A light elastic string has natural length \(2 a\) and modulus of elasticity \(4 m g\). One end of the elastic string is attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the elastic string.
The particle \(P\) hangs freely in equilibrium at the point \(E\), which is vertically below \(O\)
Find the length \(O E\).
Particle \(P\) is now pulled vertically downwards to the point \(A\), where \(O A = 4 a\), and released from rest. The resistance to the motion of \(P\) is a constant force of magnitude \(\frac { 1 } { 4 } m g\).
Find, in terms of \(a\) and \(g\), the speed of \(P\) after it has moved a distance \(a\).
Particle \(P\) is now held at \(O\) Particle \(P\) is released from rest and reaches its maximum speed at the point \(B\). The resistance to the motion of \(P\) is again a constant force of magnitude \(\frac { 1 } { 4 } m g\).