10.5 Angular Momentum and Its Conservation

Angular momentum is the rotational equivalent of linear momentum. Just as linear momentum describes how hard it is to stop something moving in a straight line, angular momentum describes how hard it is to stop something spinning. Understanding this concept explains why a figure skater spins faster when they pull their arms in, and why planets maintain stable orbits.

The core principle here is conservation: when no external torque acts on a system, its total angular momentum stays constant. This single idea connects phenomena ranging from collapsing stars to diving athletes tucking mid-air.

Angular Momentum

Angular vs Linear Momentum

Angular momentum ($L$) and linear momentum ($p$) follow a parallel structure. Each quantity in the linear world has a rotational counterpart:

• Linear momentum: $p = mv$, where $m$ is mass and $v$ is velocity. A bullet fired from a gun has large linear momentum because of its high velocity.
• Angular momentum: $L = I\omega$, where $I$ is the moment of inertia and $\omega$ is angular velocity. A spinning flywheel stores angular momentum the same way a moving truck stores linear momentum.

Both are vector quantities, but their directions work differently. Linear momentum points in the direction of motion. Angular momentum points perpendicular to the plane of rotation, and you find its direction using the right-hand rule: curl the fingers of your right hand in the direction of rotation, and your thumb points along $L$.

The role that mass plays in linear momentum is played by moment of inertia in angular momentum. Moment of inertia depends not just on how much mass an object has, but on how that mass is distributed relative to the rotation axis. A hula hoop and a solid disk can have the same mass, but the hoop has a larger moment of inertia because all its mass sits far from the center.

Conservation principles apply to both types of momentum:

• No external forces → linear momentum is conserved (colliding billiard balls on a frictionless table)
• No external torques → angular momentum is conserved (a planet orbiting the Sun)

Torque Effects on Angular Momentum

Torque ($\tau$) is to rotation what force is to linear motion. Just as a net force changes linear momentum, a net torque changes angular momentum:

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

This equation says that torque is the rate of change of angular momentum over time. If you push a door, you apply a torque that increases (or decreases) its angular momentum.

The magnitude of torque depends on three things:

$\tau = rF\sin\theta$

• $r$ is the distance from the axis of rotation to where the force is applied
• $F$ is the magnitude of the applied force
• $\theta$ is the angle between the position vector and the force vector

This is why pushing a merry-go-round at its edge (large $r$) is more effective than pushing near the center, and why pushing perpendicular to the radius ($\theta = 90°$, so $\sin\theta = 1$) gives maximum torque.

Torque is also a vector. Its direction is perpendicular to the plane formed by the position vector and the force vector, again determined by the right-hand rule.

Key takeaway: If the net external torque on a system is zero, $\frac{dL}{dt} = 0$, which means $L$ is constant. That's exactly the conservation of angular momentum.

Angular impulse is the rotational analog of linear impulse. It equals the torque applied over a time interval and represents the total change in angular momentum: $\Delta L = \tau \Delta t$ (for constant torque).

Conservation of Angular Momentum

Conservation of Angular Momentum

When no external torque acts on a system, total angular momentum is conserved:

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

For a system with multiple objects, the total angular momentum is the vector sum of each object's angular momentum:

$L_{\text{total}} = L_1 + L_2 + \cdots + L_n$

The most important application involves a system where the moment of inertia changes. Since $L = I\omega$ must stay constant, any change in $I$ forces a compensating change in $\omega$:

$I_1\omega_1 = I_2\omega_2$

Here's how this plays out for a figure skater:

1. The skater begins spinning with arms extended (large $I$, moderate $\omega$)
2. The skater pulls their arms inward, moving mass closer to the rotation axis ($I$ decreases)
3. To keep $L$ constant, $\omega$ must increase, so the skater spins faster

You can rearrange to find the new angular velocity: $\omega_2 = \frac{I_1}{I_2}\omega_1$. If the skater cuts their moment of inertia in half, their spin rate doubles.

For rotational collisions where two objects collide and stick together, conservation of angular momentum gives:

$I_1\omega_1 + I_2\omega_2 = (I_1 + I_2)\omega_f$

This is analogous to a perfectly inelastic collision in linear mechanics. For example, if a rotating disk drops onto a stationary turntable, the combined system spins at a lower angular velocity than the original disk alone, because the total moment of inertia increased.

Conservation holds even when internal forces or torques are present. In a binary star system, the two stars exert gravitational torques on each other, but these are internal to the system. With no external torque, the system's total angular momentum stays constant.

Rotational Dynamics

These related concepts come up frequently alongside angular momentum:

• Rotational inertia (moment of inertia) measures an object's resistance to changes in its rotational motion, analogous to how mass resists changes in linear motion. It depends on both the total mass and how far that mass is from the axis of rotation.
• Angular displacement is the angle (in radians) through which an object rotates about its axis. It's the rotational equivalent of linear displacement.
• Rotational kinetic energy is the kinetic energy due to spinning: $KE_{\text{rot}} = \frac{1}{2}I\omega^2$. Note that when a skater pulls their arms in and spins faster, their rotational kinetic energy actually increases. The extra energy comes from the work the skater does pulling their arms inward against centripetal acceleration.
• Centripetal force is the net inward force that keeps an object moving in a circular path. It's directed toward the center of rotation and equals $F_c = \frac{mv^2}{r}$ for uniform circular motion.