Beams, hinges and cables

5 minAS Level4.2.14.2.2

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

By the end of this lesson

By the end you can solve equilibrium problems with angled forces: find a cable's tension using moments, then the force at a hinge by resolving.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$H_x$	horizontal component of the force from the hinge
$H_y$	vertical component of the force from the hinge

Three unknowns, one clever pivot

Shop signs, drawbridges, cranes: a beam sticks out from a wall, a cable holds it up, and a hinge stops it falling. The tension and the hinge force (size and direction) are all unknown.

One clever choice of point makes the hinge force vanish completely.

The plan

Moments about the hinge. The hinge force acts at the hinge, so it has no moment there. Only the tension is unknown.
Resolve horizontally. The hinge's horizontal component balances the horizontal component of the tension.
Resolve vertically. The hinge's vertical component makes up whatever upward force the cable doesn't supply.
Combine the two components with Pythagoras and $\tan^{-1}$ if you need the hinge force itself.

Worked example: Step 1: the tension

A uniform beam $2.0\ \text{m}$ long, of weight $80\ \text{N}$ , is hinged to a wall and held horizontal by a cable attached to its far end at $30°$ to the beam. A $120\ \text{N}$ load hangs from the far end.

Find the tension $T$ .

Have a go first?

Quick check

Multiple choiceQuestion 1 · Recall

Multiple choiceQuestion 2 · Apply

Answer the 3 questions above to continue.