13 Particle $A$, of mass $m \mathrm {~kg}$, lies on the plane $\Pi _ { 1 }$ inclined at an angle of $\tan ^ { - 1 } \frac { 3 } { 4 }$ to the horizontal.\\
Particle $B$, of $4 m \mathrm {~kg}$, lies on the plane $\Pi _ { 2 }$ inclined at an angle of $\tan ^ { - 1 } \frac { 4 } { 3 }$ to the horizontal.\\
The particles are attached to the ends of a light inextensible string which passes over a smooth pulley at $P$.\\
The coefficient of friction between particle $A$ and $\Pi _ { 1 }$ is $\frac { 1 } { 3 }$ and plane $\Pi _ { 2 }$ is smooth.\\
Particle $A$ is initially held at rest such that the string is taut and lies in a line of greatest slope of each plane.

This is shown on the diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-10_398_844_868_306}
\begin{enumerate}[label=(\alph*)]
\item Show that when $A$ is released it accelerates towards the pulley at $\frac { 7 g } { 15 } \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
\item Assuming that $A$ does not reach the pulley, show that it has moved a distance of $\frac { 1 } { 4 } \mathrm {~m}$ when its speed is $\sqrt { \frac { 7 g } { 30 } } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\[0pt]
[2]
\end{enumerate}

\hfill \mbox{\textit{OCR H240/03  Q13 [8]}}