03 Rods with hinges, strings and smooth pegs
- A smooth hinge exerts two unknown components Rx, Ry (three unknowns with the direction being arbitrary); take moments about the hinge to eliminate both.
- A smooth peg or a light string over it transmits the same tension on both sides and the reaction is along the bisector of the angle between the two parts.
- A smooth wall / floor gives a reaction perpendicular to the surface only.
Worked example — hinged rod with a string
A uniform rod AB of weight W and length 2 m is hinged at A and held by a vertical string at B inclined 30° to the horizontal wall. Moments about A: T × 2 cos 30° = W × 1 ⟹ T = W/(2 cos 30°) = 0.577 W; then Ry = W − T and Rx = 0 for a vertical string.
Worked example — limiting position
A rod of length L rests with one end on a rough floor and leans against a smooth wall at angle θ. The wall reaction is horizontal, the floor gives R (vertical) and F (horizontal, F ≤ μR). Equilibrium gives F = Rwall, R = W, and moments about the floor end: Rwall·L sin θ = W·(L/2)cos θ ⟹ Rwall = (W/2)cot θ. Slipping impends when (W/2)cot θ = μW ⟹ tan θ = 1/(2μ).