A car moves round a circular racing track of radius 100 m , which is banked at an angle of \(4 ^ { \circ }\) to the horizontal.
Show that when its speed is \(8.28 \mathrm {~ms} ^ { - 1 }\), there is no sideways force acting on the car.
(4 marks)
When the speed of the car is \(12.5 \mathrm {~ms} ^ { - 1 }\), find the smallest value of the coefficient of friction between the car and the track which will prevent side-slip.
The diagram shows a particle \(P\) of mass \(m \mathrm {~kg}\) moving on the inner surface of a smooth fixed hemispherical bowl of radius \(r \mathrm {~m}\) which is fixed with its axis vertical. \(P\) moves at a constant speed in a horizontal circle, at a depth \(h \mathrm {~m}\) below the top of the bowl.
\includegraphics[max width=\textwidth, alt={}, center]{9699f53e-366b-4064-af9f-89992c5e93b7-2_254_431_786_1528}
Show that the force \(R\) exerted on \(P\) by the bowl has magnitude \(\frac { m g r } { h } \mathrm {~N}\).
Find, in terms of \(g , h\) and \(r\), the constant speed of \(P\).
The bowl is now inverted and \(P\) moves on the smooth outer surface at a height \(h\) above the plane face under the action of a force of magnitude \(m g\) applied tangentially as shown. The reaction of the
\includegraphics[max width=\textwidth, alt={}, center]{9699f53e-366b-4064-af9f-89992c5e93b7-2_209_355_1224_1636}
surface of the sphere on \(P\) now has magnitude \(S \mathrm {~N}\).
Given that \(r = 2 h\), prove that \(S < \frac { 1 } { 6 } R\).