When the net external torque on a system is zero, the total angular momentum of that system remains constant:
Equivalently, L_i = L_f. This is the rotational analog of conservation of linear momentum. The condition is the same structure: replace "no net external force" with "no net external torque." Internal torques between parts of the system always cancel by Newton's Third Law and never affect the total L.
For a nonrigid system that changes shape while isolated (no external torque), L is constant but I can change. Since L = Iω and L is fixed:
The key insight: I and ω are inversely proportional when L is conserved. Halve I (pull mass toward axis) and ω doubles. Double I (extend mass outward) and ω halves. The product Iω never changes.
Set the initial rotational inertia and angular velocity of a spinning system, then change I to a new value. L stays constant — watch omega respond. Note whether KE is conserved (it usually is not).
Pull I_final below I_initial — omega increases. Notice KE increases too when I decreases: the system does internal work (muscle energy for the skater). L is conserved but KE is not. This is a critical AP distinction.
A skater spins with arms extended (I = 4.0 kg·m²) at 2.0 rad/s. She pulls her arms in, reducing I to 1.0 kg·m². Find her new angular velocity and compare kinetic energies before and after.
When a spinning object makes contact with a stationary one — a disk landing on another disk, a ball dropped onto a turntable — angular momentum is shared between them after the collision. This is the rotational analog of a perfectly inelastic linear collision.
The system's total L before equals total L after. If the objects end up rotating together at the same final omega:
A spinning disk (I_1 = 0.8 kg·m², omega_1 = 10 rad/s CCW) has a stationary disk (I_2 = 0.4 kg·m²) dropped onto it. Friction quickly brings them to a common angular velocity. Find omega_f.
Conservation of angular momentum only holds for a system with no net external torque. The choice of system determines whether the law applies. This is why system selection is an AP scientific skill tested explicitly in FRQs.
Internal torques (ball on table, table on ball) cancel. No external torque about the vertical axis. L is conserved. L_i = L_f applies.
The ball exerts an external torque on the turntable sub-system. Net external torque ≠ 0. L of the turntable alone is NOT conserved.
When two rotating objects interact, the angular impulse one exerts on the other is equal in magnitude and opposite in sign. This is Newton's Third Law applied to rotation.
This is why internal torques always cancel within a system. Every torque object A exerts on object B is precisely cancelled by the torque B exerts on A. The net contribution of all internal torques to the total angular momentum is exactly zero — leaving only external torques to change L.
Disk A (I = 1.0 kg·m², omega = 6 rad/s) contacts disk B (I = 2.0 kg·m², omega = 0). After friction acts for 2 s, they reach a common omega. Find the torque each disk exerts on the other.