Equilibrium of a Particle

A particle in equilibrium has zero resultant force. Resolve in two directions, set each total to zero, and you have two equations for two unknowns.

Cambridge 9709 Paper 4Edexcel 9MA0 Paper 3AQA 7357 Paper 2≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Know that a particle is in equilibrium when the resultant force on it is zero.
Resolve in two perpendicular directions and solve the resulting equations.
Find unknown tensions in strings supporting a particle.
Use a closed triangle of forces for three-force problems.

Pause and think

A heavy picture hangs from a single nail by a cord. Would the cord be more or less likely to snap if the nail were moved so the two halves of the cord were almost horizontal?

Section 2 of 14

Core idea

Start hereThe hanging lamp

A lamp hangs from two cables attached to the ceiling at different angles. It's perfectly still, so the forces balance. But how hard does each cable pull? If you know the weight and the angles, resolving gives two equations and the tensions drop out.

The big idea

A particle is in equilibrium when the resultant force is zero: the components in each of two perpendicular directions sum to zero. Drawn tip-to-tail, the forces form a closed polygon.

1. The method

Solving any forces problem

Drawing diagram…

The same routine solves equilibrium, friction, slopes, connected particles and lifts.

A 50 N weight held by two strings

Three forces on the knot: two tensions along the strings and the weight. The strings make

45^\circ

and

60^\circ

with the ceiling.

Worked exampleA 50 N weight on two strings

Strings at $45^\circ$ (tension $T_1$ ) and $60^\circ$ ( $T_2$ ) to the ceiling hold a 50 N weight. Find the tensions.

Goal: $T_1$ , $T_2$ .
Strategy: Resolve horizontally and vertically.

$\rightarrow:\ T_1\cos45^\circ=T_2\cos60^\circ\Rightarrow T_1=0.7071T_2$
Horizontal components cancel.
$\uparrow:\ T_1\sin45^\circ+T_2\sin60^\circ=50$
Vertical: up = down.
$0.5T_2+0.866T_2=50\Rightarrow T_2=36.6$
Substitute.
$T_1=25.9$
Back-substitute.

Answer

T_1=25.9

T_2=36.6

✓ CheckSteeper string (60°) carries more of the weight ✓.

The three forces drawn head to tail close up

No gap in the triangle means no resultant: equilibrium.

Worked exampleOn a smooth slope

A 4 kg particle on a smooth $30^\circ$ slope is held by a horizontal force $P$ . Find $P$ and $R$ .

Goal: $P$ , $R$ .
Strategy: Resolve along and perpendicular to the slope.

$\text{along: }P\cos30^\circ=40\sin30^\circ\Rightarrow P=23.1\text{ N}$
Weight $40$ N.
$\text{perp: }R=40\cos30^\circ+P\sin30^\circ=46.2\text{ N}$
$P$ also pushes into the slope.

Answer

P=23.1

R=46.2

✓ CheckHorizontally/vertically:

R\sin30=23.1=P

✓.

Pause and think

Three forces in equilibrium: 5 N and 12 N at right angles. Third?

Summary

Resultant zero in two directions.
Smart axes save work.

Equilibrium: Zero resultant force.
Tension: Pull in a string.

HistoryLami's theorem. In 1687, the same year as Newton's Principia, the French priest Bernard Lamy published a neat rule for three forces in equilibrium: each force is proportional to the sine of the angle between the other two. It is not required by Cambridge, but it is a handy check on your answers.

Fun factYou can never pull a washing line straight. If a washing line carries any weight at all, the two halves must each have an upward component. If the line were perfectly horizontal, the tension would need to be infinite. That is why even a tight line always sags a little, and why power cables sag between pylons.

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