Equilibrium in Moments

When objects can turn, like a door or a seesaw, forces can make them rotate. Equilibrium means the object is balanced: it is not moving overall and it is not turning.

Key ideas

• Moment (turning effect): the turning effect of a force about a pivot. Equation: $M = F \times d_{\perp}$. Units: newton metre (N m).
• Pivot: the point or line about which something turns (hinge on a door, centre of a seesaw).
• Direction: moments can be clockwise or anticlockwise.
• Equilibrium conditions: no resultant force and no resultant moment. For a balanced beam about a pivot: $$\sum M_{\text{clockwise}} = \sum M_{\text{anticlockwise}}$$

Perpendicular distance

$d_{\perp}$ is measured at right angles from the pivot to the line of action of the force. It is not the slanted length of the rod or where your hand happens to be.

Simple experiment

Balance a metre rule on a triangular prism at the 50 cm mark (pivot). Hang a 2 N weight at 30 cm (20 cm from pivot). Find where a 1 N weight must hang on the other side to balance. You will find it balances at 70 cm (also 20 cm from pivot) if forces are equal, or further out if the force is smaller, showing $F \times d_{\perp}$ is the same on both sides when balanced.

Common misconceptions

• Using a slanted distance instead of the perpendicular distance to the line of action.
• Thinking equal forces always balance: distance matters.
• Mixing mass and weight: use weight (in newtons) in moment calculations.
• Forgetting that equilibrium needs no resultant force and no resultant moment.
• Wrong units: moments are in N m (not N/m).