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  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 1/14

    Tutorialassignment2partADue:12:20pmonThursday,October9,2014

    Youwillreceivenocreditforitemsyoucompleteaftertheassignmentisdue.GradingPolicy

    MomentofaForceVectorFormulation

    LearningGoal:

    Tousethevectorcrossproducttocalculatethemomentproducedbyaforce,orforces,aboutaspecifiedpointonamember.

    Themomentofaforce aboutthemomentaxispassingthroughOandperpendiculartotheplanecontainingOandcanbeexpressedusingthevectorcrossproduct, .InaproperlyconstructedCartesiancoordinatesystem,thevectorcrossproductcanbecalculatedusingamatrixdeterminant:

    Noticethattheorderoftheelementsofthematrixdeterminantisimportantswitchingrows2and3ofthedeterminantwouldchangethesignofthemomentfrompositivetonegative(orviceversa.)

    PartAMomentduetoaforcespecifiedbymagnitudeandendpoints

    Asshown,amemberisfixedattheorigin,pointO,andhasanappliedforce ,thetensionintherope,appliedatthefreeend,pointB.

    Theforcehasmagnitude =180 andisdirectedasshown.Thedimensionsare

    =0.250 , =1.70 , =2.30 ,and=1.10 .

    Whatisthemomentabouttheoriginduetotheappliedforce ?

    ExpresstheindividualcomponentsoftheCartesianvectortothreesignificantfigures,separatedbycommas.

    Hint1.Howtocalculatethemomentabouttheorigin

    WeneedtoexpresstheforceandthepositionvectorofthefreeendofthememberasCartesianvectorsthenusethevectorcrossproducttofindthemoment.

    Hint2.ExpresstheforceasaCartesianvector

    Theforcehasmagnitude =180Nandisdirectedasshown.ExpresstheforceasaCartesianvector.

    ExpresstheindividualcomponentsoftheCartesianvectortothreesignificantfigures,separatedbycommas.

    ANSWER:

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  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 2/14

    Hint3.ExpressthepositionvectorofthememberasaCartesianvector

    ThememberisanchoredatpointOandhasitsfreeendatB.Expressthepositionvector asaCartesianvector.

    ExpresstheindividualcomponentsoftheCartesianvectortothreesignificantfigures,separatedbycommas.

    ANSWER:

    Hint4.Identifythedeterminantofa3by3matrix

    Whichofthefollowingcorrectlyexpressesthedeterminantofthematrix?

    Hint1.Expansionbyminors

    Themethodofexpansionbyminorscanbeusedtocalculatedeterminantsoflargematricesbyreducingthemtoalinearcombinationofdeterminantsof2by2matrices.Forcalculatingmoments,thebestchoiceofminorsisthetoprowbecauseithastheunitdirectionvectors.Thecomponentsarethengivenbythedeterminantofthe2by2matrixleftafter"crossingout"therowandcolumncontainingtheminor:

    Thedeterminantofa2by2matrixisgivenby

    andeachterminthesumwillalsocontainaterm ,where and aretheindexoftherowandcolumnoftheminor,respectively.

    ANSWER:

    99.9,135,64.6' < J KL> /

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    0.250,2.30,1.100# < J KL> N

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  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 3/14

    ANSWER:

    Correct

    PartBMomentduetoaforcespecifiedasaCartesianvector

    Asshown,amemberisfixedattheorigin,pointO,andhasanappliedforce ,thetensionintherope,appliedatthefreeend,point .

    Theforceisgivenby.Thedimensions

    are =1.35m, =1.80m,and =1.05m.

    Whatisthemomentabouttheoriginduetotheappliedforce ?

    ExpresstheindividualcomponentsoftheCartesianvectortothreesignificantfigures,separatedbycommas.

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  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 4/14

    Hint1.Howtosetupandcalculatethemoment

    TheforceisalreadyexpressedasaCartesianvector.WeneedtoexpressthepositionvectorofthefreeendofthememberasaCartesianvectorthenusethecrossproducttofindthemoment.

    Hint2.ExpressthepositionvectorofBasaCartesianvector

    Basedonthedimensionsgiven,expressthevectorfromOtoBasaCartesianvector.

    ExpresstheindividualcomponentsoftheCartesianvectortothreesignificantfigures,separatedbycommas.

    ANSWER:

    ANSWER:

    Correct

    PartCMomentduetotwoforces

    Asshown,amemberisfixedattheorigin,pointO,andhastwoappliedforces, and ,appliedatthefreeend,pointB.

    Theforcesaregivenbyand has

    magnitude155 anddirectionangles ,,and .Thedimensionsare

    =1.30 , =1.85 ,and =1.20 .

    Whatisthemomentabouttheoriginduetotheappliedforces?

    ExpresstheindividualcomponentsoftheCartesianvectortothreesignificantfigures,separatedbycommas.

    Hint1.Howtosetupandcalculatethemoment

    Force isalreadyexpressedasaCartesianvector.BecausethepositionofpointBisgivenintermsofcomponents,wealsohave .Weneedtoexpresstheforce asaCartesianvector.Thenetmomentabout duetobothforcesisthevectorsumofthemomentsduetoeachforce.Calculateindividualmomentsbysettingupadeterminantforeachforcethensumtheresultscomponentwisetofindthenetmoment.

    1.35,1.80,1.050# < J KL> N

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  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 5/14

    Hint2.Expresstheforce asaCartesianvector

    Force hasmagnitude155 anddirectionangles , ,and .ExpressthisforceasaCartesianvector.

    ExpresstheindividualcomponentsoftheCartesianvectortothreesignificantfigures,separatedbycommas.

    Hint1.Thedirectionangles

    Thedirectionangle istheanglebetweenthevectorandthepositivexaxis.Thedirectionangle istheanglebetweenthevectorandthepositiveyaxis.Thedirectionangle istheanglebetweenthevectorandthepositivezaxis.

    ANSWER:

    Hint3.CalculatethemomentduetoforceF1

    Whatisthemomentabouttheoriginduetotheappliedforce ?

    ExpresstheindividualcomponentsoftheCartesianvectortothreesignificantfigures,separatedbycommas.

    ANSWER:

    Hint4.CalculatethemomentduetoforceF2

    Whatisthemomentabouttheoriginduetotheappliedforce ?

    ExpresstheindividualcomponentsoftheCartesianvectortothreesignificantfigures,separatedbycommas.

    ANSWER:

    ANSWER:

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    124,82.1,44.3

  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 6/14

    Correct

    Therearetwowaystoapproachthisproblem.One,themethodusedinthistutorial,istofindthenetmomentasasumofmoments,i.e.,calculateamomentforeachappliedforcethenusevectoradditiontofindthenetmoment.Equivalently,wecouldhavefoundthevectorsumoftheforcesthencalculatedasingledeterminantusingtheresultantnetforceandthepositionvectortofindthenetmoment.Thesetwomethodsproducethesamenetmoment.(Youmightfinditaninterestingexercisetoconfirmthis.)

    PrincipleofMoments

    LearningGoal:

    Toapplytheprincipleofmomentsandtheprincipleoftransmissibility.

    Asshown,aropeisattachedtoa highshedthatistoberelocated.AmanpullsontheendoftheropeatpointAtheropeisattachedtotheshedatpointB.Asthemanpullsontherope,itcreatesanangle withthehorizontal.Theendoftheropeislocatedat fromtheshedand offtheground.Themanpullsontheropewithaforceofmagnitude .

    PartA

    Whatis ,thecontributiontothemomentaboutpointOmadebythexcomponentoftheforce atpointA?Whatis ,thecontributiontothemomentaboutpointOmadebytheycomponentoftheforce atpointA?Whatisthetotalmoment duetotheforce aboutpointO?AssumethatmomentsactingcounterclockwiseaboutpointOarepositivewhereasmomentsactingclockwisearenegative.

    Expressyouranswersnumericallyinpoundfeettothreesignificantfiguresseparatedbycommas.

    Hint1.Howtoapproachtheproblem

    Theprincipleofmomentssimplifiestheprocessofcalculatingamomentbyseparatingaforceintoitscomponents.Tocalculatethetotalmoment,addthemomentsthatresultfromtheseparatecomponents:

    Here, and arethexandycomponentsoftheforce .AssumethatmomentsactingcounterclockwiseaboutpointOarepositivewhereasmomentsactingclockwisearenegative.

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  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 7/14

    Hint2.Findanexpressionfor

    WhatisthecontributiontothetotalmomentaboutpointOfromthexcomponentoftheforceatpointAintermsoftheforce ,theangle ,andthedistances and ?

    Expressyouranswerintermssomeorallofthevariables , , ,and .

    Hint1.Findanexpressionfor

    Whatis ,thexcomponentoftheforce,intermsoftheforce'smagnitude andtheangle ?

    Expressyouranswerintermsof and .

    ANSWER:

    Hint2.Findthemomentarmofthexcomponentoftheforce

    Whatisthemomentarmfor ,thexcomponentoftheforce ?

    ANSWER:

    ANSWER:

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  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 8/14

    Hint3.Findanexpressionfor

    WhatisthecontributiontothetotalmomentaboutpointOfromtheycomponentoftheforceatpointAintermsoftheforce ,theangle ,andthedistances and ?

    Expressyouranswerintermsofsomeorallofthevariables , , ,and .

    Hint1.Findanexpressionfor

    Whatistheycomponentoftheforce intermsoftheforce'smagnitude andtheangle ?

    Expressyouranswerintermsof and .

    ANSWER:

    Hint2.Findthemomentarmoftheycomponentoftheforce

    Whatisthemomentarmfor ,theycomponentoftheforce ?

    ANSWER:

    ANSWER:

    5

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    TJOJ4

  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 9/14

    Hint4.Find and

    Thevaluesof and canbecalculatedusingthegivenlengths.Whatare and ?

    Expressyouranswersnumericallytofoursignificantfiguresseparatedbyacomma.

    Hint1.Findanexpressionfor

    Whatis intermsoftheknownlengths , ,and ?

    Expressyouranswerintermsof , ,and .

    Hint1.Rightangletrigonometry

    Recallthatthefunctionssine,cosine,andtangentrelatethesidesofarighttriangle.

    Therefore , , ,and .

    ANSWER:

    Hint2.Findanexpressionfor

    Whatis intermsoftheknownlengths , ,and ?

    Expressyouranswerintermsof , ,and .

    Hint1.Rightangletrigonometry

    Recallthatthefunctionssine,cosine,andtangentrelatethesidesofarighttriangle.

    Therefore , , ,and .

    ANSWER:

    TJOJ DPTJ

    TJOJ DPTJ TJOJ DPTJ

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    4 5 (

    TJOJ 5$ DPTJ 4$ UBOJ 54 $

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    TJOJ 5$ DPTJ 4$ UBOJ 54 $

    4

    5

  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 10/14

    ANSWER:

    ANSWER:

    Correct

    PartB

    Usingtheprincipleoftransmissibility,theforce canbeslidalongitslineofactionfrompointAtoanyotherpointalongitslineofaction.Whatpointsalongthelineofactionoftheforce wouldsimplifythecalculationof ,themagnitudeofthetotalmomentduetotheforce aboutpointO?

    Checkallthatapply.

    Hint1.Thecrossproductofcollinearvectors

    Whentwovectorsarecollinear(i.e.,theanglebetweenthetwovectorsiseither0degreesor180degrees)thecrossproductofthetwovectorsiszero.

    ANSWER:

    =DPTJ4

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    4

    , = 0.8575,0.5145TJOJ DPTJ

    , , = 103,386,489

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  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 11/14

    Correct

    Thecalculationofthemoment aboutpointOduetotheforce canbesimplifiedbyslidingtheforcealongitslineofactionanotherpoint.AlthougheitherpointBorEisavalidchoice,herethemoment willbecalculatedbyslidingtheforce topointB.

    PartC

    Usingtheprincipleoftransmissibility,theforce canbeslidalongitslineofactiontopointB.AtpointB,theforcecreatesanangle withtheshed.Whatis ,the

    contributiontothemomentaboutpointOmadebythexcomponentoftheforce atpointB?Whatis ,thecontributiontothemomentaboutpointOmadebytheycomponentoftheforce atpointB?Whatisthetotalmoment duetotheforce aboutpointO?AssumethatmomentsactingcounterclockwiseaboutpointOarepositivewhereasmomentsactingclockwisearenegative.

    Expressyouranswersnumericallyinpoundfeettothreesignificantfiguresseparatedbycommas.

    Hint1.Howtoapproachtheproblem

    Theprincipleoftransmissibilitycanbeusedtorelocateaforceand,therefore,simplifythemomentcalculation.Theprincipleofmomentscanthenbeusedagaintocalculatethemoment.AssumethatmomentsactingcounterclockwiseaboutpointOarepositivewhereasmomentsactingclockwisearenegative.

    Hint2.Find

    WhatisthecontributiontothetotalmomentaboutpointOfrom ,theycomponentoftheforce atpointB?

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  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 12/14

    Expressyouranswernumericallyinpoundfeettothreesignificantfigures.

    Hint1.Thecrossproductofcollinearvectors

    Whentwovectorsarecollinear(i.e.,theanglebetweenthevectorsiseither0degreesor180degrees),thecrossproductofthetwovectorsiszero.

    ANSWER:

    Hint3.Findanexpressionfor

    WhatisthecontributiontothetotalmomentaboutpointOfrom ,thexcomponentoftheforce atpointB,intermsoftheforce'smagnitude ,theangle ,andtheheightoftheshed ?

    Expressyouranswerintermsof , ,and .

    = 0.000

    5

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    ] (

    ] (

  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 13/14

    Hint1.Findanexpressionfor

    Whatis ,thexcomponentoftheforce intermsoftheforce'smagnitude andtheangle ?

    Expressyouranswerintermsof and .

    Hint1.Rightangletrigonometry

    Recallthatthefunctionssine,cosine,andtangentrelatethesidesofarighttriangle.

    Therefore , , ,and .

    ANSWER:

    ANSWER:

    Hint4.Find

    Thevalueof canbecalculatedusingthegivenlengths.Whatis ?

    Expressyouranswernumericallytofoursignificantfigures.

    Hint1.Findanexpressionfor

    Whatis intermsoftheknownlengths , ,and ?

    Expressyouranswerintermsof , ,and .

    Hint1.Rightangletrigonometry

    Recallthatthefunctionssine,cosine,andtangentrelatethesidesofarighttriangle.

    4

    4

    ' ]

    ]

    TJOJ 5$ DPTJ 4$ UBOJ 54 $

    4

    5

    =4

    TJO]

    =

    4

    TJO](

    TJO]

    TJO] TJO]

    TJO]

    TJO] 4 5 (

    4 5 (

  • 10/14/2014 Tutorialassignment2partA

    http://session.masteringengineering.com/myct/assignmentPrintView?assignmentID=1206301 14/14

    Therefore , , ,and .

    ANSWER:

    ANSWER:

    ANSWER:

    Correct

    TheapproachesusedinPartsAandCtocalculatethetotalmomentoftheforceaboutpointOyieldthesameresult.Bymovingtheforcealongitslineofaction,thecalculationcanbesimplifiedbyeliminatingonecontributiontothemomentfromacomponentoftheforce.Analternativeapproachistoslidetheforcealongitslineofactiontotheground.ThisresultsinthexcomponentofforcecreatingazeromomentaboutpointOandallowingthemomenttobecalculatedentirelyfromtheycomponentoftheforce.

    ScoreSummary:Yourscoreonthisassignmentis107%.Youreceived21.4outofapossibletotalof20points.

    TJOJ 5$ DPTJ 4$ UBOJ 54 $

    4

    5

    =TJO]4

    4

    (5

    = 0.5145TJO]

    , , = 489,0,489

    4

    5

    MC GU