# Torque

This article is about the analogue, from linear motion to angular motion, of:force

## Contents

## Definition

### Due to a single force, about a point

Suppose is a force and is the radial vector from a point to the point of application of . The **torque** due to about is defined as the cross product of and . In other words, it is defined as:

The torque is a vector quantity. (More properly, torque is an alternating 2-tensor, and is treated as a vector via a (non-canonical) identification of with the space of alternating 2-tensors on ).

The magnitude of torque is given by:

where is the angle between the radial vector and the force vector.

Note that the torque, as a vector quantity, depends only on the point , the *line of action* of , and the magnitude of . Thus, two forces of the same magnitude with the same line of action (Even though they may act at different points) generate the same torque.

### Due to a single force, about an axis

Suppose is a force and is the vector from (and perpendicular to) a line to the point of application of . The **torque** due to about is defined as the cross product of and . In other words, it is defined as:

The torque is a vector quantity. (More properly, torque is an alternating 2-tensor, and is treated as a vector via a (non-canonical) identification of with the space of alternating 2-tensors on ).

The magnitude of torque is given by:

where is the angle between the radial vector and the force vector.

### Due to multiple forces

The torque doe to multiple forces is defined as the vector sum of the torques due to each of the forces. When there is a continuum of forces, it is defined as a suitable integral. *Fill this in later*

## Units and dimensions

Question | Answer |
---|---|

Scalar or vector? | Vector |

Instantaneous or time-cumulative? | Instantaneous |

MLT dimensions | : MLT;1;2;-2 |

SI units | (Joules) |