5.

A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre $O$ and radius $5 a$.\\
A uniform rod $A B$, of length $16 a$ and weight $W$, rests in equilibrium on the hemisphere with end $A$ on the ground. The rod rests on the hemisphere at the point $C$, where $A C = 12 a$ and angle $C A O = \alpha$, as shown in Figure 1.

Points $A , C , B$ and $O$ all lie in the same vertical plane.
\begin{enumerate}[label=(\alph*)]
\item Explain why $A O = 13 a$

The normal reaction on the rod at $C$ has magnitude $k W$
\item Show that $k = \frac { 8 } { 13 }$

The resultant force acting on the rod at $A$ has magnitude $R$ and acts upwards at $\theta ^ { \circ }$ to the horizontal.
\item Find
\begin{enumerate}[label=(\roman*)]
\item an expression for $R$ in terms of $W$
\item the value of $\theta$\\
(8)

5

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
  \includegraphics[alt={},max width=\textwidth]{0762451f-b951-4d66-9e01-61ecb7b30d95-16_426_1001_125_475}
\end{center}
\end{figure}

. T a and angle $C A O = \alpha$, as shown in Figure 1.\\
Points $A , C , B$ and $O$ all lie in the same vertical plane.\\
(a) Explain why $A O = 13 a$
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{Edexcel M2 2022 Q5 [12]}}