10.2 Angular Momentum and Its Conservation

Angular Momentum Fundamentals

Angular momentum describes how much rotational motion an object has. It plays the same role for spinning objects that linear momentum plays for objects moving in a straight line: it tells you how hard it is to stop that motion. The core equation is $L = I\omega$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.

The conservation of angular momentum explains why figure skaters spin faster when they pull their arms in, and why planets maintain stable orbits. Getting comfortable with this principle is essential for solving rotational problems throughout the course.

Definition of angular momentum

Angular momentum ($L$) quantifies rotational motion the same way linear momentum ($p = mv$) quantifies translational motion. The formula is:

$L = I\omega$

• $I$ (moment of inertia) measures how resistant an object is to changes in its rotation. It depends on both mass and how that mass is distributed relative to the axis of rotation. Mass farther from the axis means a larger $I$.
• $\omega$ (angular velocity) is how fast the object rotates, measured in radians per second.
• The units of angular momentum are $\text{kg} \cdot \text{m}^2/\text{s}$.

Think of $I$ as the rotational equivalent of mass, and $\omega$ as the rotational equivalent of velocity. A heavy flywheel spinning slowly can have the same angular momentum as a light disc spinning quickly.

Conservation of angular momentum

The total angular momentum of a system stays constant as long as no external torques act on it. Written out:

$L_{\text{initial}} = L_{\text{final}}$

An "isolated system" here means one with zero net external torque. A planet orbiting the sun is a good example: gravity acts along the line connecting the planet to the sun, producing no torque about the sun, so the planet's angular momentum is conserved throughout its orbit.

Problem-solving steps:

1. Define your system and confirm that no external torques act on it (or that they're negligible).
2. Identify the initial and final states of the system.
3. Calculate the initial angular momentum using $L = I\omega$.
4. Set $L_{\text{initial}} = L_{\text{final}}$ and solve for the unknown.

This principle applies to objects that change shape mid-rotation (a gymnast tucking during a somersault), collisions between rotating objects, and angular momentum transfer in systems like meshing gears.

Moment of inertia vs. angular velocity

Because $L = I\omega$ must stay constant in an isolated system, $I$ and $\omega$ have an inverse relationship:

$I_1\omega_1 = I_2\omega_2$

If the moment of inertia increases, the angular velocity must decrease by the same factor, and vice versa.

• Arms out while spinning on a stool: You move mass farther from the axis, increasing $I$, so $\omega$ drops and you spin slower.
• Diver tucking into a ball: Pulling limbs in close to the body decreases $I$, so $\omega$ increases and the diver rotates faster.
• Ice skater pull-in: A skater starts a spin with arms extended, then pulls them tight. The decrease in $I$ produces a visibly faster spin, with no external torque needed.

The key insight in all these cases is the same: no one is adding or removing angular momentum. The object is just redistributing its own mass.

Angular momentum in collisions

Angular momentum is conserved during collisions and explosions as long as no external torques act on the system. The approach mirrors how you use linear momentum conservation, but with rotational quantities.

• Elastic collisions conserve both angular momentum and rotational kinetic energy.
• Inelastic collisions conserve angular momentum but not kinetic energy. Some energy is lost to deformation, heat, or sound.
• Explosions work in reverse: a single rotating object breaks apart, but the total angular momentum of all the pieces equals the angular momentum of the original object.

Analysis steps:

1. Calculate the total angular momentum of the system before the collision or explosion.
2. Determine the angular momenta of the individual components afterward.
3. Set the total before equal to the total after and solve for unknowns.