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GCE JUN 2009 : AS, M3: Mechanics 3

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ADVANCED General Certificate of Education 2009 Mathematics assessing Module M3: Mechanics 3 AMM31 Assessment Unit M3 [AMM31] MONDAY 15 JUNE, AFTERNOON TIME 1 hour 30 minutes. INSTRUCTIONS TO CANDIDATES Write your Centre Number and Candidate Number on the Answer Booklet provided. Answer all six questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. You are permitted to use a graphic or scientific calculator in this paper. INFORMATION FOR CANDIDATES The total mark for this paper is 75 Figures in brackets printed down the right-hand side of pages indicate the marks awarded to each question or part question. Answers should include diagrams where appropriate and marks may be awarded for them. Take g = 9.8ms 2, unless specified otherwise. A copy of the Mathematical Formulae and Tables booklet is provided. Throughout the paper the logarithmic notation used is ln z where it is noted that ln z loge z 4165 Answer all six questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 Three particles of mass 1 kg, 2 kg and 3 kg are situated at fixed points whose coordinates, where a 0, are shown in Table 1 below. Table 1 Mass (kg) 1 2 3 Point (1, a2, 3) (4, a, a) (3, 1, 1) The centre of mass of the system is (X, Y, Z ). (i) Find X. (ii) Given that Y = 0, find a. [4] (iii) Hence find Z. 4165 [4] [3] 2 [Turn over 2 A and B are two points on a fixed horizontal beam one metre apart. A particle P of weight 50 N is attached to two light elastic strings. The other ends of the strings are attached to A and B. P hangs in equilibrium with AP stretched to 0.6 m and BP stretched to 0.8 m as shown in Fig. 1 below. A B 1.0 m 0.6m 0.8 m P 50 N Fig. 1 (i) By resolving in the direction of AP, or otherwise, show that the tension in AP is 40 N. [4] The natural length of the string AP is 0.5 m. (ii) Find the modulus of elasticity of AP. [2] The modulus of elasticity of the string BP is 50 N. (iii) Find the natural length of BP. 4165 [4] 3 [Turn over 3 The equation of motion of a particle moving in a straight line with S.H.M. of amplitude a is given by = 2x x where x is the displacement from the centre of oscillation. A particle P moving with S.H.M. has velocity 8 m s 1 when x is 3 m and has velocity 6 m s 1 when x is 4 m. (i) Find a and for the motion of P. [7] A second particle Q also moving with S.H.M. has a maximum speed of 10 m s 1 and an acceleration of maximum magnitude 50 m s 2 (ii) Find a and for the motion of Q. 4 [4] Three forces F1, F2 and F3 newtons move a particle of mass 1 kg from A to B, where AB is measured in metres and AB = 3 6 2 () (i) Given that 4 F1 = 3 12 () show that the work done by F1 is not |F1| |AB| [5] The other two forces F2 and F3 are: F2 = 0 8 6 () 2a a 2a () F3 = (ii) Find, in terms of a, the resultant of the three forces. [1] The particle has speeds of 24 m s 1 at A and 26 m s 1 at B. (iii) Use the Work Energy Principle to find a. 4165 4 [6] [Turn over 5 Fig. 2 below shows part of the River Nagal where the banks are parallel and 100 m apart. The water in the river is flowing at 2 m s 1 5 left bank 100m 2 m s 1 5 P right bank Fig. 2 Jean swims only on backstroke at 1 m s 1 3 She is at the point P on the right bank. Jean wishes to cross the river as quickly as possible. (i) At what angle to the bank should she swim? [1] (ii) Find the time taken to cross the river. [2] (iii) Find how far downstream she will be when she reaches the left bank. [2] Jean has returned to P and now wants to cross so that she lands on the left bank a shorter distance, d metres, downstream. She now swims at an angle to the bank as shown in Fig. 3 below. d 2 m s 1 5 1 ms 1 3 P Fig. 3 20(6 5 cos ) (iv) Show that d = sin (v) Find d when cos = 5 6 [2] (vi) How could you check that this is a minimum value of d ? 4165 [8] [1] 5 [Turn over 6 A particle P of mass 0.25 kg is moving towards the right along a straight horizontal track. The distance of P from a fixed point O on the track is x metres. Two forces act on P: a constant frictional force of 2 N and a variable force F newtons, towards the right, where F = 20 6 x 0 x 16 The graph of the resultant R of the two forces is shown in Fig. 4 below. R A B O x Fig. 4 The intercepts on the axes are A and B. (i) State the coordinates of the points A and B. (ii) Find the work done by the resultant force R for 0 [3] x 16 [5] The speed of P at O is 12 m s 1 (iii) Use the Work Energy Principle to find the speed of the particle when x = 16 [3] (iv) Find the maximum speed reached by the particle during this motion. [4] THIS IS THE END OF THE QUESTION PAPER 4165 6 [Turn over AMM3S9 4165 7 [Turn over S 12/07 938-014-1 [Turn over

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Additional Info : Gce Mathematics June 2009 Assessment Unit M1 Module M1 : Mechanics 3
Tags : General Certificate of Education, A Level and AS Level, uk, council for the curriculum examinations and assessment, gce exam papers, gce a level and as level exam papers , gce past questions and answer, gce past question papers, ccea gce past papers, gce ccea past papers

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