Science and Mathematics for Engineering Why is knowledge of science and mathematics important in engineering? Acareerinanyengineeringfieldwillrequirebothbasicandadvancedmathematicsandscience.Withoutmathematics andsciencetodetermineprinciples,calculatedimensionsandlimits,explorevariations,proveconcepts,andsoon, there would be no mobile telephones, televisions, stereo systems, video games, microwave ovens, computers, or virtually anything electronic. There would be no bridges, tunnels, roads, skyscrapers, automobiles, ships, planes, rocketsormostthingsmechanical.Therewouldbenometalsbeyondthecommonones,suchasironandcopper,no plastics, no synthetics.In fact,society wouldmostcertainlybe less advancedwithoutthe use ofmathematicsand sciencethroughoutthecenturiesandintothefuture. Electricalengineersrequiremathematicsandsciencetodesign,develop,test, orsupervisethemanufacturingand installationofelectricalequipment,components,orsystemsforcommercial,industrial,military,orscientificuse. Mechanicalengineersrequiremathematicsandsciencetoperformengineeringdutiesinplanninganddesigningtools, engines,machines,andothermechanicallyfunctioningequipment;theyoverseeinstallation,operation,maintenance, andrepairofsuchequipmentascentralisedheat,gas,water,andsteamsystems. Aerospace engineers require mathematics and science to perform a variety of engineering work in designing, constructing, and testing aircraft, missiles, and spacecraft; they conduct basic and applied research to evaluate adaptabilityofmaterialsandequipmenttoaircraftdesignandmanufactureandrecommendimprovementsintesting equipmentandtechniques. Nuclearengineersrequiremathematicsandsciencetoconductresearchonnuclearengineeringproblemsorapply principlesandtheoryofnuclearsciencetoproblemsconcernedwithrelease,control,andutilisationofnuclearenergy andnuclearwastedisposal. Petroleum engineers require mathematics and science to devise methods to improve oil and gas well production anddeterminetheneedfornewormodifiedtooldesigns;theyoverseedrillingandoffertechnicaladvicetoachieve economicalandsatisfactoryprogress. Industrialengineersrequiremathematicsandsciencetodesign,develop,test,andevaluateintegratedsystemsfor managingindustrialproductionprocesses,includinghumanworkfactors,qualitycontrol,inventorycontrol,logistics andmaterialflow,costanalysis,andproductioncoordination. Environmental engineers require mathematics and science to design, plan, or perform engineering duties in the prevention,control, and remediationof environmentalhealth hazards, using variousengineeringdisciplines;their workmayincludewastetreatment,siteremediation,orpollutioncontroltechnology. Civilengineersrequiremathematicsandscienceinalllevelsincivilengineering–structuralengineering,hydraulics and geotechnical engineering are all fields that employ mathematical tools such as differential equations, tensor analysis,fieldtheory,numericalmethodsandoperationsresearch. Knowledgeofmathematicsandscienceisthereforeneededbyeachoftheengineeringdisciplineslistedabove. It is intended that this text – Science and Mathematics for Engineering – will provide a step by step approachto learningfundamentalmathematicsandscienceneededforyourengineeringstudies. John Bird is the former Head of Applied Electronics in the Faculty of Technology at Highbury College, Portsmouth,U.K.Morerecently,hehascombinedfreelancelecturingattheUniversityofPortsmouth,withExaminer responsibilitiesforAdvancedMathematicswithCityandGuildsandexaminingforInternationalBaccalaureate.He has some 45 years experience of successfully teaching, lecturing, instructing, training, educating and planning of trainee engineersstudyprogrammes.He is the authorof 135textbookson engineeringand mathematicalsubjects withworldwidesalesofoveronemillioncopies.Heisacharteredengineer,acharteredmathematician,achartered scientistandaFellowofthreeprofessionalinstitutions.HeiscurrentlylecturingattheDefenceCollegeofMarine Engineeringinthe DefenceCollegeofTechnicalTrainingatH.M.S.Sultan,Gosport,Hampshire,U.K,oneofthe largesttechnicaltrainingestablishmentsinEurope. Science and Mathematics for Engineering Sixth Edition John Bird BSc(Hons), CEng, CSci, CMath, FIET, FIMA, FCollT Sixtheditionpublished2020 byRoutledge 2ParkSquare,MiltonPark,Abingdon,Oxon,OX144RN andbyRoutledge 52VanderbiltAvenue,NewYork,NY10017 RoutledgeisanimprintoftheTaylor&FrancisGroup,aninformabusiness ©2020JohnBird TherightofJohnBirdtobeidentifiedasauthorofthisworkhasbeenassertedbyhiminaccordancewithsections77and78 oftheCopyright,DesignsandPatentsAct1988. Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedorutilisedinanyformorbyanyelectronic,mechanical, orothermeans,nowknownorhereafterinvented,includingphotocopyingandrecording,orinanyinformationstorageor retrievalsystem,withoutpermissioninwritingfromthepublishers. Trademarknotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andareusedonlyforidentification andexplanationwithoutintenttoinfringe. FirsteditionpublishedbyElsevier1995 FiftheditionpublishedbyRoutledge2015 BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressCataloging-in-PublicationData Names:Bird,J.O.,author. Title:Scienceandmathematicsforengineering/JohnBird. Othertitles:Scienceforengineering Description:Sixthedition.|BocaRaton:Taylor&Francis,aCRCtitle,partoftheTaylor&Francisimprint,amemberofthe Taylor&FrancisGroup,theacademicdivisionofT&FInforma,plc,2020.|Includesindex.|Revisededition:Sciencefor engineering.5thed.London;NewYork:Routledge,2015. Identifiers:LCCN2019020085|ISBN9780367204747(pbk)|ISBN9780367204754(hbk)|ISBN9780429261701(ebk) Subjects:LCSH:Engineering.|Science.|Engineeringmathematics–Examinations,questions,etc. Classification:LCCTA145.B532020|DDC500.2024/62–dc23 LCrecordavailableathttps://lccn.loc.gov/2019020085 ISBN:978-0-367-20475-4(hbk) ISBN:978-0-367-20474-7(pbk) ISBN:978-0-429-26170-1(ebk) TypesetinTimes byServisFilmsettingLtd,Stockport,Cheshire Visitthecompanionwebsite:www.routledge.com/cw/bird To Sue Contents Preface xiii 5 Basicalgebra 50 5.1 Introduction 51 SECTION1 APPLIEDMATHEMATICS 1 5.2 Basicoperations 51 1 Basicarithmetic 3 5.3 Lawsofindices 53 1.1 Introduction 3 5.4 Brackets 54 1.2 Revisionofadditionandsubtraction 4 5.5 Factorisation 55 1.3 Revisionofmultiplicationanddivision 5 5.6 Lawsofprecedence 56 1.4 Highestcommonfactorsandlowest 6 Solvingsimpleequations 58 commonmultiples 7 6.1 Introduction 58 1.5 Orderofoperationandbrackets 8 6.2 Solvingequations 59 2 Fractions,decimalsandpercentages 10 6.3 Practicalproblemsinvolvingsimple 2.1 Fractions 11 equations 62 2.2 Ratioandproportion 15 2.3 Decimals 16 RevisionTest3: Algebraandsimpleequations 65 2.4 Percentages 19 7 Transposingformulae 66 RevisionTest1: Arithmetic,fractions, 7.1 Introduction 66 decimalsandpercentages 22 7.2 Transposingformulae 66 7.3 Furthertransposingofformulae 68 3 Indices,units,prefixesandengineeringnotation 24 7.4 Moredifficulttransposingofformulae 70 3.1 Powersandroots 25 3.2 Lawsofindices 26 8 Solvingsimultaneousequations 72 3.3 Introductiontoengineeringunits 28 8.1 Introduction 72 3.4 SIunits 28 8.2 Solvingsimultaneousequationsintwo unknowns 73 3.5 Commonprefixes 30 8.3 Furthersolvingofsimultaneousequations 75 3.6 Standardform 31 8.4 Practicalproblemsinvolving 3.7 Engineeringnotation 32 simultaneousequations 76 3.8 Metricconversions 34 3.9 Metric–US/Imperialconversions 37 9 Logarithmsandexponentialfunctions 78 9.1 Introductiontologarithms 79 4 Calculationsandevaluationofformulae 42 9.2 Lawsoflogarithms 79 4.1 Introduction 42 9.3 Indicialequations 81 4.2 Useofcalculator 42 9.4 Graphsoflogarithmicfunctions 82 4.3 Evaluationofformulae 46 9.5 Exponentialfunctions 82 RevisionTest2: Indices,units,calculatorand 9.6 Graphsofexponentialfunctions 83 evaluationofformulae 48 9.7 Napierianlogarithms 84 9.8 Lawsofgrowthanddecay 86 ScienceandMathematicsforEngineering.978-0-367-20475-4,©JohnBird.PublishedbyTaylor&Francis.Allrightsreserved. viii Contents 14.4 Calculatingmorecomplexvolumes RevisionTest4: Transpositionofformulae, simultaneousequations,logarithmsand andsurfaceareas 150 exponentialfunctions 90 14.5 Volumesofsimilarshapes 155 RevisionTest6: Areasandvolumes 157 10 Straightlinegraphs 91 10.1 Introductiontographs 91 Multiple-choicequestionson AppliedMathematics 160 10.2 Axes,scalesandco-ordinates 92 10.3 Straightlinegraphs 93 10.4 Gradients,interceptsandequationof SECTIONII MECHANICAL agraph 96 APPLICATIONS 167 10.5 Practicalproblemsinvolvingstraight 15 SIunitsanddensity 169 linegraphs 100 15.1 SIunits 169 11 Introductiontotrigonometry 105 15.2 Density 171 11.1 Introduction 105 16 Atomicstructureofmatter 175 11.2 ThetheoremofPythagoras 105 16.1 Elements,atoms,moleculesand 11.3 Sines,cosinesandtangents 108 compounds 175 11.4 Evaluatingtrigonometricratiosof 16.2 Mixtures,solutions,suspensionsand acuteangles 110 solubility 176 11.5 Solvingright-angledtriangles 112 16.3 Crystals 178 11.6 Graphsoftrigonometricfunctions 115 16.4 Metals 178 11.7 Sineandcosinerules 116 11.8 Areaofanytriangle 116 17 Speedandvelocity 181 17.1 Speed 181 11.9 Workedproblemsonthesolutionof trianglesandtheirareas 116 17.2 Distance/timegraph 182 11.10 Practicalsituationsinvolving 17.3 Speed/timegraph 185 trigonometry 118 17.4 Velocity 186 18 Acceleration 189 RevisionTest5: Straightlinegraphsand 18.1 Introductiontoacceleration 190 trigonometry 121 18.2 Velocity/timegraph 190 18.3 Free-fallandequationofmotion 191 12 Areasofcommonshapes 123 12.1 Introduction 123 19 Force,massandacceleration 195 12.2 Commonshapes 124 19.1 Introduction 196 12.3 Calculatingareasofcommonshapes 126 19.2 Newton’slawsofmotion 196 12.4 Areasofsimilarshapes 134 19.3 Centripetalacceleration 199 13 Thecircle 136 RevisionTest7: SIunits,density,speedand 13.1 Introduction 136 velocity,force,massandacceleration 202 13.2 Propertiesofcircles 136 13.3 Radiansanddegrees 138 20 Forcesactingatapoint 203 13.3 Arclengthandareasofcirclesand 20.1 Introduction 203 sectors 139 20.2 Scalarandvectorquantities 204 14 Volumesofcommonsolids 143 20.3 Centreofgravityandequilibrium 204 14.1 Introduction 143 20.4 Forces 205 14.2 Calculatingvolumesandsurface 20.5 Theresultantoftwocoplanarforces 205 areasofcommonsolids 143 20.6 Triangleofforcesmethod 206 14.3 Summaryofvolumesandsurface 20.7 Theparallelogramofforcesmethod 207 areasofcommonsolids 150 Contents ix 20.8 Resultantofcoplanarforcesby 26 Theeffectsofforcesonmaterials 270 calculation 208 26.1 Introduction 271 20.9 Resultantofmorethantwocoplanar 26.2 Forces 271 forces 209 26.3 Tensileforce 271 20.10 Coplanarforcesinequilibrium 211 26.4 Compressiveforce 272 20.11 Resolutionofforces 212 26.5 Shearforce 272 20.12 Summary 215 26.6 Stress 272 26.7 Strain 273 21 Work,energyandpower 218 26.8 Elasticity,limitofproportionalityand 21.1 Introduction 219 elasticlimit 275 21.2 Work 219 26.9 Hooke’slaw 276 21.3 Energy 223 26.10 Ductility,brittlenessandmalleability 279 21.4 Power 225 21.5 Potentialandkineticenergy 228 27 Linearmomentumandimpulse 282 27.1 Introduction 283 22 Simplysupportedbeams 234 27.2 Linearmomentum 283 22.1 Introduction 234 27.3 Impulseandimpulsiveforces 285 22.2 Themomentofaforce 235 22.3 Equilibriumandtheprincipleof RevisionTest10: Forcesonmaterialsand moments 236 linearmomentumandimpulse 289 22.4 Simplysupportedbeamshavingpoint loads 238 28 Torque 290 RevisionTest8: Forcesactingatapoint,work, 28.1 Introduction 290 energyandpowerandsimplysupportedbeams 243 28.2 Coupleandtorque 291 28.3 Workdoneandpowertransmittedby aconstanttorque 291 23 Linearandangularmotion 244 28.4 Kineticenergyandmomentofinertia 293 23.1 Introduction 244 28.5 Powertransmissionandefficiency 296 23.2 Theradian 244 23.3 Linearandangularvelocity 245 29 Pressureinfluids 300 23.4 Linearandangularacceleration 246 29.1 Pressure 300 23.5 Furtherequationsofmotion 247 29.2 Fluidpressure 301 23.6 Relativevelocity 249 29.3 Atmosphericpressure 303 29.4 Archimedes’principle 304 24 Friction 253 29.5 Measurementofpressure 306 24.1 Introductiontofriction 253 29.6 Barometers 306 24.2 Coefficientoffriction 254 29.7 Absoluteandgaugepressure 308 24.3 Applicationsoffriction 255 29.8 Themanometer 308 25 Simplemachines 258 29.9 TheBourdonpressuregauge 309 25.1 Machines 258 29.10 Vacuumgauges 310 25.2 Forceratio,movementratioand efficiency 258 30 Heatenergyandtransfer 314 25.3 Pulleys 261 30.1 Introduction 315 25.4 Thescrew-jack 262 30.2 Heatandtemperature 315 25.5 Geartrains 263 30.3 Themeasurementoftemperature 316 25.6 Levers 265 30.4 Specificheatcapacity 316 30.5 Changeofstate 318 RevisionTest9: Linearandangularmotion, 30.6 Latentheatsoffusionandvaporisation 319 frictionandsimplemachines 269 30.7 Asimplerefrigerator 321