To find the hinge force on a balanced beam, take moments about the hinge first. The hinge force acts at the pivot, so its moment about that point is zero. This lets you calculate the other support force.
Then balance the vertical forces to find the vertical component of the hinge reaction. If the only other forces are vertical, the horizontal component of the hinge reaction is zero.
- Mark each force and its perpendicular distance from the hinge.
- Balance clockwise and anticlockwise moments about the hinge.
- Use vertical force balance to find the hinge reaction and its direction.
Worked example | A beam supported at its far end
A uniform 4.0 m horizontal beam weighs 60 N and is hinged at its left end. A 90 N load hangs 3.0 m from the hinge. A vertical cable at the far end holds the beam in equilibrium.
The beam’s own weight acts 2.0 m from the hinge. Taking moments about the hinge gives cable tension × 4.0 = 60 × 2.0 + 90 × 3.0. The tension is 97.5 N.
The total downward force from the weights is 150 N. The cable supplies 97.5 N upwards. The hinge therefore exerts 52.5 N upwards on the beam.
Can the hinge push downwards?
Move the vertical cable to 2.0 m from the hinge and keep the weights where they were. The clockwise moment stays at 390 N m, so the cable tension rises to 195 N.
The cable now pulls upwards harder than the combined weights pull downwards. A hinge that can exert force in either direction must supply 45 N downwards to balance the beam.
Move the load
Use the original beam with the cable at 4.0 m. Move the 90 N load to 1.0 m from the hinge. Find the cable tension and the hinge force.
Answer. The total clockwise moment is 60 × 2.0 + 90 × 1.0 = 210 N m. The cable tension is 210 ÷ 4.0 = 52.5 N. The hinge force is 150 − 52.5 = 97.5 N upwards.
Frequently asked questions
Why does the hinge force disappear from the moment equation?
Its line of action passes through the pivot, so its perpendicular distance from that pivot is zero.
Can I take moments about another point?
Yes. The moments balance about any chosen point when the beam is in equilibrium. Choose the hinge to eliminate the unknown hinge force from the first equation.
Official information
SEAB, 2027 SEC G3 Physics syllabus (K323), Turning Effect of Forces. Syllabus checked 11 November 2026. The examples are original.