OCR M3 (Mechanics 3) 2007 June

1 A particle \(P\) is moving with simple harmonic motion in a straight line. The period is 6.1 s and the amplitude is 3 m . Calculate, in either order,

1. the maximum speed of \(P\),
2. the distance of \(P\) from the centre of motion when \(P\) has speed \(2.5 \mathrm {~ms} ^ { - 1 }\).

2 A tennis ball of mass 0.057 kg has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball receives an impulse of magnitude 0.6 N s which reduces the speed of the ball to \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Using an impulse-momentum triangle, or otherwise, find the angle the impulse makes with the original direction of motion of the ball.

3 A particle \(P\) of mass 0.2 kg is projected horizontally with speed \(u \mathrm {~ms} ^ { - 1 }\) from a fixed point \(O\) on a smooth horizontal surface. \(P\) moves in a straight line and, at time \(t \mathrm {~s}\) after projection, \(P\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is \(x \mathrm {~m}\) from \(O\). The only force acting on \(P\) has magnitude \(0.4 v ^ { 2 } \mathrm {~N}\) and is directed towards \(O\).

1. Show that \(\frac { 1 } { v } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 2\).
2. Hence show that \(v = u \mathrm { e } ^ { - 2 x }\).
3. Find \(u\), given that \(x = 2\) when \(t = 4\).

4 \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-2_332_995_1375_575} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 4 kg and 3 kg respectively. They are moving on a horizontal surface, and they collide. Immediately before the collision, \(A\) is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the line of centres, where \(\sin \alpha = 0.8\), and \(B\) is moving along the line of centres with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). The coefficient of restitution between the spheres is 0.5 . Find the speed and direction of motion of each sphere after the collision.
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5 \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-3_549_447_253_849} Two uniform rods \(A B\) and \(B C\), each of length 1.4 m and weight 80 N , are freely jointed to each other at \(B\), and \(A B\) is freely jointed to a fixed point at \(A\). They are held in equilibrium with \(A B\) at an angle \(\alpha\) to the horizontal, and \(B C\) at an angle of \(60 ^ { \circ }\) to the horizontal, by a light string, perpendicular to \(B C\), attached to \(C\) (see diagram).

1. By taking moments about \(B\) for \(B C\), calculate the tension in the string. Hence find the horizontal and vertical components of the force acting on \(B C\) at \(B\).
2. Find \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-3_691_665_1370_740} A circus performer \(P\) of mass 80 kg is suspended from a fixed point \(O\) by an elastic rope of natural length 5.25 m and modulus of elasticity \(2058 \mathrm {~N} . P\) is in equilibrium at a point 5 m above a safety net. A second performer \(Q\), also of mass 80 kg , falls freely under gravity from a point above \(P\). \(P\) catches \(Q\) and together they begin to descend vertically with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\) (see diagram). The performers are modelled as particles.

1. Show that, when \(P\) is in equilibrium, \(O P = 7.25 \mathrm {~m}\).
2. Verify that \(P\) and \(Q\) together just reach the safety net.
3. At the lowest point of their motion \(P\) releases \(Q\). Prove that \(P\) subsequently just reaches \(O\).
4. State two additional modelling assumptions made when answering this question.

7 \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-4_588_629_274_758} A particle \(P\) of mass 0.8 kg is attached to a fixed point \(O\) by a light inextensible string of length 0.4 m . A particle \(Q\) is suspended from \(O\) by an identical string. With the string \(O P\) taut and inclined at \(\frac { 1 } { 3 } \pi\) radians to the vertical, \(P\) is projected with speed \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the string so as to strike \(Q\) directly (see diagram). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 7 }\).

1. Calculate the tension in the string immediately after \(P\) is set in motion.
2. Immediately after \(P\) and \(Q\) collide they have equal speeds and are moving in opposite directions. Show that \(Q\) starts to move with speed \(0.15 \mathrm {~ms} ^ { - 1 }\).
3. Prove that before the second collision between \(P\) and \(Q , Q\) is moving with approximate simple harmonic motion.
4. Hence find the time interval between the first and second collisions of \(P\) and \(Q\).