11 Two uniform small smooth spheres A and B have equal radii and equal masses. The spheres are on a smooth horizontal surface. Sphere A is moving at an acute angle \(\alpha\) to the line of centres, when it collides with B, which is stationary. After the impact A is moving at an acute angle \(\beta\) to the line of centres. The coefficient of restitution between A and B is \(\frac { 1 } { 3 }\).

1. Show that \(\tan \beta = 3 \tan \alpha\).
2. Explain why the assumption that the contact between the spheres is smooth is needed in answering part (a). It is given that A is deflected through an angle \(\gamma\).
3. Determine, in terms of \(\alpha\), an expression for \(\tan \gamma\).
4. Determine the maximum value of \(\gamma\). You do not need to justify that this value is a maximum. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-09_488_903_264_258} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows a hemispherical bowl. The rim of this bowl is a circle with centre O and radius \(r\). The bowl is fixed with its rim horizontal and uppermost. A particle P , of mass \(m\), is connected by a light inextensible string of length \(l\) to the lowest point A on the bowl and describes a horizontal circle with constant angular speed \(\omega\) on the smooth inner surface of the bowl. The string is taut, and AP makes an angle \(\alpha\) with the vertical.
5. Show that the normal contact force between P and the bowl is of magnitude \(m g + 2 m r \omega ^ { 2 } \cos ^ { 2 } \alpha\).
6. Deduce that \(g < r \omega ^ { 2 } \left( k _ { 1 } + k _ { 2 } \cos ^ { 2 } \alpha \right)\), stating the value of the constants \(k _ { 1 }\) and \(k _ { 2 }\).