A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end is attached to a fixed point \(O\). The particle moves in a horizontal circle with constant angular speed \(\omega\), and the string is inclined at angle \(\theta\) to the downward vertical. Given \(\tan\theta=\frac43\), find \(\omega\) in terms of \(a\) and \(g\).