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 Video by Zeeshan Zamurred Applications of Forces

Edexcel A Level Maths: 7.4 Ladders and Beams (Statics of Rigid Bodies)Watch the full walkthrough 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

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 $\sum F = 0$ (both directions) and $\sum M = 0$ .

\sum F_{x} = 0, \sum F_{y} = 0, \sum M = 0

Three equations for a rigid body.

1Take moments about the base.

2The floor reaction and friction then have no moment, simplifying the equation.

AnswerAbout 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

\sum F_{x} = 0, \sum F_{y} = 0

Force equilibrium.

\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

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