Equilibrium

A body is in equilibrium when the resultant force and the resultant moment acting on it are both zero. This deceptively simple statement is the foundation of all structural analysis: every bridge, building frame, machine bracket and support structure must satisfy equilibrium under the loads it carries. The challenge lies not in the equations themselves, which follow directly from the vector tools developed in the preceding part, but in correctly identifying which forces act on which body and in which direction.

We begin with the free body diagram, the single most important modelling tool in mechanics. By isolating a body from its surroundings and replacing every contact, support and connection with the forces and moments it transmits, we convert a physical situation into a well-posed mathematical problem. We catalogue the standard support types, from cables and pins to rigid connections and rollers, and show how each constrains motion and introduces corresponding reaction forces.

The worked examples that follow apply equilibrium to progressively more complex structures, from single beams with two supports to multi-body systems requiring several coupled free body diagrams. Each example follows the same workflow: construct the free body diagram, count unknowns against available equations, set up the equilibrium system symbolically, solve it with SymPy, and interpret the results physically. We close with friction, which introduces inequality constraints into the equilibrium picture and marks the boundary between statics and dynamics.