tutorial assignment 2 - part a.pdf
TRANSCRIPT
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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:
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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.
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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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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:
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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
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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:
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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 .
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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.
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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.
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