7.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{5a2cf693-d966-4787-8778-ecc8a79a6265-24_191_1136_255_406}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}

A non-uniform beam $A B$, of mass 60 kg and length $8 a$ metres, rests in equilibrium in a horizontal position on two vertical supports. One support is at $C$, where $A C = a$ metres and the other support is at $D$, where $D B = 2 a$ metres, as shown in Figure 2.

The magnitude of the normal reaction between the beam and the support at $D$ is three times the magnitude of the normal reaction between the beam and the support at $C$.

By modelling the beam as a non-uniform rod whose centre of mass is at a distance $x$ metres from $A$,
\begin{enumerate}[label=(\alph*)]
\item find an expression for $x$ in terms of $a$.

A box of mass $M \mathrm {~kg}$ is placed on the beam at $E$, where $A E = 2 a$ metres.\\
The beam remains in equilibrium in a horizontal position.\\
The magnitude of the normal reaction between the beam and the support at $C$ is now equal to the magnitude of the normal reaction between the beam and the support at $D$.

By modelling the box as a particle,
\item find the value of $M$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M1 2021 Q7 [10]}}