 4 to 2 kg·m² while angular velocity increases from 2 to 4 rad/s. Angular momentum remains 8 kg·m²/s; internal work increases rotational kinetic energy.

Illustrative arms-in rotation: I decreases from 4 to 2 kg·m² while ω increases from 2 to 4 rad/s, keeping L = 8 kg·m²/s. K increases from 8 to 16 J through internal work.
Illustrative arms-in rotation: I decreases from 4 to 2 kg·m² while ω increases from 2 to 4 rad/s, keeping L = 8 kg·m²/s. K increases from 8 to 16 J through internal work.

Mass redistribution at constant angular momentum

The same seated person appears with arms extended and folded around a common vertical axis. Still poses are not an animation or measured angular velocities.With negligible external torque about the rotation axis, pulling mass inward reduces rotational inertia and increases angular speed. Internal work can increase kinetic energy while angular momentum stays constant.

Mass redistribution at constant angular momentum

Mass redistribution at constant angular momentum. With negligible external torque about the rotation axis, pulling mass inward reduces rotational inertia and increases angular speed. Internal work can increase kinetic energy while angular momentum stays constant.With negligible external torque about the rotation axis, pulling mass inward reduces rotational inertia and increases angular speed. Internal work can increase kinetic energy while angular momentum stays constant.

The Conservation Law

This law holds for any isolated system: no matter how the parts rearrange themselves, the total angular momentum of the system cannot change unless an external torque acts on it.

Torque and \( L \): The Connection

Net Torque Rate of Change of L If \( \tau = 0 \) \( \rightarrow L = C \)

Linear vs Angular Momentum

Linear Angular
Quantity \( p = mv \) \( L = I\omega \)
Inertia Mass \( m \) Moment of inertia \( I \)
Velocity Linear \( v \) Angular \( \omega \)
Cause of change Force \( F \) Torque \( \tau \)
Conservation condition \( \sum F_{\text{ext}} = 0 \) \( \sum \tau_{\text{ext}} = 0 \)

How to Apply the Law

1
Define the system. Identify all rotating objects and the rotation axis.
2
Check for external torques. If \( \sum \tau_{\text{ext}} = 0 \), angular momentum is conserved.
3
Write the conservation equation. Set \( I_i \omega_i = I_f \omega_f \) (or sum over parts).
4
Identify knowns and unknowns. Typically you know three of the four values.
5
Solve algebraically for the unknown, then check units and reasonableness.

Example 1: Ice Skater

A figure skater spins with arms outstretched at \( \omega_i = 2 \;\text{rad/s} \) with moment of inertia \( I_i = 4.0 \;\text{kg}\cdot\text{m}^2 \). She pulls her arms in, reducing her moment of inertia to \( I_f = 1.0 \;\text{kg}\cdot\text{m}^2 \). Find her new angular velocity.

Example 2: Merry-Go-Round

A solid disk merry-go-round (\( I_{\text{disk}} = 250 \;\text{kg}\cdot\text{m}^2 \)) spins at \( \omega_i = 3.0 \;\text{rad/s} \). A child (\( m = 30 \;\text{kg} \)) jumps on at the edge, \( r = 2.0 \;\text{m} \) from the center. Find the new angular velocity.

Adding mass increased the total moment of inertia, so the system slowed down to conserve angular momentum.

Example 3: Collapsing Star

A star with \( I_i = 5.0 \times 10^{45} \;\text{kg}\cdot\text{m}^2 \) rotates once every 25 days. It collapses to a neutron star with \( I_f = 1.0 \times 10^{38} \;\text{kg}\cdot\text{m}^2 \). Find its new rotation period.

The star goes from one rotation every 25 days to ~23 rotations per second: this is how pulsars are born.

When Is \( L \) Not Conserved?

External torque present
Friction at the axle, a brake pad, or a force applied off-axis will change \( L \).
System boundary error
If you exclude part of the interacting system, the "external" torques won't cancel.

Point Mass Angular Momentum

O \( r \) \( m \) \( v \) \( \theta \)

Equivalently, \( L = mvr_\perp \), where \( r_\perp = r\sin\theta \) is the perpendicular distance from the axis to the velocity line.

Multi-Object Systems

When a system has multiple rotating parts, the total angular momentum is the sum of each part's contribution:

1
Collisions on a turntable: A ball of clay hits a spinning disk. Sum both momenta before and after.
2
Two disks coupling: A free disk drops onto a spinning one. They reach a common \( \omega_f \).
3
Person on turntable: Turning a wheel one way makes the person spin the other way.

Common Mistakes

✕ Confusing Force and Torque
Zero net force does not guarantee zero net torque. A couple (equal, opposite, offset forces) has zero net force but non-zero torque.
✕ Forgetting Sign Conventions
Angular momentum is a vector. Assign + for counterclockwise and − for clockwise (or vice versa) and stay consistent throughout the problem.
✕ Assuming KE Is Conserved
Angular momentum is conserved, but rotational kinetic energy usually is not. When \( I \) changes, \( KE = \frac{1}{2}I\omega^2 \) changes too.

Summary Cheat Sheet

Scenario \( I \) Changes? \( \omega \) Changes? Result
Pull mass inward Decreases Increases Spins faster
Extend mass outward Increases Decreases Spins slower
Mass lands on disk Increases Decreases System slows
Mass flies off tangent Decreases Increases Remainder speeds up
No external torque, no \( I \) change Constant Constant No change

Question 1 of 5

A spinning figure skater pulls her arms in, reducing her moment of inertia by half. What happens to her angular velocity?

Question 2 of 5

What condition must be met for angular momentum to be conserved?

Question 3 of 5

A disk with \( I = 4.0 \;\text{kg}\cdot\text{m}^2 \) spins at \( 6.0 \;\text{rad/s} \). A second disk with \( I = 2.0 \;\text{kg}\cdot\text{m}^2 \) (initially at rest) is dropped on top. What is the final angular velocity?

Question 4 of 5

When a spinning skater pulls her arms in, her rotational kinetic energy:

Question 5 of 5

A point mass \( m \) moves in a straight line with velocity \( v \) past a fixed point at perpendicular distance \( d \). Its angular momentum about that point is: