Question

Suppose you place a sphere of mass ‘m’ and radius ‘r’ inside a smooth, heavy hemispherical bowl of radius of 37r placed on a horizontal table. If the sphere is given a small displacement, what is its period of oscillation?

(a) 2π√(m/37rg)

(b) 2π√(m/rg)

(c) 12π√(r/g)

(d) 2π√(r/g)

The question was posed to me in an online quiz.

My question is taken from Velocity and Acceleration of a Particle Moving with Simple Harmonic Motion topic in portion Simple Harmonic Motion of Machine Kinematics

Answer 1

Right answer is (c) 12π√(r/g)

Easy explanation: The arrangement depicted in this question is similar to that of a simple pendulum. Instead of the usual string, you have a concave surface to confine the bob (sphere) to its path along the arc of a circle. The usual expression for the period, T = 2π√(L/g) holds here also, where the length L = 36r since the length of the pendulum is measured from the centre of gravity of the bob. The point of ‘suspension’ is evidently at the centre of the hemispherical bowl. The correct option is 12π√(r/g).