A light elastic string, of natural length \(3 a\) and modulus of elasticity \(6 m g\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(2 m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\), vertically below \(A\).
Find the distance \(A O\).
The particle is now raised to point \(C\) vertically below \(A\), where \(A C > 3 a\), and is released from rest.
Show that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { } \left( \frac { a } { g } \right)\).
It is given that \(O C = \frac { 1 } { 4 } a\).
Find the greatest speed of \(P\) during the motion.
The point \(D\) is vertically above \(O\) and \(O D = \frac { 1 } { 8 } a\). The string is cut as \(P\) passes through \(D\), moving upwards.
Find the greatest height of \(P\) above \(O\) in the subsequent motion.