5. A uniform circular disc has mass \(m\) and radius \(a\). The disc can rotate freely about an axis that is in the same plane as the disc and tangential to the disc at a point \(A\) on its circumference. The disc hangs at rest in equilibrium with its centre \(O\) vertically below \(A\). A particle \(P\) of mass \(m\) is moving horizontally and perpendicular to the disc with speed \(\sqrt { } ( k g a )\), where \(k\) is a constant. The particle then strikes the disc at \(O\) and adheres to it at \(O\). Given that the disc rotates through an angle of \(90 ^ { \circ }\) before first coming to instantaneous rest, find the value of \(k\).
(Total 10 marks)