Mechanics: statics, equilibrium and moments
Statics tests whether you can prove a rigid body is at rest under forces, moments and friction. Typical questions involve ladders, hinged rods, three-force bodies, inclined planes and toppling.
Equilibrium conditions
A body in equilibrium satisfies ΣF_x = 0, ΣF_y = 0 and ΣM = 0 about any point. A moment is F × perpendicular distance; choose one sign for anticlockwise and keep it. A couple is two equal, opposite, parallel forces separated by d, with moment Fd regardless of the pivot.
Three-force bodies
If a rigid body is in equilibrium under three non-parallel forces, their lines of action are concurrent. This lets you draw a force triangle or locate an unknown reaction line rather than taking moments blindly.
Friction and inclines
At limiting friction F = μR, with R perpendicular to the contact surface. On an incline of angle θ, the weight has components mg sinθ down the slope and mg cosθ into the slope, so R = mg cosθ when no other perpendicular force acts.
Worked example
A uniform ladder of length 5 m and mass 20 kg rests against a smooth vertical wall, with its foot 3 m from the wall. Find the wall reaction. The wall is smooth, so its reaction N_w is horizontal. The ladder top is sqrt(5² − 3²) = 4 m above the ground. Weight 20g = 196 N acts at the midpoint, 1.5 m horizontally from the foot. Taking moments about the foot: N_w × 4 = 196 × 1.5, so N_w = 294/4 = 73.5 N.
Traps
Using the ladder's slant length instead of the perpendicular height; putting the weight anywhere but the midpoint of a uniform ladder; assuming wall friction when the wall is smooth; omitting a force from a moment equation; confusing a couple with a single force.