5 A particle \(P\) of mass \(m\) lies on a smooth horizontal surface. \(A\) and \(B\) are fixed points on the surface, where \(A B = 10 a\). A light elastic string, of natural length \(2 a\) and modulus of elasticity \(8 m g\), joins \(P\) to \(A\). Another light elastic string, of natural length \(4 a\) and modulus of elasticity \(16 m g\), joins \(P\) to \(B\). Show that when \(P\) is in equilibrium, \(A P = 4 a\). The particle is held at rest at the point \(C\) between \(A\) and \(B\) on the line \(A B\) where \(A C = 3 a\). The particle is now released.

1. Show that the subsequent motion of \(P\) is simple harmonic with period \(\pi \sqrt { } \left( \frac { a } { 2 g } \right)\).
2. Find the maximum speed of \(P\).