6. The diagram shows a slide, \(A B C\), at a water park. The shape of the slide may be modelled by two circular arcs, \(A B\) and \(B C\), in the same vertical plane. Arc \(A B\) has radius 7 m and subtends an angle \(\alpha\) at its centre \(D\), where \(\cos \alpha = \frac { 9 } { 10 }\). Arc \(B C\) has radius 5 m and subtends an angle of \(45 ^ { \circ }\) at its centre, \(O\). The straight line \(D B O\) is vertical.
\includegraphics[max width=\textwidth, alt={}, center]{b6b77d92-acb6-4cc3-8328-1899ed4a87cd-7_1171_940_630_552} Users of the slide are required to sit in a rubber ring and are released from rest at point \(A\). A girl decides to use the slide. The combined mass of the girl and the rubber ring is 50 kg .

1. When the rubber ring is at a point \(P\) on the circular arc \(B C\), its speed is \(v \mathrm {~ms} ^ { - 1 }\) and \(O P\) makes an angle \(\theta\) with the upward vertical.
   1. Show that \(v ^ { 2 } = 111.72 - 98 \cos \theta\).
   2. Find, in terms of \(\theta\), the reaction between the rubber ring and the slide at \(P\).
   3. Show that, according to this model, the rubber ring loses contact with the slide before reaching \(C\).
   4. In reality, there will be resistive forces opposing the motion of the rubber ring. Explain how this fact will affect your answer to (iii).
2. Show that the rubber ring will remain in contact with the slide along the arc \(A B\).