Torque
| English | Chinese | Pinyin |
|---|---|---|
| torque | 力矩 | lì jǔ |
| lever arm | 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.
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.
Force times distance
Move the force farther from the pivot or make it bigger, and watch the torque grow.
A $20\ \text{N}$ force acts at right angles, $0.4\ \text{m}$ from a bolt. What is the torque, in $\text{N·m}$?
$\tau = rF\sin 90^\circ = 0.4 \times 20 \times 1 = 8\ \text{N·m}$.
The perpendicular distance from the pivot to the force line is called the ____ arm.
The lever arm $r\sin\theta$ is what sets the torque, alongside the force.
The same $20\ \text{N}$ force now acts at $0.8\ \text{m}$ (still perpendicular). What is the torque, in $\text{N·m}$?
$\tau = 0.8 \times 20 = 16\ \text{N·m}$ — doubling the arm doubles the torque.
Select all changes that increase the torque a force produces about a pivot.
More force, more distance, or a more perpendicular angle all raise $\tau = rF\sin\theta$. Aiming at the pivot gives zero.
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.
A force is aimed straight through the pivot. What torque does it produce?
$\theta = 0$ means $\sin\theta = 0$, so $\tau = 0$ — a force through the pivot cannot turn it.
For a given force and distance, the torque is largest when the force is perpendicular to the arm.
At $\theta = 90^\circ$, $\sin\theta = 1$, giving the maximum torque $\tau = rF$.
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.
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 $r\sin\theta$.
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.
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.