7. In this question you may use, without proof, the formula for the centre of mass of a uniform sector of a circle, as given in the formulae book.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-20_444_625_354_662}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}

The uniform lamina $A B C D E$, shown shaded in Figure 3, is formed by joining a rectangle to a sector of a circle.

\begin{itemize}
  \item The rectangle $A B C E$ has $A B = E C = a$ and $A E = B C = d$
  \item The sector $C D E$ has centre $C$ and radius $a$
  \item Angle $E C D = \frac { \pi } { 3 }$ radians
\end{itemize}

The centre of mass of the lamina lies on EC.
\begin{enumerate}[label=(\alph*)]
\item Show that $a = \sqrt { 3 } d$

The lamina is freely suspended from $B$ and hangs in equilibrium with $B C$ at an angle $\beta$ radians to the downward vertical.
\item Find the value of $\beta$
\end{enumerate}

\hfill \mbox{\textit{Edexcel M2 2021 Q7 [11]}}