Question

A small particle is kept on a fixed smooth concave glass surface of radius of curvature R placed in the vertical plane. Show that if the particle is displaced from the equilibrium position and left, it executes simple harmonic motion, and find its period. (displacement being very small compared to R). 

Answer 1

O is the centre of curvature of the glass. The equilibrium position of the particle is at A with OA as the vertical. If displaced, it will oscillate to and fro about the equilibrium position.

Let B be the position of the particle at any time, so that AB = x and ∠BOA = θ. The forces acting on the particle will be 

(a) its weight mg acting vertically downwards

(b) the normal reaction N along BO Resolving the weight mg into components along the normal to the surface and perpendicular to it, the component mg sin θ directed towards the mean position provides the restoring force.