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 the same point.

$\sum \text{clockwise moments} = \sum \text{anticlockwise moments}$

Applying the principle

1. Choose or identify the pivot.
2. Calculate the clockwise moments (force x perpendicular distance for each force producing a clockwise turn).
3. Calculate the anticlockwise moments.
4. Set them equal and solve for the unknown.

Worked example

A uniform beam of length 2.0 m is balanced on a pivot at its centre. A 40 N weight is placed 0.80 m from the pivot on the left side. Where must a 50 N weight be placed on the right side to balance the beam?

Taking moments about the pivot:

Anticlockwise moment = $40 \times 0.80 = 32$ N m

For balance: clockwise moment = anticlockwise moment

$50 \times d = 32$ $d = \frac{32}{50} = 0.64\text{ m}$

The 50 N weight must be placed 0.64 m from the pivot on the right side.

Multiple forces

When more than two forces act, add all clockwise moments together and all anticlockwise moments together before setting them equal.

Worked example: three forces

A beam balances on a pivot. On the left: 10 N at 0.40 m and 20 N at 0.60 m from the pivot. Find the single force needed on the right at 0.50 m from the pivot.

Anticlockwise moments = $(10 \times 0.40) + (20 \times 0.60) = 4.0 + 12 = 16$ N m

$F \times 0.50 = 16$ $F = 32\text{ N}$

Common errors and how to correct them

Measuring distances from the wrong point. All distances must be from the same pivot. Check which point the beam rotates about.

Forgetting the weight of the beam itself. If the beam is not described as “light” or weightless, its weight acts at its centre of gravity (the midpoint for a uniform beam) and contributes a moment unless the pivot is at that point.

How to apply this in an exam

Draw a simple sketch of the beam, mark the pivot, and label every force with its distance. This prevents sign errors and missed forces.