2
\includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-2_460_725_731_708} Two uniform rods, \(A B\) and \(B C\), are freely jointed to each other at \(B\), and \(C\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(A\) resting on a rough horizontal surface. This surface is 1.5 m below the level of \(B\) and the horizontal distance between \(A\) and \(B\) is 3 m (see diagram). The weight of \(A B\) is 80 N and the frictional force acting on \(A B\) at \(A\) is 14 N .

1. Write down the horizontal component of the force acting on \(A B\) at \(B\) and show that the vertical component of this force is 33 N upwards.
2. Given that the force acting on \(B C\) at \(C\) has magnitude 50 N , find the weight of \(B C\).
   \includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-2_421_949_1793_598} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 4 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision both spheres have speed \(3 \mathrm {~ms} ^ { - 1 }\). The spheres are moving in opposite directions, each at \(60 ^ { \circ }\) to the line of centres (see diagram). After the collision \(A\) moves in a direction perpendicular to the line of centres.
3. Show that the speed of \(B\) is unchanged as a result of the collision, and find the angle that the new direction of motion of \(B\) makes with the line of centres.
4. Find the coefficient of restitution between the spheres.