OCR M3 (Mechanics 3) 2006 January

1
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_246_693_278_731} A particle \(P\) of mass 0.4 kg moving in a straight line has speed \(8.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). An impulse applied to \(P\) deflects it through \(45 ^ { \circ }\) and reduces its speed to \(5.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Calculate the magnitude and direction of the impulse exerted on \(P\).
\(2 \quad O\) is a fixed point on a horizontal straight line. A particle \(P\) of mass 0.5 kg is released from rest at \(O\). At time \(t\) seconds after release the only force acting on \(P\) has magnitude \(\left( 1 + k t ^ { 2 } \right) \mathrm { N }\) and acts horizontally and away from \(O\) along the line, where \(k\) is a positive constant.

1. Find the speed of \(P\) in terms of \(k\) and \(t\).
2. Given that \(P\) is 2 m from \(O\) when \(t = 1\), find the value of \(k\) and the time taken by \(P\) to travel 20 m from \(O\).

3 A light elastic string has natural length 3 m . One end is attached to a fixed point \(O\) and the other end is attached to a particle of mass 1.6 kg . The particle is released from rest in a position 5 m vertically below \(O\). Air resistance may be neglected.

1. Given that in the subsequent motion the particle just reaches \(O\), show that the modulus of elasticity of the string is 117.6 N .
2. Calculate the speed of the particle when it is 4.5 m below \(O\).

4
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_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.

5
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-3_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.

6
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-3_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.

7
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-4_122_1009_265_571} As shown in the diagram, \(A\) and \(B\) are fixed points on a smooth horizontal table, where \(A B = 3 \mathrm {~m}\). A particle \(Q\) of mass 1.2 kg is attached to \(A\) by a light elastic string of natural length 1 m and modulus of elasticity \(180 \mathrm {~N} . Q\) is attached to \(B\) by a light elastic string of natural length 1.2 m and modulus of elasticity 360 N .

1. Verify that when \(Q\) is in equilibrium \(B Q = 1.5 \mathrm {~m}\).
   \(Q\) is projected towards \(B\) from the equilibrium position with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Subsequently \(Q\) oscillates with simple harmonic motion.
2. Show that the period of the motion is 0.314 s approximately.
3. Show that \(u \leqslant 6\).
4. Given that \(u = 6\), find the time taken for \(Q\) to move from the equilibrium position to a position 1.3 m from \(A\) for the first time.

10 JANUARY 2006 Afternoon
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
List of Formulae (MF1) TIME
1 hour 30 minutes

• Write your name, centre number and candidate number in the spaces provided on the answer booklet.
• Answer all the questions.
• Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate.
• The acceleration due to gravity is denoted by \(\mathrm { g } \mathrm { m } \mathrm { s } ^ { - 2 }\). Unless otherwise instructed, when a numerical value is needed, use \(g = 9.8\).
• You are permitted to use a graphical calculator in this paper.

• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
• Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.
• You are reminded of the need for clear presentation in your answers.