4. \begin{figure}[h] \end{figure} A non-uniform beam \(A B\) has length 8 m and mass \(M \mathrm {~kg}\). The centre of mass of the beam is \(d\) metres from \(A\). The beam is supported in equilibrium in a horizontal position by two vertical light ropes. One rope is attached to the beam at \(C\), where \(A C = 2.5 \mathrm {~m}\) and the other rope is attached to the beam at \(D\), where \(D B = 2 \mathrm {~m}\), as shown in Figure 2. A gymnast, of mass 64 kg , stands on the beam at the point \(X\), where \(A X = 1.875 \mathrm {~m}\), and the beam remains in equilibrium in a horizontal position but is now on the point of tilting about \(C\). The gymnast then dismounts from the beam. A second gymnast, of mass 48 kg , now stands on the beam at the point \(Y\), where \(Y B = 0.5 \mathrm {~m}\), and the beam remains in equilibrium in a horizontal position but is now on the point of tilting about \(D\). The beam is modelled as a non-uniform rod and the gymnasts are modelled as particles. Find the value of \(M\).