Angular Momentum and Angular Impulse · 角动量与角冲量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| angular momentum/ˈæŋɡjʊlə məʊˈmentəm/ | 角动量 | jiǎo dòng liàng |
Why a spinning top refuses to fall over
- A stationary top topples at once; spin it and it stands, defying gravity for a surprisingly long time.
- Spinning things resist changes to their rotation — they have angular momentum 角动量.
- It is the rotational cousin of ordinary (linear) momentum.
- To change it, you must apply a torque over time — an angular impulse.
为什么旋转的陀螺不肯倒下
- 静止的陀螺立刻倒下;转起来它就立着,在出人意料的长时间里抗拒重力。
- 旋转的东西抵抗其转动状态的改变——它们有角动量。
- 它是普通(线)动量的转动表亲。
- 要改变它,你必须在一段时间里施加力矩——一个角冲量。
Angular momentum
- Angular momentum is $L = I\omega$ — rotational inertia times angular velocity.
- It is the twin of $p = mv$, with $I$ for mass and $\omega$ for velocity.
- Its units are $\text{kg}\cdot\text{m}^2/\text{s}$, and it points along the spin axis.
- The faster the spin or the larger the $I$, the more angular momentum.
角动量
- 角动量是 $L = I\omega$——转动惯量乘以角速度。
- 它是 $p = mv$ 的孪生,以 $I$ 代替质量、$\omega$ 代替速度。
- 它的单位是 $\text{kg}\cdot\text{m}^2/\text{s}$,并沿旋转轴方向。
- 旋转越快或 $I$ 越大,角动量越大。
Spin carries angular momentum · 自旋携带角动量
Change the spin to feel how angular momentum L = Iω builds with faster rotation. · 改变自旋以感受角动量 L = Iω 如何随旋转加快而增大。
A disk with $I = 2\ \text{kg}\cdot\text{m}^2$ spins at $\omega = 5\ \tfrac{\text{rad}}{\text{s}}$. What is its angular momentum, in $\text{kg}\cdot\text{m}^2/\text{s}$? · 一个具有$I = 2\ \text{kg}\cdot\text{m}^2$的圆盘以$\omega = 5\ \tfrac{\text{rad}}{\text{s}}$旋转。其角动量是多少，单位为$\text{kg}\cdot\text{m}^2/\text{s}$？
$L = I\omega = 2 \times 5 = 10\ \text{kg}\cdot\text{m}^2/\text{s}$.
Angular momentum is the rotational twin of linear momentum $p = mv$. Which is its formula? · 角动量是线性动量$p = mv$的转动对应量。其公式是什么？
$L = I\omega$ — inertia times angular velocity, the twin of $mv$. · $L = I\omega$ ——惯性乘以角速度，$mv$ 的对应物。
Angular impulse changes it
- An angular impulse is a torque acting over time: $\tau\,\Delta t$.
- It equals the change in angular momentum: $\tau\,\Delta t = \Delta L$.
- This is the rotational twin of $F\,\Delta t = \Delta p$.
- A long-lasting torque, or a large one, produces a big change in spin.
角冲量改变它
- 角冲量是一段时间内作用的力矩:$\tau\,\Delta t$。
- 它等于角动量的变化:$\tau\,\Delta t = \Delta L$。
- 这是 $F\,\Delta t = \Delta p$ 的转动孪生。
- 持续很久的力矩,或很大的力矩,会产生旋转的大变化。
An angular impulse is a torque acting over ____. · 角冲量是在____上作用的力矩。
Angular impulse $= \tau\,\Delta t$, and it equals the change in angular momentum. · 角冲量$= \tau\,\Delta t$，且等于角动量的变化。
A torque of $4\ \text{N·m}$ acts for $2\ \text{s}$. By how much does the angular momentum change, in $\text{kg}\cdot\text{m}^2/\text{s}$? · 一个大小为$4\ \text{N·m}$的力矩作用了$2\ \text{s}$。角动量改变了多少，单位为$\text{kg}\cdot\text{m}^2/\text{s}$？
$\Delta L = \tau\,\Delta t = 4 \times 2 = 8\ \text{kg}\cdot\text{m}^2/\text{s}$.
Angular momentum resists tipping
- A large angular momentum makes an object "gyroscopically" stubborn — it holds its axis.
- This steadies spinning tops, bicycle wheels, footballs thrown with spin, and satellites.
- To turn the axis you must supply a torque; without one, the axis stays put.
- The bigger $L$ is, the harder the object is to knock off its spin.
角动量抵抗倾倒
- 大的角动量使物体"陀螺般"固执——它保持自己的轴。
- 这让陀螺、自行车轮、带旋转投出的橄榄球以及卫星保持稳定。
- 要转动轴,你必须提供力矩;没有力矩,轴就保持不动。
- $L$ 越大,物体越难被打乱它的旋转。
Angular momentum · 角动量 $I\omega$ and rotational kinetic energy $\tfrac12 I\omega^2$ are the same quantity. · 角动量$I\omega$和转动动能$\tfrac12 I\omega^2$是同一物理量。
They differ: $L$ is linear in $\omega$, energy is quadratic. They are distinct quantities. · 它们不同：$L$ 在 $\omega$ 上是线性的，能量是二次的。它们是不同的量。
Select all · 所有 effects that come from a large angular momentum. · 选择所有由大角动量引起的效应。
Large $L$ makes spinning objects hold their axis. Falling speed is unrelated to angular momentum. · 大的$L$使旋转物体保持其轴线。下落速度与角动量无关。
Angular momentum $L = I\omega$ is not the same as rotational kinetic energy $\tfrac12 I\omega^2$. One is linear in $\omega$, the other quadratic. In a collision or a skater's spin, angular momentum is the conserved quantity — energy may change.
角动量 $L = I\omega$ 与转动动能 $\tfrac12 I\omega^2$ 不相同。一个与 $\omega$ 成一次方,另一个成平方。在碰撞或滑冰者的旋转中,角动量才是守恒量——能量可能改变。
A disk of $I = 2\ \text{kg}\cdot\text{m}^2$ spins at $\omega = 5\ \tfrac{\text{rad}}{\text{s}}$.
- $L = I\omega = 2 \times 5 = 10\ \text{kg}\cdot\text{m}^2/\text{s}$.
A torque of $4\ \text{N}\cdot\text{m}$ applied for $2\ \text{s}$ adds $\tau\Delta t = 8$, raising $L$ to $18\ \text{kg}\cdot\text{m}^2/\text{s}$.
一个 $I = 2\ \text{kg}\cdot\text{m}^2$ 的圆盘以 $\omega = 5\ \tfrac{\text{rad}}{\text{s}}$ 旋转。
- $L = I\omega = 2 \times 5 = 10\ \text{kg}\cdot\text{m}^2/\text{s}$。
一个 $4\ \text{N}\cdot\text{m}$ 的力矩作用 $2\ \text{s}$,增加 $\tau\Delta t = 8$,把 $L$ 提高到 $18\ \text{kg}\cdot\text{m}^2/\text{s}$。
Angular momentum is $L = I\omega$, the rotational twin of $p = mv$ (units $\text{kg}\cdot\text{m}^2/\text{s}$). An angular impulse $\tau\,\Delta t$ changes it: $\tau\,\Delta t = \Delta L$. A large $L$ makes a spinning object resist tipping.
角动量是 $L = I\omega$,$p = mv$ 的转动孪生(单位 $\text{kg}\cdot\text{m}^2/\text{s}$)。角冲量 $\tau\,\Delta t$ 改变它:$\tau\,\Delta t = \Delta L$。大的 $L$ 使旋转物体抵抗倾倒。