A light inextensible string is threaded through a fixed smooth ring \(R\) which is at a height \(h\) above a smooth horizontal surface. 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 \(\frac{6}{7} m\). The particle \(A\) moves in a horizontal circle on the surface. The particle \(B\) hangs in equilibrium below the ring and above the surface (see diagram).

When \(A\) has constant angular speed \(\omega\), the angle between \(A R\) and \(B R\) is \(\theta\) and the normal reaction between \(A\) and the surface is \(N\).

When \(A\) has constant angular speed \(\frac{3}{2} \omega\), the angle between \(A R\) and \(B R\) is \(\alpha\) and the normal reaction between \(A\) and the surface is \(\frac{1}{2} N\).
(a) Show that \(\cos \theta=\frac{4}{9} \cos \alpha\).

(b) Find \(N\) in terms of \(m\) and \(g\) and find the value of \(\cos \alpha\).