Torque · 力矩
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| torque/tɔːk/ | 力矩 | lì jǔ |
| lever arm/ˈliːvə ɑːm/ | 力臂 | lì bì |
Why a longer wrench frees a stuck bolt
- A stuck bolt won't move with a short spanner, but slip a pipe over the handle and it turns easily.
- You pushed with the same force — yet the longer arm made all the difference.
- What turns things is not force alone, but torque 力矩: force applied at a distance.
- Torque is to rotation what force is to straight-line motion.
为什么更长的扳手能拧开卡死的螺栓
- 卡死的螺栓用短扳手拧不动,但在手柄上套根管子,它就轻松转开了。
- 你用了同样的力——可更长的力臂却带来了全部的差别。
- 让东西转动的不是力本身,而是力矩:施加在一定距离处的力。
- 力矩之于转动,就如力之于直线运动。
Defining torque
- Torque is $\tau = r F \sin\theta$ — force $F$, distance $r$ from the pivot, angle $\theta$ between them.
- The distance $r\sin\theta$ is the lever arm 力臂: the perpendicular distance from the pivot to the force line.
- It is measured in newton-metres ($\text{N}\cdot\text{m}$).
- A bigger force, a longer arm, or a more perpendicular push all give more torque.
力矩的定义
- 力矩是 $\tau = r F \sin\theta$——力 $F$、到支点的距离 $r$、两者之间的夹角 $\theta$。
- 距离 $r\sin\theta$ 是力臂:从支点到力作用线的垂直距离。
- 它以牛顿·米($\text{N}\cdot\text{m}$)量度。
- 更大的力、更长的力臂,或更垂直的推,都给出更大的力矩。
Force times distance · 力乘以距离
Move the force farther from the pivot or make it bigger, and watch the torque grow. · 将力移离支点更远或增大它，观察力矩如何增长。
A $20\ \text{N}$ force acts at right angles, $0.4\ \text{m}$ from a bolt. What is the torque, in $\text{N·m}$? · 一个$20\ \text{N}$的力垂直作用于距螺栓$0.4\ \text{m}$处。产生的力矩是多少，单位为$\text{N·m}$？
$\tau = rF\sin 90^\circ = 0.4 \times 20 \times 1 = 8\ \text{N·m}$.
The perpendicular distance from the pivot to the force line is called the ____ arm. · 从支点到力作用线的垂直距离称为____臂。
The lever arm $r\sin\theta$ is what sets the torque, alongside the force. · 杠杆臂$r\sin\theta$是与力一起决定力矩的关键因素。
The same $20\ \text{N}$ force now acts at $0.8\ \text{m}$ (still perpendicular). What is the torque, in $\text{N·m}$? · 同样的 $20\ \text{N}$ 力现在作用在 $0.8\ \text{m}$ 处（仍垂直）。力矩是多少，单位为 $\text{N·m}$？
$\tau = 0.8 \times 20 = 16\ \text{N·m}$ — doubling the arm doubles the torque. · $\tau = 0.8 \times 20 = 16\ \text{N·m}$ — 臂长加倍，力矩加倍。
Select all · 所有 changes that increase the torque a force produces about a pivot. · 选择所有能增加力对支点产生力矩的改变方式。
More force, more distance, or a more perpendicular angle all raise $\tau = rF\sin\theta$. Aiming at the pivot gives zero. · 更大的力、更远的距离或更垂直的角度都会提高 $\tau = rF\sin\theta$。瞄准支点则给出零。
Push at a right angle
- Torque is largest when the force is perpendicular to the arm ($\theta = 90^\circ$, $\sin\theta = 1$).
- A force pointing straight at the pivot ($\theta = 0$) makes zero torque — it cannot turn anything.
- That is why you push a door at the handle, at right angles, far from the hinge.
- Push near the hinge, or straight into it, and the door barely moves.
垂直地推
- 当力垂直于力臂时力矩最大($\theta = 90^\circ$,$\sin\theta = 1$)。
- 直指支点的力($\theta = 0$)产生零力矩——它无法转动任何东西。
- 这就是为什么你在门把手处、垂直地、远离铰链地推门。
- 靠近铰链推,或直冲铰链推,门几乎不动。
A force is aimed straight through the pivot. What torque does it produce? · 一个力直指支点方向。它产生多大的力矩？
$\theta = 0$ means $\sin\theta = 0$, so $\tau = 0$ — a force through the pivot cannot turn it. · $\theta = 0$ 意味着 $\sin\theta = 0$，因此 $\tau = 0$ ——通过支点的力无法使其转动。
For a given force and distance, the torque is largest when the force is perpendicular to the arm. · 对于给定的力和距离，当力与臂垂直时，力矩最大。
At $\theta = 90^\circ$, $\sin\theta = 1$, giving the maximum torque $\tau = rF$. · 在$\theta = 90^\circ$、$\sin\theta = 1$处，产生最大力矩$\tau = rF$。
Direction: clockwise or anticlockwise
- Torque has a turning direction — clockwise or anticlockwise about the pivot.
- We usually call anticlockwise positive and clockwise negative.
- Opposing torques can cancel (a balanced seesaw) or add (both turning the same way).
- Keeping the signs straight is the key to rotational problems.
方向:顺时针或逆时针
- 力矩有一个转动方向——绕支点顺时针或逆时针。
- 我们通常把逆时针取为正,顺时针取为负。
- 相反的力矩可以相消(平衡的跷跷板)或相加(两者朝同一方向转)。
- 弄清符号是解决转动问题的关键。
Torque depends on the lever arm, not just the force. A force aimed straight through the pivot ($\theta = 0$) gives zero torque, no matter how hard you push. Always use the perpendicular distance $r\sin\theta$.
力矩取决于力臂,而不仅是力。直穿支点($\theta = 0$)的力产生零力矩,无论你推得多用力。始终使用垂直距离 $r\sin\theta$。
A force of $20\ \text{N}$ is applied at right angles, $0.4\ \text{m}$ from a bolt.
- $\tau = rF\sin 90^\circ = 0.4 \times 20 \times 1 = 8\ \text{N}\cdot\text{m}$.
Double the handle length to $0.8\ \text{m}$ and the same force gives $16\ \text{N}\cdot\text{m}$ — twice the turning effect.
一个 $20\ \text{N}$ 的力垂直地施加在距螺栓 $0.4\ \text{m}$ 处。
- $\tau = rF\sin 90^\circ = 0.4 \times 20 \times 1 = 8\ \text{N}\cdot\text{m}$。
把手柄长度加倍到 $0.8\ \text{m}$,同样的力给出 $16\ \text{N}\cdot\text{m}$——两倍的转动效果。
Torque is the turning effect of a force: $\tau = rF\sin\theta$ (in $\text{N}\cdot\text{m}$), where $r\sin\theta$ is the lever arm. It is greatest for a perpendicular push far from the pivot, and zero for a force aimed at the pivot. Torque has a clockwise/anticlockwise sign.
力矩是力的转动效果:$\tau = rF\sin\theta$(单位 $\text{N}\cdot\text{m}$),其中 $r\sin\theta$ 是力臂。当垂直地在远离支点处推时最大,直指支点的力则为零。力矩有顺时针/逆时针的符号。