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.
\begin{enumerate}[label=(\alph*)]
\item 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 $t \mathrm {~s}$.
\item \begin{enumerate}[label=(\roman*)]
\item Write down, in terms of $x , m , g$ and $l$, an expression for the tension in the string.
\item Hence, show that the particle is moving with Simple Harmonic Motion which satisfies the differential equation,

$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 14 } { m l } x$$
\item 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.
\end{enumerate}\item Given that $m = 0.5 , l = 0.7$ and that the particle is pulled to the position where $x = 0.2$ before being released,
\begin{enumerate}[label=(\roman*)]
\item find the maximum speed of the particle,
\item determine the time taken for the particle to reach $x = 0.15$ for the first time.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{WJEC Further Unit 6 2019 Q3 [14]}}