Raymond A. Serway Chris Vuille...Raymond A. Serway Chris Vuille Chapter Seven Rotaonal Mo9on and The...
Transcript of Raymond A. Serway Chris Vuille...Raymond A. Serway Chris Vuille Chapter Seven Rotaonal Mo9on and The...
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