Physics 9702 · Paper 2

Three Forces: Resolve or Close the Triangle

Three Forces: Resolve, or Close the Triangle

Many equilibrium questions have exactly three forces: a weight, a pull at an angle, and one more force.
Two tools solve all of them: resolving into components, and the triangle of forces.

Tool 1: resolve and balance

Split every force into horizontal and vertical parts with $F\cos\theta$ and $F\sin\theta$ : the resolving skill from lesson 1.9, used exactly as before.
Equilibrium then gives two separate equations: horizontal parts balance, and vertical parts balance.

Worked example

Smallest case: a stone on two strings

A stone hangs from two strings. String 1 pulls with 1.00 N at 60° above the horizontal, string 2 with 0.58 N at 30° above the horizontal, on the other side. Show the horizontal parts balance, and find the stone's weight (9702-style).

Horizontal: string 1 gives $1.00\cos 60^\circ = 0.50 \text{ N}$ ; string 2 gives $0.58\cos 30^\circ = 0.50 \text{ N}$ the other way. They cancel.
Vertical: the two upward parts hold the weight: $W = 1.00\sin 60^\circ + 0.58\sin 30^\circ = 0.87 + 0.29 = 1.16 \text{ N}$ .

Answer

W = 1.16 \text{ N}

Three forces on the sphere. Resolve: the wire's vertical part T sin 53° holds the weight; its horizontal part T cos 53° is balanced by the string.

Worked example

Exam version: sphere on a wire and a string

A sphere of weight 2.4 N hangs from a wire at 53° to the horizontal. A horizontal string pulls it to one side. Find the tension T in the wire and the force F in the string. (9702/21/M/J/20 Q3)

Vertical balance

Only the wire has an upward part: $T\sin 53^\circ = 2.4$ , so $T = \dfrac{2.4}{\sin 53^\circ} = 3.0 \text{ N}$ .

Horizontal balance

The string balances the wire's sideways part: $F = T\cos 53^\circ = 3.0 \times \cos 53^\circ = 1.8 \text{ N}$ .

Answer

T = 3.0 \text{ N}, \quad F = 1.8 \text{ N}

Tool 2: the triangle of forces

A small spider hanging from a single thread that leans at an angle because the wind pushes it sideways. — Three forces hold the spider in equilibrium: its weight (straight down), the tension in the thread (along the thread), and the wind's push (sideways). Because the spider stays still, the three forces add to zero — drawn head-to-tail they form a closed triangle.

Draw the three forces tip to tail, each with its true direction and length.
If the object is in equilibrium, the three arrows come back to the start: a closed triangle. Closed triangle means the forces add to zero.
This works in reverse too. Told the object is in equilibrium? Then the triangle must close, and you can find a missing force from the triangle's geometry.

The same three forces, two ways. Left: the free-body diagram. Right: drawn tip to tail they close into a triangle, so the resultant is zero and the stone is in equilibrium.

The multiple-choice paper shows four candidate triangles for a kite or a sign and asks which one is correct: check that the arrows run tip to tail all the way round (9702/12/O/N/24 Q12).
Paper 2 sometimes asks for a scale drawing of the triangle. State your scale, and draw it large: both points are marked (9702/22/M/J/25 Q2(b)(ii)).

Resolving is the safer everyday tool. Use the triangle when the question shows one, or asks for a vector diagram or scale drawing.

PreviousThe Two Conditions for Equilibrium NextForces on a Slope

Three Forces: Resolve or Close the Triangle