6. A light spring of natural length \(L\) has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The particle is moving vertically. As it passes through the point \(B\) below \(A\), where \(A B = L\), its speed is \(\sqrt { } ( 2 g L )\). The particle comes to instantaneous rest at a point \(C , 4 L\) below \(A\).

1. Show that the modulus of elasticity of the spring is \(\frac { 8 m g } { 9 }\). At the point \(D\) the tension in the spring is \(m g\).
2. Show that \(P\) performs simple harmonic motion with centre \(D\).
3. Find, in terms of \(L\) and \(g\),
   1. the period of the simple harmonic motion,
   2. the maximum speed of \(P\).
      (5)