Physics - Rotation Question with Solution | TestHub

The center of a solid sphere of radius $r$ and mass $m$ is attached to a spring of spring constant $k$, inside a ring of radius $R>r$ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the sphere are in the same vertical plane. The sphere can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the sphere is at the bottom of the ring. Assuming small displacement of the sphere, the time period of oscillation of center of mass of the sphere is written as $T=\frac{2\pi }{\omega }$. The correct expression for $\omega$ is :- ( $g$ is the acceleration due to gravity)