Moment of a force

A force can make an object turn about a fixed point called a pivot. The moment of a force measures the turning effect of the force about that pivot. Depending on the direction of the turning effect, the moment is either clockwise or anticlockwise.

The magnitude of the moment depends on the force and its perpendicular distance from the pivot:

With force in newtons ($$\mathrm{N}$$) and distance in metres ($$\mathrm{m}$$), the moment is measured in newton metres ($$\mathrm{N,m}$$).

Moment and perpendicular distance

The line of action is the straight line along which the force acts. The perpendicular distance is measured from the pivot to this line, at right angles to it. It is not necessarily the distance from the pivot to the point where the force is applied.

A greater force produces a greater moment if the perpendicular distance stays the same. A greater perpendicular distance also produces a greater moment if the force stays the same. If the force's line of action passes through the pivot, the perpendicular distance is zero, so the force produces no moment about that pivot.

Moments in a door and a spanner

A door turns about its hinges. Pushing the door at a point far from the hinges, with the force perpendicular to the door, gives a greater moment than the same push close to the hinges.

A spanner turns a nut about its centre. Applying a force perpendicular to the handle produces a turning effect about the nut. For the same force, a longer perpendicular distance from the nut gives a greater moment.

Principle of moments

A force acting on an object can produce a turning effect about a pivot. This turning effect is called a moment. Its size depends on the force and on the perpendicular distance from the pivot to the force’s line of action.

A force can turn the object clockwise or anticlockwise. When there is no resultant moment about the pivot, the total clockwise moment equals the total anticlockwise moment. This is the principle of moments.

Applying the principle to a balanced beam

For each force, calculate its moment by multiplying the force by its perpendicular distance from the pivot. A moment is measured in newton metres ($$\text{N m}$$) when force is in newtons ($$\text{N}$$) and distance is in metres ($$\text{m}$$).

If one force produces a clockwise moment and another produces an anticlockwise moment, their turning effects balance when:

Here, $$F_1$$ and $$F_2$$ are the forces, and $$d_1$$ and $$d_2$$ are their respective perpendicular distances from the pivot to their lines of action.

A larger force can balance a smaller force if it acts closer to the pivot. For example, a $$10,\text{N}$$ force acting at a perpendicular distance of $$0.2,\text{m}$$ produces a moment of $$2,\text{N m}$$. It can balance a $$5,\text{N}$$ force acting in the opposite turning direction at a perpendicular distance of $$0.4,\text{m}$$, which also produces a moment of $$2,\text{N m}$$.

Equilibrium of an object

An object is in equilibrium when neither its motion as a whole nor its turning motion is accelerating. For this to happen, both the resultant force and the resultant moment acting on it must be zero.

The resultant force is the combined effect of all the forces on the object. When it is zero, the object’s centre of mass does not accelerate. Its centre of mass can therefore remain stationary or move at constant velocity.

A moment is the turning effect of a force about a point. When the resultant moment about the centre of mass is zero, there is no turning acceleration.

Why both equilibrium conditions are needed

Forces can balance without their turning effects balancing. For example, two equal forces in opposite directions have zero resultant force, but if they act along different parallel lines, they can produce a non-zero resultant moment and cause turning acceleration.

Conversely, moments can balance while a resultant force remains. A single force acting through the centre of mass produces no moment about that point, but it still accelerates the centre of mass. In neither case is the object in equilibrium.

Experiment to demonstrate the principle of moments

A weight hanging from a metre rule tends to turn the rule about its support. Weights on opposite sides tend to turn it in opposite directions. When these turning effects balance, the rule can remain horizontal and stationary.

Setting up the metre rule

Support the metre rule on a pivot at its balance point. At this point, the rule’s own weight acts through the pivot, so it produces no moment about the pivot.

Hang known weights at measured positions on either side of the pivot. Adjust their positions until the rule is horizontal and stationary.

Calculating and comparing moments

For each weight, measure the perpendicular distance from the pivot to the weight’s line of action: the vertical line along which its force acts. When the rule is horizontal, this distance is the horizontal distance from the pivot to the point where the weight hangs.

Calculate each weight’s moment about the pivot:

Use the weight as the force in newtons ($$\text{N}$$) and the distance in metres ($$\text{m}$$). The moment is then in newton metres ($$\text{N m}$$).

Identify whether each weight tends to turn the rule clockwise or anticlockwise. Add the clockwise moments, then separately add the anticlockwise moments. The two totals should be equal within experimental uncertainty. They may not be exactly equal because measurements have some uncertainty.

Equal totals mean that the clockwise and anticlockwise turning effects cancel, giving zero resultant moment about the pivot. This demonstrates the principle of moments for the rule in equilibrium.

Repeat the experiment with different arrangements, including several weights on each side. Calculate the moment of every weight and include it in the total for its direction of turning.