Angular momentum

Torque changes angular momentum

Angular momentum is the rotational form of momentum. It is conserved when external torque about the chosen origin is zero, even when a body changes shape or moves closer to the axis.

Integrate the torque law

With L = Iω for a rigid body about a fixed axis and τ = Iα, τ = I dω/dt = d(Iω)/dt when I is constant. More generally, angular momentum obeys dL/dt = τ_ext. Zero external torque therefore gives constant L, not necessarily constant ω if I changes.

A skater reduces I from 4 to 2 kg·m² while initially spinning at 3 rad/s. L = 12 kg·m²/s remains fixed, so final ω = 12/2 = 6 rad/s. Rotational kinetic energy rises because internal work changes the configuration.

Conservation depends on the origin and external torque

A shaft bearing can exert torque about the center, breaking angular-momentum conservation for the rotor alone. Include the bearing or measure its torque if the ideal model fails.

Work it: A turntable with I = 5 kg·m² spins at 2 rad/s. A person steps on and increases I to 8 kg·m². Predict the new speed and compare rotational energies.