Rotational Kinematics · 转动运动学
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| angular position/ˈæŋɡjʊlə pəˈzɪʃn/ | 角位置 | jiǎo wèi zhì |
| angular velocity/ˈæŋɡjʊlə vəˈlɒsɪti/ | 角速度 | jiǎo sù dù |
| angular acceleration/ˈæŋɡjʊlə əkˌseləˈreɪʃn/ | 角加速度 | jiǎo jiā sù dù |
Everything that spins plays by one set of rules
- A figure skater, a wheel, a planet, a hard drive -- all rotate.
- Straight-line motion had position, velocity, acceleration; spinning has angular twins of each.
- The equations look almost identical -- just swap distance for angle.
- Learn the mapping once and all of rotation opens up.
一切旋转的东西都遵守同一套规则
- 花样滑冰者、车轮、行星、硬盘——都在旋转。
- 直线运动有位置、速度、加速度;旋转则有它们各自的角版本。
- 方程看起来几乎一模一样——只需把距离换成角度。
- 学一次这套对应,整个旋转世界就打开了。
Angular quantities
- Angular position 角位置 $\theta$ is the angle turned through (in radians).
- Angular velocity 角速度 $\omega = \dfrac{d\theta}{dt}$ is how fast the angle changes.
- Angular acceleration 角加速度 $\alpha = \dfrac{d\omega}{dt}$ is how fast $\omega$ changes.
角量
- 角位置 $\theta$ 是转过的角度(以弧度计)。
- 角速度 $\omega = \dfrac{d\theta}{dt}$ 是角度改变的快慢。
- 角加速度 $\alpha = \dfrac{d\omega}{dt}$ 是 $\omega$ 改变的快慢。
Angular acceleration $\alpha$ is the rotational twin of which linear quantity? · 角加速度 $\alpha$ 是哪个线性量的旋转对应物？
$\alpha = d\omega/dt$ mirrors $a = dv/dt$ -- angular acceleration is the twin of linear acceleration. · $\alpha = d\omega/dt$ 对应 $a = dv/dt$ ——角加速度是线性加速度的对应物。
The same equations, rotated
- For constant $\alpha$, the rotational equations mirror the linear ones exactly.
- Replace $x \to \theta$, $v \to \omega$, $a \to \alpha$ and every kinematic formula still works.
- Constant angular acceleration behaves just like constant linear acceleration.
同样的方程,换成旋转
- 对于恒定的 $\alpha$,旋转方程与直线方程完全对应。
- 把 $x \to \theta$、$v \to \omega$、$a \to \alpha$ 替换,每个运动学公式仍然成立。
- 恒定的角加速度表现得就像恒定的直线加速度。
Select all · 所有 correct linear-to-rotational pairings. · 选择所有正确的线性与旋转配对。
$x\to\theta$ and $v\to\omega$ are correct. Mass maps to rotational inertia, not angle. · $x\to\theta$ 和 $v\to\omega$ 是正确的。质量映射到转动惯量，而不是角度。
When alpha is not constant
- If $\alpha$ changes with time, use calculus: $\omega = \int \alpha\,dt$ and $\theta = \int \omega\,dt$.
- Differentiate to go down the chain $\theta \to \omega \to \alpha$; integrate to go up.
- It is the same derivative chain as straight-line motion.
当 alpha 不恒定时
- 如果 $\alpha$ 随时间变化,就用微积分:$\omega = \int \alpha\,dt$,$\theta = \int \omega\,dt$。
- 求导沿链 $\theta \to \omega \to \alpha$ 向下;积分向上。
- 这与直线运动是同一条求导链。
Rotational analogs · 旋转类比
Each linear quantity has a rotational partner. Sort each description. · 每个线性量都有一个旋转对应物。对每个描述进行分类。
If angular acceleration varies with time, you find angular velocity by taking its ____. · 如果角加速度随时间变化，你可以通过对其取____来找到角速度。
$\omega = \int \alpha\,dt$ -- integrate the angular acceleration. · $\omega = \int \alpha\,dt$ ——积分角加速度。
A wheel starts at rest with constant $\alpha = 3\ \tfrac{\text{rad}}{\text{s}^2}$. Its angular velocity after $4\ \text{s}$ (in rad/s)? · 一个轮子从静止开始以恒定的 $\alpha = 3\ \tfrac{\text{rad}}{\text{s}^2}$ 旋转。经过 $4\ \text{s}$ 后其角速度是多少（单位：rad/s）？
$\omega = \omega_0 + \alpha t = 0 + 3 \times 4 = 12\ \tfrac{\text{rad}}{\text{s}}$ -- the rotational version of $v = at$. · $\omega = \omega_0 + \alpha t = 0 + 3 \times 4 = 12\ \tfrac{\text{rad}}{\text{s}}$ ——它是 $v = at$ 的旋转版本。
Period and frequency
- A rotation repeats every period $T$, and its frequency is $f = 1/T$.
- These connect to the angular velocity by $\omega = 2\pi f = \dfrac{2\pi}{T}$.
- One full turn is $2\pi$ radians, so a faster spin means a larger $\omega$.
周期与频率
- 旋转每过一个周期 $T$ 重复一次,频率是 $f = 1/T$。
- 它们通过 $\omega = 2\pi f = \dfrac{2\pi}{T}$ 与角速度相联系。
- 一整圈是 $2\pi$ 弧度,所以转得越快意味着 $\omega$ 越大。
A disk spins at $f = 4$ turns per second. What is its angular velocity (in rad/s)? · 一个磁盘以每秒 $f = 4$ 转旋转。其角速度是多少（单位：rad/s）？
$\omega = 2\pi f = 2\pi(4) \approx 25.13\ \tfrac{\text{rad}}{\text{s}}$.
In the formula $\omega = 2\pi f$, one full revolution counts as $2\pi$ radians. · 在公式 $\omega = 2\pi f$ 中，一整圈算作 $2\pi$ 弧度。
One turn is $2\pi$ radians, so $\omega = 2\pi f$ converts turns-per-second to radians-per-second. · 一圈是 $2\pi$ 弧度，因此 $\omega = 2\pi f$ 将“转/秒”转换为“弧度/秒”。
A wheel spins at $f = 5$ turns per second.
- Its angular velocity is $\omega = 2\pi f = 2\pi(5) \approx 31.4\ \tfrac{\text{rad}}{\text{s}}$.
- If $\alpha$ is constant, the rotational kinematic equations then predict its later angle and spin.
一个轮子以每秒 $f = 5$ 转旋转。
- 它的角速度是 $\omega = 2\pi f = 2\pi(5) \approx 31.4\ \tfrac{\text{rad}}{\text{s}}$。
- 如果 $\alpha$ 恒定,旋转运动学方程就能预测它之后的角度和转速。
Angular quantities use radians, not degrees, in these formulas. A common slip is plugging in degrees, or forgetting that one full revolution is $2\pi$ (not $360$) in the equation $\omega = 2\pi f$.
在这些公式里,角量使用弧度,而不是角度。常见的失误是代入角度,或忘记在 $\omega = 2\pi f$ 里一整圈是 $2\pi$(而不是 $360$)。
Rotation has angular position $\theta$, angular velocity $\omega = d\theta/dt$, and angular acceleration $\alpha = d\omega/dt$ -- exact twins of the linear quantities. For constant $\alpha$ the kinematic equations carry over; otherwise integrate. Spin rate links to period by $\omega = 2\pi f$.
旋转有角位置 $\theta$、角速度 $\omega = d\theta/dt$ 和角加速度 $\alpha = d\omega/dt$——直线量的精确孪生。对于恒定的 $\alpha$,运动学方程可以照搬;否则就积分。转速通过 $\omega = 2\pi f$ 与周期相联系。