RaymondA.SerwayChrisVuille

ChapterSevenRota9onalMo9onandTheLawofGravity

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

MoreAboutRadians

•  Comparingdegreesandradians

•  Conver9ngfromdegreestoradians

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

BankedCurves

•  Acomponentofthenormalforceaddstothefric9onalforcetoallowhigherspeeds

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

Communica9onsSatellite

•  Ageosynchronousorbit– Remainsabovethesameplaceontheearth– Theperiodofthesatellitewillbe24hr

•  r=h+RE•  S9llindependentofthemassofthesatellite

Sec9on7.6