Section 5.4: Angular Momentum
This section introduces angular momentum, its calculation, and its conservation principles in rotational systems.
Angular Momentum:
\( \vec{L} = \vec{r} \times \vec{p} = I \vec{\omega} \)
- \( \vec{r} \): Position vector from axis of rotation
- \( \vec{p} \): Linear momentum
- \( I \): Moment of inertia
- \( \vec{\omega} \): Angular velocity
Angular momentum is conserved if net external torque is zero.
Example 1
A disk of I = 2 kg·m² rotates with ω = 5 rad/s. Find angular momentum.
\( L = I \omega = 2 \times 5 = 10 \, \text{kg·m²/s} \)
Example 2
A point mass m = 3 kg moves in a circle of radius 2 m with speed v = 4 m/s. Find angular momentum.
\( L = r m v = 2 \times 3 \times 4 = 24 \, \text{kg·m²/s} \)
Practice Problems
- Disk of I = 5 kg·m², ω = 3 rad/s. Compute L.
- Point mass 2 kg, radius 1.5 m, speed 6 m/s. Find angular momentum.
- Hollow cylinder, I = 4 kg·m², ω = 10 rad/s. Find L and rotational kinetic energy.
- Two disks collide and stick. Find final ω given I1, I2, ω1, ω2.
- Sphere of mass 3 kg, radius 0.5 m rotates with ω = 8 rad/s. Compute L.