Chapter 2 Force Vectors - CTMD OUM2章力向量- CTMD烏姆

STATICS

TUTORIAL1&2–

UNIT1:ForceandEquilibrium

Chapter2:ForceVectorsIrDrKanesan

MuthusamyEBXS3103StaticsJan20051PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaSEQUENCEOFCHAPTER2IntroductionObjectives2.1Forceinaplane 2.1.1Resultantof2forcesactingonaparticle2.2Vectors2.3Vectoroperations

2.3.1Additionofvectors

2.3.2Vectorsubtraction

2.3.3Resolutionofvector2.4Additionofasystemofcoplanarforces

2.4.1Scalarnotation

2.4.2Cartesianvectornotation

2.4.3Coplanarforceresultants2.5Cartesianvectors

2.5.1Right-handedcoordinatesystem

2.5.2Cartesianunitvectors

2.5.3Cartesianvectorrepresentation

2.5.4Magnitudeofacartesianvector

2.5.5Directionofacartesianvector2.6Additionandsubtractionofcartesianvectors

2.6.1Concurrentforcesystems2.7Positionvectors

2.7.1x,y,zcoordinates

2.7.2Positionvector2.8Forcevectordirectedalongaline2.9DotproductSummary2PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaIntroductionThefirstphaseofthechapterfocusesonresultantforces,i.e.thecombinedforces(twoormoreforces)actingonparticleshavingthesameeffectastheoriginalforces.Thisresultantforceissingleforceactingontheparticle.Alltheforcesactingonagivenbodyisassumedtobeappliedatthesamepoint.3PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaObjectivesExplaintheeffectofforcesactingonparticles;andApplyforcevectorsknowledgeinsolvingproblemsrelatedtostatics.4PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.1.1Resultantof2forcesactingonaparticleAforcerepresentstheactionofonebodyonanotherandisnormallycharacterizedbyitspointofapplication,itsmagnitudeanditsdirection.Thedirectionoftheforceisalsocharacterizedbythesenseoftheforce.Forexample,asshowninFigure2.1(a)and2.1(b),twoforceshavingthesamemagnitudeandpointofapplicationbutoppositesenseofdirectionwillhavedirectlyoppositeeffectsontheparticle.

(b)Figure2.12.1ForcesinaPlane5PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.1.1Resultantof2forcesactingonaparticleTwoforcesPandQactingonaparticleA,Figure2.2(a)canbereplacedbyasingleforceRwhichhasthesameeffectontheparticle,Figure2.2(c).ThisforceRiscalledtheresultantoftheforcesPandQandthismethodforfindingtheresultantoftheforceiscalledtheparallelogramlawfortheadditionoftwoforces.2.1ForcesinaPlane(b) (c)Figure2.26PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaVectorsaredefinedasmathematicalexpressionspossessingmagnitudeanddirection,whichaddaccordingtotheparallelogramlaw.Itwilldistinguishfromscalarquantitiesinthismodulethroughtheuseofboldfacetype(P).Twovectorswithsamemagnitudeandthesamedirectionaresaidtobeequal,whetherornottheyhavethesamepointofapplication(Figure2.3).ThenegativevectorofagivenvectorPisthedefinedasavectorhavingthesamemagnitudeasPandadirectionoppositetothatofP

(Figure2.4).Figure2.3Figure2.4P+(-P)=02.2Vectors7PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.3.1AdditionofVectorsThesumoftwovectorsPandQcanbeobtainedbyattachingthetwovectorstothesamepointAandconstructingaparallelogram,usingPandQastwosidesoftheparallelogram(Figure2.5and2.6)

Figure2.5Figure2.6R=A+B=B+A

2.3VectorOperations8PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.3VectorOperations

2.3.2VectorSubtractionThevectorsubtractionisdefinedastheadditionofthecorrespondingnegativevector.(Figure2.7)ThevectorP–QrepresentsthedifferencebetweenthevectorsPandQ.

Figure2.7P–Q=P+(-Q)9PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.3VectorOperations2.3.3ResolutionofVectorAvectormayberesolvedintotwoormore“components”havingknownlinesofactionbyusingtheparallelogramlaw(Figure2.8)

Figure2.810PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaExample2.1ThetwoforcesPandQactsonaboltAasshowninFigure2.9(a).Determinetheresultants.Thetriangleisused.Twosidesandtheincludedangleareknown.Wecanapplythelawofcosines.

(b)

Figure2.911PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaExample2.112PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaWhenthesystemofmorethantwoforceshastobeobtained,itismuchmoreconvenientandeasiertofindthecomponentsofeachforcealongthespecifiedaxes,addthesecomponentsalgebraicallyandthenformtheresultantAlthoughtheaxesarehorizontalandvertical,theymaybedirectedatanyinclination,aslongastheyremainperpendiculartooneanother

Figure2.10Figure2.11Figure2.12(a)Figure2.12(b)2.4Additionofasystemofcoplanar forces13PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.4Additionofasystemofcoplanar forces2.4.1.ScalarNotationAxesxandyinFigure2.12(a)and(b)havebeendesignatedpositiveandnegativedirections.Therefore,themagnitudeanddirectionalsenseoftherectangularcomponentsofaforcecanbeexpressedintermsofalgebraicscalars.ThecomponentsFinFigure2.12(a)canberepresentedbytwopositivescalarsFxandFysincetheirsenseofdirectionisalongthepositivexandyaxisrespectively.Inasimilarway,thecomponentsofF’inFigure2.12(b)canberepresentedbyFx’and–Fy’.Thisscalarnotationistobeusedforcomputationalpurposesonly,notforgraphicalrepresentationinfigures

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OpenUniversityMalaysia2.4Additionofasystemofcoplanar forces2.4.2.CartesianVectorNotationTwovectorsofunitmagnitudecalledCartesianunitvectors

i

and

j

andaredirectedalongthexandyaxesrespectively.Thesevectorsarecalledunitvectorsandaredenotedbyiandjrespectively(Figure2.13)WecannotethattherectangularcomponentsFx

andFyofaforceFmaybeobtainedbymultiplyingrespectivelytheunitvectors

i

and

jbyappropriatescalars,Figure2.14

Figure2.13Figure2.14Fx=Fxi

and

Fy=FyjandF=Fxi

+Fyj15PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.4Additionofasystemofcoplanar forces2.4.2.CartesianVectorNotationThescalarsFx

and

Fy

arecalledthescalarcomponentsoftheforceFwhiletheactualcomponentforcesFxandFymustbereferredtoasthevectorcomponentsofF.WenotethatthescalarcomponentFxispositivewhenthevector

componentFxhasthesamesenseastheunitvector

i.DenotingbyFthemagnitudeoftheforceFandbytheanglebetweenF

andthexaxis,measuredcounterclockwisefromthepositivexaxis(Figure2.14),wemayexpressthescalarcomponentsofFas:

Fx=F

cos

Fy=Fsin

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Example2.2Aforceof800NisexertedonaboltAasshowinFigure(a).Determinethehorizontalandverticalcomponentsoftheforce.17PengenalanKepadaPangkalanData

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Example2.2ByinspectingthesignsofFxandFy(Figureb)andutilisingthetrigonometricfunctionsoftheangle,wecanwrite,ThevectorcomponentsofFarethus,

andwecanwriteFintheform18PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.4Additionofasystemofcoplanar forces2.4.3.CoplanarForceResultantsInordertodeterminetheresultantofseveralcoplanarforces,eachforceisfirstresolvedintoitsxandycomponents.Then,therespectivecomponentsareaddedusingscalaralgebrasincetheyarecollinear.Theresultantforcecanbethenformedbyaddingtheresultantsofxandycomponentsusingtheparallelogramlaw.Consider,forinstance,threeforces,F1,F2andF3actingonaparticleshowninFigure2.17(a).ThesethreeforceshavetheirrespectivecomponentsasshowninFigure2.17(b).InordertosolvethisproblemusingCartesianvectornotation,eachforcecanbefirstrepresentedasaCartesianvector,

19PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaThevectorresultantistherefore2.4.3.CoplanarForceResultants20PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaGenerally,thexandycomponentsoftheresultantofanynumberofcoplanarforcescanberepresentedsymbolicallybythealgebraicsumofthexandycomponentsofalltheforces,Positivecoordinateaxesareconsideredpositivescalars,whereasthosehavingadirectionalsensealongthenegativecoordinateaxesareconsiderednegativescalars.Oncetheresultantcomponentsaredetermined,theymaybesketchedalongthexandyaxesintheirproperdirectionsandtheresultantforcecanbedeterminedfromvectoraddition,asshowninFigure2.17(c).

2.4.3.CoplanarForceResultants21PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaFromthisfigure,themagnitudeofFRcanbethenfoundusingthePythagoreantheorem;Thedirectionanglewhichspecifiestheorientationoftheforce,isdeterminedfromthetrigonometry:2.4.3.CoplanarForceResultants22PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaExample2.6TheendofboomOinFigure2.20(a)issubjectedtothreeconcurrentandcoplanarforces.Determinethemagnitudeandorientationoftheresultantforce.AnswerEachforceisresolvedintoitsxandycomponents(Figure2.20(b)).Summingthexcomponents,wehave23PengenalanKepadaPangkalanData

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Example2.6Eachforceisresolvedintoitsxandycomponents(Figure2.20(b)).Summingthexcomponents,wehaveThenegativesignindicatesthatFRX

actstotheleft,i.e.tothenegativexdirectionasnotedbythesmallarrow.Summingtheycomponentsyields24PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaExample2.6TheresultantforceasshowninFigure2.20(c),hasamagnitudeofFromthevectoradditioninFigure2.20(c),thedirectionangleis:25PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.5.1Right-HandedCoordinateSystemACartesianorrectangularcoordinatesystemissaidtoberighthandedprovidedthethumboftherighthandpointsinthedirectionofthepositivezaxiswhentheright-handfingersarecurledaboutthisaxisanddirectedfromthepositivextowardthepositiveyaxis(Fig2.23).

2.5CartesianVectors26PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.5CartesianVectors2.5.2CartesianUnitVectorsInthreedimensions,thedirectionoftheaxesx,yandzisrepresentedbyCartesianunitvectorsofi,jandkrespectively(Fig.2.24).

27PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.5CartesianVectors2.5.3CartesianVectorRepresentationConsidervectorAasshowninFigure2.25.SincethethreecomponentsofAactinthepositivei,jandkdirectionsrespectively,onecanwriteAinCartesianvectorformas

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OpenUniversityMalaysia2.5CartesianVectors2.5.4MagnitudeofaCartesianVectorInFigure2.26,fromthestandingdarkcoloredrighttriangle,

andfromthelyingdownshadedrighttriangle,

.Combiningthesetwoequations,weobtainA

=√Ax2+Ay2+Az229PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.5.5DirectionofaCartesianVectorTheorientationofAisdefinedbythecoordinatedirectionangles(alpha),

(beta)and(gamma).,

and

canbedeterminedbyconsideringtheprojectionofAontothex,yandzaxes,

ThesenumbersarecalledthedirectioncosinesofA.

2.5CartesianVectorsAAx__cos

=__AAycos

=__AAzcos

=30PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.5CartesianVectors2.5.5DirectionofaCartesianVectorProvidedAisexpressedinCartesianvectorform,A=Axi+Ayj+Azk,then

Itcanbeseenfromtheaboveequation,thattheiandkcomponentsofuArepresentsthedirectioncosinesofA,i.e.

uA=cos

i

+cos

j+cos

k

uA==i+j+k

A

Ax

Ay

AzAAAA________andA

=√Ax2+Ay2+Az231PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.5CartesianVectors2.5.5DirectionofaCartesianVectorSincethemagnitudeofavectorisequaltothepositivesquarerootofthesumofthesquaresofthemagnitudesofitscomponentsanduAhasamagnitudeof1,thenanimportantrelationbetweenthedirectioncosinescanbeformulatedas, cos2

+cos2

+cos2=1IfthemagnitudeandcoordinatedirectionanglesofAaregiven,AcanbeexpressedinCartesianvectorformas

A=AuA

=Acos

i+Acos

j+Acos

k =Axi+Ayj+Az

k

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OpenUniversityMalaysiaA=Ax

i

+Ay

j+Az

k

andB=Bx

i

+By

j

+Bz

k,(Figure2.29),thentheresultantvectorRhascomponentsinthescalarsumsofthe

i,

j

and

k

componentsofAandB,

R=A+B =(Ax+Bx)i+(Ay+By)j

+(Az+Bz)kVectorsubtraction,simplyrequiresascalarsubtractionoftherespectivei,jandkcomponentsofeitherAorB.Forinstance,

R=A-B =(Ax-Bx)i

+(Ay-By)j+(Az-Bz)k2.6AdditionandSubtractionof CartesianVectors33PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.6AdditionandSubtractionof CartesianVectors2.6.1ConcurrentForceSystemsTheforceresultantisthevectorsumofalltheforcesinthesystemandcanbewrittenas:

FR=F =Fx

i+Fy

j+Fz

kFx,FyandFzrepresentthealgebraicsumsoftherespectivex,y,zori,j,kcomponentsofeachforceofthesystem.34PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaExample2.10DeterminethemagnitudeandthecoordinatedirectionanglesoftheresultantforceactingontheringasshowninFigure2.31(a).

Figure2.31(a) Figure2.31(b)35PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaExample2.10Theresultantforce,showninFigure2.31(b)is

FR=F=F1+F2={60j+80k}kN+{50i–100j+100k}kN ={50i–40j+180k}kN ThemagnitudeofFRisfoundtobe

FR

=

√

(50)2+(-40)2+(180)2=191kNThecoordinatedirectionanglesaredeterminedfromthecomponentsoftheunitvectoractinginthedirectionofFR.andtherefore,cos

=0.2617,

=74.8°cos

=-0.2094,=102°cos=0.9422,=19.6°uFR

==i-j+k

=0.2617i–0.2094j+0.9422k

FR

50

40180FR

191

191

191___________36PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.7.1x,y,zCoordinatesThepositivezaxisisdirectedupwardsothatitshowsthealtitudeoftheobjectoritcanalsobesaidthatitmeasurestheheightofanobject.Thex,yaxesthenlieinthehorizontalplane,Figure2.33.ThecoordinatesofpointAcanbeobtainedbystartingatOandmeasuringxA=+4malongthexaxis,yA=+2malongtheyaxisandzA=-6malongthezaxis.Therefore,A(4,2,-6).

2.7PositionVectors37PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.7.2PositionVectorThepositionvectorrisdefinedasafixedvectorthatlocatesthepointinspacerelativetoanotherpoint.Forinstance,ifrextendsfromtheoriginofcoordinatesO,topointP(x,y,z),Figure2.34(a),thenrcanbeexpressedintheCartesianvectorformas

r

=xi

+yj

+zk2.7PositionVectors38PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.7.2PositionVectorNotehowthehead-to-tailvectoradditionofthethreecomponentsyieldsvectorr,Figure2.34(b).StartingattheoriginO,onetravelsxinthepositiveidirection,thenyinthepositivejdirectionandzinthepositivekdirectiontoarriveatpointP(x,y,z).rcanalsobedesignatedasrAB.NotethatrAandrBinFigure2.35(a)arereferencedwithonlyonesubscriptsincetheyextendfromtheoriginofcoordinates.

2.7PositionVectors39PengenalanKepadaPangkalanData

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OpenUniversityMalaysia2.7.2PositionVectorFromFigure2.35(a),bythehead-to-tailvectoraddition,werequire

r

rA+r=rBSolvingforrandexpressingrAandrBinCartesianvectorformyields r=rB–rA

=(xBi+yBj+zBk)–(xAi+yAj+zAk)

orinothermanner,wecanwrite

r=(xB–xA)i+(yB

–yA)j+(zB

–

zA)k2.7PositionVectors40PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaWecanrepresentFasaCartesianvectorbyrealizingthatithasthesamedirectionandsenseasthepositionvectorrdirectedfrompointAtopointBonthecord.Thisdirectionisspecifiedbyunitvector.Hence,

F=Fu

=F[]rrrru=2.8ForceVectors41PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaExample2.12Figure2.37(a)showsthemanpullsthecordwithaforceof350N.Representthisforce,thatactsonpointAasaCartesianvectoranddetermineitsdirection.42PengenalanKepadaPangkalanData

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OpenUniversityMalaysiaExample2.12Figure2.37(b)showstheforce

F.Thedirectionofthisvector,

uisdeterminedfromthepositionvector,

r

whichextendsfromAtoB.

ThecoordinatesofA(0,0,7.5m)andB(3m,-2m,1.5m)areshowninFigure2.37(a).Thepositionvectorcanbeformedbysubtractingthecorrespondingx,yandzcoordinatesofAfromthoseofB.4