Angular Momentum and Conservation

Angular Momentum Basics

A spinning skater pulls her arms in and speeds up with no one pushing her. With no external torque, the product $L = I\omega$ is locked. Shrink $I$ and $\omega$ must rise.

Angular momentum is the rotational analogue of linear momentum:

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

Here, $I$ is rotational inertia and $\omega$ is angular velocity.

Newton's second law (rotational form) connects torque and momentum:

$$\vec{\tau}_{\text{net}} = \frac{d\vec{L}}{dt}$$

If net external torque is zero, conservation of angular momentum applies:

$$\vec{L}_i = \vec{L}_f \quad\Longrightarrow\quad I_i\omega_i = I_f\omega_f$$

Classic example: An ice skater pulls their arms in. $I$ decreases as mass moves closer to the axis, forcing $\omega$ to increase to keep $L$ constant.

Energy and Analogues

Rotational kinetic energy is given by:

$$KE_{\text{rot}} = \frac{1}{2}I\omega^2$$

For rolling without slipping, $v_{\text{cm}} = R\omega$ and total KE combines translation and rotation:

$$KE_{\text{total}} = \frac{1}{2}mv_{\text{cm}}^2 + \frac{1}{2}I\omega^2$$

Common pitfall: Conserving $L$ does not conserve rotational kinetic energy. The skater’s $KE = L^{2}/2I$ increases as she pulls her arms in because her muscles supply extra work. One conservation law never implies another.