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 |
| angular impulse/ˈæŋɡjʊlə ˈɪmpʌls/ | 角冲量 | jiǎo chōng liàng |
The spin that won't quit
- A spinning top stays upright and keeps turning.
- A figure skater glides through a spin for many seconds.
- Something about the motion resists being stopped.
- Spinning objects carry a stubborn "quantity of turning."
停不下来的旋转
- 旋转的陀螺保持直立,不停地转。
- 花样滑冰运动员的旋转能持续好几秒。
- 这种运动里有某种东西抗拒被停下来。
- 旋转的物体携带着一份顽固的"旋转的量"。
Angular momentum
- Angular momentum 角动量 measures how much turning an object carries:
- Rotational inertia times angular speed.
- It is the rotational twin of linear momentum $p = mv$.
角动量
- 角动量衡量物体携带了多少旋转:
- 转动惯量乘以角速度。
- 它是直线动量 $p = mv$ 的旋转孪生。
A flywheel with $I = 5\ \text{kg}\cdot\text{m}^2$ spins at $\omega = 4\ \text{rad/s}$. Its angular momentum (in kg·m²/s)? · 一个转动惯量为$I = 5\ \text{kg}\cdot\text{m}^2$的飞轮以角速度$\omega = 4\ \text{rad/s}$旋转。其角动量（单位 kg·m²/s）是多少？
$L = I\omega = 5 \times 4 = 20\ \text{kg}\cdot\text{m}^2/\text{s}$.
Angular momentum · 角动量 $L = I\omega$ is the rotational twin of which linear quantity? · 角动量$L = I\omega$是哪个线性量的转动对应量？
$I$ mirrors $m$ and $\omega$ mirrors $v$, so $L = I\omega$ mirrors $p = mv$. · $I$镜像$m$，$\omega$镜像$v$，因此$L = I\omega$镜像$p = mv$。
A mass moving in a circle
- A point mass $m$ at radius $r$ moving at speed $v$ has:
- Farther out or faster means more angular momentum.
- This is why pulling in your arms changes a skater's spin.
沿圆周运动的质量
- 半径 $r$ 处、以速度 $v$ 运动的质点 $m$ 有:
- 越远或越快,角动量越大。
- 这正是收拢双臂能改变滑冰者旋转的原因。
A $2\ \text{kg}$ ball moves at $3\ \text{m/s}$ in a circle of radius $4\ \text{m}$. Its angular momentum (in kg·m²/s)? · 一个质量为$2\ \text{kg}$的球以速度$3\ \text{m/s}$在半径为$4\ \text{m}$的圆周上运动。其角动量（单位 kg·m²/s）是多少？
$L = mvr = 2 \times 3 \times 4 = 24\ \text{kg}\cdot\text{m}^2/\text{s}$.
Angular impulse changes L
- A torque acting for a time delivers angular impulse 角冲量:
- It is the rotational twin of $F\,\Delta t = \Delta p$.
- Equivalently $\tau = \dfrac{dL}{dt}$: torque is the rate of change of angular momentum.
角冲量改变 L
- 力矩作用一段时间,传递角冲量:
- 它是 $F\,\Delta t = \Delta p$ 的旋转孪生。
- 等价地 $\tau = \dfrac{dL}{dt}$:力矩是角动量的变化率。
Angular momentum · 角动量
Angular momentum is rotational inertia times angular speed. · 角动量等于转动惯量乘以角速度。
A torque of $3\ \text{N}\cdot\text{m}$ acts for $5\ \text{s}$. The change in angular momentum (in kg·m²/s)? · 力矩$3\ \text{N}\cdot\text{m}$作用了时间$5\ \text{s}$。角动量的变化量（单位 kg·m²/s）是多少？
$\Delta L = \tau\,\Delta t = 3 \times 5 = 15\ \text{kg}\cdot\text{m}^2/\text{s}$.
Torque equals the rate of change of angular momentum, $\tau = dL/dt$. · 力矩等于角动量的变化率，$\tau = dL/dt$。
This is Newton's second law for rotation in momentum form. · 这是牛顿第二定律的转动形式（动量形式）。
A wheel with $I = 3\ \text{kg}\cdot\text{m}^2$ spins at $\omega = 2\ \text{rad/s}$.
- Angular momentum: $L = I\omega = 3 \times 2 = 6\ \text{kg}\cdot\text{m}^2/\text{s}$.
- A steady torque $\tau = 4\ \text{N}\cdot\text{m}$ for $3\ \text{s}$ adds $\Delta L = 12$, so $L$ rises to $18$.
一个 $I = 3\ \text{kg}\cdot\text{m}^2$ 的轮子以 $\omega = 2\ \text{rad/s}$ 旋转。
- 角动量:$L = I\omega = 3 \times 2 = 6\ \text{kg}\cdot\text{m}^2/\text{s}$。
- 稳定力矩 $\tau = 4\ \text{N}\cdot\text{m}$ 作用 $3\ \text{s}$,增加 $\Delta L = 12$,于是 $L$ 升到 $18$。
To state an object's angular momentum meaningfully, you must also give the... · 要有意义地描述物体的角动量，还必须给出...
Angular momentum is defined relative to an axis; different axes give different $L$. · 角动量是相对于轴定义的；不同的轴给出不同的$L$。
Angular momentum depends on the axis you measure it about -- always state the axis. And $L = mvr$ uses the perpendicular distance from the axis to the line of motion; if the velocity is not perpendicular to $r$, keep only the perpendicular component ($L = mv_\perp r$).
角动量取决于你绕哪个轴来测量——一定要说明轴。而 $L = mvr$ 用的是从轴到运动直线的垂直距离;若速度不垂直于 $r$,就只取垂直分量($L = mv_\perp r$)。
Angular momentum $L = I\omega$ (or $L = mvr$ for a point mass) is the rotational twin of $p = mv$. A torque acting over time is an angular impulse that changes it: $\tau\,\Delta t = \Delta L$, or $\tau = dL/dt$. Always name the axis, and use the perpendicular distance.
角动量 $L = I\omega$(质点则为 $L = mvr$)是 $p = mv$ 的旋转孪生。力矩作用一段时间就是角冲量,会改变它:$\tau\,\Delta t = \Delta L$,即 $\tau = dL/dt$。始终指明轴,并使用垂直距离。