6. A light elastic string, of natural length \(4 a\) and modulus of elasticity \(8 m g\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\).

1. Find the distance \(A O\).
   (2) The particle is now pulled down to a point \(C\) vertically below \(O\), where \(O C = d\). It is released from rest. In the subsequent motion the string does not become slack.
2. Show that \(P\) moves with simple harmonic motion of period \(\pi \sqrt { \left( \frac { 2 a } { g } \right) }\). The greatest speed of \(P\) during this motion is \(\frac { 1 } { 2 } \sqrt { } ( g a )\).
3. Find \(d\) in terms of \(a\).
   (3) Instead of being pulled down a distance \(d\), the particle is pulled down a distance \(a\). Without further calculation,
4. describe briefly the subsequent motion of \(P\).
   (2)