Section 5.5: Angular Momentum
Angular momentum is the rotational analogue of linear momentum. It describes the quantity of rotation an object has, taking into account its mass, shape, and rotational speed.
\( L \) = angular momentum (kg·m²/s)
\( I \) = moment of inertia (kg·m²)
\( \omega \) = angular velocity (rad/s)
Angular momentum is a vector quantity, pointing along the axis of rotation according to the right-hand rule. If no external torque acts on a system, angular momentum is conserved.
Example 1: Rotating Disc
A disc of mass 2 kg and radius 0.5 m rotates at 10 rad/s. Calculate its angular momentum. Assume the disc is solid and rotates about its central axis.
Moment of inertia of solid disc: \( I = \frac{1}{2} m R^2 = \frac{1}{2} (2)(0.5^2) = 0.25 \, \text{kg·m²} \)
Angular momentum: \( L = I \omega = 0.25 \times 10 = 2.5 \, \text{kg·m²/s} \)
Example 2: Spinning Figure Skater
A figure skater spins with arms extended at 2 rad/s. She pulls her arms in, reducing her moment of inertia from 5 kg·m² to 2 kg·m². Find her new angular speed.
Angular momentum conserved: \( L_i = L_f \Rightarrow I_i \omega_i = I_f \omega_f \)
\( \omega_f = \frac{I_i \omega_i}{I_f} = \frac{5 \times 2}{2} = 5 \, \text{rad/s} \)
Practice Problems
- A solid cylinder of mass 4 kg and radius 0.3 m rotates at 6 rad/s. Find its angular momentum.
- A flywheel of moment of inertia 10 kg·m² rotates at 15 rad/s. Calculate its angular momentum.
- A particle of mass 0.5 kg moves in a circle of radius 2 m with speed 3 m/s. Find its angular momentum about the center.
- A figure skater has moment of inertia 6 kg·m² and spins at 3 rad/s. She reduces her radius so that \( I = 2 \, \text{kg·m²} \). Find new angular speed.
- A disc of radius 0.4 m and mass 3 kg rotates at 8 rad/s. Find its angular momentum about the central axis.