7. A particle of mass \(m \mathrm {~kg}\) is attached to one end of an elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). The other end of the string is attached to a fixed point \(O\). The particle hangs in equilibrium at a point \(C\).

1. 1. Prove that if the particle is slightly displaced in a vertical direction, it will perform simple harmonic motion about \(C\).
   2. Find the period, \(T \mathrm {~s}\), of the motion in terms of \(l , m\) and \(\lambda\).
   3. Explain the significance of the term 'slightly' as used in (i) above. When an additional mass \(M\) is attached to the particle, it is found that the system oscillates about a point \(D\), at a distance \(d\) below \(C\), with period \(T _ { 1 } \mathrm {~s}\).
2. 1. Write down an expression for \(T _ { 1 }\) in terms of \(l , m , M\) and \(\lambda\).
   2. Hence show that \(T _ { 1 } ^ { 2 } - T ^ { 2 } = \frac { 4 \pi ^ { 2 } d } { g }\).