Back to course
4.3MechanicsStretch

Equilibrium

A rigid body is in equilibrium when it has no tendency to move or rotate. This requires the resultant force to be zero AND the resultant moment about any point to be zero — giving equations to solve.

26 min Moments
Edexcel A Level Maths: 4.3 Equilibrium (Moments, Resultant Force)Curated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

What you'll be able to do

  • State the conditions for equilibrium
  • Resolve forces to zero
  • Take moments to zero about a chosen point
  • Solve beam and plank problems
1

Two conditions

For equilibrium: (1) the resultant force is zero (resolve horizontally and vertically), and (2) the resultant moment about any point is zero.

∑F=0and∑M=0\sum F = 0 \quad \text{and} \quad \sum M = 0
Both must hold.
2

Choosing a pivot

Take moments about a point where an unknown force acts — that force then has zero moment and drops out, simplifying the equation.

Example 1A uniform beam rests on two supports. How do you find a support reaction?
1Take moments about the OTHER support so its reaction vanishes.
2Solve for the wanted reaction.
Answer
Moments about the other support isolates the unknown.

Tip — Take moments about an unknown force to eliminate it from the equation.

Formula recap

∑F=0\sum F = 0
No resultant force.
∑M=0\sum M = 0
No resultant moment.

Common mistakes to avoid

Only checking forces, not moments.
Both ΣF = 0 and ΣM = 0 are required.
Forgetting the weight acts at the centre of a uniform beam.
A uniform beam’s weight acts at its midpoint.

Key takeaways

  • Equilibrium: ΣF = 0 and ΣM = 0.
  • Take moments about an unknown force to eliminate it.
  • A uniform beam’s weight acts at its midpoint.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. For equilibrium of a rigid body:

    • ΣF = 0 and ΣM = 0
    • ΣF = 0 only
    • ΣM = 0 only
    • ΣF = ΣM
    Show answer and explanation

    Answer: ΣF = 0 and ΣM = 0

    Both conditions.

  2. To eliminate an unknown force, take moments:

    • about where it acts
    • about the centre
    • about the far end
    • anywhere but there
    Show answer and explanation

    Answer: about where it acts

    Its moment is then zero.

  3. The weight of a uniform beam acts at:

    • its midpoint
    • one end
    • the support
    • the heaviest end
    Show answer and explanation

    Answer: its midpoint

    Centre of mass.

  4. The resultant moment in equilibrium is:

    • zero
    • maximum
    • equal to the weight
    • clockwise
    Show answer and explanation

    Answer: zero

    No rotation.

  5. Resolving forces in equilibrium gives:

    • ΣF = 0 horizontally and vertically
    • one equation only
    • ΣM = 0
    • nothing useful
    Show answer and explanation

    Answer: ΣF = 0 horizontally and vertically

    Two force equations.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 5 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.