Conservation of Angular Momentum

The law

When the net external torque on a system is zero:

$\vec{L} = \text{constant}$

$I_1\omega_1 = I_2\omega_2$

Derivation

From $\vec{\tau} = \frac{d\vec{L}}{dt}$:

If $\vec{\tau}_{ext} = 0$, then $\frac{d\vec{L}}{dt} = 0$, so $\vec{L} = \text{constant}$.

Key consequence: changing $I$ changes $\omega$

When a rotating body changes its moment of inertia (by redistributing mass), its angular velocity changes to conserve $L$:

$I_1\omega_1 = I_2\omega_2 \implies \omega_2 = \frac{I_1}{I_2}\omega_1$

• Decrease $I$ (pull mass inward) → increase $\omega$ (spin faster)
• Increase $I$ (extend mass outward) → decrease $\omega$ (spin slower)

Classic examples

Figure skater spinning:

A skater starts spinning with arms extended (large $I$) at angular velocity $\omega_1$. She pulls her arms in (smaller $I_2$):

$\omega_2 = \frac{I_1}{I_2}\omega_1 > \omega_1$

She spins faster. The angular momentum is the same; the moment of inertia decreased, so angular velocity increased.

Diver from springboard:

A diver leaves the board extended (large $I$, slow rotation). Tucking into a ball (small $I$) increases rotation rate. Opening up before entry slows the rotation again.

Collapsing star (pulsar):

A massive star collapses into a neutron star, reducing its radius from millions of km to about 10 km. The dramatic decrease in $I$ causes it to spin hundreds of times per second — a pulsar.

Rotating stool with weights:

A person sits on a freely rotating stool holding weights at arm's length. Pulling the weights in reduces $I$ and increases $\omega$ — a classic demonstration.

Conservation in a collision

When two rotating bodies interact (a bullet hitting a door, a person jumping onto a carousel):

$L_{initial} = L_{final}$

$I_1\omega_1 + I_2\omega_2 = (I_1 + I_2)\omega_f \quad \text{(if they stick together)}$

Angular momentum vs linear momentum

A body can have $F_{ext} \neq 0$ but $\tau_{ext} = 0$ (force through the axis — centripetal force), conserving $L$ even as the body accelerates.