7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a2cf693-d966-4787-8778-ecc8a79a6265-24_191_1136_255_406}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\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 \(A\),
- 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, - find the value of \(M\).