 # Chris Fox's Engineering Section

## Torques

"The moment of a force about an axis is equal to the sum of the moments of its components about that axis."

Principle of Moments

A force may cause a body to rotate instead of moving it along the force's line of action. This effect is known as the moment of the force and is defined about an axis. The moment (M) is given by the equation M=Fd, where d is the perpendicular distance from the axis of the force's line of action.

By convention, it is usually the clockwise moments that are positive.

A force that produces a moment may be resolved into perpendicular components. The magnitudes of these components are F cos θ and F sin θ, and it can be shown graphically that their perpendicular distances from the axis are d cos θ and d sin θ respectively. Adding together the moments they exert and simplifying gives Fd(sin²θ+cos²θ). This leads to Fd, the moment exerted by the original force.

A couple consists of two forces equal in magnitude, opposite in direction and having perpendicular separation d. It too has a tendency to rotate a body. To find its effect, consider the moments exerted by the individual forces about a point o. If the force further from the point produces a clockwise moment and is separated from the point by a distance a+d, and the other force produces an anticlockwise moment and is separated from the point by a distance a, adding together the moments gives Mo=F(a+d)-Fa. This simplifies to Mo=Fd. Mo is not a function of a and is therefore constant over the whole body.

A free vector, a couple is represented by a curved arrow wrapped round a dot.

Moments and couples are collectively called torques. Torque is denoted by the letter T.

A force that induces a moment may be replaced by a force acting at the pivot, plus a couple. Start by adding two equal, opposite forces at the pivot, parallel to the existing force. Then replace the opposite, non-collinear forces with a couple acting at the pivot. This procedure can be used in reverse to replace a force and couple with a single force.

The resultant of a system of forces and couples can be found as follows. To find the magnitude and direction of the resultant, simply add together the individual forces. Then find the sum of the moments about a chosen point; this is done by adding together the moments produced by the individual forces about that point, then adding on the couples. The sum of the moments is equal to the moment produced by the resultant; this can be used to find the resultant force's line of action.

The SI unit of distance is the metre (m), and that of torque is the newton metre (Nm).