Angular Momentum and Impulse

For a rigid body rotating about a fixed axis, angular momentum is $L=I\omega$. If the resultant external torque about that axis is zero, total angular momentum remains constant. Internal forces can redistribute mass without changing the system’s total angular momentum.

The statement applies to a chosen system and axis. A skater who pulls in their arms reduces I and increases ω. A diver uses changes of body shape to alter rotation during flight while external torque about their centre of mass is small. Friction with a support or air can introduce an external torque.

A constant net external torque acting for a time produces an angular impulse equal to the change in angular momentum. For a changing torque, use the signed area under a torque–time graph. Angular momentum has units kg m² s⁻¹, and torque impulse has units N m s.

Choose positive rotation before using signs. A torque opposing the rotation gives a negative change in angular momentum. If a machine is brought to rest, the final angular momentum is zero, so the impulse is the negative of the initial value.

With angular momentum fixed, rotational kinetic energy can be written $E_k=L^2/(2I)$. A smaller I therefore gives a larger kinetic energy. In an athlete this extra energy comes from muscular work; conservation of angular momentum does not imply unchanged kinetic energy.

When two rotating objects couple together, conserve total angular momentum if external torque is negligible, but some kinetic energy can become internal energy during the coupling. Write the angular momenta of both objects with their signs before combining them.

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.