Angular Momentum

For a single particle, angular momentum $\vec{L}$ is a vector given by the cross product of the displacement vector from the rotation axis with the linear momentum of the particle

$$\vec{L} = \vec{r} \times p \vec{}$$

$$L = \Sigma r_{i} m_{i} v_{i} = \Sigma m_{i} r_{i}^2 \omega$$ $$\therefore$$ $$L = I\omega$$ $$\vec{L} = I\vec{\omega}$$ Units: $[\vec{L}] = [I][\vec{\omega}] = [kg][m^2][s^{-1}]$

The rotational analog of impulse is: $$\vec{\tau} = \frac{d \vec{L}}{dt}$$

Similarly to linear momentum, angular momentum changes only if there is a net torque acting on the system. When net torque is zero, angular momentum is conserved.

The vector quantity is also conserved: $\vec{L}{final} = \vec{L}{initial}$