Particles \(A\) and \(B\), of masses \(m\) and \(3 m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles with speed \(v\). Particles \(A\) and \(B\) are at the same horizontal level. The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram).
(a) Show that \(\cos \theta=\frac{1}{3}\).

(b) Find an expression for \(v\) in terms of \(a\) and \(g\).