A light elastic string, of natural length \(l\) and modulus of elasticity \(2 m g\), has one end attached to a fixed point \(A\) and the other end attached to a particle \(P\) of mass \(m\). The particle \(P\) hangs in equilibrium at the point \(O\).
Show that \(A O = \frac { 3 l } { 2 }\)
The particle \(P\) is pulled down vertically from \(O\) to the point \(B\), where \(O B = l\), and released from rest.
Air resistance is modelled as being negligible.
Using the model,
prove that \(P\) begins to move with simple harmonic motion about \(O\) with period \(\pi \sqrt { \frac { 2 l } { g } }\)
The particle \(P\) first comes to instantaneous rest at the point \(C\).
Using the model,
find the length \(B C\) in terms of \(l\),
find, in terms of \(l\) and \(g\), the exact time it takes \(P\) to move directly from \(B\) to \(C\).