\includegraphics{figure_5}

A small ring $R$ of mass $m$ is free to slide on a smooth straight wire which is fixed at an angle of $30°$ to the horizontal. The ring is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $\lambda$. The other end of the string is attached to a fixed point $A$ of the wire, as shown in Fig. 5. The ring rests in equilibrium at the point $B$, where $AB = \frac{a}{2}$.

\begin{enumerate}[label=(\alph*)]
\item Show that $\lambda = 4mg$. [3]
\end{enumerate}

The ring is pulled down to the point $C$, where $BC = \frac{1}{4}a$, and released from rest. At time $t$ after $R$ is released the extension of the string is $(\frac{1}{4}a + x)$.

\begin{enumerate}[label=(\alph*)]\setcounter{enumi}{1}
\item Obtain a differential equation for the motion of $R$ while the string remains taut, and show that it represents simple harmonic motion with period $\pi\sqrt{\left(\frac{a}{g}\right)}$. [6]
\item Find, in terms of $g$, the greatest magnitude of the acceleration of $R$ while the string remains taut. [2]
\item Find, in terms of $a$ and $g$, the time taken for $R$ to move from the point at which it first reaches maximum speed to the point where the string becomes slack for the first time. [5]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M3 2001 Q7 [16]}}