Principle of Moments

The principle of moments states that for an object in rotational equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about that same point.

The condition for balance

sum of clockwise moments = sum of anticlockwise moments

A moment is the turning effect of a force: moment = force x perpendicular distance from the pivot.

Worked example

A uniform beam of length 2 m is balanced on a pivot at its centre. A 30 N weight hangs 0.4 m from the pivot on the left. Where should a 20 N weight be placed on the right to balance the beam?

Clockwise moment = anticlockwise moment 20 x d = 30 x 0.4 20d = 12 d = 0.6 m from the pivot

When the beam has weight

If the beam is uniform, its weight acts at the centre of mass (the midpoint). If the pivot is at the centre, the beam’s weight creates no moment because its distance from the pivot is zero.

If the pivot is not at the centre, include the beam’s weight in the calculation. The weight of the beam acts at the midpoint, and its moment is calculated using the distance from the midpoint to the pivot.

Multiple forces

When several forces act on each side, calculate each moment separately and add them:

sum of clockwise moments = F1 x d1 + F2 x d2 sum of anticlockwise moments = F3 x d3 + F4 x d4

Set these sums equal and solve for the unknown.

Conditions for equilibrium

For complete equilibrium (not just rotational), two conditions must be met:

1. The sum of clockwise moments equals the sum of anticlockwise moments (no net rotation).
2. The sum of upward forces equals the sum of downward forces (no net translation).