Lecture3:Equilibriumof
ParticlesandRigidBodies
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Objectives
• Tointroducetheconceptof“Free-body
diagram”forparticlesandrigidbodies.
• Toshowhowtosolveequilibriumproblems
usingtheequationsofequilibrium.
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ConditionsfortheEquilibriumofa
Particle
Remember the definition for aparticle:Particle hasamass,but
negligible size(e.g.,earth orbiting the sun)
Necessaryandsufficientconditionforthe
equilibriumofaparticle:
ΣF=ma=0
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TheFree-BodyDiagram(FBD)
• FBDisadrawingthatshowstheΣFonthe
particle.
– Thinkoftheparticleasisolatedandfreefromits
surroundings.
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ProcedureforDrawingaFree-Body
Diagram
1. Drawtheoutlinedshape.
2. Showallforces.
3. Identifyeachforce.
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ConnectionsinParticleEquilibrium
Problems
Spring
Cableintension
Cableshavenegligible
weightandcannotstretch.
Forafrictionlesspulley,Tis
constantalongthecable.
F=ks
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CoplanarForceSystems
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3DForceSystems
Determinetheforceineachcable
usedtosupportthe40lbcrate.
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ConditionsfortheEquilibriumofa
RigidBody
Generalrigidbodymotion:3translation+3
rotation(6degreesoffreedom,DOF)
Planarrigidbodymotion:2translation+1
rotationà 3independentquantities(3DOF)
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ConditionsforEquilibriumforRigid
Bodies
• Necessaryandsufficientconditionsfor
equilibrium:
– ΣFR =ma=0
– Σ(MR)o =0
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RigidBodies- Beams
Structuralelements thatarecapableofwithstandingloadprimarilybyresistingagainstbending
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RigidBodies- Beams
Typesofbeamsdependingonhowtheyaresupported
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SupportReactions
Supportsimposerestrictions(constraints)on
themotionofbodiesinspecificdirections.
GeneralRules:
1. Ifasupportpreventsthetranslationofa
bodyinagivendirection,thenaforceis
developedonthebodyinthatdirection.
2. Ifarotationisprevented,acouplemomentis
exertedonthebody.
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SupportReactions
Mostcommonsupportswewillencounterin
thisclass:
1. Rollersupport:
3.Fixed(cantilever)support:
2. Pinsupport:
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SupportReactions
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SupportReactions
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FreeBodyDiagramsforRigidBodies
• Establishthex,ycoordinateaxesinasuitableorientation.
• Drawanoutlinedshapeofthebody.
• Representalltheknownandunknownexternalforcesonthe
body,sothatequilibriumconditionscanbeapplied.
MR
F1
F2
FRy
FRx
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FreeBodyDiagramExercises
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EquationsofEquilibrium
• OncetheFBDisdrawnusetheequationsof
equilibriumforrigidbodiestofindthe
reactionforces.
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EquationsofEquilibrium
Determine thehorizontaland
verticalcomponentsof
reactiononthebeamcaused
bythepinatBandtherocker
atAasshowninthefigure.
Neglecttheweightofthe
beam.
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Free BodyDiagram Exercises
1. DrawaFBDofthebar,which
hasnoweight,hassmooth
pointsofcontactatA,Band
C.
2. Determinethenormal
reactionsatthepointsof
contactA,BandC.(Hint:
reorientthecoordinateaxis).
Justwritedownthe
equilibriumequations.You
donotneedtosolvethem.
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EquilibriumofBodieswithmorethan
OneMember
• Moststructuresconsistofmorethanone
member.
• Ifthewholestructureisinequilibrium,then
allmemberscomprisingthatstructurearealso
inequilibrium.Apply:
– Globalequilibrium
– Localequilibrium
tosolveforunknownforces
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TwoForceMembers
Someequilibriumproblemscan
besimplified byrecognizing
membersthataresubjecttoonly
twoforces.
Checkpoints:
*Pinsatbothends
*No(external)forces
inbetweenthepins
Two-force
member
Attention:
Atwo-forcemember
cannothaveamoment
appliedtoit
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RecognizingTwoForceMembers
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RecognizingTwoForceMembers
Themassof700kgissuspendedfroma
trolleywhichmovesalongacranerailata
distanceofdfromthesupportA.
1. DeterminetheforcealongthepinconnectedkneestrutBC(shortlink)
2. Determinethemagnitudeofforceat
pinAasafunctionofpositiond.
Takeg=10m/s2
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BackupSlides
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ThreeForceMembers
Momentequilibrium canbe
satisfiedaslongasthethree
forcesformaconcurrentor
parallelforcesystem.
Three-force
member
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Whatwehavecoveredlasttime
Mo=Fd (Scalar
Definition)
Mo=Fd =Fxy - Fyx
VectorDefinition
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Whatwehavecoveredlasttime
Momentaboutanaxis
CoupleMoment
M=Fd (Magnitude).
Orientationgivenbytherighthandrule.
M =r xF(Vectordefinition)
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Whatwehavecoveredlasttime
SimplificationofaForceand
CoupleSystem
CoplanarDistributedLoading
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