Conservation of Angular Momentum
Introduced| English |
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| conservation of angular momentum/ˌkɒnsəˈveɪʃn ɒv ˈæŋɡjʊlə məʊˈmentəm/ |
The skater's secret, explained at last
- We met the spinning skater who speeds up by pulling her arms in. Now we can say why.
- No one gives her a push — so her angular momentum cannot change.
- If $L = I\omega$ must stay fixed and she shrinks her $I$, then $\omega$ must rise.
- This is conservation of angular momentum 角动量守恒, one of nature's great rules.
The conservation law
- With no external torque, an object's or system's angular momentum stays constant.
- $L = I\omega$ before equals $L = I\omega$ after: $I_1\omega_1 = I_2\omega_2$.
- It is the rotational twin of conservation of linear momentum.
- Internal changes (pulling in, reshaping) can shift $I$ and $\omega$, but never their product $L$.
Angular momentum is conserved when there is no external:
With no external torque, $L$ cannot change: $I_1\omega_1 = I_2\omega_2$.
Conservation of angular momentum for a reshaping object: $I_1\omega_1 = I_2\_\_$ (fill in the symbol).
$I_1\omega_1 = I_2\omega_2$ — the product $I\omega$ is unchanged.
Skaters, divers, and stars
- A skater pulls in: $I$ drops, $\omega$ jumps — a faster spin.
- A diver tucks to spin quickly, then opens out to slow down before entry.
- A collapsing star (or forming planet) spins up dramatically as it shrinks.
- All of them keep $I\omega$ fixed while trading inertia for spin rate.
A skater with $I_1 = 6\ \text{kg}\cdot\text{m}^2$ spins at $2\ \tfrac{\text{rad}}{\text{s}}$, then pulls in to $I_2 = 2\ \text{kg}\cdot\text{m}^2$. What is her new $\omega$, in $\tfrac{\text{rad}}{\text{s}}$?
$I_1\omega_1 = I_2\omega_2 \Rightarrow 12 = 2\omega_2 \Rightarrow \omega_2 = 6\ \tfrac{\text{rad}}{\text{s}}$.
Pulling mass toward the axis (lowering $I$) makes a freely spinning object rotate faster.
To keep $L = I\omega$ constant, a smaller $I$ must be matched by a larger $\omega$.
Select all examples explained by conservation of angular momentum.
Each spins faster as $I$ shrinks, keeping $I\omega$ fixed. A resting book is not spinning.
Momentum stays, energy can change
- When the skater pulls in, her rotational kinetic energy actually increases.
- That extra energy comes from the work her muscles do pulling her arms against the spin.
- Angular momentum is conserved; kinetic energy is not automatically conserved.
- Never assume energy is unchanged just because angular momentum is.
Conservation of angular momentum
With no external torque, a spinning skater speeds up by pulling in.
When a skater pulls her arms in, her rotational kinetic energy:
Angular momentum is conserved, but she does work pulling in, so her kinetic energy increases.
Angular momentum is conserved only when there is no external torque. And conserving $L$ does not mean conserving energy — a skater pulling in gains rotational kinetic energy from the work her muscles do. Keep the two ideas separate.
A skater has $I_1 = 6\ \text{kg}\cdot\text{m}^2$ spinning at $\omega_1 = 2\ \tfrac{\text{rad}}{\text{s}}$. She pulls in to $I_2 = 2\ \text{kg}\cdot\text{m}^2$.
- Conserve $L$: $I_1\omega_1 = I_2\omega_2 \Rightarrow 6 \times 2 = 2 \times \omega_2$.
- $\omega_2 = \dfrac{12}{2} = 6\ \tfrac{\text{rad}}{\text{s}}$ — three times faster.
Conservation of angular momentum: with no external torque, $L = I\omega$ stays constant, so $I_1\omega_1 = I_2\omega_2$. Shrinking $I$ (pulling mass in) speeds the spin up. Momentum is conserved, but kinetic energy can change (the skater's muscles do work).