The principle of moments, moments and torque

The principle of moments says that when an object is balanced, the total clockwise moment about a pivot equals the total anticlockwise moment about that pivot. A moment is the turning effect of a force, given by the moment equation M = Fd: moment (N m) = force (N) × perpendicular distance from the pivot to the line of the force (m). So a 300 N child sitting 1.5 m from the middle of a seesaw is balanced by a 450 N child sitting 1.0 m from the middle on the other side, because 300 × 1.5 = 450 × 1.0.

The moment equation

AQA's GCSE specification defines the moment this way, with d as the perpendicular distance from the pivot to the line of action of the force, and its equations sheet gives moment of a force = force × distance (normal to direction of force), M = Fd. A bigger force, or the same force further from the pivot, gives a bigger turning effect. That is why door handles are far from the hinges, and why a long spanner loosens a tight nut more easily.

Rearranging the moment equation

• Force: F = M ÷ d
• Distance: d = M ÷ F

The principle of moments

AQA's GCSE specification states it as: if an object is balanced, the total clockwise moment about a pivot equals the total anticlockwise moment about that pivot. To use it:

1. Choose a pivot. In a balance problem, any point works, but choosing the point where an unknown force acts removes that force from the equation.
2. For each force, work out its moment, Fd, and decide whether it turns the object clockwise or anticlockwise.
3. Set the total clockwise moment equal to the total anticlockwise moment, and solve.

For an object to be fully balanced, in equilibrium, the forces must balance too: the total upward force equals the total downward force. At A-level, AQA's specification lists the principle of moments with moments about a point, couples and centre of mass, in the first-year content.

Principle of moments worked examples

Example 1: a single moment

A force of 30 N pushes on a door handle 0.40 m from the hinge, at right angles to the door. Find the moment.

Example 2: balancing a seesaw

A child of weight 300 N sits 1.5 m from the pivot of a seesaw. Where must a child of weight 450 N sit to balance it?

The heavier child sits 1.0 m from the pivot, on the other side. This is the seesaw in the diagram.

Example 3: finding a force

A nut needs a moment of 40 N m to loosen it. The spanner is 0.25 m long. What force is needed at the end of it?

A spanner twice as long would need only half the force, 80 N.

Example 4: a force at an angle

A 0.30 m spanner is pushed with a force of 50 N at 60° to the handle, instead of at right angles. Find the moment.

Only the perpendicular distance counts. The perpendicular distance from the pivot to the line of the force is 0.30 × sin 60°:

Pushing at right angles would give 50 × 0.30 = 15 N m, the most this force can manage.

Example 5: a beam with its own weight

A uniform plank 5.0 m long weighs 150 N. It rests on a pivot 1.5 m from its left end. What weight hung from the left end will balance it?

A uniform object's weight acts at its centre of mass, its middle, which is 2.5 m from the left end, so 1.0 m to the right of the pivot. Taking moments about the pivot:

Example 6: a beam on two supports (A-level)

A uniform 6.0 m beam of weight 300 N rests on supports at each end. A 500 N load sits 2.0 m from the left end. Find the force from each support.

Take moments about the left support, so R₁ has no moment. Clockwise: the load and the beam's weight. Anticlockwise: R₂.

The upward forces must add up to the downward forces, 800 N, so R₁ = 800 − 316.7 = 483 N. The support nearer the load takes more of it.

The Moments Calculator works out moments, missing forces and distances, and forces at an angle, and draws the beam so you can see each moment.

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Moments Calculator

Balance moments and find the torque about a pivot

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Moment equation vs torque equation

Torque is another name for the moment of a force, used especially for things that spin, such as engines, motors and wheels. The torque equation is the same:

where r is the perpendicular distance from the axis of rotation. AQA's A-level data booklet gives T = Fr in the engineering physics option, alongside T = Iα, which links torque to angular acceleration the way F = ma links force to acceleration. A car engine's torque is quoted in N m for the same reason a spanner's moment is.

Couples

A couple is a pair of equal and opposite forces whose lines of action are a distance apart, such as two hands turning a steering wheel. AQA defines the moment of a couple as one of the forces × the perpendicular distance between their lines of action. Two 20 N forces on a steering wheel, 0.15 m apart, give a moment of 20 × 0.15 = 3.0 N m. A couple turns an object without moving it sideways, because the two forces cancel.

Levers, gears and centre of mass

• Levers use the principle of moments to turn a small force into a big one. Push far from the pivot, and a big force acts close to it. AQA's GCSE specification asks students to explain how levers and gears transmit the rotational effects of forces.
• Gears pass a moment from one wheel to another. A bigger gear driven by a smaller one turns more slowly but with a bigger moment.
• Centre of mass: the point where the whole weight of an object can be taken to act. For a uniform, regular object it is at the centre. An object tips over when its centre of mass moves beyond its base, because the weight then gives a moment that turns it further.

Common mistakes

• Using the distance along a slanted line. d must be the perpendicular distance to the line of the force.
• Distances in cm. The answer is then in N cm. Convert to metres for N m, or keep every distance in cm consistently.
• Forgetting the object's own weight. A uniform beam's weight acts at its middle and has a moment too.
• Mixing up clockwise and anticlockwise. Imagine the pivot is a pin and ask which way each force on its own would turn the object.
• Writing the unit as J. A moment is in N m. It is not energy, even though the units multiply out the same way.

Practise forces in a short timed paper

GCSE Physics Higher and A-level Physics: Year 1 each give you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so moments turn up regularly but not in every paper.

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Physics · MechanicsTorque calculator: τ = Fd, angles and balanceFind torque (moment) from force and distance, including forces at an angle, balance a seesaw with the principle of moments, and find the net torque.