Serway & Jewett — Oscillatory Motion: Problem 18

A particle of mass \( m \) slides without friction inside a hemispherical bowl of radius \( R \). Show that if the particle starts from rest with a small displacement from equilibrium, it moves in simple harmonic motion with an angular frequency equal to that of a simple pendulum of length \( R \). That is, \( \omega = \sqrt{g/R} \).