Question 2 - A-Level Maths

Fig. 2 shows a car of mass 800 kg moving at constant speed in a horizontal circle with centre C and radius 45 m , on a road which is banked at an angle of \(18 ^ { \circ }\) to the horizontal. The forces shown are the weight \(W\) of the car, the normal reaction, \(R\), of the road on the car and the frictional force \(F\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-3_286_970_402_561} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
1. Given that the frictional force is zero, find the speed of the car.
2. Given instead that the speed of the car is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the frictional force and the normal reaction.
One end of a light inextensible string is attached to a fixed point O , and the other end is attached to a particle P of mass \(m \mathrm {~kg}\). Starting with the string taut and P vertically below \(\mathrm { O } , \mathrm { P }\) is set in motion with a horizontal velocity of \(7 \mathrm {~ms} ^ { - 1 }\). It then moves in part of a vertical circle with centre O . The string becomes slack when the speed of P is \(2.8 \mathrm {~ms} ^ { - 1 }\). Find the length of the string. Find also the angle that OP makes with the upward vertical at the instant when the string becomes slack.