Torque
Introduced| English | Meaning |
|---|---|
| torque/tɔːk/ | The turning effect of a force, given by the force multiplied by the perpendicular distance of its line of action from the pivot. |
| lever arm/ˈliːvə ɑːm/ | The perpendicular distance from a pivot or axis to a force's line of action. |
| vector product/ˈvektə ˈprɒdʌkt/ | The cross product a×b, with magnitude ab sin θ and direction by the right-hand rule. |
| net torque/net tɔːk/ | The combined turning effect of all forces acting on a body about a specified axis. |
Why a long wrench beats a short one
- A stuck bolt won't budge with a stubby spanner, but a long one cracks it loose easily.
- Same force from your hand -- yet a completely different result.
- The difference is where and how you push relative to the pivot.
- That twisting effect has a name and a formula.
Torque is a twist
- Torque 力矩 measures how strongly a force turns something about an axis:
- $r$ is the distance from the axis, $\theta$ the angle between the force and that line.
- More distance, or a more perpendicular push, means more twist.
A $20\ \text{N}$ force pushes perpendicular to a wrench $0.25\ \text{m}$ from the bolt. What torque results (in N·m)?
$\tau = rF\sin 90^\circ = 0.25 \times 20 \times 1 = 5\ \text{N}\cdot\text{m}$.
The same $20\ \text{N}$ at $0.25\ \text{m}$ now acts at $30^\circ$ to the wrench. The torque (in N·m)?
$\tau = rF\sin 30^\circ = 0.25 \times 20 \times 0.5 = 2.5\ \text{N}\cdot\text{m}$ -- only the perpendicular part counts.
The lever arm
- The lever arm 力臂 is the perpendicular distance from the axis to the line of the force.
- Torque equals force times lever arm -- a longer lever arm wins.
- That is exactly why the long wrench works and pushing near the hinge does not.
The perpendicular distance from the axis to the line of a force is called the lever ____.
Torque = force times lever arm; a longer lever arm gives more torque.
Torque as a vector
- More fully, torque is a vector product 矢量积 $\vec{\tau} = \vec{r}\times\vec{F}$.
- Its direction lies along the axis, found by the right-hand rule.
- The magnitude is still $rF\sin\theta$.
Torque and the principle of moments
Torque is force times distance from the pivot. Balance the turning effects on each side.
Select all true statements about torque.
The first two define torque. A force along the radius gives zero torque, not maximum.
Net torque
- The net torque 净力矩 about an axis is the sum of all the torques on the body.
- Counterclockwise and clockwise torques carry opposite signs.
- A push far from the axis and perpendicular to $r$ gives the most torque.
A force pushing straight toward the axis produces...
With $\theta = 0$, $\sin 0 = 0$, so the torque is zero -- pushing along the radius cannot spin it.
To get the most torque from a fixed force, apply it...
Torque $rF\sin\theta$ is largest for big $r$ and $\theta = 90^\circ$ -- far out and perpendicular.
You push a door with $10\ \text{N}$ at the handle, $0.8\ \text{m}$ from the hinge, perpendicular to the door.
- Torque: $\tau = rF\sin 90^\circ = 0.8 \times 10 \times 1 = 8\ \text{N}\cdot\text{m}$.
- Push at $30^\circ$ instead and it drops to $0.8 \times 10 \times 0.5 = 4\ \text{N}\cdot\text{m}$.
Only the perpendicular part of a force makes torque. A push straight toward the axis ($\theta = 0$) gives $\sin 0 = 0$ -- zero torque, no matter how hard you shove. Pushing along the radius does nothing to spin it.
Torque $\tau = rF\sin\theta$ (or the vector product $\vec{r}\times\vec{F}$) measures a force's turning effect. It equals force times the lever arm (perpendicular distance to the axis). The net torque sums all torques -- push far out and perpendicular for the most twist.