Angular Momentum and Conservation

A spinning skater pulls her arms in and inexplicably speeds up — nobody pushed her. With no external torque, the product $L = I\omega$ is locked; shrink $I$ and $\omega$ must rise. Angular momentum conservation shapes everything from figure skating to why neutron stars spin hundreds of times a second.

Angular momentum is the rotational analogue of linear momentum:

$$\vec{L} = \vec{r}\times\vec{p} = I\vec{\omega} \quad\text{(for rigid bodies)}$$

Newton's second law (rotational form)

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

If the net external torque is zero, angular momentum is conserved:

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

Classic example — An ice skater pulls their arms in:

• $I$ decreases (mass moves closer to axis).
• $\omega$ increases to conserve $L$.
• Rotational $KE = \tfrac{1}{2}I\omega^{2}$ increases — the skater does work by pulling inward.

Rotational kinetic energy

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

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

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

Summary of analogues

Key insight: Angular momentum conservation explains phenomena from spinning galaxies to gyroscopic stability — any system with no external torque.

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