Moments equation for A-Level Physics explained
The moment of a force is the turning effect produced by that force about a pivot. It is calculated using the equation moment = force × perpendicular distance from the pivot to the line of action of the force. The unit is the newton metre (Nm), and the direction is described as either clockwise or anticlockwise.
This guide explains the equation properly, walks through the principle of moments, covers couples and the moment of a couple, and then shows the worked examples and exam mistakes that come up in AQA A-Level Physics Paper 1 every year.
Moment = F × d
Force in newtons multiplied by the perpendicular distance in metres. The unit is the newton metre (Nm), never confused with joules.
Principle of moments
For an object in equilibrium, the sum of clockwise moments about any pivot equals the sum of anticlockwise moments about that same pivot.
Perpendicular distance matters
The distance must be measured perpendicular to the line of action of the force, not along the object. This is where most marks are dropped.
Defining a moment properly
A moment is the turning effect of a force about a pivot. The mark-scheme definition is: Moment = force × perpendicular distance from the pivot to the line of action of the force. The phrase "perpendicular distance" and the phrase "line of action" both need to be in your answer for full marks.
The SI unit is the newton metre (Nm). It is not a joule, even though both have base units of kg m² s⁻². The two quantities measure different things: A joule is energy transferred, a newton metre is a turning effect. Examiners look for the right unit and will dock marks if you write Nm as J.
Newton metres are not joules Both units share base SI dimensions, but they represent different physical quantities. Always write Nm for moments, torques and couples, and J for energy. Mixing them up is one of the cheapest mark losses on Paper 1.
The moment equation in detail
The full equation is M = F × d, where M is the moment in newton metres, F is the force in newtons, and d is the perpendicular distance from the pivot to the line of action of the force.
When the force is not perpendicular to the lever, you have two equivalent options. You can resolve the force into the component perpendicular to the lever, then multiply by the distance along the lever. Or you can keep the full force and use the perpendicular distance from the pivot to the line of action. Both give the same answer.
| Symbol | Quantity | Unit |
|---|---|---|
| M | Moment of the force about the pivot | Nm |
| F | Applied force | N |
| d | Perpendicular distance from pivot to line of action | m |
| θ | Angle between the force and the lever | degrees or radians |
The principle of moments
For a rigid body in rotational equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about that same point. This is the principle of moments, and it is the workhorse equation for beam and seesaw questions at A-Level.
The principle works about any pivot, not just the one the problem gives you. Choosing the pivot cleverly, often through an unknown force, is the standard trick for reducing the number of unknowns in a question.
Pick your pivot to kill an unknown If a beam question gives you two support forces and asks for one, take moments about the other support. That unknown then has zero distance from the pivot, so its moment is zero and it drops out of the equation.
Worked example: A horizontal beam in equilibrium
A uniform horizontal beam of length 4.0 m and weight 200 N rests on two supports, one at each end. A 600 N load hangs 1.0 m from the left support. Find the upward force at each support.
Step 1: Take moments about the left support. Clockwise moments are 600 × 1.0 plus 200 × 2.0 (the beam weight acts at its centre). That gives 600 + 400 = 1000 Nm clockwise.
Step 2: The anticlockwise moment about the left support is the right support force times 4.0 m. Setting them equal: R × 4.0 = 1000, so R = 250 N upward at the right support.
Step 3: Use vertical equilibrium to find the left support. Total upward = total downward, so L + 250 = 600 + 200, giving L = 550 N upward at the left support.
Couples and the moment of a couple
A couple is a pair of equal and opposite parallel forces that do not act along the same line. A couple produces a pure turning effect with no net translational force, which is why a steering wheel turns without sliding sideways.
The moment of a couple, sometimes called the torque, is calculated as one of the forces multiplied by the perpendicular distance between them. The equation is τ = F × d, where d is the separation of the two forces, not the distance to a pivot.
Couples versus single-force moments For a single force about a pivot, d is the distance from the pivot to the line of action. For a couple, d is the perpendicular distance between the two parallel forces. Read the question carefully to spot which one you are dealing with.
Forces at an angle: Resolving for moments
When a force acts at an angle to a lever, only the perpendicular component contributes to the moment. If the force F acts at angle θ to a lever of length L, the moment about the pivot is M = F × L × sin(θ).
This appears in questions about hinged signs, ladders against walls, and crane arms. Drawing the force, marking the angle, and resolving into perpendicular and parallel components on the lever itself is the cleanest approach. The parallel component does not contribute to the moment because its line of action passes through the pivot.
Where students lose marks on moments questions
AQA examiner reports flag the same pattern of mistakes year after year. The physics is rarely the problem. The marks are lost on units, signs, and the perpendicular distance.
Common mistakes that cost easy marks Using distance along the lever rather than perpendicular distance to the line of action. Writing the unit as joules instead of newton metres. Forgetting that the weight of a uniform beam acts at its centre of mass. Taking moments about the wrong pivot and ending up with too many unknowns. Mixing up the formula for a couple (F × separation) with the formula for a single moment (F × distance from pivot).
Key facts to memorise for the exam
- Moment = force × perpendicular distance from pivot to line of action of the force
- Unit: Newton metre (Nm), never joules
- Principle of moments: Sum of clockwise moments = sum of anticlockwise moments about the same pivot
- For forces at angle θ to a lever of length L, moment = F × L × sin(θ)
- Moment of a couple = one force × perpendicular distance between the two parallel forces
- The weight of a uniform beam acts at its geometric centre
- Always state whether each moment is clockwise or anticlockwise
- Pick your pivot to eliminate an unknown force from the equation
