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GCE JUN 2006 : AS, M3: Mechanics 3

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ADVANCED General Certificate of Education 2006 Mathematics assessing Module M3: Mechanics 3 AMM31 Assessment Unit M3 [AMM31] THURSDAY 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 a 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 = 9.8 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 AMM3S6 994 BLANK PAGE AMM3S6 994 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 A particle P of mass 0.5 kg is projected with speed 2 m s 1 from a point O across a horizontal surface. The distance of P from O at time t seconds is x metres. When x = d, P is at rest. (i) Find the kinetic energy lost by P. [2] P is subject to a variable resistive force F of magnitude e2x N. (ii) Find, in terms of d, the work done by F when x = d (iii) Given that the surface is smooth find the value of d. 2 [4] [3] Shown in Fig. 1 below is a symmetrical uniform lamina ABCD formed by cutting a triangular lamina ADC of height 2L metres from the triangular lamina ABC of height 3L metres. AC = 4L metres. B D 3L 2L A 4L C Fig. 1 (i) Find the distance of the centre of mass of the lamina ABCD from AC. [7] The lamina is freely suspended from C. (ii) Find the angle CA makes with the vertical. AMM3S6 994 3 [4] [Turn over 3 Shown in Fig. 2 below is a particle A, mass 4m kg, resting on a smooth plane inclined at 30 to the horizontal. A is attached to one end of a light inextensible string which passes over a smooth pulley B fixed at the top of the plane. The other end of this string is attached to a second particle C, mass M1 kg. One end of a second light inextensible string is attached to C. This string passes over a fixed smooth pulley D and a particle E, mass M2 kg, is attached to its other end. The section of string from B to C makes an angle of 60 with the vertical. The section of string from C to D makes an angle of with the vertical, where tan = 3 2 D B 60 A C E 30 Fig. 2 The system is in equilibrium and A, B, C, D and E are in the same vertical plane. (i) Draw a diagram showing all the forces acting on the three particles. (ii) Find M1 and M2 in terms of m. AMM3S6 994 [3] [10] 4 [Turn over 4 One end of a light elastic spring, modulus of elasticity 2.5mg N, is attached to a fixed point O. The other end is attached to a particle P of mass m kg which is held vertically below O with OP = 0.8 m. P is released from rest and is observed to move vertically in simple harmonic motion between the levels of point A and point B where A and B are 0.6 m and 0.8 m vertically below O as shown in Fig. 3 below. O 0.6 A 0.2 P B Fig. 3 (i) Write down the distance of the centre of motion from O. [1] (ii) Show that the natural length l of the spring is 0.5 m. [4] The periodic time of the oscillations is 2 2l seconds. 5g (iii) Find the distance travelled by P in the first second after it is released from rest. AMM3S6 994 5 [8] [Turn over 5 A smooth rod AB is bent at its midpoint O with AOB = 90 AOB is fixed in a vertical plane with AO vertical. E is the point on AO such that OE = 0.1 m as shown in Fig. 4 below. One end of a light elastic string of modulus g newtons and natural length 0.1 m is attached to E. The other end of the string is attached to a smooth ring R of mass 0.1 kg which is fed onto BO. The ring is projected from O along OB with a speed of 6 m s 1 A E 0.1 O R B Fig. 4 T is the tension in the string when the speed of the ring is 1 m s 1 (i) Show that T = 7 N. [7] The bent rod AOB is now rotated clockwise and fixed with OA inclined at an angle of above the horizontal as shown in Fig. 5 below. A E O R B Fig. 5 The ring is now projected from O along OB but with a speed of 0.5 m s 1 When the length of the string ER is 0.125 m the velocity of the ring is 1 m s 1 (ii) Find . AMM3S6 994 [5] 6 [Turn over 6 Fig. 6 below shows three forklift trucks T1, T2 and T3 in a rectangular builder s yard ABCD. A B T2 T1 T3 D C Fig. 6 ABCD is 8L metres by 6L metres. The point O is the centre of the yard, M is the point on BA such that BM = L metres and N is the point on CD such that CN = L metres as shown in Fig. 7 below. L A M B 6L O D N 7L L C Fig. 7 At time t = 0: T1 is at the point O and moving with a constant speed of 2V m s 1 in the direction OB; T2 is at the point M and moving with a constant speed of 5V m s 1 in the direction MA; T3 is at the point N and moving with a constant speed of 5V m s 1 in the direction ND. (i) Find the velocity of T1 relative to T2 [7] (ii) Find, in terms of L and V, the time that elapses from when T1 is closest to T3 until T1 is closest to T2 AMM3S6 994 7 [10] [Turn over S 6/05 1400 302507(57) [Turn over

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Additional Info : Gce Mathematics June 2006 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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