9 [In this question you may use the facts that for a uniform solid right circular cone of height \(h\) and base radius \(r\) the volume is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) and the centre of mass is \(\frac { 1 } { 4 } h\) above the base on the line from the centre of the base to the vertex.]
\includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-07_677_963_395_248} The diagram shows the shaded region S bounded by the curve \(\mathrm { y } = \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { x } }\) for \(0 \leqslant x \leqslant 2\), the x -axis, the y -axis, and the line \(\mathrm { y } = \frac { 1 } { 4 } \mathrm { e } ( 6 - \mathrm { x } )\). The line \(\mathrm { y } = \frac { 1 } { 4 } \mathrm { e } ( 6 - \mathrm { x } )\) meets the curve \(\mathrm { y } = \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { x } }\) at the point A with coordinates (2,e).
The region S is rotated through \(2 \pi\) radians about the x -axis to form a uniform solid of revolution T .

1. Show that the x -coordinate of the centre of mass of T is \(\frac { 3 \left( 5 \mathrm { e } ^ { 2 } + 1 \right) } { 7 \mathrm { e } ^ { 2 } - 3 }\). Solid T is freely suspended from A and hangs in equilibrium.
2. Determine the angle between AO , where O is the origin, and the vertical.
   \includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-08_725_1541_251_248} A small toy car runs along a track in a vertical plane.
   The track consists of three sections: a curved section AB , a horizontal section BC which rests on the floor, and a circular section that starts at C with centre O and radius r m . The section BC is tangential to the curved section at B and tangential to the circular section at C , as shown in the diagram. The car, of mass mkg , is placed on the track at A , at a height hm above the floor, and released from rest. The car runs along the track from A to C and enters the circular section at C . It can be assumed that the track does not obstruct the car moving on to the circular section at C . The track is modelled as being smooth, and the car is modelled as a particle P .
3. Show that, while P remains in contact with the circular section of the track, the magnitude of the normal contact force between P and the circular section is
   \(\mathrm { mg } \left( 3 \cos \theta - 2 + \frac { 2 \mathrm {~h} } { \mathrm { r } } \right) \mathrm { N }\),
   where \(\theta\) is the angle between OC and OP .
4. Hence determine, in terms of r , the least possible value of h so that P can complete a vertical circle.
5. Apart from not modelling the car as a particle, state one refinement that would make the model more realistic.
   \includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-09_668_695_258_251} The diagram shows two small identical uniform smooth spheres, A and B, just before A collides with B. Sphere B is at rest on a horizontal table with its centre vertically above the edge of the table. Sphere A is projected vertically upwards so that, just before it collides with B, the speed of A is \(\mathrm { U } \mathrm { m } \mathrm { s } ^ { - 1 }\) and it is in contact with the vertical side of the table. The point of contact of A with the vertical side of the table and the centres of the spheres are in the same vertical plane.
6. Show that on impact the line of centres makes an angle of \(30 ^ { \circ }\) with the vertical. The coefficient of restitution between A and B is \(\frac { 1 } { 2 }\). After the impact B moves freely under gravity.
7. Determine, in terms of U and g , the time taken for B to first return to the table.
   \includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-10_867_1045_255_255} The diagram shows a uniform square lamina ABCD , of weight W and side-length \(a\). The lamina is in equilibrium in a vertical plane that also contains the point O . The vertex A rests on a smooth plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The vertex B rests on a smooth plane inclined at an angle of \(60 ^ { \circ }\) to the horizontal. OA is a line of greatest slope of the plane inclined at \(30 ^ { \circ }\) to the horizontal and OB is a line of greatest slope of the plane inclined at \(60 ^ { \circ }\) to the horizontal. The side AB is inclined at an angle \(\theta\) to the horizontal and the lamina is kept in equilibrium in this position by a clockwise couple of magnitude \(\frac { 1 } { 8 } \mathrm { aW }\).
8. By resolving horizontally and vertically, determine, in terms of W, the magnitude of the normal contact force between the plane and the lamina at B .
9. By taking moments about A , show that \(\theta\) satisfies the equation $$2 ( \sqrt { 3 } + 2 ) \sin \theta - 2 \cos \theta = 1 .$$
10. Verify that \(\theta = 22.4 ^ { \circ }\), correct to 1 decimal place.