Equilibrium

A body is in equilibrium when it has no acceleration, either of its centre of mass or of its rotation. That requires two separate conditions, and both must hold.

First condition: the resultant force is zero — the forces balance in every direction, so the centre of mass does not accelerate. Equilibrium therefore includes bodies moving at constant velocity, not only stationary ones.

Second condition: the resultant moment about any point is zero — the total clockwise moment equals the total anticlockwise moment, so the body does not start to rotate.

Example: a ladder leaning against a wall can have perfectly balanced forces and still slide, because the moments are unbalanced. Both conditions are genuinely independent, and a question that gives you a rigid body almost always needs both.

The principle of moments — for a body in equilibrium, taking moments about any point:

sum of clockwise moments = sum of anticlockwise moments

Choosing where to take moments — you may take moments about any point, and choosing well removes work. Taking them about the line of action of an unknown force makes that force's moment zero, eliminating it from the equation.

Example: for a plank on two supports with both reaction forces unknown, take moments about one support. That support's reaction has zero moment about itself, leaving a single equation in a single unknown.

Three coplanar forces — if three non-parallel forces hold a body in equilibrium, two things follow: their lines of action all pass through a single point, and drawing them nose to tail produces a closed triangle. The closed triangle is the graphical statement that the resultant is zero, and it is often the quickest way to find an unknown force.