BKCAR.NET BKCAR.NET Automotive Science and Mathematics BKCAR.NET This page intentionally left blank BKCAR.NET Automotive Science and Mathematics Allan Bonnick AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Butterworth-Heinemann is an imprint of Elsevier BKCAR.NET Butterworth-Heinemann is an imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First edition 2008 Copyright © 2008, Allan Bonnick. Published by Elsevier Ltd. All rights reserved The right of Allan Bonnick to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@elsevier.com. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-7506-8522-1 For information on all Butterworth-Heinemann publications visit our web site at books.elsevier.com Printed and bound in Hungary 080910 10987654321 Working together to grow libraries in developing countries www.elsevier.com | www.bookaid.org | www.sabre.org BKCAR.NET Contents Preface xvii Units and symbols xviii Glossary xix 1 Arithmetic 1 1.1 Terminology of number systems 1 1.2 The decimal system 1 Addition and subtraction of decimals 2 Multiplication and division – decimals 2 1.3 Degrees of accuracy 4 Rounding numbers 4 1.4 Accuracy in calculation 4 1.5 Powers and roots and standard form 4 General rules for indices 5 1.6 Standard form 5 Multiplying and dividing numbers in standard form 5 1.7 Factors 6 1.8 Fractions 6 Addition and subtraction 6 Fractions and whole numbers 6 Combined addition and subtraction 7 Multiplication and division of fractions 7 Order of performing operations in problems involving fractions 7 1.9 Ratio and proportion. Percentages 8 Examples of ratios in vehicle technology 8 1.10 The binary system 10 Most significant bit (MSB) 10 Hexadecimal 10 Converting base 10 numbers to binary 10 Uses of binary numbers in vehicle systems 10 1.11 Directed numbers 11 Rules for dealing with directed numbers 11 1.12 Summary of main points 12 1.13 Exercises 12 BKCAR.NET vi Contents 2 Statistics – An introduction 16 2.1 Definition 16 2.2 Collecting and sorting raw data 17 2.3 Making sense of data 17 Discrete variables 17 Continuous variables 17 2.4 Descriptive statistics – pictographs 18 Pie charts 18 2.5 Interpreting data. Statistical inference 19 Frequency and tally charts 19 The tally chart and frequency distribution 19 2.6 Importance of the shape of a frequency distribution 20 The histogram 20 The frequency polygon 20 Cumulative frequency 21 2.7 Interpreting statistics 22 Sampling 22 2.8 Features of the population that are looked for in a sample 22 Average 22 2.9 The normal distribution 23 Importance of the normal distribution 24 Other ways of viewing frequency distributions – quartiles, deciles, percentiles 25 2.10 Summary of main points 26 2.11 Exercises 26 3 Algebra and graphs 29 3.1 Introduction 29 3.2 Formulae 29 3.3 Evaluating formulae 29 3.4 Processes in algebra 30 Brackets 30 3.5 Algebraic expressions and simplification 30 Expression 30 3.6 Factorising 31 3.7 Equations 31 Solving equations 31 3.8 Transposition of formulae 33 3.9 Graphs 34 Variables 34 Scales 34 Coordinates 35 3.10 Graphs and equations 36 The straight-line graph 36 3.11 Summary of main points 37 BKCAR.NET Contents vii 3.12 Exercises 38 Exercises – Section 3.3 38 Exercises – Section 3.4 38 Exercises – Section 3.5 38 Exercises – Section 3.6 38 Exercises – Section 3.7 38 Exercises – Section 3.8 38 Exercises – Section 3.10 38 4 Geometry and trigonometry 41 4.1 Angles 41 Angular measurement 41 Angles and rotation 41 4.2 Examples of angles in automotive work 42 Angles and lines 43 Adding and subtracting angles 43 4.3 Types of angle 44 Adjacent angles 44 Opposite angles 44 Corresponding angles 44 Alternate angles 44 Supplementary angles 44 Complementary angles 44 4.4 Types of triangle 45 Acute angled triangle 45 Obtuse angled triangle 45 Equilateral triangle 45 Isosceles triangle 45 Scalene triangle 45 Right angled triangle 46 Labelling sides and angles of a triangle 46 Sum of the three angles of a triangle 46 4.5 Pythagoras’ theorem 46 4.6 Circles 46 Ratio of diameter and circumference  47 Length of arc 47 4.7 Timing marks 47 4.8 Wheel revolutions and distance travelled 48 4.9 Valve opening area 48 4.10 Trigonometry 48 4.11 Using sines, cosines and tangents 49 Sines 49 Cosines 50 Tangents 51 4.12 Summary of formulae 52 BKCAR.NET viii Contents 4.13 Exercises 52 Exercises – Section 4.2 53 Exercises – Section 4.3 53 Exercises – Section 4.4 54 Exercises – Section 4.5 55 Exercises – Section 4.6 55 5 Forces 58 5.1 Force 58 5.2 Types of force – examples 58 5.3 Describing forces 58 5.4 Graphical representation of a force 58 5.5 Addition of forces 59 5.6 Parallelogram of forces 60 5.7 Triangle of forces 60 5.8 Resolution of forces 61 5.9 Mass 62 5.10 Equilibrium 62 5.11 Pressure 62 5.12 Pressure in hydraulic systems 63 5.13 Hooke’s law 64 5.14 Practical applications 65 5.15 Summary 65 5.16 Exercises 65 6 Materials – Stress, strain, elasticity 68 6.1 Introduction 68 6.2 Stress 68 Types of stress 68 6.3 Tensile test 69 6.4 Examples of stress and strain 70 6.5 Stress raisers 71 6.6 Strain 72 Shear strain 72 6.7 Elasticity 73 Stress, strain, elasticity 73 6.8 Tensile strength 73 6.9 Factor of safety 74 6.10 Torsional stress 74 6.11 Strain energy 75 6.12 Strength of materials 75 6.13 Other terms used in describing materials 75 6.14 Non-ferrous metals 76 6.15 Non-metallic materials 76 Kevlar 76 6.16 Recycling of materials 77 BKCAR.NET Contents ix 6.17 Summary of main formulae 77 6.18 Exercises 77 7 Levers and moments, torque and gears 79 7.1 Levers 79 7.2 Principles of leverage 79 7.3 The principle of moments 79 7.4 The bell crank lever 81 A practical application of the bell crank lever 81 7.5 Axle loadings 82 7.6 Torque 83 7.7 Engine torque 83 7.8 Leverage and gears 84 Torque multiplication 84 Drivers and driven 85 7.9 Gear trains: calculating gear ratios 85 Spur gear ratios 85 7.10 Couples 85 7.11 Summary of main points 85 7.12 Exercises 86 8 Work energy, power and machines 89 8.1 Work 89 8.2 Power 89 8.3 Work done by a torque 90 8.4 Work done by a constantly varying force 90 Mid-ordinate method for calculating work done 91 8.5 Energy 92 Potential energy 92 Chemical energy 92 Conservation of energy 92 Energy equation 92 Kinetic energy 92 Energy of a falling body 93 Kinetic energy of rotation 93 8.6 Machines 94 Mechanical advantage 94 Velocity ratio (movement ratio) 95 Efficiency of a machine 95 Work done against friction 95 A steering mechanism as a machine 95 8.7 Summary of formulae 97 8.8 Exercises 98 [...]... and other training providers BK C AR One of the main aims of this book is to provide a course of study of science and mathematics that constantly demonstrates the links between these disciplines and the everyday work of technicians in the automotive field The subject matter has been chosen to provide full cover for the Science and Mathematics of the BTEC and IMI National Certificates and Diplomas and. .. 1.14 88 = 2 × 44 = 11 × 8 The factors here are 2 and 44, or 11 and 8 1.8 Fractions N ET 6 1/3 + 5/6 = Highest common factor (HCF) BK C Take two numbers such as 40 and 60 The factors of 40 can be 5 and 8, or 10 and 4, or 2 and 20 The factors of 60 can be 5 and 12, 15 and 4, 30 and 2, 20 and 3 One number, 20, is the largest factor that is common to 40 and 60 The highest common factor (HCF) of a set of... standard form When numbers are written in standard form the rules of indices can be used to facilitate multiplication and division Automotive science and mathematics Example 1.13 Simplify: 25 × 103 × 3 × 102 25 × 103 × 3 × 102 = 25 × 3 103 × 102 = 75 × 10 3+2 = 7 5 × 106 Simplify: 8 × 104 2 × 102 8 × 104 8 = × 104−2 = 4 × 102 2 2 × 10 2 1.7 Factors Numbers such as 24 can be made up from 6 × 4 So 6 and. .. follows 1 Brackets – deal with brackets first 2 Divide and multiply and deal with of 3 Add and subtract Some people find the acronym BODMAS (brackets, of, divide, multiply, add, subtract) useful as a reminder of the order in which operations on fractions are performed Automotive science and mathematics State this as a ratio in the form m : n and m/n In the form m:n the ratio is 500 : 50, which can... 567 = 5×102 +6×101 +7×100 , or 5×100 +6× 10 + 7 × 1 That is, 5 hundreds, 6 tens and 7 ones The decimal point is used to indicate the position in a number, after which the digits represent fractional parts of the number For example, 567.423 means 567 plus 4/10 + 2/100 + 3/1000 Automotive science and mathematics Addition and subtraction of decimals When adding or subtracting decimals the numbers must... basic level and the coverage should provide the necessary skill for trainees and students to demonstrate competence in key skills The coverage of some topics, such as vehicle dynamics and heat engines (thermodynamics), is at the advanced end of National Level 3 and will be found helpful by HNC/HND and Foundation Degree students Answers are provided to assist those who may be studying privately and a set... means that the result is +16 The plus sign is normally omitted and the absence of a sign in front of the number shows that it is a plus number Automotive science and mathematics Example Find the value of: −3 − 2 − 4 Add the numbers together to give 9; the sign is minus, so −3 − 2 − 4 = −9 The justification for this is: N ET 12 −12 = −4 3 and Addition of directed numbers where the signs are different... related Technical Certificates and NVQs up to and including Level 3 The needs of students in the 14 to 19 age group who may be following a scheme of vocational studies have been borne in mind during the writing of the book It is hoped that these students and their teachers will find the links between theory and practice that are demonstrated in the text N ET Preface .N ET Units and symbols SI units Quantity... journal bearing 9.8 Summary of formulae 9.9 Exercises 10 Velocity and acceleration, speed 10.1 Speed and velocity 10.2 Acceleration 10.3 Velocity–time graph Uniform velocity Uniform acceleration 10.4 Equations of motion and their application to vehicle technology 10.5 Force, mass and acceleration Newton’s laws of motion 10.6 Relation between mass and weight 10.7 Inertia 10.8 Motion under gravity 10.9 Angular... calculation could continue but in many problems the degree of accuracy required does not warrant further division The procedure for making approximations and deciding how many decimal places to show in an answer is shown in Section 1.3 Automotive science and mathematics 1.3 Degrees of accuracy Rounding numbers At a particular petrol station in Britain in 2005 the price of petrol was 93.6 pence/litre Taking . BKCAR.NET BKCAR.NET Automotive Science and Mathematics BKCAR.NET This page intentionally left blank BKCAR.NET Automotive Science and Mathematics Allan Bonnick AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW. to provide full cover for the Science and Mathematics of the BTEC and IMI National Certificates and Diplo- mas and the related Technical Certificates and NVQs up to and including Level 3. The needs. indices 5 1.6 Standard form 5 Multiplying and dividing numbers in standard form 5 1.7 Factors 6 1.8 Fractions 6 Addition and subtraction 6 Fractions and whole numbers 6 Combined addition and subtraction