7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05fd9db2-2ff3-4b84-99c2-f348ff567ebd-09_337_1287_228_370}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A beam \(A B\) is supported by two vertical ropes, which are attached to the beam at points \(P\) and \(Q\), where \(A P = 0.3 \mathrm {~m}\) and \(B Q = 0.3 \mathrm {~m}\). The beam is modelled as a uniform rod, of length 2 m and mass 20 kg . The ropes are modelled as light inextensible strings. A gymnast of mass 50 kg hangs on the beam between \(P\) and \(Q\). The gymnast is modelled as a particle attached to the beam at the point \(X\), where \(P X = x \mathrm {~m} , 0 < x < 1.4\) as shown in Figure 2. The beam rests in equilibrium in a horizontal position.
- Show that the tension in the rope attached to the beam at \(P\) is \(( 588 - 350 x ) \mathrm { N }\).
- Find, in terms of \(x\), the tension in the rope attached to the beam at \(Q\).
- Hence find, justifying your answer carefully, the range of values of the tension which could occur in each rope.
Given that the tension in the rope attached at \(Q\) is three times the tension in the rope attached at \(P\),
- find the value of \(x\).
\section*{LU
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