\includegraphics{figure_3}

A particle $P$ of mass 2 kg is attached to one end of a light inextensible string. A particle $Q$ of mass 5 kg is attached to the other end of the string. The string passes over a small smooth light pulley. The pulley is fixed at a point on the intersection of a rough horizontal table and a fixed smooth inclined plane. The string lies along the table and also lies in a vertical plane which contains a line of greatest slope of the inclined plane. This plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac{3}{4}$. Particle $P$ is at rest on the table, a distance $d$ metres from the pulley. Particle $Q$ is on the inclined plane with the string taut, as shown in Figure 3. The coefficient of friction between $P$ and the table is $\frac{1}{4}$.

The system is released from rest and $P$ slides along the table towards the pulley.

Assuming that $P$ has not reached the pulley and that $Q$ remains on the inclined plane,

\begin{enumerate}[label=(\alph*)]
\item write down an equation of motion for $P$, [2]

\item write down an equation of motion for $Q$, [2]

\item 
\begin{enumerate}[label=(\roman*)]
\item find the acceleration of $P$,
\item find the tension in the string. [5]
\end{enumerate}
\end{enumerate}

When $P$ has moved a distance 0.5 m from its initial position, the string breaks. Given that $P$ comes to rest just as it reaches the pulley,

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item find the value of $d$. [7]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M1 2016 Q7 [16]}}