Rotational Kinematics · 转动运动学
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| angular velocity/ˈæŋɡjʊlə vəˈlɒsɪti/ | 角速度 | jiǎo sù dù |
| angular acceleration/ˈæŋɡjʊlə əkˌseləˈreɪʃn/ | 角加速度 | jiǎo jiā sù dù |
| angular displacement/ˈæŋɡjʊlə dɪˈspleɪsmənt/ | 角位移 | jiǎo wèi yí |
| radian/ˈreɪdɪən/ | 弧度 | hú dù |
A spinning wheel needs a new set of words
- A car drives in a straight line; a wheel just spins in place. Both are "moving", but differently.
- To describe spinning we measure angles, not distances travelled.
- Three new quantities appear — angle, angular velocity, angular acceleration.
- They mirror position, velocity and acceleration exactly, so everything you learned carries over.
旋转的轮子需要一套新词汇
- 汽车沿直线行驶;轮子只是原地旋转。两者都在"运动",但方式不同。
- 要描述旋转,我们测量角度,而不是走过的距离。
- 出现了三个新量——角度、角速度、角加速度。
- 它们恰好对应位置、速度和加速度,所以你学过的一切都能沿用。
The three angular quantities
- Angular displacement 角位移 $\theta$ — the angle turned, measured in radians 弧度.
- Angular velocity 角速度 $\omega$ — how fast the angle changes, $\omega = \dfrac{\Delta\theta}{\Delta t}$ (in $\tfrac{\text{rad}}{\text{s}}$).
- Angular acceleration 角加速度 $\alpha$ — how fast $\omega$ changes, $\alpha = \dfrac{\Delta\omega}{\Delta t}$.
- These are the spin versions of $x$, $v$ and $a$.
三个角量
- 角位移 $\theta$——转过的角度,以弧度量度。
- 角速度 $\omega$——角度变化的快慢,$\omega = \dfrac{\Delta\theta}{\Delta t}$(单位 $\tfrac{\text{rad}}{\text{s}}$)。
- 角加速度 $\alpha$——$\omega$ 变化的快慢,$\alpha = \dfrac{\Delta\omega}{\Delta t}$。
- 这些是 $x$、$v$、$a$ 的旋转版本。
A wheel goes from rest to $\omega = 12\ \tfrac{\text{rad}}{\text{s}}$ in · 入 $4\ \text{s}$. What is its angular acceleration, in $\tfrac{\text{rad}}{\text{s}^2}$? · 一个轮子从静止加速到$\omega = 12\ \tfrac{\text{rad}}{\text{s}}$用时$4\ \text{s}$。其角加速度是多少，单位为$\tfrac{\text{rad}}{\text{s}^2}$？
$\alpha = \Delta\omega/\Delta t = 12/4 = 3\ \tfrac{\text{rad}}{\text{s}^2}$.
Angular velocity is the rate of change of the ____. · 角速度是____的变化率。
$\omega = \Delta\theta/\Delta t$ — the rate of change of the angle · 角度. · $\omega = \Delta\theta/\Delta t$ — 即角度的变化率。
Radians, the natural angle
- A radian is the angle for which the arc length equals the radius.
- A full circle is $2\pi$ radians ($\approx 6.28$), so $180^\circ = \pi$ radians.
- Radians make the linking formulas (next lesson) clean and simple.
- Always work in radians for rotational physics, not degrees.
弧度,天然的角度单位
- 弧度是弧长等于半径时所对的角。
- 一整圈是 $2\pi$ 弧度($\approx 6.28$),所以 $180^\circ = \pi$ 弧度。
- 弧度让联系公式(下一节)变得干净简单。
- 旋转物理中永远用弧度,而不是度。
Sweep an angle in radians · 扫过以弧度为单位的角度
Change the angle and radius to feel how radians measure the turn and the arc length. · 改变角度和半径，感受弧度如何衡量转动幅度和弧长。
How many radians are there in one complete turn? · 一整圈有多少弧度？
One full circle is $2\pi$ radians ($\approx 6.28$). · 一整圈是 $2\pi$ 弧度（$\approx 6.28$）。
A half turn ($180^\circ$) is how many radians? Use $\pi = 3.14$. · 半圈（$180^\circ$）等于多少弧度？请使用$\pi = 3.14$。
$180^\circ = \pi$ radians $\approx 3.14\ \text{rad}$. · $180^\circ = \pi$弧度对应$\approx 3.14\ \text{rad}$。
The same equations, new symbols
- With constant angular acceleration, the SUVAT equations reappear, symbol for symbol.
- $\omega = \omega_0 + \alpha t$ mirrors $v = u + at$.
- $\theta = \omega_0 t + \tfrac12 \alpha t^2$ mirrors $s = ut + \tfrac12 a t^2$.
- Learn the analogy once and every rotation problem becomes a translation exercise.
同样的方程,新的符号
- 在匀角加速下,SUVAT 方程逐个符号地重现。
- $\omega = \omega_0 + \alpha t$ 对应 $v = u + at$。
- $\theta = \omega_0 t + \tfrac12 \alpha t^2$ 对应 $s = ut + \tfrac12 a t^2$。
- 学会这个类比一次,每个旋转问题就变成一道翻译题。
Match each rotational quantity to its linear twin. · 将每个转动量与其对应的直线量匹配。
$\theta \leftrightarrow x$, $\omega \leftrightarrow v$, $\alpha \leftrightarrow a$ — the rotational quantities mirror the linear ones. · $\theta \leftrightarrow x$、$\omega \leftrightarrow v$、$\alpha \leftrightarrow a$ — 这些转动量与直线量一一对应。
Use radians, not degrees, in rotational formulas. A relation like $v = r\omega$ only works when $\omega$ is in $\tfrac{\text{rad}}{\text{s}}$. Mixing in degrees is one of the most common rotational-motion mistakes.
在旋转公式中用弧度,而不是度。像 $v = r\omega$ 这样的关系只有当 $\omega$ 以 $\tfrac{\text{rad}}{\text{s}}$ 为单位时才成立。混入度是旋转运动最常见的错误之一。
Rotational formulas like $v = r\omega$ require $\omega$ to be in radians per second. · 旋转公式如 $v = r\omega$ 要求 $\omega$ 以弧度/秒为单位。
These relations are derived using radians; using degrees gives wrong answers. · 这些关系式是基于弧度推导的；使用度会导致错误答案。
A wheel starts from rest and reaches $\omega = 12\ \tfrac{\text{rad}}{\text{s}}$ in $4\ \text{s}$ at constant angular acceleration.
- $\alpha = \dfrac{\Delta\omega}{\Delta t} = \dfrac{12 - 0}{4} = 3\ \tfrac{\text{rad}}{\text{s}^2}$.
This is exactly like finding a linear acceleration — only the symbols changed.
一个轮子从静止出发,在匀角加速下于 $4\ \text{s}$ 内达到 $\omega = 12\ \tfrac{\text{rad}}{\text{s}}$。
- $\alpha = \dfrac{\Delta\omega}{\Delta t} = \dfrac{12 - 0}{4} = 3\ \tfrac{\text{rad}}{\text{s}^2}$。
这与求线加速度完全一样——只是符号变了。
Rotation is described by angular displacement $\theta$, angular velocity $\omega$ and angular acceleration $\alpha$ — the spin twins of $x$, $v$, $a$. Measure angles in radians ($2\pi$ per turn). With constant $\alpha$, the SUVAT equations reappear with $\theta,\omega,\alpha$.
旋转由角位移 $\theta$、角速度 $\omega$ 和角加速度 $\alpha$ 描述——它们是 $x$、$v$、$a$ 的旋转孪生。角度用弧度量度(每圈 $2\pi$)。在匀 $\alpha$ 下,SUVAT 方程以 $\theta,\omega,\alpha$ 重现。