6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b896c631-00a0-46c5-bce9-16d65f6e3095-11_600_969_127_491}
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\caption{Figure 3}
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A non-uniform beam \(A D\) has weight \(W\) newtons and length 4 m . It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. The ropes are attached to two points \(B\) and \(C\) on the beam, where \(A B = 1 \mathrm {~m}\) and \(C D = 1 \mathrm {~m}\), as shown in Figure 3. The tension in the rope attached to \(C\) is double the tension in the rope attached to \(B\). The beam is modelled as a rod and the ropes are modelled as light inextensible strings.
- Find the distance of the centre of mass of the beam from \(A\).
A small load of weight \(k W\) newtons is attached to the beam at \(D\). The beam remains in equilibrium in a horizontal position. The load is modelled as a particle.
Find
- an expression for the tension in the rope attached to \(B\), giving your answer in terms of \(k\) and \(W\),
- the set of possible values of \(k\) for which both ropes remain taut.