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Rota9onalMo9on
• Animportantpartofeverydaylife– Mo9onoftheEarth– Rota9ngwheels
• Angularmo9on– Expressedintermsof
• Angularspeed• Angularaccelera9on• Centripetalaccelera9on
Introduc9on
Gravity
• Rota9onalmo9oncombinedwithNewton’sLawofUniversalGravityandNewton’sLawsofmo9oncanexplainaspectsofspacetravelandsatellitemo9on
• Kepler’sThreeLawsofPlanetaryMo9on– Formedthefounda9onofNewton’sapproachtogravity
Introduc9on
AngularMo9on
• Willbedescribedintermsof– Angulardisplacement,Δθ– Angularvelocity,ω– Angularaccelera9on,α
• Analogoustothemainconceptsinlinearmo9on
Sec9on7.1
TheRadian
• Theradianisaunitofangularmeasure
• Theradiancanbedefinedasthearclengthsalongacircledividedbytheradiusr
•
Sec9on7.1
AngularDisplacement
• Axisofrota9onisthecenterofthedisk
• Needafixedreferenceline
• During9met,thereferencelinemovesthroughangleθ
• Theangle,θ,measuredinradians,istheangularposi,on
Sec9on7.1
RigidBody
• Everypointontheobjectundergoescircularmo9onaboutthepointO
• Allpartsoftheobjectofthebodyrotatethroughthesameangleduringthesame9me
• Theobjectisconsideredtobearigidbody– Thismeansthateachpartofthebodyisfixedinposi9onrela9vetoallotherpartsofthebody
Sec9on7.1
AngularDisplacement,cont.• Theangulardisplacementis
definedastheangletheobjectrotatesthroughduringsome9meinterval
• • Theunitofangular
displacementistheradian• Eachpointontheobject
undergoesthesameangulardisplacement
Sec9on7.1
AverageAngularSpeed
• Theaverageangularspeed,ω,ofarota9ngrigidobjectisthera9ooftheangulardisplacementtothe9meinterval
Sec9on7.1
AngularSpeed,cont.
• Theinstantaneousangularspeedisdefinedasthelimitoftheaveragespeedasthe9meintervalapproacheszero
• SIunit:radians/sec– rad/s
• Speedwillbeposi9veifθisincreasing(counterclockwise)
• Speedwillbenega9veifθisdecreasing(clockwise)• Whentheangularspeedisconstant,theinstantaneousangularspeedisequaltotheaverageangularspeed
Sec9on7.1
AverageAngularAccelera9on
• Anobject’saverageangularaccelera9onαavduring9meintervalΔtisthechangeinitsangularspeedΔωdividedbyΔt:
Sec9on7.1
AngularAccelera9on,cont
• SIunit:rad/s²• Posi9veangularaccelera9onsareinthecounterclockwisedirec9onandnega9veaccelera9onsareintheclockwisedirec9on
• Whenarigidobjectrotatesaboutafixedaxis,everypor9onoftheobjecthasthesameangularspeedandthesameangularaccelera9on– Thetangen9al(linear)speedandaccelera9onwilldependonthedistancefromagivenpointtotheaxisofrota9on
Sec9on7.1
AngularAccelera9on,final
• Theinstantaneousangularaccelera9onisdefinedasthelimitoftheaverageaccelera9onasthe9meintervalapproacheszero
Sec9on7.1
AnalogiesBetweenLinearandRota9onalMo9on
• Therearemanyparallelsbetweenthemo9onequa9onsforrota9onalmo9onandthoseforlinearmo9on
• Everyterminagivenlinearequa9onhasacorrespondingtermintheanalogousrota9onalequa9ons
Sec9on7.2
Rela9onshipBetweenAngularandLinearQuan99es
• Displacements s=θr• Speeds
vt=ωr
• Accelera9ons at=αr
• Everypointontherota9ngobjecthasthesameangularmo9on
• Everypointontherota9ngobjectdoesnothavethesamelinearmo9on
Sec9on7.3
CentripetalAccelera9on
• Anobjecttravelinginacircle,eventhoughitmoveswithaconstantspeed,willhaveanaccelera9on
• Thecentripetalaccelera9onisduetothechangeinthedirec,onofthevelocity
Sec9on7.4
CentripetalAccelera9on,cont.
• Centripetalrefersto“center-seeking”
• Thedirec9onofthevelocitychanges
• Theaccelera9onisdirectedtowardthecenterofthecircleofmo9on
Sec9on7.4
CentripetalAccelera9on,final
• Themagnitudeofthecentripetalaccelera9onisgivenby
– Thisdirec9onistowardthecenterofthecircle
Sec9on7.4
CentripetalAccelera9onandAngularVelocity
• Theangularvelocityandthelinearvelocityarerelated(v=rω)
• Thecentripetalaccelera9oncanalsoberelatedtotheangularvelocity
Sec9on7.4
TotalAccelera9on
• Thetangen9alcomponentoftheaccelera9onisduetochangingspeed
• Thecentripetalcomponentoftheaccelera9onisduetochangingdirec9on
• Totalaccelera9oncanbefoundfromthesecomponents
Sec9on7.4
VectorNatureofAngularQuan99es
• Angulardisplacement,velocityandaccelera9onareallvectorquan99es
• Direc9oncanbemorecompletelydefinedbyusingtherighthandrule– Grasptheaxisofrota9on
withyourrighthand– Wrapyourfingersinthe
direc9onofrota9on– Yourthumbpointsinthe
direc9onofω
Sec9on7.4
VelocityDirec9ons,Example
• Ina,thediskrotatescounterclockwise,thedirec9onoftheangularvelocityisoutofthepage
• Inb,thediskrotatesclockwise,thedirec9onoftheangularvelocityisintothepage
Sec9on7.4
Accelera9onDirec9ons
• Iftheangularaccelera9onandtheangularvelocityareinthesamedirec9on,theangularspeedwillincreasewith9me
• Iftheangularaccelera9onandtheangularvelocityareinoppositedirec9ons,theangularspeedwilldecreasewith9me
Sec9on7.4
ForcesCausingCentripetalAccelera9on
• Newton’sSecondLawsaysthatthecentripetalaccelera9onisaccompaniedbyaforce– FC=maC– FCstandsforanyforcethatkeepsanobjectfollowingacircularpath• Tensioninastring• Gravity• Forceoffric9on
Sec9on7.4
CentripetalForceExample
• Apuckofmassmisafachedtoastring
• Itsweightissupportedbyafric9onlesstable
• Thetensioninthestringcausesthepucktomoveinacircle
Sec9on7.4
CentripetalForce
• Generalequa9on
• Iftheforcevanishes,theobjectwillmoveinastraightlinetangenttothecircleofmo9on
• Centripetalforceisaclassifica9onthatincludesforcesac9ngtowardacentralpoint– Itisnotaforceinitself– Acentripetalforcemustbesuppliedbysomeactual,physicalforce
Sec9on7.4
ProblemSolvingStrategy
• Drawafreebodydiagram,showingandlabelingalltheforcesac9ngontheobject(s)
• Chooseacoordinatesystemthathasoneaxisperpendiculartothecircularpathandtheotheraxistangenttothecircularpath– Thenormaltotheplaneofmo9onisalsoogenneeded
Sec9on7.4
ProblemSolvingStrategy,cont.
• Findthenetforcetowardthecenterofthecircularpath(thisistheforcethatcausesthecentripetalaccelera9on,FC)– Thenetradialforcecausesthecentripetalaccelera9on
• UseNewton’ssecondlaw– Thedirec9onswillberadial,normal,andtangen9al– Theaccelera9onintheradialdirec9onwillbethecentripetalaccelera9on
• Solvefortheunknown(s)
Sec9on7.4
Applica9onsofForcesCausingCentripetalAccelera9on
• Manyspecificsitua9onswilluseforcesthatcausecentripetalaccelera9on– Levelcurves– Bankedcurves– Horizontalcircles– Ver9calcircles
Sec9on7.4
LevelCurves
• Fric9onistheforcethatproducesthecentripetalaccelera9on
• Canfindthefric9onalforce,µ,orv
Sec9on7.4
Ver9calCircle
• Lookattheforcesatthetopofthecircle
• Theminimumspeedatthetopofthecirclecanbefound
Sec9on7.4
ForcesinAccelera9ngReferenceFrames
• Dis9nguishrealforcesfromfic99ousforces• “Centrifugal”forceisafic99ousforce– Itmostogenistheabsenceofanadequatecentripetalforce
– Arisesfrommeasuringphenomenainanoniner9alreferenceframe
Sec9on7.4
Newton’sLawofUniversalGravita9on
• Iftwopar9cleswithmassesm1andm2areseparatedbyadistancer,thenagravita9onalforceactsalongalinejoiningthem,withmagnitudegivenby
Sec9on7.5
UniversalGravita9on,2
• Gistheconstantofuniversalgravita9onal• G=6.673x10-11Nm²/kg²• Thisisanexampleofaninversesquarelaw• Thegravita9onalforceisalwaysafrac9ve
Sec9on7.5
UniversalGravita9on,3• Theforcethatmass1exerts
onmass2isequalandoppositetotheforcemass2exertsonmass1
• TheforcesformaNewton’sthirdlawac9on-reac9on
Sec9on7.5
UniversalGravita9on,4
• Thegravita9onalforceexertedbyauniformsphereonapar9cleoutsidethesphereisthesameastheforceexertediftheen9remassofthespherewereconcentratedonitscenter– ThisiscalledGauss’Law
Sec9on7.5
Gravita9onConstant
• Determinedexperimentally
• HenryCavendish– 1798
• Thelightbeamandmirrorservetoamplifythemo9on
Sec9on7.5
Applica9onsofUniversalGravita9on
• Accelera9onduetogravity
• gwillvarywithal9tude
• Ingeneral,
Sec9on7.5
Gravita9onalPoten9alEnergy
• PE=mghisvalidonlyneartheearth’ssurface
• Forobjectshighabovetheearth’ssurface,analternateexpressionisneeded
– Withr>Rearth– Zeroreferencelevelis
infinitelyfarfromtheearth
Sec9on7.5
EscapeSpeed
• Theescapespeedisthespeedneededforanobjecttosoaroffintospaceandnotreturn
• Fortheearth,vescisabout11.2km/s• Note,visindependentofthemassoftheobject
Sec9on7.5
VariousEscapeSpeeds
• Theescapespeedsforvariousmembersofthesolarsystem
• Escapespeedisonefactorthatdeterminesaplanet’satmosphere
Sec9on7.5
Kepler’sLaws
• Allplanetsmoveinellip9calorbitswiththeSunatoneofthefocalpoints.
• AlinedrawnfromtheSuntoanyplanetsweepsoutequalareasinequal9meintervals.
• Thesquareoftheorbitalperiodofanyplanetispropor9onaltocubeoftheaveragedistancefromtheSuntotheplanet.
Sec9on7.6
Kepler’sLaws,cont.
• Basedonobserva9onsmadebyBrahe• Newtonlaterdemonstratedthattheselawswereconsequencesofthegravita9onalforcebetweenanytwoobjectstogetherwithNewton’slawsofmo9on
Sec9on7.6
Kepler’sFirstLaw
• Allplanetsmoveinellip9calorbitswiththeSunatonefocus.– Anyobjectboundtoanotherbyaninversesquarelawwillmoveinanellip9calpath
– Secondfocusisempty
Sec9on7.6
Kepler’sSecondLaw
• AlinedrawnfromtheSuntoanyplanetwillsweepoutequalareasinequal9mes– AreafromAtoBandCtoDarethesame
– TheplanetmovesmoreslowlywhenfartherfromtheSun(AtoB)
– TheplanetmovesmorequicklywhenclosesttotheSun(CtoD) Sec9on7.6
Kepler’sThirdLaw
• Thesquareoftheorbitalperiodofanyplanetispropor9onaltocubeoftheaveragedistancefromtheSuntotheplanet.– Tistheperiod,the9merequiredforonerevolu9on
– T2=Ka3– FororbitaroundtheSun,K=KS=2.97x10-19s2/m3– Kisindependentofthemassoftheplanet
Sec9on7.6
Kepler’sThirdLaw,cont
• CanbeusedtofindthemassoftheSunoraplanet
• WhentheperiodismeasuredinEarthyearsandthesemi-majoraxisisinAU,Kepler’sThirdLawhasasimplerform– T2=a3
Sec9on7.6
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