Stability of bodies in equilibrium — A/L Combined Mathematics

The equilibrium of a body subjected to conservative forces is classified into three states based on its behavior following an infinitesimal displacement: (1) Stable Equilibrium: The body generates restoring forces that return it to its original position (potential energy V is a minimum, dV/dθ = 0 and d²V/dθ² > 0); (2) Unstable Equilibrium: The body moves further away from equilibrium (V is a maximum, dV/dθ = 0 and d²V/dθ² < 0); (3) Neutral Equilibrium: The body remains in equilibrium in the adjacent displaced position (V is constant, d²V/dθ² = 0). For a body resting on a fixed spherical surface, stability depends on the height h of its centre of mass above the contact point: stable if 1/h > 1/r₁ + 1/r₂.

Check before you answer (A/L Applied Maths Paper 2)

1. Potential Energy Formulation: Formulating the total potential energy V(θ) as a function of the angular displacement θ with respect to a fixed datum.
2. First and Second Derivative Tests: Differentiating to verify dV/dθ = 0 at the equilibrium position and evaluating the sign of d²V/dθ².
3. Curvature Criterion: Correctly applying the geometric stability criterion h < (r₁ r₂) / (r₁ + r₂) for bodies rolling on curved surfaces.