5. Two smooth uniform spheres \(A\) and \(B\) have equal radii. Sphere \(A\) has mass \(m\) and sphere \(B\) has mass \(k m\). The spheres are at rest on a smooth horizontal table. Sphere \(A\) is then projected along the table with speed \(u\) and collides with \(B\). Immediately before the collision, the direction of motion of \(A\) makes an angle of \(60 ^ { \circ }\) with the line joining the centres of the two spheres. The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\).

1. Show that the speed of \(B\) immediately after the collision is \(\frac { 3 u } { 4 ( k + 1 ) }\).
   (6) Immediately after the collision the direction of motion of \(A\) makes an angle arctan \(( 2 \sqrt { 3 } )\) with the direction of motion of \(B\).
2. Show that \(k = \frac { 1 } { 2 }\).
3. Find the loss of kinetic energy due to the collision.
   (4) \section*{6.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{fe647e21-9c4f-4035-b28f-b12a00087692-4_515_1077_301_502}
   \end{figure} A smooth wire with ends \(A\) and \(B\) is in the shape of a semi-circle of radius \(a\). The mid-point of \(A B\) is \(O\). The wire is fixed in a vertical plane and hangs below \(A B\) which is horizontal. A small ring \(R\), of mass \(m \sqrt { 2 }\), is threaded on the wire and is attached to two light inextensible strings. The other end of each string is attached to a particle of mass \(\frac { 3 m } { 2 }\). The particles hang vertically under gravity, as shown in Figure 1.
4. Show that, when the radius \(O R\) makes an angle \(2 \theta\) with the vertical, the potential energy, \(V\), of the system is given by $$V = \sqrt { } 2 m g a ( 3 \cos \theta - \cos 2 \theta ) + \text { constant }$$
5. Find the values of \(\theta\) for which the system is in equilibrium.
6. Determine the stability of the position of equilibrium for which \(\theta > 0\).