A hollow cone with a smooth inner surface is fixed with its vertex \(O\) downwards. The semi-vertical angle of the cone is \(\alpha\), where \(\tan\alpha=\dfrac34\). A light inextensible string has a particle \(A\) of mass \(m\) attached to one end and a particle \(B\) of mass \(m\) attached to the other end. The string passes through a small hole in the cone at \(O\).

Particle \(B\) hangs in equilibrium below \(O\). Particle \(A\) is on the inner surface of the cone at a height \(h\) above the level of \(O\) and moves in horizontal circles with constant angular speed \(\omega\).

Find \(\omega\) in terms of \(g\) and \(h\).