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GCE JUN 2008 : AS, M4: Mechanics 4

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ADVANCED General Certificate of Education 2008 Mathematics assessing Module M4: Mechanics 4 AMM41 Assessment Unit M4 [AMM41] MONDAY 16 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 AMM4S8 3040 BLANK PAGE AMM4S8 3040 2 [Turn over Answer all six questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 Fig. 1 below shows a framework of four light pin jointed rods AB, BC, BD and DC. The rods all lie in the same vertical plane. The lengths of the rods are: AB = 3 m, BC = BD = 5m, DC = 6 m The framework is freely hinged to a vertical wall at A and D. The rods AB and DC are horizontal. A mass of 20kg is suspended from C. 6 D 5 Wall C 5 20 kg 3 A B Fig. 1 (i) Explain briefly why the reaction of the hinge at A must be horizontal. [1] (ii) Find the magnitudes of the reactions at A and D. [8] (iii) Find the force in the rod BC, identifying it as a thrust or a tension. [3] AMM4S8 3040 3 [Turn over 2 The mass of a major planet is M kg. One of its satellites has mass m kg and moves in a circle of radius r metres around the planet. (i) Show that the period, T, of the satellite is r3 GM T = 2 where G is the universal gravitational constant. [6] The mass of Saturn is 5.688 1026 kg. Its largest satellite, Titan, moves in a circle of radius of 1.222 109 m. (ii) Hence find, in days, the period of Titan. Take G as 6.673 10 11 m3 kg 1 s 2 3 [3] The moment of inertia I of a body is used in rotational dynamics. The kinetic energy E of a body when rotating with angular velocity is E= I 2 2 (i) Given that [ ] = [T] 1, show that the dimensions of I are [M][L]2 [4] A rod of length l and moment of inertia I is freely pivoted about its upper end and hangs vertically. When the lower end of the rod is struck by an impulse J the rod just makes complete circles. (ii) Find x, y and z if J = kI xgyl z where k is a dimensionless constant. AMM4S8 3040 [7] 4 [Turn over 4 A particle A of mass 1kg is moving with speed 4m s 1 directly towards a second particle B of mass 3kg which is at rest on a smooth horizontal surface. The coefficient of restitution between the particles is 0.6 After the particles collide the velocity of A is v1 and that of B is v2 (i) Find v1 and v2 [7] (ii) Find the total loss in kinetic energy due to the collision. [5] 5 y y = 2+ x 2 O 4 x Fig. 2 Fig. 2 above shows the face of a steel plate having a density of kg m 2 The plate forms part of a bulkhead for the cargo ship Normandic. Take the plate to be a lamina bounded by the x- and y-axes, the line x = 4 and the curve y = 2 + x Both x and y are measured in metres. Find the coordinates of the plate s centre of mass, relative to the origin O. AMM4S8 3040 5 [13] [Turn over 6 David is riding his motorcycle at v metres per second in a horizontal circle of radius r metres as he negotiates a bend banked at to the horizontal, as shown in Fig. 3 below. Fig. 3 The normal reaction between the road and the motorcycle is N newtons and the frictional force is F newtons. (i) By modelling David and his motorcycle as a particle show that v 2 N tan + F = rg N F tan [9] The coefficient of friction between the tyres of the motorcycle and the road is 0.7 Take = 10 and r = 80 (ii) Find the maximum speed at which David can negotiate the bend. AMM4S8 3040 6 [4] [Turn over David is negotiating the bend at maximum speed. He needs to incline his motorcycle at from the normal to the road surface as shown in Fig. 4 below. G C Fig. 4 The centre of mass of rider and motorcycle is G. The contact with the road surface is at C. The distance GC is h metres. (iii) By taking moments, show that if David s motorcycle does not overturn, then is approximately 35 AMM4S8 3040 7 [5] [Turn over S 12/06 530-090-1 [Turn over

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Additional Info : Gce Mathematics June 2008 Assessment Unit M2 Module M2 : Mechanics 4
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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