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

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