7. A non-uniform rod \(A B\) has length 6 m and mass 8 kg . The rod rests in equilibrium, in a horizontal position, on two smooth supports at \(C\) and at \(D\), where \(A C = 1 \mathrm {~m}\) and \(D B = 1 \mathrm {~m}\), as shown in Figure 3. The magnitude of the reaction between the rod and the support at \(D\) is twice the magnitude of the reaction between the rod and the support at \(C\). The centre of mass of the rod is at \(G\), where \(A G = x \mathrm {~m}\).

1. Show that \(x = \frac { 11 } { 3 }\). The support at \(C\) is moved to the point \(F\) on the rod, where \(A F = 2 \mathrm {~m}\). A particle of mass 3 kg is placed on the rod at \(A\). The rod remains horizontal and in equilibrium. The magnitude of the reaction between the rod and the support at \(D\) is \(k\) times the magnitude of the reaction between the rod and the support at \(F\).
2. Find the value of \(k\).
   7. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{04b73f81-3316-4f26-ad98-a7be3a4b738f-20_223_1262_127_338}
   \end{figure}