Section 5.5: Conservation of Angular Momentum

This section covers the principle of conservation of angular momentum and applications in rotating systems when no external torque is present.

Conservation of Angular Momentum:

If no external torque acts, total angular momentum is conserved:

\( L_\text{initial} = L_\text{final} \) or \( I_1 \omega_1 = I_2 \omega_2 \)

Example 1

A figure skater has I1 = 4 kg·m² and spins at ω1 = 2 rad/s. She pulls arms in to reduce I2 = 2 kg·m². Find ω2.

\( \omega_2 = \frac{I_1 \omega_1}{I_2} = \frac{4 \times 2}{2} = 4 \, \text{rad/s} \)

A rotating disk of I1 = 3 kg·m², ω1 = 6 rad/s, another disk I2 = 2 kg·m² is dropped onto it. Find final ω.

\( \omega_f = \frac{I_1 \omega_1}{I_1 + I_2} = \frac{3 \times 6}{3+2} = 3.6 \, \text{rad/s} \)

Skater I1 = 5 kg·m², ω1 = 3 rad/s, reduces to I2 = 2 kg·m². Find ω2.
Two disks, I1 = 4 kg·m², I2 = 1 kg·m², ω1 = 5 rad/s. Find ω_f when combined.
Sphere spinning I1 = 3 kg·m², ω1 = 6 rad/s. Moment of inertia doubles. Find ω2.
Disk I1 = 6 kg·m², ω1 = 2 rad/s. Mass is added changing I2 = 8 kg·m². Find ω2.
Rod rotating with ω1 = 4 rad/s, I1 = 2 kg·m². Ends clamped to reduce I2 = 1 kg·m². Find ω2.