Moment of a Force

A force can make an object turn about a point. The moment of the force measures its turning effect about that point. A larger force or a greater perpendicular distance from the point produces a larger moment.

Here, $$M$$ is the moment, $$F$$ is the force, and $$d$$ is the perpendicular distance from the chosen point to the force’s line of action. Moment is measured in newton metres, written $$\mathrm{N,m}$$.

The line of action is the straight line along which the force acts. To find $$d$$, measure from the chosen point to this line at a right angle. The distance is not necessarily the distance to the place where the force is applied.

A moment is described as clockwise or anticlockwise about the chosen point, according to the direction in which the force tends to turn the object. If the force’s line of action passes through that point, the perpendicular distance is zero, so the force has zero moment about it.

For example, a downward force of $$10,\mathrm{N}$$ applied $$0.30,\mathrm{m}$$ to the right of a pivot has a moment of $$3.0,\mathrm{N,m}$$ clockwise about the pivot.

Couple

A couple is a pair of forces that produces a turning effect without a resultant force. The forces are equal in magnitude and opposite in direction. They act in the same plane along different, parallel lines of action.

Because the forces are equal and opposite, their resultant force is zero. However, they do not act along the same line, so their turning effects do not cancel. The couple therefore has a non-zero moment.

Moment of a couple

The moment of a couple depends on the magnitude of either force and the perpendicular distance between the two lines of action:

Here, $$M$$ is the moment of the couple, $$F$$ is the magnitude of either one of the forces, and $$d$$ is the perpendicular distance between their lines of action. If $$F$$ is measured in newtons and $$d$$ in metres, the moment is measured in newton metres.

Independence from the reference point

The moment of a couple is the same whichever point is chosen for the calculation. For example, choose a point on the line of action of one force. That force has zero moment about the point, while the other force has a perpendicular distance $$d$$ from it, giving a moment of $$Fd$$.

If a different point is chosen, the individual moments may change. Their changes cancel because the forces are equal and opposite, so their combined moment remains $$Fd$$.

Principle of Moments

A force can turn a body about a chosen point. Its moment measures this turning effect. The moment depends on the force and on the perpendicular distance from the point to the force’s line of action:

The line of action is the line along which the force acts. A force produces a clockwise or an anticlockwise moment about the chosen point.

Balancing moments

The principle of moments states that, for a body in equilibrium, the total clockwise moment about any point equals the total anticlockwise moment about that point. The turning effects therefore balance.

The same balance can be expressed using signs. If clockwise moments are taken as positive and anticlockwise moments as negative, their sum is zero. This sum is the resultant moment. The opposite sign convention gives the same result, provided it is used consistently.

Moments and complete equilibrium

A zero resultant moment about a chosen point does not by itself establish complete equilibrium: the resultant force must also be zero. For example, a single force whose line of action passes through the chosen point has zero moment about that point, but the force is still unbalanced.

For a body in complete equilibrium, both the resultant force and the resultant moment are zero; the moments balance about any point.

Centre of Mass

The centre of mass is the point that represents the mean position of a body’s mass. Its position depends on how the mass is distributed: regions containing more mass have a greater influence on where the centre of mass lies.

In a uniform gravitational field, gravity has the same strength and direction throughout the body. The body's weight can then be treated as a single force acting at its centre of mass.

Centre of mass of a uniform regular solid

A solid is uniform when its mass is distributed evenly throughout it. For a uniform regular solid, the centre of mass is at its geometric centre. For example, the centre of mass of a uniform cuboid is at the point halfway along its length, width and height.