\includegraphics{figure_10}

Particles $A$ and $B$ of masses $2m$ and $m$, respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley $P$. The particle $A$ rests in equilibrium on a rough plane inclined at an angle $\alpha$ to the horizontal, where $\alpha \leqslant 45°$ and $B$ is above the plane. The vertical plane defined by $APB$ contains a line of greatest slope of the plane, and $PA$ is inclined at angle $2\alpha$ to the horizontal (see diagram).

\begin{enumerate}[label=(\alph*)]
\item Show that the normal reaction R between A and the plane is mg(2$\cos\alpha - \sin\alpha$). [3]
\item Show that R $\geqslant \frac{1}{2}mg\sqrt{2}$. [3]
\end{enumerate}

The coefficient of friction between $A$ and the plane is $\mu$. The particle is about to slip down the plane.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Show that $0.5 < \tan\alpha \leqslant 1$. [3]
\item Express $\mu$ as a function of $\tan\alpha$ and deduce its maximum value as $\alpha$ varies. [3]
\end{enumerate}

\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2019 Q10 [12]}}