3 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and each has mass \(m\). Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\). Sphere \(B\) is initially at a distance \(d\) from a fixed smooth vertical wall which is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 3 }\).

1. Show that the speed of \(B\) after its collision with the wall is \(\frac { 5 } { 18 } u\).
2. Find the distance of \(B\) from the wall when it collides with \(A\) for the second time.
   \includegraphics[max width=\textwidth, alt={}, center]{b10d2991-abff-4d2b-b470-1df844d1c7ee-08_743_673_258_737} A uniform rod \(A B\) of length \(3 a\) and weight \(W\) is freely hinged to a fixed point at the end \(A\). The end \(B\) is below the level of \(A\) and is attached to one end of a light elastic string of natural length 4a. The other end of the string is attached to a point \(O\) on a vertical wall. The horizontal distance between \(A\) and the wall is \(5 a\). The string and the rod make angles \(\theta\) and \(2 \theta\) respectively with the horizontal (see diagram). The system is in equilibrium with the rod and the string in the same vertical plane. It is given that \(\sin \theta = \frac { 3 } { 5 }\) and you may use the fact that \(\cos 2 \theta = \frac { 7 } { 25 }\).