OCR M3 (Mechanics 3)

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\includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-02_283_711_1754_722} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres, and \(B\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.

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\includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-03_462_1109_283_569} Two uniform rods \(A B\) and \(B C\) have weights 64 N and 40 N respectively. The rods are freely jointed to each other at \(B\). The rod \(A B\) is freely jointed to a fixed point on horizontal ground at \(A\) and the rod \(B C\) rests against a vertical wall at \(C\). The rod \(B C\) is 1.8 m long and is horizontal. A particle of weight 9 N is attached to the rod \(B C\) at the point 0.4 m from \(C\). The point \(A\) is 1.2 m below the level of \(B C\) and 3.8 m from the wall (see diagram). The system is in equilibrium.

1. Show that the magnitude of the frictional force at \(C\) is 27 N .
2. Calculate the horizontal and vertical components of the force exerted on \(A B\) at \(B\).
3. Given that friction is limiting at \(C\), find the coefficient of friction between the \(\operatorname { rod } B C\) and the wall.

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\includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-03_598_839_1480_706} One end of a light inextensible string of length 0.5 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.3 kg is attached to the other end of the string. With the string taut and at an angle of \(60 ^ { \circ }\) to the upward vertical, \(P\) is projected with speed \(2 \mathrm {~ms} ^ { - 1 }\) (see diagram). \(P\) begins to move without air resistance in a vertical circle with centre \(O\). When the string makes an angle \(\theta\) with the upward vertical, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 8.9 - 9.8 \cos \theta\).
2. Find the tension in the string in terms of \(\theta\).
3. \(P\) does not move in a complete circle. Calculate the angle through which \(O P\) turns before \(P\) leaves the circular path.