4 Fig. 4 shows a uniform beam of length $2 a$ and weight $W$ leaning against a block of weight $2 W$ which is on a rough horizontal plane. The beam is freely hinged to the plane at O and makes an angle $\theta$ with the horizontal. The contact between the beam and the block is smooth. The beam and block are in equilibrium, and it may be assumed that the block does not topple.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-4_350_830_461_246}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}

Let

\begin{itemize}
  \item $S$ be the normal contact force between the beam and the block,
  \item $R$ be the normal contact force between the plane and the block,
  \item $F$ be the frictional force between the plane and the block.
\end{itemize}

Partially complete force diagrams showing the beam and the block separately are given in the Printed Answer Booklet.
\begin{enumerate}[label=(\alph*)]
\item Add the forces listed above to these diagrams.

It is given that $\theta = 30 ^ { \circ }$.
\item Determine the minimum possible value of the coefficient of friction between the block and the plane.
\item In each case explain, with justification, how your answer to part (b) would change (assuming the rest of the system remained unchanged) if
\begin{enumerate}[label=(\roman*)]
\item $\theta < 30 ^ { \circ }$,
\item the contact between the beam and the block were rough.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics A AS 2020 Q4 [9]}}