A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The string is held taut with \(O P\) making an angle \(\alpha\) with the downward vertical, where \(\cos \alpha=\frac{2}{3}\). The particle \(P\) is projected perpendicular to \(O P\) in an upwards direction with speed \(\sqrt{3 a g}\). It then starts to move along a circular path in a vertical plane.

Find the cosine of the angle between the string and the upward vertical when the string first becomes slack.