5 Fig. 5.1 shows a particle P, of mass 5 kg , and a particle Q, of mass 11 kg , which are attached to the ends of a light, inextensible string. The string is taut and passes over a small smooth pulley fixed to the ceiling.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Fig. 5.1}
  \includegraphics[alt={},max width=\textwidth]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-5_367_707_495_251}
\end{center}
\end{figure}

When a force of magnitude $H \mathrm {~N}$, acting at an angle $\theta$ to the upward vertical, is applied to Q the particles hang in equilibrium, with the part of the string connecting the pulley to Q making an angle of $40 ^ { \circ }$ with the upward vertical. It is given that the force acts in the same vertical plane in which the string lies.
\begin{enumerate}[label=(\alph*)]
\item Determine the values of $H$ and $\theta$.

Particle Q is now removed. The string is instead attached to one end of a uniform beam B of length 3 m and mass 7 kg . The other end of B is in contact with a rough horizontal floor. The situation is shown in Fig. 5.2.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Fig. 5.2}
  \includegraphics[alt={},max width=\textwidth]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-5_504_978_1557_251}
\end{center}
\end{figure}

With B in equilibrium, at an angle $\phi$ to the horizontal, the part of the string connecting the pulley to B makes an angle of $30 ^ { \circ }$ with the upward vertical.

It is given that the string and B lie in the same vertical plane.
\item Determine the smallest possible value for the coefficient of friction between B and the floor.
\item Determine the value of $\phi$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics Minor 2023 Q5 [12]}}