Equilibrium of a Rigid Body

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Calculate the moment of a force about a point (for coplanar forces only)
  2. Use the result that the effect of gravity on a rigid body is equivalent to a single force acting at the centre of mass of the body, and identify the position of the centre of mass of a uniform body using considerations of symmetry
  3. Use given information about the position of the centre of mass of a triangular lamina and other simple shapes
  4. Determine the position of the centre of mass of a composite body by considering an equivalent system of particles
  5. Use the principle that if a rigid body is in equilibrium under the action of coplanar forces then the vector sum of the forces is zero and the sum of the moments of the forces about any point is zero, and the converse of this
  6. Solve problems involving the equilibrium of a single rigid body under the action of coplanar forces, including those involving toppling or sliding

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What is a Rigid Body?

A rigid body is an object that does not bend, stretch, or change shape when forces are applied to it. Think of it as a solid object that stays exactly the same shape no matter what. Examples include a metal rod, a wooden plank, or a door.

In reality, all objects bend a little bit when forces act on them, but for many problems we can treat them as if they don't bend at all. This makes the mathematics much simpler.

What is Equilibrium?

An object is in equilibrium when it is not moving and not rotating. This means:

  • It's either completely still (stationary), or
  • It's moving at a constant speed in a straight line with no rotation

For an object in equilibrium, all the forces and all the turning effects (called moments) must balance out perfectly.

1. Moment of a Force

What is a Moment?

A moment (also called a turning effect or torque) is the rotational effect that a force has around a point. When you push a door to open it, you're creating a moment about the hinges.

Calculating Moments

The moment of a force about a point is calculated using:

Moment = Force × Perpendicular distance from the point

In symbols: M = F × d

Where:

  • M = moment (measured in newton-metres, N m)
  • F = force (measured in newtons, N)
  • d = perpendicular distance from the point to the line of action of the force (measured in metres, m)

Important: The distance must be perpendicular (at right angles) to the direction of the force.

Direction of Moments

Moments can turn things in two directions:

  • Clockwise moments turn to the right (like clock hands)
  • Anticlockwise moments turn to the left (opposite to clock hands)

We usually take anticlockwise moments as positive and clockwise moments as negative (or vice versa, as long as you're consistent).

Example 1: Simple Moment Calculation

A force of 20 N acts on a rod at a distance of 0.5 m from a pivot point. The force acts perpendicular to the rod. Calculate the moment about the pivot.

Solution:

  • F = 20 N
  • d = 0.5 m
  • M = F × d = 20 × 0.5 = 10 N m

Example 2: Force at an Angle

If a force doesn't act perpendicular to the rod, you need to find the perpendicular distance first.

A force of 30 N acts at an angle of 60° to a horizontal rod, at a distance of 2 m from a pivot. Calculate the moment.

Solution:

  • The perpendicular distance = 2 × sin 60° = 2 × (√3/2) = √3 m
  • M = 30 × √3 = 30√3 N m or approximately 52.0 N m

Alternatively, you could use the perpendicular component of the force:

  • Perpendicular component = 30 × sin 60° = 30 × (√3/2) = 15√3 N
  • M = 15√3 × 2 = 30√3 N m

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