Formula of Torque: τ = rF sin θ, Moment of Force and Examples

The formula of torque is $\tau = rF\sin\theta$ : the distance from the axis to the point where the force acts, times the force, times the sine of the angle between those two directions. In vector form it is $\vec{\tau} = \vec{r} \times \vec{F}$ . Torque is also called the moment of force, and its SI unit is the newton metre (N m).

Torque measures how well a force turns something. Push a door near its hinge and it barely moves; push at the handle with the same force and it swings easily. The larger r, the larger the turning effect. Push along the door toward the hinge (θ = 0) and nothing turns at all, which is where the sine comes in. Torque belongs to the rotation section, and its twin page derives the dimensional formula of torque.

Quick facts

Formula
$\tau = rF\sin\theta$
Vector form
$\vec{\tau} = \vec{r} \times \vec{F}$
Lever arm form
$\tau = F \times d_{\perp}$
Newton's second law
$\tau = I\alpha$
SI unit
newton metre (N m)
Dimensional formula
[M¹ L² T⁻²]
Type
Vector along the axis, right hand rule

Reading the formula of torque in physics symbol by symbol

Symbols in τ = rF sin θ
Symbol	Name	SI unit
$\tau$	Torque, or moment of force	N m
$r$	Distance from the axis (pivot) to the point where F acts	m
$F$	Size of the applied force	N
$\theta$	Angle between the vectors $\vec{r}$ and $\vec{F}$	degree or rad

You can group the terms in two useful ways. $r\sin\theta$ is how far the line of action of the force passes from the axis, measured at right angles of the force, called the lever arm or moment arm $d_{\perp}$ . Then $\tau = Fd_{\perp}$ . Or $F\sin\theta$ is the part of the force at right angles to r, so $\tau = rF_{\perp}$ . Same answer, three routes.

Solving a problem with the formula of torque

Mark the axis or pivot. Torque has no meaning without one.
Find r from the pivot to the point where the force acts, in metres.
Find θ between r and F. If the force is at right angles to the bar, sin θ = 1.
Multiply $rF\sin\theta$ . Give a sign: anticlockwise positive and clockwise negative is the usual convention.
For balance, set the sum of torques to zero; for turning, use the sum in τ = Iα.

Worked examples

Example 1: tightening a nut

A mechanic pulls with 80 N at right angles to a spanner, 25 cm from the nut. Find the torque.

r = 0.25 m, θ = 90°. $\tau = 0.25 \times 80 \times \sin 90^\circ = 20\ \text{N m}$

Example 2: pushing a door at an angle

A 40 N push acts 0.9 m from the hinges at 30° to the door. Find the torque.

$\tau = 0.9 \times 40 \times \sin 30^\circ = 0.9 \times 40 \times 0.5 = 18\ \text{N m}$

Pushing at right angles would give 36 N m, twice as much.

Example 3: seesaw balance

A 30 kg child sits 2.0 m from the pivot of a seesaw. Where must a 40 kg child sit to balance it?

Balance: $30g \times 2.0 = 40g \times d$ , so g cancels.

$d = \frac{30 \times 2.0}{40} = 1.5\ \text{m}$

This is the principle of moments: clockwise moments equal anticlockwise moments.

Moment of force formula and rotational dynamics

Newton's second law for rotation reads $\tau_{net} = I\alpha$ . A net torque produces an angular acceleration, just as a net force produces a linear one. The stand in for mass here is the moment of inertia. Torque is also the rate of change of angular momentum: $\tau = \frac{dL}{dt}$ . Zero torque means L stays fixed.

A couple is two equal and opposite forces whose lines of action are a distance d apart. Its torque is $Fd$ about any point, and it turns a body without moving its center of mass. Turning a steering wheel with both hands is a couple.

N m for torque versus joules for work

Torque and work share the dimensions [M L² T⁻²], since each is a force multiplied by a length. But they are different quantities. In work the force and displacement are along the same line, while in torque the force and the lever arm are at right angles. So write torque in N m and work or energy in joules, never torque in J.

A common board question asks whether torque and work are the same. Answer: same dimensions, different physical meaning, and torque is a vector while work is a scalar.

Mistakes examiners see

Measuring r to the wrong point, such as the edge of the object instead of the pivot.
Using cos θ where sin θ is needed. Check: force along the bar gives zero torque.
Leaving r in centimetres.
Forgetting the direction. Anticlockwise and clockwise torques have opposite signs and must be subtracted.

Frequently asked questions

What is the formula of torque?

Torque is τ = rF sin θ, where r is the distance from the axis, F the force and θ the angle between them. In vector form τ = r × F.

How do you write the formula of moment of force?

Moment of force is another name for torque, so the formula is the same: moment = force × perpendicular distance from the pivot to the line of action, or rF sin θ.

In what unit is torque measured?

In newton metres, N m. It has dimensions [M¹ L² T⁻²], matching work, yet you should not write it in joules.

Is torque a vector quantity?

Yes. Lay your right palm so the fingers follow r, sweep them toward F, and the thumb shows the torque direction, along the turning axis.

What is the torque formula with moment of inertia?

Net torque equals moment of inertia times angular acceleration, τ = Iα. It is the rotational copy of F = ma.