Bird's Basic Engineering Mathematics

481 Pages·2021·41.707 MB·English

by  John Bird

Bird’s Basic Engineering Mathematics
Whyisknowledgeofmathematicsimportantinengineering?
A career in any engineering or scientific field will concerned with release, control and utilisation of
requirebothbasicandadvancedmathematics.Without nuclearenergyandnuclearwastedisposal.
mathematics to determine principles, calculate dimen-
Petroleum engineers require mathematics to devise
sionsandlimits,explorevariations,proveconcepts,and
methods to improve oil and gas well production and
soon,therewouldbenomobiletelephones,televisions,
determine the need for new or modified tool designs;
stereo systems, video games, microwave ovens, com-
they oversee drilling and offer technical advice to
putersorvirtuallyanythingelectronic.Therewouldbe
achieveeconomicalandsatisfactoryprogress.
no bridges, tunnels, roads, skyscrapers, automobiles,
ships,planes,rocketsormostthingsmechanical.There Industrial engineers require mathematics to design,
wouldbenometalsbeyondthecommonones,suchas develop,testandevaluateintegratedsystemsforman-
ironandcopper,noplastics,nosynthetics.Infact,soci- agingindustrialproductionprocesses,includinghuman
etywouldmostcertainlybelessadvancedwithoutthe work factors, quality control, inventory control, logis-
use of mathematics throughout the centuries and into tics and material flow, cost analysis and production
thefuture. coordination.
Electrical engineers require mathematics to design,
Environmental engineers require mathematics to
develop,test,orsupervisethemanufacturingandinstal-
design, plan, or perform engineering duties in the
lation of electrical equipment, components or systems
prevention, control and remediation of environmen-
forcommercial,industrial,militaryorscientificuse.
tal health hazards, using various engineering disci-
Mechanicalengineersrequiremathematicstoperform plines; their work may include waste treatment, site
engineering duties in planning and designing tools, remediationorpollutioncontroltechnology.
engines,machinesandothermechanicallyfunctioning
Civil engineers require mathematics in all levels in
equipment;theyoverseeinstallation,operation,mainte-
civil engineering – structural engineering, hydraulics
nanceandrepairofsuchequipmentascentralisedheat,
andgeotechnicalengineeringareallfieldsthatemploy
gas,waterandsteamsystems.
mathematical tools such as differential equations, ten-
Aerospace engineers require mathematics to perform soranalysis,fieldtheory,numericalmethodsandoper-
a variety of engineering work in designing, construct- ationsresearch.
ing and testing aircraft, missiles, and spacecraft; they
conduct basic and applied research to evaluate adapt- Knowledgeofmathematicsisthereforeneededbyeach
abilityofmaterialsandequipmenttoaircraftdesignand oftheengineeringdisciplineslistedabove.
manufacture and recommend improvements in testing
Itisintendedthatthistext– Bird’sBasicEngineering
equipmentandtechniques.
Mathematics–willprovideastepbystepapproachto
Nuclear engineers require mathematics to conduct learningalltheearly,fundamentalmathematicsneeded
research on nuclear engineering problems or apply foryourfutureengineeringstudies.
principles and theory of nuclear science to problems
Now in its eighth edition, Bird’s Basic Engineer- John Bird, BSc (Hons), CEng, CMath, CSci, FIMA,
ing Mathematics has helped thousands of students FIET,FCollT,istheformerHeadofAppliedElectron-
to succeed in their exams. Mathematical theories ics in the Faculty of Technology at Highbury College,
are explained in a straightforward manner, supported Portsmouth,UK.Morerecently,hehascombinedfree-
by practical engineering examples and applications lance lecturing at the University of Portsmouth, with
to ensure that readers can relate theory to practice. Examiner responsibilities for Advanced Mathematics
Some1,000engineeringsituations/problemshavebeen with City and Guilds and examining for the Inter-
‘flagged-up’tohelpdemonstratethatengineeringcan- national Baccalaureate Organisation. He has over 45
not be fully understood without a good knowledge of years’ experience of successfully teaching, lecturing,
mathematics. instructing, training, educating and planning trainee
The extensive and thorough coverage makes this a engineers study programmes. He is the author of 146
great text for introductory level engineering courses – textbooks on engineering, science and mathematical
such as for aeronautical, construction, electrical, elec- subjects, with worldwide sales of over one million
tronic, mechanical, manufacturing engineering and copies. He is a chartered engineer, a chartered math-
vehicle technology – including for BTEC First, ematician, a chartered scientist and a Fellow of three
NationalandDiplomasyllabuses,City&GuildsTech- professional institutions. He has recently retired from
nicianCertificateandDiplomasyllabuses,andevenfor lecturing at the Defence College of Marine Engineer-
GCSErevision. ing in the Defence College of Technical Training at
Its companion website provides extra materials for H.M.S. Sultan, Gosport, Hampshire, UK, one of the
students and lecturers, including full solutions for all largestengineeringtrainingestablishmentsinEurope.
1,700 further questions, lists of essential formulae,
multiple choice tests, and illustrations, as well as full
solutionstorevisiontestsforcourseinstructors.
To Sue

Bird’s Basic Engineering Mathematics
Eighth Edition
John Bird
Eightheditionpublished2021
byRoutledge
2ParkSquare,MiltonPark,Abingdon,Oxon,OX144RN
andbyRoutledge
52VanderbiltAvenue,NewYork,NY10017
RoutledgeisanimprintoftheTaylor&FrancisGroup,aninformabusiness
©2021JohnBird
TherightofJohnBirdtobeidentifiedasauthorofthisworkhasbeenassertedbyhiminaccordancewithsections77and78of
theCopyright,DesignsandPatentsAct1988.
Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedorutilisedinanyformorbyanyelectronic,mechanical,
orothermeans,nowknownorhereafterinvented,includingphotocopyingandrecording,orinanyinformationstorageor
retrievalsystem,withoutpermissioninwritingfromthepublishers.
Trademarknotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andareusedonlyforidentification
andexplanationwithoutintenttoinfringe.
FirsteditionpublishedbyNewnes1999
SeventheditionpublishedbyRoutledge2017
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
LibraryofCongressCataloging-in-PublicationData
Acatalogrecordhasbeenrequestedforthisbook
ISBN:9780367643706(hbk)
ISBN:9780367643676(pbk)
ISBN:9781003124214(ebk)
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Visitthecompanionwebsite:www.routledge.com/cw/bird
Contents
Preface xi 6.3 Directproportion 50
6.4 Inverseproportion 54
Acknowledgements xiii
1 Basicarithmetic 1
7 Powers,rootsandlawsofindices 57
1.1 Introduction 1
7.1 Introduction 57
1.2 Revisionofadditionandsubtraction 2
7.2 Powersandroots 57
1.3 Revisionofmultiplicationanddivision 3
7.3 Lawsofindices 59
1.4 Highestcommonfactorsandlowest
commonmultiples 5
1.5 Orderofoperationandbrackets 7 8 Units,prefixesandengineeringnotation 64
8.1 Introduction 64
2 Fractions 10 8.2 SIunits 64
2.1 Introduction 10 8.3 Commonprefixes 65
2.2 Addingandsubtractingfractions 11 8.4 Standardform 68
2.3 Multiplicationanddivisionoffractions 13 8.5 Engineeringnotation 70
2.4 Orderofoperationwithfractions 15 8.6 Metricconversions 72
8.7 Metric-US/Imperialconversions 76
RevisionTest1 18
RevisionTest3 82
3 Decimals 19
3.1 Introduction 19
3.2 Convertingdecimalstofractionsand 9 Basicalgebra 83
viceversa 19 9.1 Introduction 83
3.3 Significantfiguresanddecimalplaces 21 9.2 Basicoperations 84
3.4 Addingandsubtractingdecimal 9.3 Lawsofindices 87
numbers 22
3.5 Multiplyinganddividingdecimal
10 Furtheralgebra 91
numbers 23
10.1 Introduction 91
10.2 Brackets 91
4 Usingacalculator 26
10.3 Factorisation 93
4.1 Introduction 26
10.4 Lawsofprecedence 94
4.2 Adding,subtracting,multiplyingand
dividing 26
4.3 Furthercalculatorfunctions 28 11 Solvingsimpleequations 97
4.4 Evaluationofformulae 32 11.1 Introduction 97
11.2 Solvingequations 97
5 Percentages 38 11.3 Practicalproblemsinvolvingsimple
5.1 Introduction 38 equations 101
5.2 Percentagecalculations 39
5.3 Furtherpercentagecalculations 40
RevisionTest4 107
5.4 Morepercentagecalculations 42
RevisionTest2 46
12 Transposingformulae 108
12.1 Introduction 108
6 Ratioandproportion 47 12.2 Transposingformulae 108
6.1 Introduction 47 12.3 Furthertransposingofformulae 110
6.2 Ratios 48 12.4 Moredifficulttransposingofformulae 113
viii Contents
13 Solvingsimultaneousequations 118 18 Graphsreducingnon-linearlawstolinearform 185
13.1 Introduction 118 18.1 Introduction 185
13.2 Solvingsimultaneousequationsintwo 18.2 Determinationoflaw 185
unknowns 118 18.3 Revisionoflawsoflogarithms 188
13.3 Furthersolvingofsimultaneousequations 120 18.4 Determinationoflawsinvolving
13.4 Solvingmoredifficultsimultaneous logarithms 189
equations 122
19 Graphicalsolutionofequations 194
13.5 Practicalproblemsinvolving
19.1 Graphicalsolutionofsimultaneous
simultaneousequations 124
equations 194
13.6 Solvingsimultaneousequationsinthree
19.2 Graphicalsolutionofquadraticequations 196
unknowns 128
19.3 Graphicalsolutionoflinearand
quadraticequationssimultaneously 200
RevisionTest5 131 19.4 Graphicalsolutionofcubic
equations 200
20 Graphswithlogarithmicscales 203
14 Solvingquadraticequations 132
20.1 Logarithmicscalesandlogarithmic
14.1 Introduction 132
graphpaper 203
14.2 Solutionofquadraticequationsby
20.2 Graphsoftheformy=axn 204
factorisation 133
14.3 Solutionofquadraticequationsby 20.3 Graphsoftheformy=abx 207
‘completingthesquare’ 135 20.4 Graphsoftheformy=aekx 208
14.4 Solutionofquadraticequationsby
formula 137 RevisionTest7 211
14.5 Practicalproblemsinvolvingquadratic
equations 138 21 Anglesandtriangles 213
14.6 Solutionoflinearandquadratic 21.1 Introduction 213
equationssimultaneously 141 21.2 Angularmeasurement 213
21.3 Triangles 219
15 Logarithms 143 21.4 Congruenttriangles 223
15.1 Introductiontologarithms 143 21.5 Similartriangles 225
15.2 Lawsoflogarithms 145 21.6 Constructionoftriangles 227
15.3 Indicialequations 147
22 Introductiontotrigonometry 230
15.4 Graphsoflogarithmicfunctions 149
22.1 Introduction 230
22.2 ThetheoremofPythagoras 230
16 Exponentialfunctions 151 22.3 Sines,cosinesandtangents 233
16.1 Introductiontoexponentialfunctions 151 22.4 Evaluatingtrigonometricratiosofacute
16.2 Thepowerseriesforex 152 angles 235
16.3 Graphsofexponentialfunctions 154 22.5 Solvingright-angledtriangles 238
16.4 Napierianlogarithms 156 22.6 Anglesofelevationanddepression 241
16.5 Lawsofgrowthanddecay 159
RevisionTest8 245
RevisionTest6 164
23 Trigonometricwaveforms 247
23.1 Graphsoftrigonometricfunctions 247
23.2 Anglesofanymagnitude 248
17 Straightlinegraphs 165
23.3 Theproductionofsineandcosinewaves 251
17.1 Introductiontographs 165
23.4 Terminologyinvolvedwithsineand
17.2 Axes,scalesandco-ordinates 165
cosinewaves 251
17.3 Straightlinegraphs 167
23.5 Sinusoidalform:Asin(!t(cid:6)(cid:11)) 254
17.4 Gradients,interceptsandequations
ofgraphs 170
17.5 Practicalproblemsinvolvingstraight
linegraphs 177
Contents ix
24 Non-right-angledtrianglesandsomepractical 29 Irregularareasandvolumesandmeanvalues 318
applications 258 29.1 Areasofirregularfigures 318
24.1 Thesineandcosinerules 258 29.2 Volumesofirregularsolids 321
24.2 Areaofanytriangle 259 29.3 Meanoraveragevaluesofwaveforms 322
24.3 Workedproblemsonthesolutionof
trianglesandtheirareas 259
RevisionTest11 327
24.4 Furtherworkedproblemsonthesolution
oftrianglesandtheirareas 261
30 Vectors 329
24.5 Practicalsituationsinvolving
30.1 Introduction 329
trigonometry 262
30.2 Scalarsandvectors 329
24.6 Furtherpracticalsituationsinvolving
30.3 Drawingavector 330
trigonometry 264
30.4 Additionofvectorsbydrawing 331
30.5 Resolvingvectorsintohorizontaland
25 Cartesianandpolarco-ordinates 268
verticalcomponents 333
25.1 Introduction 268
30.6 Additionofvectorsbycalculation 334
25.2 ChangingfromCartesiantopolar
30.7 Vectorsubtraction 338
co-ordinates 268
30.8 Relativevelocity 339
25.3 ChangingfrompolartoCartesian
30.9 i,jandknotation 340
co-ordinates 270
25.4 UseofPol/Recfunctionson
calculators 271 31 Methodsofaddingalternating
waveforms 343
31.1 Combiningtwoperiodicfunctions 343
RevisionTest9 273
31.2 Plottingperiodicfunctions 344
31.3 Determiningresultantphasorsby
drawing 345
26 Areasofcommonshapes 274
31.4 Determiningresultantphasorsbythe
26.1 Introduction 274
sineandcosinerules 347
26.2 Commonshapes 274
31.5 Determiningresultantphasorsby
26.3 Areasofcommonshapes 277
horizontalandverticalcomponents 348
26.4 Areasofsimilarshapes 285
RevisionTest12 352
27 Thecircleanditsproperties 287
27.1 Introduction 287
27.2 Propertiesofcircles 287 32 Presentationofstatisticaldata 354
27.3 Radiansanddegrees 289 32.1 Somestatisticalterminology 355
27.4 Arclengthandareaofcirclesand 32.2 Presentationofungroupeddata 356
sectors 290 32.3 Presentationofgroupeddata 359
27.5 Theequationofacircle 294
33 Mean,median,modeandstandarddeviation 367
33.1 Measuresofcentraltendency 367
RevisionTest10 297
33.2 Mean,medianandmodefordiscrete
data 368
28 Volumesandsurfaceareasofcommonsolids 299 33.3 Mean,medianandmodeforgrouped
28.1 Introduction 299 data 369
28.2 Volumesandsurfaceareasofcommon 33.4 Standarddeviation 370
shapes 299 33.5 Quartiles,decilesandpercentiles 372
28.3 Summaryofvolumesandsurfaceareas
ofcommonsolids 306 34 Probability 375
28.4 Morecomplexvolumesandsurface 34.1 Introductiontoprobability 376
areas 306 34.2 Lawsofprobability 377
28.5 Volumesandsurfaceareasoffrustaof
pyramidsandcones 312 RevisionTest13 384
28.6 Volumesofsimilarshapes 316