6. A light elastic string, of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\frac { m g } { 2 }\) newtons, has one end fastened to a fixed point \(O\). A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to the other end of the string. \(P\) hangs in equilibrium at the point \(E\), vertically below \(O\), where \(O E = ( l + e ) \mathrm { m }\)

1. Find the numerical value of the ratio \(e : l\).
   \(P\) is now pulled down a further distance \(\frac { 3 l } { 2 } \mathrm {~m}\) from \(E\) and is released from rest.
   In the subsequent motion, the string remains taut. At time \(t \mathrm {~s}\) after being released, \(P\) is at a distance \(x \mathrm {~m}\) below \(E\).
2. Write down a differential equation for the motion of \(P\) and show that the motion is simple harmonic.
3. Write down the period of the motion.
4. Find the speed with which \(P\) first passes through \(E\) again.
5. Show that the time taken by \(P\) after it is released to reach the point \(A\) above \(E\), where $$A E = \frac { 3 l } { 4 } \mathrm {~m} , \text { is } \frac { 2 \pi } { 3 } \sqrt { \frac { 2 l } { g } } \mathrm {~s} .$$