# What is torque

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

Torque is the moment of a force about a point.

Torque is a vector (strictly, a pseudovector, like angular momentum and any other moment of a vector), and is measured in units of newton metres ($N m$ or $N.m$). By comparison, energy is a scalar, and is measured in units of newton metres or joules ($J$).

Net torque about a point equals rate of change of angular momentum about that point (rotational version of Newton's second law).

"A torque" is also the name of a pair of equal-and-opposite forces which are not in-line, and so have a purely rotational effect.

A system of forces may be combined into a net force and a net torque.

Equations

Torque equals distance "cross" force:

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

Magnitude of torque equals distance times sine times magnitude of force, equals perpendicular distance ("lever arm") times magnitude of force:

$$\tau\ =\ (r\,sin\theta)\,F$$

Net torque equals rate of change of angular momentum:

$$\mathbf{\tau}_{net}\ =\ \frac{d\mathbf{L}}{dt}$$

Work done in circular motion:

$$W\ =\ \tau\,\theta$$

Extended explanation

Torque is measured about a point. This contrasts with moment of inertia, which must be measured about an axis.

However, torque about a point is a vector, and so it has a component in the direction of any axis through that point, and so we can describe that component as the (scalar) torque about that axis.

"Torque" vs. "moment":

The words "torque" and "moment" (of force) mean the same.

However, "torque" tends to be used when there is an axle or pivot to be turned around, while "moment" tends to be used in essentially non-rotational situations, such as analysis of forces on a beam.

Lever arm:

Lever arm is the name given to the perpendicular distance from the line of a force to the point about which moments are taken.

Net torque of a system of forces:

Although force is usually thought of as a vector, it is strictly a vector acting along a particular line.

When "adding" forces, therefore, if they are only added as vectors, the result will be only the linear part of the net force: to obtain the rotational part also, their moments (the torques) must be added.

$$\mathbf{F}_{net}\ =\ \sum_i\mathbf{F}_i\ \ \ \ \ \ \mathbf{\tau}_{net}\ =\ \sum_i\mathbf{r}_i\,\times\mathbf{F}_i$$

Torques about different points:

Net torque about Q equals net torque about P plus QP "cross" net force:

$$\mathbf{\tau}_{net(Q)}\ =\ \sum_i(\mathbf{r}_i\,-\,\mathbf{q})\times\mathbf{F}_i\ =\ \sum_i(\mathbf{r}_i\,-\,\mathbf{p})\times\mathbf{F}_i\ +\ (\mathbf{p}\,-\,\mathbf{q})\times\sum_i\mathbf{F}_i\ =\ \mathbf{\tau}_{net(P)}\ +\ (\mathbf{p}\,-\,\mathbf{q})\times\mathbf{F}_{net}$$

In particular, if the net force is zero, then the net torque is the same about any point.

Torque as a pair of forces:

Net torque may be considered as a pair of equal-and-opposite forces which are not in-line.

The distance between them, and the magnitude of the forces, do not matter, so long as their product equals the magnitude of the torque.

Example:

If a brake is applied to one side of a flywheel, the combination of the friction force from the brake and the reaction force from the axle is a pair of forces which cancel out linearly, leaving only a torque.

Work done:

Work done in circular motion equals torque times angle (in radians):

$$W\ =\ F r \theta\ =\ \tau\,\theta$$

This is because, for circular motion, displacement is arc-length, and so the work done by a torque is the magnitude of force times arc-length, which is torque times arc-length divided by radius, which is simply torque times angle.

Units:

In the SI system, in which there is no dimensional unit of angle, torque has the same unit as energy (the newton metre).

In an extended system in which the radian was a dimensional unit, torque would be replaced by torque per radian, measured in newton metres per radian, and arc-length would be measured in metre radians, so that energy and work done would still be measured in newton metres (joules).

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