# Torque

## Definition

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

Suppose $\overline{F}$ is a force and $\overline{r}$ is the radial vector from a point $A$ to the point of application of $\overline{F}$. The torque due to $\overline{F}$ about $A$ is defined as the cross product of $\overline{r}$ and $\overline{F}$. In other words, it is defined as:

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

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 $\R^3$ with the space of alternating 2-tensors on $\R^3$).

The magnitude of torque is given by:

$\tau = |rF \sin \theta|$

where $\theta$ is the angle between the radial vector and the force vector.

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

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

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 $\R^3$ with the space of alternating 2-tensors on $\R^3$).

The magnitude of torque is given by:

$\tau = |rF \sin \theta|$

where $\theta$ 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

MLT dimensions $ML^2T^{-2}$: MLT;1;2;-2
SI units $J$ (Joules)