Torque

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

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

The magnitude of torque is given by:

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

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

The magnitude of torque is given by:

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

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

The dimensions of torque are $M{L}^2{T}^{-2}$, which are the same as those of energy. We can take the units to be $J$ (Joules) though it is more customary to take the units as $J/rad$ (Joules per radian).