Mechanics of particles and systems Definition

Torque

Definition

Let a force $F \in \R^3$ act at a point with position $p \in \R^3$ , and let $O \in \R^3$ be a reference point. The torque, or moment, of $F$ about $O$ is $\tau_O = (p - O) \times F,$ where $\times$ is the cross product. For forces $F_1, \dots, F_k$ acting at points $p_1, \dots, p_k$ , the total torque about $O$ is $\sum_{i=1}^{k} (p_i - O) \times F_i$ .

Remarks

The size of the torque is $\abs{\tau_O} = \abs{p - O}\, \abs{F} \sin\varphi$ , where $\varphi$ is the angle between $p - O$ and $F$ . Its direction is the axis of the turning it tends to cause, by the right-hand rule.

Example: $O = 0$ , $p = (0.8, 0, 0)$ and $F = (0, 10, 0)$ give $\tau_O = (0, 0, 8)$ , a turning about the vertical axis.

Non-example: a force along the line through $O$ and $p$ , such as $F = (-10, 0, 0)$ above, has zero torque about $O$ , although it is not zero. Changing the reference point gives $\tau_{O'} = \tau_O + (O - O') \times \sum_i F_i$ , so the total torque does not depend on the reference point when the total force is zero.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Euclidean vector concept
Cross product definition
Torque definition

Tag	`EBXFJNKGKQ`
URI	`m2://EBXFJNKGKQ`
Kind	definition
Writing	W0 stub
MSC	70C20, 70A05
Defines	Torque
Uses	Cross product
Changed	3 November 2026

Torque

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here