Learn7408Paper 1 · Paper 1Mechanics and materialsMoments, couples and equilibrium

Mechanics and materials · Subtopic 5 of 11

Moments, Couples and Equilibrium

The moment of a force is force times the perpendicular distance from the pivot to its line of action. A couple is a pair of equal, opposite, parallel forces: it turns without moving the object along, with torque $=$ one force $\times$ the separation. An object is in equilibrium when the resultant force and the resultant moment are both zero; then its forces form a closed vector polygon.

Cambridge 9702 Paper 1Cambridge 9702 Paper 2Edexcel YPH11 WPH11AQA 7408 Paper 1≈ 40 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Calculate moments using the perpendicular distance.
Calculate the torque of a couple.
Apply the principle of moments, including to beams on two supports.
Use the two conditions for equilibrium and the triangle of forces.

Pause and think

You turn a car's steering wheel with both hands on opposite sides. Why doesn't the wheel slide sideways as well as turning?

Section 2 of 14

Core idea

Start hereWhy a shop sign doesn't fall

A sign hangs from a rod held by a slanting cable. Nothing moves, so the forces balance and nothing turns, so the turning effects balance. Two conditions, and together they find every unknown force.

The big idea

Equilibrium: $\sum F=0$ and $\sum\tau=0$ about any point. Couple: equal, opposite, parallel forces; torque $Fd$ , no resultant force.

1. Moments

A uniform beam held by two supports

Moments about the left support:

200(4)+100(6)=R_2(8)

, so

R_2=175

N; then

R_1=300-175=125

Worked exampleBeam on two supports

8 m uniform beam (200 N), supports at the ends; 100 N at 6 m.

Goal: Reactions.
Strategy: Moments about the left support.

$200(4)+100(6)=8R_2$
Clockwise = anticlockwise.
$R_2=175\text{ N}$
Solve.
$R_1=300-175=125\text{ N}$
Vertical balance.

Answer

125

N and

175

✓ CheckLoad is right of centre, so

R_2>R_1

✓.

2. Couples

A couple: turning a steering wheel

Torque of a couple

=Fd

. The forces cancel, so the wheel turns without moving sideways.

Torque of a couple is the same about any point

$\tau=Fx+F(d-x)$
Point a distance $x$ from one force, between them.
$=Fd$
$x$ cancels.

3. Three forces

Worked exampleA hanging sign

Light horizontal rod, hinged at the wall, length $L$ ; 40 N sign at the far end; cable from the far end to the wall at $30^\circ$ to the rod.

Goal: Tension; hinge force.
Strategy: Moments about the hinge removes the hinge force.

$T\sin30^\circ\times L=40\times L$
Only the perpendicular component of $T$ turns.
$T=80\text{ N}$
Solve.
$H=80\cos30^\circ=69.3,\ V=40-40=0$
Resolve.

Answer

T=80

N; hinge force

69.3

N horizontally.

✓ CheckThree forces meet at one point (the far end) ✓.

Key takeawayThree-force rule Three non-parallel forces in equilibrium are concurrent and form a closed triangle.

Pause and think

Why is a door handle far from the hinge?

Summary

$\sum F=0$ , $\sum\tau=0$ .
Couple: torque $Fd$ .

Torque: Moment of a force or couple.
Couple: Pair of equal, opposite, parallel forces.

HistoryArchimedes' law of the lever. Around

250

BC Archimedes proved that magnitudes balance at distances inversely proportional to their weights. He's said to have moved a fully loaded ship single-handedly using a system of levers and pulleys, astonishing the king of Syracuse.

Fun factWhy tightrope walkers carry poles. A long, heavy pole spreads mass far from the walker's body, increasing their moment of inertia so they tip more slowly and have time to correct. Bending the pole down at the ends also lowers the centre of gravity.

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