Torque and Work · 力矩与功
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| rotational work/rəʊˈteɪʃənl wɜːk/ | 转动功 | zhuǎn dòng gōng |
| rotational power/rəʊˈteɪʃənl ˈpaʊə/ | 转动功率 | zhuǎn dòng gōng lǜ |
Turning effort into spin
- Crank a winch and the drum speeds up.
- Push a merry-go-round around and around, and you feel the effort add up.
- The longer you push through the turn, the faster it spins.
- A turning force acting through an angle changes a spin's energy.
把力气变成旋转
- 摇动绞盘,卷筒就转得越来越快。
- 一圈圈推旋转木马,你感到力气不断累积。
- 你推着转过的角度越多,它转得越快。
- 转动的力作用在一段角度上,会改变旋转的能量。
Work done by a torque
- The rotational work 转动功 done by a constant torque is:
- Torque times the angle $\theta$ turned, in radians.
- It mirrors $W = Fd$, with torque for force and angle for distance.
力矩做的功
- 恒定力矩做的转动功是:
- 力矩乘以转过的角度 $\theta$(用弧度)。
- 它与 $W = Fd$ 对应,力矩代替力,角度代替距离。
A constant torque of $6\ \text{N}\cdot\text{m}$ turns a wheel through $\theta = 4\ \text{rad}$. The work done (in J)? · 恒定力矩$6\ \text{N}\cdot\text{m}$使轮子转过角度$\theta = 4\ \text{rad}$。所做的功（单位 J）是多少？
$W = \tau\theta = 6 \times 4 = 24\ \text{J}$.
Match each rotational quantity to its linear twin. · 将每个转动量与其对应的直线量匹配。
$W = \tau\theta$ mirrors $W = Fd$; $P = \tau\omega$ mirrors $P = Fv$. · $W = \tau\theta$镜子$W = Fd$；$P = \tau\omega$镜子$P = Fv$。
Work becomes spin energy
- Net rotational work equals the change in rotational kinetic energy:
- Do positive work and the object spins faster.
- Same idea as the linear work-energy theorem.
功变成旋转的能量
- 净转动功等于转动动能的变化:
- 做正功,物体就转得更快。
- 与直线运动的功能定理是同一个道理。
Positive net rotational work makes a spinning object speed up. · 正的净转动功会使旋转物体加速。
$W_{net} = \Delta K_{rot}$; positive work raises the rotational kinetic energy, so $\omega$ rises. · $W_{net} = \Delta K_{rot}$；正功增加了转动动能，因此$\omega$上升。
Rotational power
- Rotational power 转动功率 is the rate of doing rotational work:
- Torque times angular speed -- the twin of $P = Fv$.
- A motor's power sets how quickly it spins a load up.
转动功率
- 转动功率是做转动功的快慢:
- 力矩乘以角速度——$P = Fv$ 的孪生。
- 电机的功率决定它把负载转起来有多快。
Work done by a torque · 力矩所做的功
A constant torque turning through an angle does work in proportion to that angle. · 恒定力矩转过一定角度所做的功与该角度成正比。
A motor delivers $\tau = 4\ \text{N}\cdot\text{m}$ to a shaft spinning at $\omega = 10\ \text{rad/s}$. The power (in W)? · 电动机向以 $\tau = 4\ \text{N}\cdot\text{m}$ 旋转的轴输送 $\omega = 10\ \text{rad/s}$。功率（单位：瓦特）？
$P = \tau\omega = 4 \times 10 = 40\ \text{W}$.
A motor applies $\tau = 5\ \text{N}\cdot\text{m}$ while a wheel turns through $\theta = 2\pi\ \text{rad}$ (one full turn).
- Work: $W = \tau\theta = 5 \times 2\pi \approx 31.4\ \text{J}$.
- If that turn takes $2\ \text{s}$ at steady speed, average power $\approx 15.7\ \text{W}$.
一台电机施加 $\tau = 5\ \text{N}\cdot\text{m}$,同时轮子转过 $\theta = 2\pi\ \text{rad}$(整整一圈)。
- 功:$W = \tau\theta = 5 \times 2\pi \approx 31.4\ \text{J}$。
- 如果这一圈以稳定速度花 $2\ \text{s}$,平均功率 $\approx 15.7\ \text{W}$。
If the torque changes with angle, the work done is... · 如果力矩随角度变化，所做的功是...
$W = \int \tau\, d\theta$ -- the area under $\tau$ versus $\theta$. · $W = \int \tau\, d\theta$——$\tau$相对于$\theta$图线下的面积。
In $W = \tau\theta$, the angle must be measured in ____. · 在$W = \tau\theta$中，角度必须以____为单位测量。
Rotational SI formulas need the angle in radians. · SI 转动公式需要角度单位为弧度。
Keep $\theta$ in radians for $W = \tau\theta$, and use the torque component perpendicular to the lever arm. The formula $W = \tau\theta$ is for a constant torque; if torque changes with angle, integrate $W = \int \tau\, d\theta$ -- the area under a torque-angle graph.
在 $W = \tau\theta$ 中 $\theta$ 要用弧度,并且要用垂直于力臂的力矩分量。公式 $W = \tau\theta$ 只适用于恒定力矩;若力矩随角度变化,就积分 $W = \int \tau\, d\theta$——即力矩-角度图下的面积。
A torque acting through an angle does rotational work $W = \tau\theta$ (the twin of $W = Fd$), and that work changes the spin energy: $W_{net} = \Delta K_{rot}$. The rate of doing it is rotational power $P = \tau\omega$, mirroring $P = Fv$. Keep angles in radians, and integrate when the torque varies.
力矩作用在一段角度上做转动功 $W = \tau\theta$($W = Fd$ 的孪生),这份功改变旋转能量:$W_{net} = \Delta K_{rot}$。做功的快慢是转动功率 $P = \tau\omega$,与 $P = Fv$ 对应。角度用弧度,力矩变化时要积分。