# Difference between revisions of "Torque"

(Created page with '{{angular analogue of|force}} ==Definition== ===Due to a single force=== Suppose <math>\overline{F}</math> is a force and <math>\overline{r}</math> is the radial vector fr…') |
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==Definition== | ==Definition== | ||

− | ===Due to a single force=== | + | ===Due to a single force, about a point=== |

Suppose <math>\overline{F}</math> is a [[force]] and <math>\overline{r}</math> is the radial vector from a point <math>A</math> to the point of application of <math>\overline{F}</math>. The '''torque''' due to <math>\overline{F}</math> about <math>A</math> is defined as the cross product of <math>\overline{r}</math> and <math>\overline{F}</math>. In other words, it is defined as: | Suppose <math>\overline{F}</math> is a [[force]] and <math>\overline{r}</math> is the radial vector from a point <math>A</math> to the point of application of <math>\overline{F}</math>. The '''torque''' due to <math>\overline{F}</math> about <math>A</math> is defined as the cross product of <math>\overline{r}</math> and <math>\overline{F}</math>. In other words, it is defined as: | ||

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Note that the torque, as a vector quantity, depends only on the point <math>A</math>, the ''line of action'' of <math>\overline{F}</math>, and the magnitude of <math>\overline{F}</math>. 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. | Note that the torque, as a vector quantity, depends only on the point <math>A</math>, the ''line of action'' of <math>\overline{F}</math>, and the magnitude of <math>\overline{F}</math>. 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 <math>\overline{F}</math> is a [[force]] and <math>\overline{r}</math> is the vector from (and perpendicular to) a line <math>\ell</math> to the point of application of <math>\overline{F}</math>. The '''torque''' due to <math>\overline{F}</math> about <math>A\ell</math> is defined as the cross product of <math>\overline{r}</math> and <math>\overline{F}</math>. In other words, it is defined as: | ||

+ | |||

+ | <math>\overline{\tau} = \overline{r} \times \overline{F}</math> | ||

+ | |||

+ | 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 <math>\R^3</math> with the space of alternating 2-tensors on <math>\R^3</math>). | ||

+ | |||

+ | The magnitude of torque is given by: | ||

+ | |||

+ | <math>\tau = |rF \sin \theta|</math> | ||

+ | |||

+ | where <math>\theta</math> is the angle between the radial vector and the force vector. | ||

===Due to multiple forces=== | ===Due to multiple forces=== |

## Revision as of 23:56, 19 January 2010

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

The dimensions of torque are