Torque

What torque is

Torque is the rotational effect of a force — it is what causes angular acceleration, just as force causes linear acceleration.

$\vec{\tau} = \vec{r} \times \vec{F}$

In magnitude:

$\tau = rF\sin\theta$

where $r$ is the distance from the axis to the point of application, $F$ is the force, and $\theta$ is the angle between $\vec{r}$ and $\vec{F}$.

The cross product

The torque vector $\vec{\tau} = \vec{r} \times \vec{F}$ has:

• Magnitude: $|\vec{r}||\vec{F}|\sin\theta = rF\sin\theta$
• Direction: perpendicular to both $\vec{r}$ and $\vec{F}$, given by the right-hand rule

Right-hand rule: point fingers along $\vec{r}$, curl toward $\vec{F}$, thumb points along $\vec{\tau}$.

The lever arm

$\tau = rF\sin\theta$ can be rewritten as:

$\tau = F \cdot (r\sin\theta) = F \cdot l$

where $l = r\sin\theta$ is the lever arm (perpendicular distance from the axis to the line of action of the force).

Or:

$\tau = r \cdot (F\sin\theta) = r \cdot F_\perp$

where $F_\perp = F\sin\theta$ is the component of force perpendicular to $\vec{r}$.

Both interpretations give the same result.

When is torque zero?

$\tau = rF\sin\theta = 0$ when:

• $r = 0$: force applied at the axis — no lever arm, no torque
• $F = 0$: no force
• $\theta = 0°$ or $180°$: force along the line from axis to point of application — no perpendicular component

Units

$[\tau] = [r][F] = \text{m} \cdot \text{N} = \text{N} \cdot \text{m}$

Note: $\text{N} \cdot \text{m}$ for torque is kept distinct from $\text{J}$ (Joule) for energy, even though numerically they are the same. Torque and energy are different physical quantities.

Sign convention

Counterclockwise torque: positive (by convention, in 2D) Clockwise torque: negative

In 3D, the sign is determined by the direction of $\vec{\tau}$ relative to the chosen axis direction.

Torque and door

A door is more easily opened by pushing far from the hinges (large $r$) and perpendicular to the door (maximum $\sin\theta = 1$). Pushing near the hinge requires much more force for the same torque — this is why door handles are placed at the far edge.