Conservation of Angular Momentum · 角动量守恒
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| conservation of angular momentum/ˌkɒnsəˈveɪʃn ɒv ˈæŋɡjʊlə məʊˈmentəm/ | 角动量守恒 | jiǎo dòng liàng shǒu héng |
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.
滑冰者的秘密,终于揭晓
- 我们见过收臂就加快的旋转滑冰者。现在我们能说出为什么了。
- 没有人给她推力——所以她的角动量无法改变。
- 如果 $L = I\omega$ 必须保持不变,而她缩小了 $I$,那么 $\omega$ 就必须增大。
- 这就是角动量守恒,自然界的伟大规则之一。
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 $L$.
守恒定律
- 在没有外部力矩时,物体或系统的角动量保持恒定。
- 之前的 $L = I\omega$ 等于之后的 $L = I\omega$:$I_1\omega_1 = I_2\omega_2$。
- 它是线动量守恒的转动孪生。
- 内部改变(收拢、变形)可以改变 $I$ 和 $\omega$,但绝不改变它们的乘积 $L$。
Angular momentum is conserved when there is no external: · 当没有外部____时，角动量守恒：
With no external torque, $L$ cannot change: $I_1\omega_1 = I_2\omega_2$. · 在没有外部力矩的情况下，$L$不会改变：$I_1\omega_1 = I_2\omega_2$。
Conservation of angular momentum for a reshaping object: $I_1\omega_1 = I_2\_\_$ (fill in the symbol). · 变形物体的角动量守恒：$I_1\omega_1 = I_2\_\_$（填入符号）。
$I_1\omega_1 = I_2\omega_2$ — the product $I\omega$ is unchanged. · $I_1\omega_1 = I_2\omega_2$ ——乘积 $I\omega$ 保持不变。
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.
滑冰者、跳水者和恒星
- 滑冰者收臂:$I$ 下降,$\omega$ 跃升——旋转更快。
- 跳水者团身以快速旋转,然后展开以在入水前减速。
- 坍缩的恒星(或形成中的行星)在收缩时急剧加速旋转。
- 它们全都在用惯量换旋转速率的同时保持 $I\omega$ 不变。
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 $\tfrac{\text{rad}}{\text{s}}$? · 一位转动惯量为 $I_1 = 6\ \text{kg}\cdot\text{m}^2$ 的滑冰者以 $2\ \tfrac{\text{rad}}{\text{s}}$ 旋转，然后收拢手臂至 $I_2 = 2\ \text{kg}\cdot\text{m}^2$。她的新 $\omega$ 是多少，单位为 $\tfrac{\text{rad}}{\text{s}}$？
$I_1\omega_1 = I_2\omega_2 \Rightarrow 12 = 2\omega_2 \Rightarrow \omega_2 = 6\ \tfrac{\text{rad}}{\text{s}}$.
Pulling mass toward the axis (lowering $I$) makes a freely spinning object rotate faster. · 将质量移向轴心（降低$I$）会使自由旋转的物体转得更快。
To keep $L = I\omega$ constant, a smaller $I$ must be matched by a larger $\omega$. · 为了保持$L = I\omega$不变，较小的$I$必须由较大的$\omega$来匹配。
Select all · 所有 examples explained by conservation of angular momentum. · 选择所有由角动量守恒解释的例子。
Each spins faster as $I$ shrinks, keeping $I\omega$ fixed. A resting book is not spinning. · 随着$I$减小，每个物体都转得更快，同时保持$I\omega$固定。静止的书本并未旋转。
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.
动量守恒,能量却可以改变
- 当滑冰者收臂时,她的转动动能其实增大了。
- 那份额外能量来自她的肌肉对抗旋转、收拢手臂所做的功。
- 角动量守恒;动能并不自动守恒。
- 绝不要仅因角动量守恒就假设能量不变。
Conservation of angular momentum · 角动量守恒
With no external torque, a spinning skater speeds up by pulling in. · 在没有外力矩的情况下，旋转的滑冰者通过收拢手臂加速。
When a skater pulls her arms in, her rotational kinetic energy: · 当滑冰者收拢手臂时，她的转动动能：
Angular momentum is conserved, but she does work pulling in, so her kinetic energy increases. · 角动量守恒，但她收拢时做了功，因此她的动能增加。
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.
角动量只有在没有外部力矩时才守恒。而且守恒 $L$ 不意味着守恒能量——收臂的滑冰者从肌肉所做的功中获得转动动能。要把这两个概念分开。
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.
一位滑冰者 $I_1 = 6\ \text{kg}\cdot\text{m}^2$,以 $\omega_1 = 2\ \tfrac{\text{rad}}{\text{s}}$ 旋转。她收拢到 $I_2 = 2\ \text{kg}\cdot\text{m}^2$。
- 守恒 $L$:$I_1\omega_1 = I_2\omega_2 \Rightarrow 6 \times 2 = 2 \times \omega_2$。
- $\omega_2 = \dfrac{12}{2} = 6\ \tfrac{\text{rad}}{\text{s}}$——快了三倍。
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).
角动量守恒:在没有外部力矩时,$L = I\omega$ 保持恒定,所以 $I_1\omega_1 = I_2\omega_2$。缩小 $I$(把质量收拢)使旋转加快。动量守恒,但动能可以改变(滑冰者的肌肉做了功)。