Conservation of Angular Momentum
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| conservation of angular momentum/ˌkɒnsəˈveɪʃn ɒv ˈæŋɡjʊlə məʊˈmentəm/ | Bảo toàn động lượng góc |
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:
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
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).
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$I_1\omega_1 = I_2\omega_2$ — 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 $\tfrac{\text{rad}}{\text{s}}$?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$I_1\omega_1 = I_2\omega_2 \Rightarrow 12 = 2\omega_2 \Rightarrow \omega_2 = 6\ \tfrac{\text{rad}}{\text{s}}$.
Pulling mass toward the axis (lowering $I$) makes a freely spinning object rotate faster.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
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:
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
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).
Write from a new spin of 4 rad/s
- Rotational inertia falls from 8 to 2. The initial spin is 1 rad/s. No external torque acts.
- Find the new spin.
An isolated skater's rotational inertia drops from 8 to 2 kg m squared. The spin starts at 1 rad/s. What is the new angular speed?
8 divided by 2 is 4.
What is the new angular speed?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The product stays 8, so the new speed is 4 rad/s.