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Mechanics · Moments

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Equilibrium of rigid bodies

A rigid body in equilibrium needs both zero resultant force and zero resultant moment — using these two conditions to find the reaction forces on a supported beam, and taking moments about a support to eliminate an unknown.

Mechanics · Moments

Equilibrium of rigid bodies

A rigid body in equilibrium needs both zero resultant force and zero resultant moment — using these two conditions to find the reaction forces on a supported beam, and taking moments about a support to eliminate an unknown.

Why it works

For a particle, equilibrium just means the forces balance: ∑F=0\sum F = 0. But a rigid body — a beam, a plank, a see-saw — can also rotate, so balancing the forces isn't enough. A body in equilibrium must satisfy two conditions:
  1. No resultant force: the forces balance in each direction (∑F=0\sum F = 0), so it
doesn't move off.
  1. No resultant moment: the turning effects balance (∑moments=0\sum \text{moments} = 0
about any point), so it doesn't spin.R₁R₂WThe standard problem is a uniform beam resting on two supports. "Uniform" means its weight acts at the centre. You have two unknown support reactions, and the two equilibrium conditions give you two equations to find them.

The key trick: take moments about a support. Because a force acting through a point has no moment about that point, taking moments about one support makes its reaction vanish from the equation — leaving a single equation in the other reaction. That's far quicker than carrying both unknowns. Then resolve vertically (∑F=0\sum F = 0) to get the remaining reaction.

A neat check: the two support reactions must add up to the total weight (R1+R2=WR_1 + R_2 = W), since the vertical forces balance.