3. A light elastic string, of natural length \(l \mathrm {~m}\) and modulus of elasticity 14 N , is hanging vertically with its upper end fixed and a particle of mass \(m \mathrm {~kg}\) attached to the lower end. The particle is initially in equilibrium and air resistance is to be neglected.

1. Find, in terms of \(m , g\) and \(l\), the extension, \(e\), of the string when the particle is in equilibrium. The particle is pulled vertically downwards a further distance from its equilibrium position and released. In its subsequent motion, the string remains taut. Let \(x \mathrm {~m}\) denote the extension of the string from the equilibrium position at time \(t \mathrm {~s}\).
2. 1. Write down, in terms of \(x , m , g\) and \(l\), an expression for the tension in the string.
   2. Hence, show that the particle is moving with Simple Harmonic Motion which satisfies the differential equation, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 14 } { m l } x$$
   3. State the maximum distance that the particle could be pulled vertically downwards from its equilibrium position and still move with Simple Harmonic Motion. Give a reason for your answer.
3. Given that \(m = 0.5 , l = 0.7\) and that the particle is pulled to the position where \(x = 0.2\) before being released,
   1. find the maximum speed of the particle,
   2. determine the time taken for the particle to reach \(x = 0.15\) for the first time.