Problem

A light inextensible string of length $12a$ is threaded through a fixed smooth ring $R$. One end of the string is attached to a particle $A$ of mass $m$. The other end of the string is attached to a particle $B$ of mass $0.5m$. Particle $A$ hangs in equilibrium vertically below the ring. Particle $B$ moves with constant angular speed $\omega$ in a horizontal circle with particle $A$ at its centre. The angle between $AR$ and $BR$ is $\theta$ (see diagram).

Express $\omega$ in terms of $g$ and $a$.