Chapter 8

RaymondA.Serway
ChrisVuille
ChapterEight
Rota:onalEquilibriumand
Rota:onalDynamics
Applica:onofForces
•  Thepointofapplica:onofaforceisimportant
–  Thiswasignoredintrea:ngobjectsaspoint
par:cles
•  Theconceptsofrota:onalequilibriumand
rota:onaldynamicsareimportantinmany
fieldsofstudy
•  Angularmomentummaybeconserved
•  Angularmomentummaybechangedby
exer:ngatorque
Introduc:on
Forcevs.Torque
•  Forcescauseaccelera:ons
•  Torquescauseangularaccelera:ons
•  Forceandtorquearerelated
Sec:on8.1
Torque
•  ThedoorisfreetorotateaboutanaxisthroughO
•  Therearethreefactorsthatdeterminetheeffec:venessofthe
forceinopeningthedoor:
–  Themagnitudeoftheforce
–  Theposi-onoftheapplica:onoftheforce
–  Theangleatwhichtheforceisapplied
Sec:on8.1
Torque,cont
•  Torque,τ,isthetendencyofaforcetorotate
anobjectaboutsomeaxis
–  τ =rF
•  τisthetorque
–  SymbolistheGreektau
•  risthelengthoftheposi:onvector
•  Fistheforce
•  SIunitisNewton.meter(N.m)
Sec:on8.1
Direc:onofTorque
•  Torqueisavectorquan:ty
–  Thedirec:onisperpendiculartotheplane
determinedbytheposi:onvectorandtheforce
–  Iftheturningtendencyoftheforceis
counterclockwise,thetorquewillbeposi:ve
–  Iftheturningtendencyisclockwise,thetorque
willbenega:ve
Sec:on8.1
Mul:pleTorques
•  Whentwoormoretorquesareac:ngonan
object,thetorquesareadded
–  Asvectors
•  Ifthenettorqueiszero,theobject’srateof
rota:ondoesn’tchange
Sec:on8.1
GeneralDefini:onofTorque
•  Theappliedforceisnotalwaysperpendicular
totheposi:onvector
•  Thecomponentoftheforceperpendicularto
theobjectwillcauseittorotate
Sec:on8.1
GeneralDefini:onofTorque,cont
•  Whentheforceisparallelto
theposi:onvector,no
rota:onoccurs
•  Whentheforceisatsome
angle,theperpendicular
componentcausesthe
rota:on
Sec:on8.1
GeneralDefini:onofTorque,final
•  Takingtheangleintoaccountleadstoamore
generaldefini:onoftorque:
–  τ = rFsinθ
•  ristheposi:onvector
•  Fistheforce
•  θistheanglebetweentheforceandtheposi:on
vector
Sec:on8.1
LeverArm
•  Theleverarm,d,istheperpendiculardistancefromtheaxisof
rota:ontoalinedrawnalongthedirec:onoftheforce
•  d=rsinθ
•  Thisalsogivesτ=rFsinθ
Sec:on8.1
TorqueandAxis
•  Thevalueofthetorquedependsonthe
chosenaxisofrota:on
•  Torquescanbecomputedaroundanyaxis
–  Theredoesn’thavetobeaphysicalrota:onaxis
present
•  Onceapointischosen,itmustbeused
consistentlythroughoutagivenproblem
Sec:on8.1
RightHandRule
•  Pointthefingersinthe
direc:onoftheposi:on
vector
•  Curlthefingerstoward
theforcevector
•  Thethumbpointsinthe
direc:onofthetorque
Sec:on8.1
NetTorque
•  Thenettorqueisthesumofallthetorques
producedbyalltheforces
–  Remembertoaccountforthedirec:onofthe
tendencyforrota:on
•  Counterclockwisetorquesareposi:ve
•  Clockwisetorquesarenega:ve
Sec:on8.2
TorqueandEquilibrium
•  FirstCondi:onofEquilibrium
–  Thenetexternalforcemustbezero
–  Thisisastatementoftransla:onalequilibrium
•  TheSecondCondi:onofEquilibriumstates
–  Thenetexternaltorquemustbezero
–  Thisisastatementofrota:onalequilibrium
Sec:on8.2
Selec:nganAxis
•  It’susuallybesttochooseanaxisthatwill
makeatleastonetorqueequaltozero
–  Thiswillsimplifythetorqueequa:on
Sec:on8.2
EquilibriumExample
•  Thewoman,massm,sitson
theleZendofthesee-saw
•  Theman,massM,sits
wherethesee-sawwillbe
balanced
•  ApplytheSecondCondi:on
ofEquilibriumandsolvefor
theunknowndistance,x
Sec:on8.2
CenterofGravity
•  Theforceofgravityac:ngonanobjectmust
beconsidered
•  Infindingthetorqueproducedbytheforceof
gravity,alloftheweightoftheobjectcanbe
consideredtobeconcentratedatasingle
point
Sec:on8.3
Calcula:ngtheCenterofGravity
•  Theobjectisdividedupinto
alargenumberofvery
smallpar:clesofweight
(mg)
•  Eachpar:clewillhaveaset
ofcoordinatesindica:ngits
loca:on(x,y)
Sec:on8.3
Calcula:ngtheCenterofGravity,cont.
•  Weassumetheobjectisfreetorotateabout
itscenter
•  Thetorqueproducedbyeachpar:cleabout
theaxisofrota:onisequaltoitsweight:mes
itsleverarm
–  Forexample,τ1 = m1gx1
Sec:on8.3
Calcula:ngtheCenterofGravity,cont.
•  Wewishtolocatethepointofapplica:onof
thesingleforcewhosemagnitudeisequalto
theweightoftheobject,andwhoseeffecton
therota:onisthesameasalltheindividual
par:cles.
•  Thispointiscalledthecenterofgravityofthe
object
Sec:on8.3
CoordinatesoftheCenterofGravity
•  Thecoordinatesofthecenterofgravitycan
befoundfromthesumofthetorquesac:ng
ontheindividualpar:clesbeingsetequalto
thetorqueproducedbytheweightofthe
object
Sec:on8.3
CenterofGravityandCenterofMass
•  Thethreeequa:onsgivingthecoordinatesof
thecenterofgravityofanobjectareiden:cal
totheequa:onsgivingthecoordinatesofthe
centerofmassoftheobject
•  Thecenterofgravityandthecenterofmass
oftheobjectarethesameifthevalueofg
doesnotvarysignificantlyovertheobject
Sec:on8.3
CenterofGravityofaUniformObject
•  Thecenterofgravityofahomogenous,
symmetricbodymustlieontheaxisof
symmetry
•  OZen,thecenterofgravityofsuchanobject
isthegeometriccenteroftheobject
Sec:on8.3
ExperimentallyDeterminingtheCenter
ofGravity
•  Thewrenchishungfreely
fromtwodifferentpivots
•  Theintersec:onofthelines
indicatesthecenterof
gravity
•  Arigidobjectcanbe
balancedbyasingleforce
equalinmagnitudetoits
weightaslongastheforce
isac:ngupwardthrough
theobject’scenterof
gravity
Sec:on8.3
NotesAboutEquilibrium
•  Azeronettorquedoesnotmeantheabsence
ofrota:onalmo:on
–  Anobjectthatrotatesatuniformangularvelocity
canbeundertheinfluenceofazeronettorque
•  Thisisanalogoustothetransla:onalsitua:onwherea
zeronetforcedoesnotmeantheobjectisnotin
mo:on
Sec:on8.4
SolvingEquilibriumProblems
•  Diagramthesystem
–  Includecoordinatesandchooseaconvenientrota:onaxis
•  Drawafreebodydiagramshowingalltheexternal
forcesac:ngontheobject
–  Forsystemscontainingmorethanoneobject,drawa
separatefreebodydiagramforeachobject
Sec:on8.4
ProblemSolving,cont.
•  ApplytheSecondCondi:onofEquilibrium
–  Thiswillyieldasingleequa:on,oZenwithoneunknown
whichcanbesolvedimmediately
•  ApplytheFirstCondi:onofEquilibrium
–  Thiswillgiveyoutwomoreequa:ons
•  Solvetheresul:ngsimultaneousequa:onsforallof
theunknowns
–  Solvingbysubs:tu:onisgenerallyeasiest
Sec:on8.4
ExampleofaFreeBodyDiagram
(Forearm)
•  Isolatetheobjecttobeanalyzed
•  Drawthefreebodydiagramforthatobject
–  Includealltheexternalforcesac:ngontheobject
Sec:on8.4
ExampleofaFreeBodyDiagram
(Ladder)
•  Thefreebodydiagramshowsthenormalforceandtheforceof
sta:cfric:onac:ngontheladderattheground
•  Thelastdiagramshowstheleverarmsfortheforces
Sec:on8.4
ExampleofaFreeBodyDiagram
(Beam)
•  Thefreebodydiagram
includesthedirec:ons
oftheforces
•  Theweightsactthrough
thecentersofgravityof
theirobjects
Sec:on8.4
TorqueandAngularAccelera:on
•  Whenarigidobjectissubjecttoanettorque
(Στ≠0),itundergoesanangularaccelera:on
•  Theangularaccelera:onisdirectly
propor:onaltothenettorque
–  Therela:onshipisanalogousto∑F=ma
•  Newton’sSecondLaw
Sec:on8.5
MomentofIner:a
•  Theangularaccelera:onisinversely
propor:onaltotheanalogyofthemassina
rota:ngsystem
•  Thismassanalogiscalledthemomentof
iner-a,I,oftheobject
–  SIunitsarekgm2
Sec:on8.5
Newton’sSecondLawforaRota:ng
Object
•  Theangularaccelera:onisdirectly
propor:onaltothenettorque
•  Theangularaccelera:onisinversely
propor:onaltothemomentofiner:aofthe
object
Sec:on8.5
MoreAboutMomentofIner:a
•  Thereisamajordifferencebetweenmoment
ofiner:aandmass:themomentofiner:a
dependsonthequan:tyofmagerandits
distribu:onintherigidobject
•  Themomentofiner:aalsodependsuponthe
loca:onoftheaxisofrota:on
Sec:on8.5
MomentofIner:aofaUniformRing
•  Imaginethehoopis
dividedintoanumber
ofsmallsegments,m1…
•  Thesesegmentsare
equidistantfromthe
axis
Sec:on8.5
OtherMomentsofIner:a
Sec:on8.5
Example,Newton’sSecondLawfor
Rota:on
•  Drawfreebodydiagramsof
eachobject
•  Onlythecylinderisrota:ng,
soapplyΣτ=Iα
•  Thebucketisfalling,butnot
rota:ng,soapplyΣF=ma
•  Rememberthata=αrand
solvetheresul:ng
equa:ons
Sec:on8.5
Rota:onalKine:cEnergy
•  Anobjectrota:ngaboutsomeaxiswithan
angularspeed,ω,hasrota:onalkine:cenergy
KEr=½Iω2
•  Energyconceptscanbeusefulforsimplifying
theanalysisofrota:onalmo:on
Sec:on8.6
Conserva:onofEnergy
•  Conserva:onofMechanicalEnergy
–  Remember,thisisforconserva:veforces,no
dissipa:veforcessuchasfric:oncanbepresent
–  Poten:alenergiesofanyotherconserva:ve
forcescouldbeadded
Sec:on8.6
Work-EnergyinaRota:ngSystem
•  Inthecasewheretherearedissipa:veforces
suchasfric:on,usethegeneralizedWorkEnergyTheoreminsteadofConserva:onof
Energy
•  Wnc=ΔKEt+ΔKEr+ΔPE
Sec:on8.6
ProblemSolvingHintsforEnergy
Methods
•  Choosetwopointsofinterest
–  Onewhereallthenecessaryinforma:onisgiven
–  Theotherwhereinforma:onisdesired
•  Iden:fytheconserva:veandnonconserva:ve
forces
Sec:on8.6
ProblemSolvingHintsforEnergy
Methods,cont
•  Writethegeneralequa:onfortheWorkEnergytheoremiftherearenonconserva:ve
forces
–  UseConserva:onofEnergyifthereareno
nonconserva:veforces
•  Usev=rωtoeliminateeitherωorvfromthe
equa:on
•  Solvefortheunknown
Sec:on8.6
AngularMomentum
•  Similarlytotherela:onshipbetweenforce
andmomentuminalinearsystem,wecan
showtherela:onshipbetweentorqueand
angularmomentum
•  Angularmomentumisdefinedas
–  L=Iω
–  and
Sec:on8.7
AngularMomentum,cont
•  Ifthenettorqueiszero,theangularmomentum
remainsconstant
•  Conserva-onofAngularMomentumstates:LetLi
andLfbetheangularmomentaofasystemattwo
different:mes,andsupposethereisnonetexternal
torquesoΣτ=0,thenangularmomentumissaidto
beconserved
Sec:on8.7
Conserva:onofAngularMomentum
•  Mathema:cally,when
–  Appliestomacroscopicobjectsaswellasatoms
andmolecules
Sec:on8.7
Conserva:onofAngularMomentum,
Example
•  Withhandsandfeet
drawnclosertothe
body,theskater’s
angularspeedincreases
–  Lisconserved,I
decreases,ωincreases
Sec:on8.7
Conserva:onofAngularMomentum,
Example,cont
•  Comingoutofthespin,
armsandlegsare
extendedandrota:on
isslowed
–  Lisconserved,I
increases,ωdecreases
Sec:on8.7
Conserva:onofAngularMoment,
AstronomyExample
•  CrabNebula,resultofsupernova
•  Centerisaneutronstar
–  Asthestar’smomentofiner:adecreases,itsrota:onal
speedincreases
Sec:on8.7