FURTHER MATHEMATICS Paper 3 Mechanics

FURTHER MATHEMATICS Paper 3 Mechanics Wednesday 14 June 2023 Afternoon Time allowed: 2 hours Materials  You must have the AQA Formulae and statistical tables booklet for A‑level Mathematics and A‑level Further Mathematics.  You should have a graphical or scientific calculator that meets the requirements of the specification.  You must ensure you have the other optional Question Paper/Answer Book for which you are entered (either Discrete or Statistics). You will have 2 hours to complete both papers. Instructions  Use black ink or black ball‑point pen. Pencil should only be used for drawing.  Fill in the boxes at the top of this page.  Answer all questions.  You must answer each question in the space provided for that question. If you require extra space for your answer(s), use the lined pages at the end of this book. Write the question number against your answer(s).  Do not write outside the box around each page or on blank pages.  Show all necessary working; otherwise marks for method may be lost.  Do all rough work in this book. Cross through any work that you do not want to be marked. Information  The marks for questions are shown in brackets.  The maximum mark for this paper is 50. Advice  Unless stated otherwise, you may quote formulae, without proof, from the booklet.  You do not necessarily need to use all the space provided. PB/KL/Jun23/E5 7367/3M Answer all questions in the spaces provided. 1 State the dimensions of power. Circle your answer. ML2T —3 ML3T —3 ML3T —2 ML2T —2 [1 mark] box 2 The force (3i þ 4j) N acts at the point with coordinates (0, 2) The unit vectors i and j are directed along the x-axis and the y-axis respectively. Calculate the magnitude of the moment of this force about the origin. Circle your answer. [1 mark] 6Nm 8 Nm 10 Nm 14 N m 3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, ABCD, has mass 4 kg, width 8 cm and length 20 cm The disc is fixed to the lamina to form a composite body as shown in the diagram below. A B outside the box 8 cm D C 20 cm The sides AB, AD and CD are tangents to the disc. Calculate the distance of the centre of mass of the composite body from AD Circle your answer. [1 mark] 4 cm 5.6 cm 6.4 cm 8.8 cm 4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres. The car has a constant speed of 12 m s—1 on the bend. Calculate the magnitude of the resultant force acting on the car. [2 marks] Turn over 5 A region bounded by the curve with equation y 4 x2 shown below. , the x-axis and the y-axis is outside the box y O x The region is rotated through 360° around the x-axis to create a uniform solid. 5 (a) Show that the distance of the centre of mass of the solid from the circular face is 5 8 [5 marks] box 5 (b) The solid is suspended in equilibrium from a point on the edge of the circular face. Find the angle between the circular face and the horizontal, giving your answer to the nearest degree. [2 marks] Turn over 6 In this question use g ¼ 10 m s—2 A sphere of mass 0.8 kg is attached to one end of a string of length 2 metres. The other end of the string is attached to a fixed point O The sphere is released from rest with the string taut and at an angle of 30° to the vertical, as shown in the diagram below. box O 30° 6 (a) Find the speed of the sphere when it is directly below O [3 marks] 6 (b) State one assumption that you made about the string. [1 mark] 6 (c) As the sphere moves, the string makes an angle y with the downward vertical. By finding an expression for the tension in the string in terms of y, show that the tension is a maximum when the sphere is directly below O box [6 marks] 6 (d) A physics student conducts an experiment and uses a device to measure the tension in the string when the sphere is directly below O They find that the tension is 9.5 newtons. Explain why this result is reasonable, showing any calculations that you make. [2 marks] Turn over (07) 7 Two particles, A and B, are moving on a smooth horizontal surface. A straight line has been marked on the surface and the particles are on opposite sides of the line. Particle A has mass 2 kg and moves with velocity 4 m s—1 at an angle of 30° to the line. Particle B has mass 3 kg and moves with velocity 6 m s—1 at an angle of 45° to the line. The particles and their velocities are shown in the diagram below. A box 4 m s–1 30° 45° 6 m s–1 B The particles collide when they reach the line and then move together as a single combined particle. 7 (a) Show that the magnitude of the impulse on particle A during the collision is 7.55 N s correct to three significant figures. [8 marks]