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7.4MechanicsStretch

Static Rigid Bodies

A rigid body such as a ladder leaning against a wall needs both force and moment equations. Resolving in two directions and taking moments about a clever point gives enough equations to solve.

28 min Applications of Forces
Edexcel A Level Maths: 7.4 Ladders and Beams (Statics of Rigid Bodies)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Apply ΣF = 0 and ΣM = 0 to a rigid body
  • Analyse ladder and beam problems
  • Include friction at contact points
  • Choose a good point for moments
1

Ladder problems

For a ladder against a wall: there is friction and reaction at the floor, a reaction (and possibly friction) at the wall, and weight at the centre. Use ∑F=0\sum F = 0 (both directions) and ∑M=0\sum M = 0.

∑Fx=0, ∑Fy=0, ∑M=0\sum F_x = 0, \ \sum F_y = 0, \ \sum M = 0
Three equations for a rigid body.
Example 1For a ladder, where is the best point to take moments?
1Take moments about the base.
2The floor reaction and friction then have no moment, simplifying the equation.
Answer
About the base, to eliminate the floor forces.

Tip — Take moments about the foot of the ladder to eliminate the floor’s reaction and friction.

Formula recap

∑Fx=0, ∑Fy=0\sum F_x = 0, \ \sum F_y = 0
Force equilibrium.
∑M=0\sum M = 0
Moment equilibrium.

Common mistakes to avoid

Using only force equations for a rigid body.
A rigid body also needs ΣM = 0.
Placing the ladder’s weight at one end.
A uniform ladder’s weight acts at its midpoint.

Key takeaways

  • Rigid body: ΣFₓ = 0, ΣF_y = 0 and ΣM = 0.
  • Include friction/reaction at each contact point.
  • Take moments about the base to remove the floor forces.

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. A rigid body in equilibrium needs:

    • ΣF = 0 and ΣM = 0
    • ΣF = 0 only
    • ΣM = 0 only
    • neither
    Show answer and explanation

    Answer: ΣF = 0 and ΣM = 0

    Both conditions.

  2. For a ladder, take moments about:

    • the base
    • the centre
    • the top only
    • a random point
    Show answer and explanation

    Answer: the base

    Removes floor forces.

  3. A uniform ladder’s weight acts at:

    • its midpoint
    • the base
    • the top
    • the wall
    Show answer and explanation

    Answer: its midpoint

    Centre of mass.

  4. How many independent equations for a rigid body in 2D?

    • three
    • two
    • one
    • four
    Show answer and explanation

    Answer: three

    Two force + one moment.

  5. At the floor a ladder usually has:

    • reaction and friction
    • only weight
    • only friction
    • no forces
    Show answer and explanation

    Answer: reaction and friction

    Rough floor contact.

Next step

Check your understanding

Work through questions on Static Rigid Bodies. Use the feedback to decide what needs another look.

2 past-paper questions · 9 marks

Practise this topic

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