6 A bungee jumper of mass 70 kg is joined to a fixed point \(O\) by a light elastic rope of natural length 30 m and modulus of elasticity 1470 N . The jumper starts from rest at \(O\) and falls vertically. The jumper is modelled as a particle and air resistance is ignored.

1. Find the distance fallen by the jumper when maximum speed is reached.
2. Show that this maximum speed is \(26.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
3. Find the extension of the rope when the jumper is at the lowest position. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{09d3e8ca-0062-4f62-8453-7acaff591db5-4_543_616_310_301} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{09d3e8ca-0062-4f62-8453-7acaff591db5-4_668_709_267_1135} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A smooth horizontal cylinder of radius 0.6 m is fixed with its axis horizontal and passing through a fixed point \(O\). A light inextensible string of length \(0.6 \pi \mathrm {~m}\) has particles \(P\) and \(Q\), of masses 0.3 kg and 0.4 kg respectively, attached at its ends. The string passes over the cylinder and is held at rest with \(P , O\) and \(Q\) in a straight horizontal line (see Fig. 1). The string is released and \(Q\) begins to descend. When the line \(O P\) makes an angle \(\theta\) radians, \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), with the horizontal, the particles have speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2).
4. By considering the total energy of the system, or otherwise, show that $$v ^ { 2 } = 6.72 \theta - 5.04 \sin \theta .$$
5. Show that the magnitude of the contact force between \(P\) and the cylinder is $$( 5.46 \sin \theta - 3.36 \theta ) \text { newtons. }$$ Hence find the value of \(\theta\) for which the magnitude of the contact force is greatest.
6. Find the transverse component of the acceleration of \(P\) in terms of \(\theta\).