Topic 6.4 Conservation of Angular Momentum: apply conservation of angular momentum to systems with no net external torque, including changes in rotational inertia.

What this topic is asking

The College Board (Topic 6.4) wants you to apply conservation of angular momentum to systems with no net external torque, including situations where the rotational inertia changes. When $\tau_{net} = 0$, the angular momentum $L = I\omega$ stays constant, so reducing $I$ increases $\omega$ and vice versa. This is the rotational twin of conservation of linear momentum from Topic 4.3.

When angular momentum is conserved

The condition is zero net external torque, exactly parallel to the "zero net external force" condition for linear momentum. Internal forces, such as a skater's muscles pulling their arms in, can rearrange the mass and change $I$, but they cannot change the total angular momentum, because they produce no external torque. This is why the spin rate changes but $I\omega$ does not.

The spinning-skater effect

This is the signature application of the topic. The numbers follow straight from $I_i\omega_i = I_f\omega_f$: solve for the new angular velocity once you know how the rotational inertia changed. The same algebra describes a person on a rotating stool pulling in weights, a planet sweeping through an elliptical orbit (Topic 6.6), and a merry-go-round that speeds up as riders walk toward the center.

Energy is not conserved when $I$ changes

A subtle and heavily tested point: when $I$ changes with $L$ fixed, the rotational kinetic energy changes. Writing $K_{rot} = \tfrac{1}{2}I\omega^2 = \tfrac{L^2}{2I}$ shows that for fixed $L$, decreasing $I$ increases $K_{rot}$. The extra energy comes from the work done by the agent that pulls the mass inward: the skater's muscles do positive work against the tendency of the spinning mass to fly outward. Conversely, letting mass drift outward lowers the kinetic energy, with the system doing work on its surroundings or storing energy elsewhere. This is the rotational analogue of the inelastic-collision lesson from Unit 4: a conservation law (here angular momentum) can hold while mechanical energy is not conserved. Rotational collisions follow the same logic: when a sliding object strikes and sticks to a pivoted rod, conserve angular momentum about the pivot to find the final angular velocity, then compute the kinetic energy lost. Recognizing which quantity is conserved, angular momentum about the pivot, not linear momentum, is the strategic key the exam rewards.

Try this

Q1. A disc spinning at $4.0$ rad/s has its rotational inertia tripled (mass moved outward) with no external torque. Calculate its new angular velocity. [2 points]

• Cue. $I_i\omega_i = I_f\omega_f$; tripling $I$ gives $\omega_f = 4.0/3 = 1.3$ rad/s.

Q2. State the condition under which a system's angular momentum is conserved. [1 point]

• Cue. The net external torque on the system is zero.