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GCE JUN 2006 : AS, M4: Mechanics 4

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ADVANCED General Certificate of Education 2006 Mathematics assessing Module M4: Mechanics 4 AMM41 Assessment Unit M4 [AMM41] MONDAY 19 JUNE, MORNING 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 a scientific calculator in this paper. 1 27/6/05EA 2 24/10/05ES 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 = 9.8 m s 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 log e z AMM4S6 993 Answer all six questions. Show clearly the full development of your answers. Answers should be given to three significant figures unless otherwise stated. 1 Shown in Fig. 1 below is a gate modelled by seven light pin-jointed rods forming a framework ABCDE. ABCD is a rectangle and E is the midpoint of DA. Angle ABE = angle ECD = 30 The gate is hinged at A and D and rests in a vertical plane. A mass of m kg is placed on top of the gate at B. A H1 30 B E DH 2 30 C Fig. 1 The horizontal components of the reactions at the hinges A and D on the gate are H1 and H2 [4] (ii) Find, in terms of m and g, the force in the rod BC. [3] 1 27/6/05EA 2 24/10/05ES (i) Find H1 and H2 in terms of m and g. AMM4S6 993 2 [Turn over 2 A small planet P of mass m orbits a much larger planet of mass M. The effective potential U of P can be given by U= H2 GmM 2 r 2 mr where H is the angular momentum of P, r is the distance of P from the larger planet and G is the gravitational constant. G has dimensions of [M 1L3T 2] (i) By using the equation above find the dimensions of (a) U (b) H [4] The length l of the semi latus rectum of this orbit varies with H, m, M and G as follows l= H x m yG z M (ii) Find the values of x, y and z 3 [7] A bead B of mass m kg is threaded onto a smooth hoop, radius r metres and centre O, which is fixed in a vertical plane. The bead moves in a vertical circle around the hoop. At rg m s 1 and the magnitude of the normal the top of the hoop the bead has a speed of 2 reaction acting upwards on the bead due to the hoop is N0 (i) Find N0 [4] When OB makes an angle of , 0 < < , with the upward vertical the magnitude of the 2 normal reaction between the bead and the hoop is again found to be N0 (ii) Find the value of . 1 27/6/05EA 2 24/10/05ES [8] AMM4S6 993 3 [Turn over 4 (i) Show that the centre of mass, G, of a uniform solid hemisphere of radius r lies on 3 its axis of symmetry r from its base. 8 [6] AB is a diameter of the circular base of the hemisphere. B rests on a rough horizontal surface and the hemisphere is held in limiting equilibrium with AB inclined at an angle to the horizontal by a horizontal force F applied at A as shown in Fig. 2 below. AB and F are in the same vertical plane. The coefficient of friction between the ground and the hemisphere is 0.5 F A G B Fig. 2 [2] (iii) Find . [8] 1 27/6/05EA 2 24/10/05ES (ii) Draw a diagram showing all the external forces acting on the hemisphere. AMM4S6 993 4 [Turn over 5 A car of mass 1600 kg is travelling around a bend banked at an angle of tan 1 5 12 to the horizontal. The car moves in a horizontal circle of radius 80 metres and is on the point of slipping outwards. The coefficient of friction between the road and the wheels is 0.5 (i) Find the total normal reaction of the road on the car. [5] (ii) Find the speed of the car. [4] The centre of mass, G, of the car is h metres above the road. The inner and outer wheels are 4h metres apart as shown in Fig. 3 below. G h 2h 2h Fig. 3 [5] 1 27/6/05EA 2 24/10/05ES (iii) By taking moments about G, or otherwise, find the total normal reaction between the road and the inner wheels. AMM4S6 993 5 [Turn over 6 A smooth sphere A of mass m1 kg moving at u1 m s 1 collides directly with a smooth sphere B of mass m2 kg moving at u2 m s 1 (u1 > u2). Both spheres are moving on a smooth horizontal surface and collide at time t = 0 s (i) Show that u1 m2 3 = = u2 m1 2 [8] At the time of collision B is d metres from a smooth vertical wall. B collides directly with this wall and subsequently collides again with A when t = T 1 The coefficient of restitution between B and the wall is 4 (ii) By modelling A and B as particles, express T in terms of d and u2 [7] 1 27/6/05EA 2 24/10/05ES THIS IS THE END OF THE QUESTION PAPER AMM4S6 993 6 [Turn over 1 27/6/05EA 2 24/10/05ES 2 24/10/05ES 1 27/6/05EA S 6/05 800 302507(66) [Turn over

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Additional Info : Gce Mathematics June 2006 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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