OCR M3 (Mechanics 3) 2011 January

1
\includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-2_476_583_258_781} A ball of mass 0.5 kg is moving with speed \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line when it is struck by a bat. The impulse exerted by the bat has magnitude 15 N s and the ball is deflected through an angle of \(90 ^ { \circ }\) (see diagram). Find

1. the direction of the impulse,
2. the speed of the ball immediately after it is struck.

2 A particle of mass 0.4 kg is attached to a fixed point \(O\) by a light inextensible string of length 0.5 m . The particle is projected horizontally with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the point 0.5 m vertically below \(O\). The particle moves in a complete circle. Find the tension in the string when

1. the string is horizontal,
2. the particle is vertically above \(O\).

3
\includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-2_586_1435_1537_354} A uniform \(\operatorname { rod } P Q\) has weight 72 N . A non-uniform \(\operatorname { rod } Q R\) has weight 54 N and its centre of mass is at \(C\), where \(Q C = 2 C R\). The rods are freely jointed to each other at \(Q\). The rod \(P Q\) is freely jointed to a fixed point of a vertical wall at \(P\) and the rod \(Q R\) rests on horizontal ground at \(R\). The rod \(P Q\) is 2.8 m long and is horizontal. The point \(R\) is 1.44 m below the level of \(P Q\) and 4 m from the wall (see diagram).

1. Find the vertical component of the force exerted by the wall on \(P Q\).
2. Hence show that the normal component of the force exerted by the ground on \(Q R\) is 90 N .
3. Given that the friction at \(R\) is limiting, find the coefficient of friction between the rod \(Q R\) and the ground.

4
\includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-3_497_1157_255_493} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.4 kg and \(B\) has mass 0.3 kg . Immediately before the collision \(A\) is moving with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, where \(\cos \theta = 0.6\), and \(B\) is moving with speed \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution between the spheres is 0.7. Find

1. the speed of \(B\) immediately after the collision,
2. the angle turned through by the direction of motion of \(A\) as a result of the collision.

5 A particle \(P\) of mass 0.05 kg is suspended from a fixed point \(O\) by a light elastic string of natural length 0.5 m and modulus of elasticity 2.45 N .

1. Show that the equilibrium position of \(P\) is 0.6 m below \(O\).
   \(P\) is held at rest at a point 0.675 m vertically below \(O\) and then released. At time \(t \mathrm {~s}\) after \(P\) is released, its downward displacement from the equilibrium position is \(x \mathrm {~m}\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 98 x\).
3. Find the value of \(x\) and the magnitude and direction of the velocity of \(P\) when \(t = 0.2\).

6
\includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-4_638_473_260_836} A particle \(P\), of mass 3.5 kg , is in equilibrium suspended from the top \(A\) of a smooth slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 40 } { 49 }\), by an elastic rope of natural length 4 m and modulus of elasticity 112 N (see diagram). Another particle \(Q\), of mass 0.5 kg , is released from rest at \(A\) and slides freely downwards until it reaches \(P\) and becomes attached to it.

1. Find the value of \(V ^ { 2 }\), where \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(Q\) immediately before it becomes attached to \(P\), and show that the speed of the combined particles, immediately after \(Q\) becomes attached to \(P\), is \(\frac { 1 } { 2 } \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The combined particles slide downwards for a distance of \(X \mathrm {~m}\), before coming instantaneously to rest at \(B\).
2. Show that \(28 X ^ { 2 } - 8 X - 5 = 0\).

7 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) and falls vertically. Air resistance of magnitude \(\frac { v ^ { 2 } } { 2000 } \mathrm {~N}\) acts upwards on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) when it has fallen a distance of \(x \mathrm {~m}\).

1. Show that \(\left( \frac { 400 v } { 3920 - v ^ { 2 } } \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1\).
2. Find \(v ^ { 2 }\) in terms of \(x\) and hence show that \(v ^ { 2 } < 3920\) for all values of \(x\).
3. Find the work done against the air resistance while \(P\) is falling, from \(O\), to the point where its downward acceleration is \(5.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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