A light inextensible string of length \(12 a\) 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 0.5 m . 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 \(A R\) and \(B R\) is \(\theta\) (see diagram).

Express \(\omega\) in terms of \(g\) and \(a\).