Statement

When the net external torque on a system is zero: $\vec{L}_{total} = \sum I_i\omega_i = \text{constant}$

Physical Interpretation

• Angular momentum is the rotational analogue of linear momentum.
• Just as $F = 0 \Rightarrow p = \text{const}$, we have $\tau = 0 \Rightarrow L = \text{const}$.

Key Examples

1. Ice skater pulling arms in: $I_i\omega_i = I_f\omega_f$; pulling in reduces $I$, so $\omega$ increases.

2. Diver tucking: Off the board, $\tau_{ext} \approx 0$ (gravity acts through CM). Tucking reduces $I$ → $\omega$ increases → faster spin.

3. Rotating turntable + person: If the person moves outward, $I$ of system increases → $\omega$ decreases.

4. Disc collision: Disc 1 (rotating) + Disc 2 (stationary) dropped coaxially: $I_1\omega_1 = (I_1 + I_2)\omega_f$ → $\omega_f < \omega_1$

Critical Distinction

Angular momentum conservation does NOT imply kinetic energy conservation.

Since $KE = L^2/(2I)$ and $L$ is constant:

• If $I$ decreases → $KE$ increases (e.g., skater)
• The extra energy comes from internal forces (muscles)

Kinetic energy ratio: $\frac{KE_f}{KE_i} = \frac{I_i}{I_f}$