Solving force triangles: two strings and a washing line

5 minAS Level4.2.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can find unknown forces when three forces in equilibrium make a triangle that isn't right-angled, using the sine rule or by resolving.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$\theta$	an angle
$\alpha$	an angle

The washing-line puzzle

Hang a wet coat in the middle of a washing line and the line sags. Pull the ends as hard as you like: it never goes perfectly straight.

The vector triangle explains why, and the answer is a little shocking.

Easy case first: two equal strings

A weight $W$ hangs from two strings, each at $\theta$ to the vertical. By symmetry the two tensions are equal.

Only the vertical components hold the weight up:

$2T\cos\theta = W \quad\Rightarrow\quad T = \frac{W}{2\cos\theta}$

Check: at $\theta = 0$ (both strings vertical) each takes half the weight, $T = \frac{W}{2}$ . ✓

Why the line can't be straight

As the strings flatten, $\cos\theta$ shrinks, so $T$ grows:

at $60°$ , each tension equals the whole weight;
at $80°$ , it's nearly 3 times the weight;
at $90°$ (perfectly straight) the strings have no vertical component at all, so no tension, however large, could hold the weight up.

That's why power lines sag and a tightrope dips under the walker.

Quick check

Write an expressionQuestion 1 · Apply

CalculateQuestion 2 · Apply

Answer the 3 questions above to continue.