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Physics1ATopic1Mechanics
FirstYearUniversityPhysics1ATopic1:Mechanics
ParticleKinematicsInOneDimension(2.12.6)DisplacementDisplacementisthedistanceanobjectisfromtheorigin,irrespectiveoftheroutetakentothatposition.Beingavector,itwillalsostipulatedirectionfromtheorigin.Thedisplacementofanobjectunderconstantvelocitycanbedeterminedasfollows: (Wherevxisconstant)VelocityandAccelerationVelocityistherateofchangeofdisplacementwithrespecttotime.Averagevelocityisthechangeindisplacementdividedbythechangeintime.Instantaneousvelocitycanbedeterminedaccordingtothefollowingequation:
lim
Accelerationistherateofchangeofvelocitywithrespecttotime.Averageaccelerationisthechangeinvelocitydividedbythechangeintime.Instantaneousaccelerationcanbedeterminedbythefollowingequation:
lim
Totalaccelerationisgivenby: MotionwithConstantAccelerationIncaseswhereaparticlemovesunderconstantacceleration,itsaverageaccelerationwillbenumericallyequaltoitsinstantaneousaccelerationatanygivenpoint/sintime.Assuch,agraphofvelocityvstimewillproduceastraightline.
12
2
NonUniformAccelerationThisoccurswhenanobjectchangesvelocityatdifferentratesatanygivenpointintime.MotioninTwoandThreeDimensions(3.13.4,4.14.6)Vectors~Vectorarrowsindicatethemagnitudeanddirectionofthevector~Thetriangleisthesimplestformofavectordiagrampossible
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ResolutionandUnitVectorsVectorsareresolvedonthebasisoftheirdirectionfromtheorigin.Aguideforthenotationisprovidedbelow.
Unitvectorsallhaveamagnitudeof1.Theyarealwaysunderlinedandindicatedbya^signabovethewordedvector(eg.)PolarCoordinateSystemsThisisbase ontheprinciplethatanypointrofcoordinates(x,y)willconsistof2dcomponent indicatingitslocation.Thesecomponentsare:s
VectorAddition
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VectorSubtraction:v=vuToworkoutvectorvuwemustreverseuandaddtov.Letusimaginethatacarismovingat20m/sNanditturnsacornersoitisnowmoving20m/sW.Itispossibletouseavectordiagramtodeterminethechangeinvelocity.
Step1:Draworiginalvectordiagram
Step2:Reverseu
Step3:Solveforx. 800Therefore 202Or28.28ms1SW(2DecimalPlaces)
EquatingVectorsTheprocessofequatingvectorsisachievedbyensuringallcomponentsareequalandthenensuringthatallforcesareequal(asforcesarevectorsinthemselves).Assuchthemagnitudeanddirectionofthevectorsshouldbeequal.EquationsofMotioninVectorForm
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ProjectileMotionProjectilemotioncanbethoughtofasconsistingofhorizontalandverticalcomponentswhenwithintheEarthsgravitationalfield.Thehorizontalcomponentinvolvedtheprojectiletravellingatconstantvelocity.Theverticalcomponentinvolvedtheprojectileexperiencingaforceofgravity,pullingitdownataconstantacceleration.Thesecomponentsareresolvedasvectorswhichareindependentofoneanother,involvingvectoradditiontodeterminemotion. Atanygiventime,thediagrambelowisindicativeofthecomponentsofprojectilemotion.ThehorizontalcomponentisequaltoVcos,whiletheverticalcomponentisequaltoVsin.Theequationsforcalculatingthedisplacement,accelerationsandvelocitiesoftheindividualcomponentsareprovidedbelow.
V Vy = V sin
Otherequationsusedinprojectilemotioninclude:
Vx = V cos
2
Whendealingwiththeprojectilemotionof1particlein3D,itispossibletoaltertheaxestocreatea2Dproblem.Thisensuresthattheprojectilefliesintheplaneofthepage,butwillnotworkinsituationsof2ormoreprojectiles.UniformCircularMotionUniformcircularmotionisachievedbyaconstantangularvelocity.Forthistoexist,theaccelerationvectormusthaveacomponentwhichisperpendiculartothepath,orinotherwords,itpointstothecentreofthecircle.Thefollowingequationsallowthedeterminationofvelocityandanglesbeingsweptout:
2
2
Notethatwhenquestionsrequestdisplacementofparticlesinuniformcircularmotioninpolarform,itisinthefollowingform: . . Whereristheradius.
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CentripetalAccelerationandPeriodofRotationThisreferstoanaccelerationwhichiscentreseeking,andiscalculatedaccordingtotheequation:
Notethatcentripetalaccelerationisneverconstantowingtothepersistentchangingofvelocity.Theperiodofrotationcanbecalculatedaccordingtotheequation:
2
ExampleAplanetravelsinahorizontalcircle,speedvandradiusr.Foragivenv,whatistherforwhichthenormalforceexertedbytheplaneonthepilotistwiceherweight.Whatisthedirectionofthisforce?
CentripetalForce:
VerticalForces:
Byeliminating,
30,TangentialandRadialAccelerationTangentialaccelerationisthecomponentwhichwillcauseachangeinthevelocityoftheparticle.Ithasparallelswithinstantaneousvelocityandisgivenby:
Radialaccelerationisderivedfromthechangesindirectionofthevelocity,andisgivenby:
Becauseradialandtangentialaccelerationareperpendicularcomponentvectorsofacceleration,themagnitudeoftotalaccelerationcanbegivenby:
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RelativeMotionAllmeasurementsaremaderelativetoaframeofreference.Whendescribingthepositingormotionofamovingobject,weneedtostateclearlytheframeofreferenceweareusingforourobservations.Thevelocityofanobject,asmeasuredbyamovingobserver,isreferredtoasrelativevelocity.RelativeVelocityisthedifferencebetweenthevelocityoftheobject,relativetotheground,andthevelocityoftheobserverrelativetotheground.Whentheobjectsaretravellinginthesamedirection,relativevelocitycanbecalculatedbytheformula:V1V2Whentheobjectsaretravellinginoppositedirections,relativevelocitycanbecalculatedby:V1+V2ParticleDynamics(5.15.8,6.1)NewtonsLawsofMotionNewtonsFirstLawofMotion(LawofInertia)Anobjectwillremainatrestortravelwithaconstantvelocityunlessacteduponbyanetforce.NewtonsSecondLawofMotionTheforcerequiredtomoveanobjectisproportionaltoitsmass. Notethatthislawonlyappliesininertialframesandisasumofallforcesactingonanobject.NewtonsThirdLawofMotionForeveryaction,thereisanequalandoppositereaction.MassThisistheamountofmatterwithinagivenobject.Akeypropertyisthatthemassisproportionaltoberesistancetheobjectofferswhenattemptingtoalteritsvelocity(forinstance,causingatennisballtoaccelerateiseasierthancausingatraintoaccelerate).Asaconsequence,theaccelerationoftheobjectisinverselyproportionaltothemassofanobjectwhenafixedforceisapplied.
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ApplicationsofNewtonsLawsNewtonslawscanbeappliedintermsofthetensionincables.Theparticleswhichareapplyingatensiontothecablecanbeinequilibriumorunderanetforce.Iftheparticleisinequilibrium, 0asthereisnoforceinthexdirection.Additionally, 0 0 ,suchthatFgisthegravitationalforceandTistheupwardforceprovidedbytension.
Iftheparticleisunderanetforce,then
,where
theforceisbeingexertedinthehorizontalplane.Thisprinciplecanbeappliedtotheyplaneiftheforceisappliedvertically.However,ifthereisnoforceappliedvertically,then: .Inotherwords,thenormalforcehasthesamemagnitudebutoppositedirectiontogravitationalforce.
0
ContactForcesWhenanobjectmoveswhileitisincontactwithanothermedium,thereisaforceexertedwhichresiststhemotion.Thisisbecauseoftheinteractionsbetweentheobjectanditssurroundings;resultantofthejaggednatureofobjects.Theseforcesareknownascontactforces. Thenormalcomponentofacontactforceisthenormalforce(N).Thecomponentwhichoccursintheplaneofcontactisthefrictionforce(Ff).
Thenormalforceisatrightanglestothesurfaceandresultsfromdeformation.Ifthereexistsrelativemotion,thereiskineticfriction(whichopposesmotion).Ifthereisnorelativemotion,thenthereisstaticfriction,whichopposesanyappliedforce.
Belowareequationswhichdisplaytheratiooffrictionforcesto
normalforces. (Assuch,thenormalforceisproportionaltothekineticfrictionforce) (Assuch,thenormalforceisgreaterthanthestaticfrictionalforce,wherefrictioncanbe0)Ageneralisedlawinvolvingfrictionstatesthatkineticandstaticfrictionareroughlyindependentofthenormalforceandofcontactarea.
Acomparativelargescaleexamplewouldbeplatetectonics.Thejaggednatureofplatescausesthemtolockperiodically,andthislockingcontinuesuntilsuchtimeasenoughforceisexertedtobreakfreeofthislocking.Thislockingcausesstaticfrictionduetothelackofrelativemotionandopposesanyappliedforcetounlocktheplates.Thebreakingfreecausesadeformationintheplatesandtheninteractswithitssurroundsresultinginkineticfriction. Whatbecomesclearistheforcerequiredtokeepanobjectmovingislessthanthattoinitiallymoveanobject.
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Examples
DynamicsofCircularMotionIncircularmotion,thereexistsaradialcomponentofacceleration,nutalsoa
tangentialcomponentwithamagnitudeof| |,thereforetheforceontheparticle
hasaradialandtangentialcomponent.Assuch, andhence ExampleAcivilengineerwishestoredesignacurvedroadwayinsuchawaythatacarwillnothavetorelyonfrictiontoroundacurvewithoutskidding.Inotherwords,acarmovingatthedesignatedspeedcannegotiatethecurveevenwhenitiscoveredinice.Sucharoadisbanked.Supposethedesignatedspeedis13.4m/sandthecurveradiusis35m.Findtheangletheroadshouldbebankedby.Onabankedroad,thenormalforcehasahorizontalcomponentwhichpointstothecentreofthecurve.However,sincetheforceofstaticfrictionis0,theonlycomponentwhichcancausecentripetalaccelerationis . . Therefore:
.
(1)
. 0. (2)
(1)/(2)=
.
. 27.6
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WorkandEnergy(7.17.4,15.1,7.57.8,8.18.2,8.5)MechanicalWorkTheworkWdoneonasystembyanagentexertingaconstantforceonthesystemisequalto: . whereistheanglebetweentheforceandthedisplacementvectors.
DeformingSprings
BasedonthediagramandHookesLaw,theworkdonebythespringontheblockis
equalto
.
Whereastheworkdoneappliedontothespringisequalto:
VectorDotProductThisisalsoknownasthescalarproduct(beingtheproductof2scalarquantities).Thisisduetothenotationforthemultiplicationofscalarpropertiesbeing(a.b),while(axb)isusedforvectors.
Becauseofthis,wecanderivescalarproductsbycomponents.Hence:
. .
. . . .
. . . Since. , . . allequal0,
.
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Thiscanbeappliedtotheproblembelow:
ariableForcesV
HookesLawHookesLawdealswiththebehaviourofproductswhichdisplaylinearelasticity.Itisbasedontheprinciplethattheforceappliedtoanobjectwillbeproportionaltoanydeformation. However, helawhasamajorflawwithit;itonlyappliestoasmallportionofthe
of
tive
hbelow:
tgraphofforcevsdeformation(orintermolecularseparation,asthisisanindicatordeformation).Indeformationbybending,someseparationsarestretched,whileothersarecompressed.Therealsoexistsaneutralpositionwherethereisnorelastretchingorcompression.Whenthisstretchingorcompressionexceedsthelimitationsoftheobject,thelawfails.Assuch,itcanonlybeusedasanapproximation.Observethegrap
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Clearlytherepulsiveforcesmustbesubstantiallystrongeroverallasitisdifficulttocompressanobjectbeyondafewpercent.Alsoquiteclearisthatatrest,forceis0.Anotherclearobservationisthatthereareseveralrangesoverwhichforceisproportionaltodeformationareverysmall(whereHookeslawwillapply).Also,eachregionhasitsownapproximationandhenceconstant.KineticEnergyUsingNewtonssecondlaw,wecanderivethefollowingequationforthenetworkonanobject:
.
.
However,
Therefore, WorkEnergyTheoremsWhenworkisdoneonasystemandtheonlychangeinthesystemisinitsspeed,thenetworkdoneonthesystemequalsthechangeinthekineticenergyofthesystem.Assuch,thistheoremindicatesthatthespeedofasystemwillincreaseifthenetworkdoneonitispositiveasthefinalkineticenergywillbegreater.Conversely,thespeedofasystemwilldecreaseiftheworkdoneonitisnegative.PotentialEnergyPotentialenergyisbasedontheconceptofgettingenergybac .Notallforcescankstoreenergyhowever;frictionenergycannotberecovered,whereasthatinacompressedspringcanberecovered(likewiseforworkdoneinagravitationalfield).Thereisaminimumofpotentialenergyattheequilibriumpoint.Foranyconservativeforce(whereworkdoneagainstisW=W(r)),itispossibleto
defineapotentialenergyUas .Thatis, .
Forinstance,usinggravity,whereanobjectofmassmisbeingslowlylifted(withnoacceleration)fromaheightyitoafinalheightofyf,itisfoundthattheworkdoneontheobjectasthedisplacementincreasesisaproductoftheappliedforceandthedisplacement,suchthat :
. . As ssiblearesultofthis,itispo todeterminethatgravitationalpotentialenergyisequalto: and enceh
ison,aspring,whichpossesseselasticpotentialenergy:
Bycompar
arbitraryandcanbeanything.Forinstance,withtheThechoiceofzeroforUis
exampleofthespring,U=0atx=0,andso
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ConservativeForcesTheseforcesarewheretheworkdonearoundaclosedloopisequaltozero.Suchforceshave2keyproperties:
Theworkdonebyaconservativeforceonaparticlemovingbetweenany2pointsisindependentofthepathtakenbyaparticle
Theworkdonebyaconservativeforceonaparticlemovingthroughanyclosedpathiszero(wherethebeginningpointandendpointareidentical)
Anexampleofsuchaforceisg n .
ravity.Basedo ,itbecomesclearthatonlytheinitialandfinalcoordinatesmatterandhenceoveranyclosedpaththeworkdonewillbezero.Thisissimilarforelasticsystems.Theworkofaconservativeforceisgenerallyrepresentedby andgenerally NonConservativeForcesTheseareforceswheretheworkdoneinaclosedlookcannot ualzero.Aneqexampleofthiswouldbefriction.Assuch,themechanicalenergyisdefinedas: (wh icenergyandUispotentialenergy).SuchforceswillereKiskinetcauseachangeinthemechanicalenergyofthesystem.Forinstancbookalonganonidealisedtable,thekineticenergyisconvertedto
ewhenslidingainternalenergy
asheat.Furthermore,thepathtakeninaclosedloopwilldeterminehowmuchtion.
onservationofMechanicalEnergy
kineticenergyisconvertedtointernalenergy;thelongerthepath,themorefricCMechanicalenergyisgenerallynotconserved.However,ifnonconservativeforcesdonowork,thereforemechanicalenergyisconserved.pplicationsA
PowerThisistherateofdoingwork.Generallyspeaking,Wisusedforwork,while| |isusedforweight.Itisworthnotingthatwhendisplacinginthehorizontaldirection,thereisnoworkbeingdoneagainstgravity.Additionally,velocityisnormally
f
constant.TheSIunitisjoulespersecond,orwatts.Positiveworkisthatdoneinthesamedirectionastheforce.Negativeforceisworkdoneoppositetothedirectionotheforce.
AveragePower:
InstantaneousPower:
Example:
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Gravitation(13.1,13.413.6)NewtonsLawofGravitationNewtonsLawofUniversalGravitationstatesthateverymassintheuniverseisttractedtoeveryothermassintheuniversebyaforceofgravitation.a
221
dGF =
MotionofPlanetsandSatellitesSlingshotEffectSatellitesmusthaveacertainvelocityinordertostayinorbit.IfnottheywilleithercrashintoEarthorgooutintospaceandneverreturn.Thisvelocitydependsontheforceofgra
m
vityactingonthesatellite.Satellitespathsofmotionareaffectedbyseenbyorbitsandtheslingshotaffect.
a Motionlawstatesthatallplanetsorbitthesuninanellipticalorbitwherethe
tedatoneofthefoci.Keplerssecondlawisaconsequenceofconservationofangularmomentumand
m
gravity,thiscanbeKeplersLawsofPlanet ryKeplersfirstsunisloca
statesthataplanetwillsweepequalareasinequalamountsoftime.Assuch:
whereLandMareconstants.
Keplersthirdlawshowstherelationshipbetweentheperiodandradiusoforbits.
GandVariationThevalueofgravityvariesasthedistanceawayfromthecentreofgravityincreases.
Assuch,thevalueofgravitycanbeseentobe:
whereristheradiusofthe
thesurface.Assuch,agravitationalfieldexists,andlthoughatlargedistances,theforceisnegligible.
planetandhistheheightaboveextendsinfinitelyintospace,a
Basedon ,
whereisaunitvectorpointingradiallyoutward
fromtheplanet,thenegativesignindicatesthefieldpointstothecentreoftheplanet.
entialEnergyGravitationalPotWhile isareasonableestimateofthepotentialenergyofanobject,itcanonlyapplyclosetothesurfaceofaplanet.Whereamoregeneralequationisrequired,
willapply.
EscapeVelocityThisiswhenthekineticenergyofanobjectallowsittoescapefromthegravitationafieldofanyobject(usuallyaplane
lt).Hence:
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OrbitsandEnergySincenonconservativeforcesdonowork,mechanicalenergyisconserved.Hence ,whereEisthemechanicalenergyofthesatellite.Assuch,
.Byremovingvelo consideringthecircularcityand
orbit:
.
Since
Therefore:
Assuch radiuswillresultinaverynegativepotentialenergyandaverylarge,asmallkineticenergy.Hencetheinnerplanetsarefasterthantheouterplanets.Basedonthis,large largeamountorbitsrequirea ofworktoreachtherequiredaltitude,whereitsvelocitywilldecreaseatthesehigheraltitudes.
mesevidentwithspacecraftinorbit.Inordertospeedupandcatch
itwilltravelfasteruntilitcatchesupwiththesecondspacecraft.Itwillthenfireitsenginesforwardtoslowdownandenceclimbuptoitsoriginal,slowerorbit.
Thisbecouptoanotherspacecraft,aspacecraftwillfireitsenginesbackwards,losingenergyanddoingnegativeworkonitself.Thishastheeffectofmorenegativemechanicalenergy,causingittofalltoalowerorbitwhere
h
MomentumandCollisions(9.1,9.39.6)ConservationofLinearMomentumTotalMomentumPrior=TotalMomentumFollowingM1U1+M2U2+=M1V1+M2V2+
MomentumisconservedincollisionsbecauseofNewtonsThirdLaw.Thisisbecausethereisanequalandoppositereactiontoeveryactiondone.Becauseofthis,thereactionforcesmustequaltheactionforces.
ThechangeinmomentumofanobjectisreferredtoasImpulse.Itisdefinedbythefollowingequation:
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CollisionsinOneDimensionElasticcollisionsin1dimensionarethosewherethetotalkineticenergy,andhencemomentum,ofthesystemisthesamebeforeandafterthecollision.Inreality,nocollisionsareperfectlyelasticbecauseobjectswilldeformslightlyduringthecollisionaswellasasmallamountofenergybeingconvertedtootherforms,suchasheatandsound.Basedonthis,thefollowingapply: and
Basedonthis,inelasticcollisionsarethosewherethetotalkineticenergyinasystemisdifferentbeforeandafteracollision,evenifmomentumisconserved.Thiscanhappenwhenobjectssticktogether(knownasaperfectlyinelasticcollision)orwhenanobjectissubstantiallydeformed(aswithaball)whenbeingbounced.Inelasticcollisionsaregenerallyhardtoanalysewithoutadditionalinformation. Inaperfectlyinelasticcollision,bothobjectswilltravelwithacommonvelocityafterimpact.Beinganisolatedsystem,momentumisconservedandassuch,thefinalvelocitycanbedeterminedbythefollowing:
.
CollisionsinTwoDimensionsIncollisionsintwodimensions,momentumisconservedinboththexandyaxesindependentlyas and
Giventheinitialycomponentofmomentumina2particlesystemiszero(giventhedirectioncanbetakenasthexaxis): cos cosWhereistheangleobject1fliesoffat,andistheangleatwhichobject2fliesoffat(whereobject1collideswithobject2),and:0 sin sin
Ifthecollisioniselastic,then:
.Kinetic
energywillnotbeconservedifthecollisionisinelastic.CentreofMassThexcoordinatesofthecentreofmassofaseriesofparticlescanbefoundbythe
followingequation:
Wherexiisthexcoordinateof particleandthetotalmassistheith ,
w thesumrunsovernparticles.Theyandzcoordinatesofthecentreofmasscanalsobederivedbysimilarequations:here
Thevectorpositionofthecentrefmassofanextendedobjectcanbeexpressedintheform:
Thecentreofmassofanysymmetricalobjectli fsymmetryesontheaxiso .Forinstance,thecentreofmassforasphereisatitsandonanyplaneofsymmetry
geometriccentre. Forallothe edeterobjects,thecentreofmasscanb rminedbysuspendingtheobjectfromanypointA,andthendrawingaverticallineAB(determinableviaaplumb bob).IftheobjectisthensuspendedfromanypointC,andagain,averticallinedrawnasCD,theintersectionofABandCDwillmarkthecentreofmass.
Physics1ATopic1Mechanics
Theworkdonebythecentreofmasscanbegivenbythechangeinkineticenergyofthecentreofmass.Example:
any
reofmassofsuchasystemisgivenby:M ParticleSystemsThevelocityofthecent
Thetotalmomentumofasystemofparticlesisgivenby: heaccT elerationofthecentreofmasscanbegivenby: Assuch,thesumofallexternalforcescanbegivenby: Rotation(10.1,10.310.6)AngularVelocityandAccelerationAngulardisplacement:
Angularvelocity:
Angularacceleration:
2.
Notethattheaboveequationsarecomparabletotheprojectileequations,however,therespectivecomponentsofdisplacement,velocityandaccelerationhavebeendividedbyrtoobtaintheirangularequivalents.AngularQuantitiesLinear Angular
. sin
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Example:Abicyclewheelhasaradiusof40cm.Whatisitsangularvelocitywhenthewheeltravelsat40kmh.
.
.
/
. 28.
RotationalKineticEnergyenergyisderived omthesumoftheindividualkineticenergiesof
idobject.
Asseenabove,rotational ineticenergyisequalto
Rotationalkinetic fr
.However,sincetheparticleswithinarig
k
,
.
Thetotalkineticenergyof rollingobjectis:a
MomentofInertiaThisisameasureoftheabilityofanobjecttoresistchangesinitsrateofrotation.Itisalsoreferredtoastherotationalanalogueofmass.Forasystemofmasses,thisisequalto:
Foracontinuousbody:
ecomes thatIwilldependonthetotalmass,distributionofmassandshape,aswe astheaxisofrotation.Whatb evidentis
llThemomen ofinertiacan sobedefinedbytheequation:t al wherenisanumber. canalsobe efinedasIt d where .Forahoop, .Otherobjects(forwhichkdoesnotneedtoberemembered):
Disc:
Solidsphere:
Itisworthnotingthat enwh anobjecthasalowermomentofinertia,thereislessrotationalkineticenergy,andhencemoretranslationalkineticenergy.TorqueTorqueistheturning omentof force.Torquethatisaboutanaxisofrotationis
distancebetweentheendofthebeamandthepoint
m aequaltotheproductofthethroughwhichtheforceisappliedandthecomponentofforceperpendiculartothebeam.
. sin
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AngularMomentum
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SystemofParticles
Example:Persononrotatingseatholdstwo2.2kgmassesatarmslengthanddrawsthemtowardstheirchest.Whatistheincreaseinagularmomentum?IsKconserved?
Physics1ATopic2ThermalPhysics
FirstYearUniversityPhysics1ATopic2:ThermalPhysics
TemperatureTemperatureandThermalEquilibriumTemperatureistheconceptbywhichanobjectisperceivedtobehotorcold.Thedefinitionoftemperatureisdependentupontheconceptsofthermalcontactandthermalequilibrium.Athighertemperatures,particleswillpossessmoreenergyandhencehavemorekineticenergyandcollisions. Thermalcontactiswheretwoobjectsareabletoexchangeenergybetweenthemselves,eitherviaheatorEMR.Thisisaconsequenceofatemperaturedifferencebetweenthetwoobjects.Whenparticlescollidewithawall,itcantransferenergytothiswall,whichinturncantransfertheenergytotheparticlesontheothersideofthewall. ThermalEquilibriumiswheretwoobjectsexchangenonetenergywhenplacedinthermalcontact.Asstatedabove,thiscontactdoesnotneedtobephysicalasenergycanbetransferredviaEMR.AkeylawrelatedtothermalequilibriumistheZerothLawofThermodynamics.ThislawstatesthatifanytwoobjectsAandBareseparatelyinthermalequilibriumwithathirdobjectC,thenAandBareinthermalequilibriumwitheachother.Consequently,ifAandBarebroughttogether,nonetenergywillbeexchanged. Temperaturecanthenbedefinedusingthermalequilibrium.Itcanbethoughtoftobeapropertywhichdetermineswhetherornotanobjectisinthermalequilibriumwithanotherobject/s.Iftwoobjectsareinthermalequilibrium,thentheyarethesametemperature.Quiteevidently,thetemperatureisnowdefinedbyenergy.MeasuringTemperatureThetemperatureofanobjectismeasuredwithathermometer.Theyarebasedontheprinciplethatsomephysicalpropertyofasystemchangedasthetemperatureofthesystemchanges.Atemperaturescalecanbebasedonanyofthefollowingproperties:
Volumeofaliquid(Mercuryoralcohol) Dimensionsofasolid(suchasintheexpansion/contractionoftraintracks) Pressureofagasataconstantvolume(Idealgases) Volumeofagasataconstantpressure(Idealgases) Electricalresistanceofaconductor
(Asheatrises,conductivitydecreasesproportionallytothetemperature) Colourofanobject(aswithBlackBodies)
Incalibratingathermometer,itmustbeplacedinthermalcontactwithsomenaturalsystemthatremainsataconstanttemperature(suchasthetriplepointofwater).TheCelsiusscaleisbasedupontheicepointofwaterbeingat0andtheboilingpointat100with100incrementsinbetweenthesepoints.
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Akeyproblemwiththermometers,especiallyliquidinglassvariants(suchasthoseinvolvingalcoholormercury),isthatthethermometermayonlyagreewiththecalibrationpoint/s.Asaconsequence,therecanbelargediscrepancieswhenthetemperatureisfarfromthesepoints(thescalebetweencalibrationpointsmaynotbelinear).Thermometersalsohavealimitedrange;forinstance,mercurycannotoperatebelow30Candalcoholcannotworkabove85C. Theconstantvolumegasthermometerisbasedupontheeffectsonthepressureofafixedvolumeofgasasthetemperaturechanges.Anincreaseinthetemperaturecausesgreaterpressure,whichpushesthemercuryout.Theheightdifferenceisproportionaltothetemperature.
Whencomparingtemperature,itisimportanttousetheKelvinscalebecausethisisatruerepresentationofthekineticenergyofanobject.Forinstance,waterat100Cisnottwiceashotaswaterat50C.However,aflameat233Cistwiceashotasanicecubeat20C.AbsoluteZeroThisconceptisbasedupontheclassicalphysicsprinciplethatat0K,particleswillhavezerokineticenergy.Itwouldfollowlogicallythatthemoleculeswouldthensettleoutonthebottomofthecontainer.However,quantumphysicsshowsthatsomeresidualenergywillremain,andthisiscalledzeropointenergy.Furthermore,theconceptofabsolutezeroisatheoreticalconcept.Thisisbecauseforasubstancetobeatabsolutezero,thecontainermustbeatabsolutezerobecausethetwoobjectswouldbeinthermalcontact.Thisproblemcontinuestocompound,andhenceforanobjecttobeabsolutezero,everythingmustbeatabsolutezero.ThermalPropertiesofMatterWhenanobjectisheated,itwillexpand.Consequently,thejointsinmanyobjectsallowroomforthisintheformofexpansionjoints.Thisexpansionisaconsequenceofthechangeinaverageseparationbetweenatomsinanobject.Ifthisexpansionissmallrelativetotheoriginaldimensions,thenagoodapproximationofthischangeindimensionsisthisisproportionaltothechangeintemperature.LinearExpansionThecoefficientoflinearexpansionisasfollows:
/
or or
WhereLiistheinitiallengthandLfisthefinallengthandhasunitsof(C)1.
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Itshouldbenotedthatsome,butnotall,materialscanexpandinonedirectionwhilecontractinanotherastemperatureincreases.Alsoasthelineardimensionschange,thesurfaceareaandvolumewillalsochange.VolumeExpansionThisisbasedontheprinciplethatthechangeinvolumeisproportionaltotheoriginalvolumeandthechangeintemperature. WhereisthecoefficientofvolumeexpansionandViistheinitialvolume.Insolids, 3.However,theformulaassumesthematerialisisotropic,orthesameinalldirections. Theprincipleofvolumeexpansioncanbeexploitedinthermometers.Thegreatervolumealiquidwillexpand,thegreatertheaccuracyofthethermometer.AreaExpansionThisisbasedontheprinciplethatthechangeinareaisproportionaltotheoriginalareaandchangeintemperature. 2WhereAiistheinitialarea.BimetallicStripsThesearestripsofmetalcontainingtwodifferentmetalsphysicallybondedtooneanother.Atacertaintemperature,thestripwillbeperfectlystraight.However,asonestripexpandsmorethantheother,itwillbendasthetemperaturechanges.Anapplicationforsuchanobjectisinthermostats.WaterWater,unlikeothersubstances,willincreaseindensityasitstemperaturerisesfrom0Cto4C.Above4C,waterwillbehavelikeanyothersubstance.Thisisaconsequenceofthehydrogenbondsbetweenmolecules.ChangesinVolumeWhilethevolumeexpansionequationrequiresaninitialvolumefortemperaturechange,thereisnoequilibriumseparationfortheatomsinagas.Inotherwords,thereisnostandardvolumeforanyfixedtemperature,andhencethevolumedependssoleonthecontainer.Consequently,thevolumeforgasesisvariable.Andthechangeinvolumeisconsidered.KineticTheoryandtheIdealGasMacroscopicPropertiesofaGasEquationofStateforaGasThisdescribeshowthevolume,pressureandtemperatureofagasofmassmarerelated.TheMoleTheamountofgaswithinagivenvolumecanbeexpressedinthenumberofmoles.OnemoleofanysubstancecontainsAvogadrosnumberofconstituentparticles.Thenumberofmolesiscalculatedby
.
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BoylesLawWhenagasiskeptataconstanttemperature,itspressureisinverselyproportionaltoitsvolume.CharlesandGayLussacssLawWhenagasiskeptataconstantpressure,itsvolumeisdirectlyproportionaltoitstemperature.IdealGasLawTheequationofstateforanidealgasis: Wherenisthenumberofmoles,RisaconstantcalledtheUniversalGasConstant(8.314J/mol.K),Tistemperature,Pispressure,Visvolume,andtheunitsforPVisJoules.Thislawisoftenstatedintermsofthetotalnumberofmoleculespresent,hence:
WherekBisBoltzmannsconstantof
and 1.38 10/.
ItiscommontocallP,VandTthethermodynamicvariablesofanidealgas. Thislawhelpsexplainthatthepressureagasexertsonthewallsofacontainerareaconsequenceofthecollisionsofgasmoleculeswiththewall.IdealGasLawAssumptions
1. Thenumberofmoleculesinthegasislargeandtheaverageseparationbetweenthemoleculesislargecomparedwiththeirdimensions.Suchmoleculesoccupyanegligiblevolumeinthecontainer.
2. ThemoleculesobeyNewtonslawsofmotion,butasawholetheymoverandomly.Consequently,anymoleculecanmoveinanydirectionwithanyspeed.Atanygivenmoment,acertainpercentageofmoleculesmoveathighspeedsandacertainpercentagewillmoveatslowspeeds.
3. Themoleculesinteractonlybyshotrangeforcesduringcollisions.Hencetherearenoattractiveorrepulsiveforcesbetweenthem.
4. Moleculesmakeelasticcollisionswiththewalls5. Thegasunderconsiderationisapuresubstance.Inotherwords,allthe
moleculesareidentical(notentirelytrueinrealityduetoisotopes)Thefirst3assumptionsarethemostimportanthowever.MolecularModeloftheIdealGasThemolecularmodelforagaswasdevelopedbyBrownin1801afterobservingthatpollensuspendedinwatermovedinanirregularpattern.Hethoughtthatthepollencontainedsomelifeforce,however,itisnowknownthatthiswasfromwatermoleculesbumpingintothepollenrandomly.Thiswasthefirstevidenceofatomisationwhichwasanobservationratherthanadeduction.KineticInterpretationofTemperaturePressureandKineticEnergyTherelationshipbetweenpressureandkineticenergyisasfollows:
Thisisinterpretedasthepressureisproportionaltothenumberofmoleculesperunitvolume(N/V)andtotheaveragetranslationalkineticenergyofthemolecules.
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Notetheuseof ,whichisthemeanvalueofthespeedsquared.Thisisaconsequenceofthelargenumberofparticlesinagas,andhenceitisimpossibletorefertoaspecificparticle.Furthermore,totalvelocityiszeroasthereasmanyvectorcomponentsinonedirectionastheother. BycomparingtheaboveequationwiththatfortheIdealGasLawwefind:
Hence,thetemperatureisadirectmeasureoftheaveragemolecularkineticenergy.Simplifyingtheequationgives:
Andgiventhiscanbeappliedinanydirection:
The
isaconsequenceofeachcomponent(x,yandz)are
oftheoverallequation.
Usingthis,itbecomesapparentthatthetranslationaldegreeoffreedomcontributesanequalamountofenergytothegas(giveneachdirectionisindependentoftheothers).ThisconceptofeachcomponentcontributingequallytotheenergyofthesystemcanbereferredtoastheTheoremoftheEquipartitionofEnergy.TotalKineticEnergyofaGasThetotalkineticenergyisjustNtimesthekineticenergyofeachmolecule:
Notingthat
Ifthegashasonlytranslationalenergy,thethisistheinternalenergyofthegas.Basedonthis,theinternalenergyofanidealhasdependssolelyontemperature.RootMeanSquare(RMS)SpeedThisisthesquarerootoftheaverageofthesquaresofthespeeds.Hence:
WhereMisthemolarmass.HeatandtheFirstLawofThermodynamicsHeatandInternalEnergyofIdealGasesIn1850,Joulediscoveredalinkbetweenthetransferofenergybyheatinthermalprocessesandthetransferofenergybyworkinmechanicalprocesses.Thisledtheconceptofenergytobegeneralisedtoincludeinternalenergy. Internalenergyisthesumofallenergiespossessedbyaparticle.Forinstance,inagas,thisincludesgravitationalpotential,vibration,rotational,randomtranslational,chemicalpotentialandrestmassenergies.ThekineticenergyduetomotionthroughspaceisNOTincluded.Internalenergycanbechangedbyboththeapplicationofheat(orflowofenergy)orwork(applyingaforce). Thermalenergyreferstothesumofgravitationalpotential,rotational,vibrationalandrandommotionkineticenergies.ThisisrepresentedbythesymbolQ. Heatcanbeinterpretedtobeaflowofenergybetweentwoormoresystems.Thisisduetoatemperaturedifferencebetweentherespectivesystems.Heatisquantifiedastheamountofenergytransferred.
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HeatCapacityThisisdefinedastheamountofenergyrequiredtoraisethetemperatureofasampleby1C.Itcanbedepictedbythefollowingequation. .SpecificHeatforSolidsandIdealGas
Specificheatistheheatcapacityperunitarea,or .Theequationforspecific
heatisgivenby .Thespecificheatisameasureofhowinsensitiveanobjectistotemperaturechangeasagreaterspecificheatwillrequiremoreenergytochangethatsubstancestemperature. Signconventionsforheatcapacityandspecificheatare:
Iftemperatureincrease,Qand Tarepositiveasenergytransfersintothesystem
Iftemperaturedecreases,QandTarenegativeasenergytransfersoutofthesystem.
Thespecificheatofwaterisratherlargecomparedtomanyothersubstances.Theconsequencesofthisarevariousweatherphenomenon,suchasmoderatedtemperaturesalongthecoastandseabreezes.CalorimetryThisisatechniqueformeasuringthespecificheatofasubstance.Itinvolvesheatingamaterial,addingittoasampleofwater,andthenrecordingthefinaltemperature.Assumingthesystemofthesampleandthewaterisisolated,conservationofenergyrequiresthattheamountofenergywhichleavesthesampletobethesameastheenergywhichentersthewater. Thisminussignisimportant,andisindicativeofthesamplelosingenergy,whichthewatergainsenergy.PhaseChangesThisreferstothechangeofphysicalstateofasubstance,suchassolidtoliquid.Duringaphasechange,thereisnochangeintemperatureofthesubstance.TheenergyrequiredtoeffectthischangeiscalledLatentHeat.LatentHeatThisistheamountofenergyrequiredtocauseasubstancetochangestate.Itisequalto wheremisthemassofthesample.Thelatentheatoffusionistheenergyrequiredtochangefromsolidtoliquid,whilethelatentheatofvaporisationistheenergyrequiredtochangefromliquidtogas.Apositivesignwillbeusedtoindicateenergybeingtransferredintothesystem.Conversely,anegativesignwillindicatethatenergyislostbythesystem.Animportantconceptincalculatinglatentheatisthatthetemperaturewillnotchangeuntilthesamplehascompletelychangedstate.
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WorkDoneonanIdealGasThestatevariableswilldescribethemacroscopicstateofasystem.Inanidealgas,thesearepressure,temperature,volumeandinternalenergy.However,thismacroscopicstatecanonlybespecifiedifthesystemisinthermalequilibrium. Transfervariablesdescribethechangesinstate.Theyarezerounlessaprocessoccurstocausethetransferofenergyacrossasystemboundary.Forexample,heatandworkaretransfervariables.Forinstance,heatcanonlybeassignedavalueifitcrossesaboundary. Theworkdoneonanidealgascanbegivenby: . . . SinceP=F/AThechangeinvolumeisgivenby . andhencetheworkdoneis . .Thetotalworkdoneisgivenby:
PVDiagramsThesearediagramsshowingthecorrelationbetweenpressureandvolumetoallowadeterminationoftheworkdoneonanidealgas.Theworkdoneonsuchadiagramisverydependentonthepathtaken.
Theabovediagramindicatesthatthevolumehasbeenreducedbeforethepressureisincreasedbyheating.
Theabovediagramshowsthatthepressurewasfirstlyincreasedbeforethevolumewasdecreased.
Theabovediagramshowsthatthepressureandvolumecontinuallychanged.While isanapplicableequationforthefirst2cases,indiagram3,theevaluationofworkrequirestheP(V)functiontobeknown.
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ConversionofWorktoThermalEnergyIfapistoncompressesagas,thekineticenergyoftheparticleswillbeincreased.Thisisthroughconservationofmomentum,wherethemovingpistonsupplieskineticenergytotheparticles,therebyincreasethermalenergy.FirstLaworThermodynamicsThisisaspecialcaseofconservationofenergytakingintoaccountthechangeininternalenergythroughenergytransfersinworkandheat.Thelawstatesthat: Akeyconsequenceofthislawisthattheremustexistaninternalenergywhichisdeterminedbythestateofthesystem.Forinfinitesimalchanges: ApplicationsoftheFirstLawofThermodynamicsTheAdiabaticprocessiswherenoenergyentersofleavesthesystembyheat.Thisisachievedbyinsulatingthesystem,orhavingthesystemproceedquicklyenoughthatnoheatcanbeexchanged.Since 0, .Ifthegasiscompressedinthismanner,thenWispositive,sointernalenergyisalsopositiveandhencetemperatureincreases. Theisobaricprocessisonewhichoccursatconstantpressure.Theworkdoneis ,wherePisconstant.
Theisovolumetricprocesstakesplaceinconstantvolume.Sincethereisnochangeinvolume,W=0.HenceininternalenergyequalsQ.Additionally,ifanyheatisadded,sincethevolumeisconstant,alloftheenergytransferredresultsinanincreaseininternalenergy. Theisothermalprocessoccursatconstanttemperature.Sincetemperaturedoesntchange,internalenergyequalszero.Anyenergythatentersmustleavethesystem.ItsPVgraphisasfollows:
Since ,theequationformsaparabola.
Also,sincethegasisanidealgasandtheprocessisquasistatic, . log
Ifthegasexpands,thenVf>Vi,andhencetheworkdoneisnegative.TheTransferofHeatInanisolatedsystem,therearenointeractionswiththesurroundingenvironment,andhence 0 .Asaconsequence,theinternalenergyofsuchasystemremainsconstant.
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Cyclicprocessesarewherethesystemstartsandendsatthesamestate.Suchaprocessisnotisolated.OnaPVdiagram,thiswouldbeindicatedbyaclosedcurve.Becausetheinternalenergyisastatevariable,thereisnochangeininternalenergy.Hence,if 0, .Insuchprocesses,thenetworkdonepercycleisequaltotheareaenclosedbythecurveonaPVdiagram. Heatistypicallytransferredbyconduction,convectionorradiation.Inconduction,particlesbecomeenergisedandcollidewithotherparticles.Anincreaseinkineticenergythereforewilltransferthroughtotheotherparticles,thusconductingheat.However,theconductionprocesscanonlytakeplaceifthereisadifferencebetweentwopartsoftheconductingmedium.Therateoftransfercanbe
givenby
whereisaconstantofthermalconductivity,Pispower,dxisthethicknessandTaandTbarethetemperaturesof2differentmaterials,wheresubstanceAishotterthanB. Convectionreferstoenergytransferbemovingasubstance.Itisoftenassociatedwithchangesindensity,suchasinair.Thisisreferredtoasnaturalconvection.Forcedconvectionisachievedwithfansorpumps.Convectionresultsfromtheheatingofair,suchthatitexpandsandrises,whilecooleraircyclesin.Thusacontinuouscurrentisestablished. Radiationdoesnotrequirephysicalcontact.ItisaresultoftheIRemissionsofawarmbody.TherateofradiationcanbegivenbyStefanslawstating: ,wherePistherateoftransfer(inWatts),isaconstant(5.6696 10),Aisthesurfacearea,eistheconstantofemissionoremissivity,andTisthetemperatureinK.
Therateatwhichanobjectradiatesheatisdeterminedbyitssurrounds,hence
.Ifanobjectisinthermalequilibriumwithitsurrounds,therewillbenonetradiation.
Physics1ATopic3Waves
FirstYearUniversityPhysics1ATopic3:Waves
OscillationsOscillatingSystemsOscillatingsystemsoftenundergoperiodicmotion,wherethemotionoftheobjectwillrepeatatregularintervals.Onesuchexampleofthisissimpleharmonicmotion.SimpleHarmonicMotionThisiswhereaforceactingonanobjectisproportionaltothepositionoftheobject,relativetosomeequilibriumposition(wherethisforceisalsodirectedsuchaposition).Inthecaseofaspring,thisforcecanbequantifiedwith . TheaccelerationofaparticleundergoingSHMisnotconstant.However,theaccelerationofthemasscanbedeterminedusingNewtonssecondlaw:
ItisthisaccelerationequationwhichdefinesasystemundergoingSHM.Fromthisequation,itisclearthattheaccelerationisproportionaltothedisplacementoftheobject.Furthermore,thedirectionofthisaccelerationisoppositetothedisplacementfromtheequilibriumpoint.Notethatwhenablockcompletesonefulloscillation,ithasmoved4A,whereAistheamplitudeoftheoscillation.Thisisbecauseitmustmovefrommaxdisplacement,toneutral,tominimumdisplacement,andthenallthewaybacktomaxdisplacement. Whenablockishungfromaverticalspring,itsweightwillcausethespringtostretchtosomeequilibriumpoint.Iftherestingpositionofthespringisdefinedasy=0,thenitisclearthat:
WhenrepresentingSHMmathematically,itisusefultochoosethexaxisas
theoneinwhichtheoscillationoccurs.ApplyingNewtonssecondlaw:
Furthermore,if ,then2
wherefisthefrequencyoftheoscillations.
Clearly,theperiodandfrequencyofthemotionareverydependentuponthemassoftheparticleandtheforceconstantofthespring.Evidently,thefrequencyisproportionaltothespringconstant,butinverselyproportionaltothemass. InrepresentingSHMgraphically,anequationwhichcandefineSHMis: . cos
. sin
. cos
. sin
asrequiredforSHM.Similarly,anotherequationwhichcandefineSHMis .
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Notethatintheaboveequations:
2
,
2
istheangularfrequency(unitsofradian/s) Aistheamplitudeofthemotion,orthemaximumdisplacementthe
particleachieves isthephaseconstant,orinitialphaseangle
Notethatthephaseofthemotionisgivenby . cos
Since ,itispossibletodeterminethevalueofAandbyapplyingtheconditiont=0totheequation. Becauseofthenatureofthesineandcosinefunctionsoscillatingbetween1and1,themaximumvaluesofvelocityandaccelerationaregivenby:
.
Anotherimportantfactisthatthevelocityfunctionistypically90outofphasewiththedisplacementfunction,whiletheaccelerationistypically180outofphasewiththedisplacementfunction.EnergyofCollisionsAssumingthatthespringmasssystemismovingonafrictionlesssurface,thekineticenergycanbefoundby:
. sin
. sin ,
Theelasticpotentialenergycanbefoundby:
. cos
Hencethetotalenergyofthesystemcanbegivenby:
.
Notethatthistotalenergygivenaboveremainsconstant,andthatitisproportionaltothesquareoftheamplitude.Theenergyisbeingcontinuouslytransferredbetweenthepotentialandkineticenergyoftheblock.
MolecularModelofSimpleHarmonicMotionIftheatomsinamoleculedonotmovetoofarapart,theforcebetweenthemcanbemodelledasiftherewerespringsbetweentheatoms.Hencethepotentialenergyactssimilartothatofanoscillatorundergoingsimpleharmonicmotion.
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UniformCircularMotionThereexistsaclearlinkbetweensimpleharmonicmotionandcircularmotion.Ifadiskwithaknobisrotatedandviewedfromabove,theknobappearstomovebackandforthasthoughitwereundergoingsimpleharmonicmotion.Thisbecomesevidentinthediagrambelow:
Theparticlemovesalongthecirclewithaconstantangularvelocity.
OPmeansananglewiththexaxis. Attimet,theangle illbew . cos
. sin . sin
Hence whicht equirementforSHM
Qu evidently,simpleh necanberepresentedby
her
ite armonicmotionalongastraightliaprojectionofuniformcircularmotionalongthediameterofareferencecircle.Thisallowsuniformcircularmotiontobeconsideredacombinationof2simpleharmonicmotions:
Onealongthexaxis Theotheralongtheyaxis
Wherethetwodifferinphaseby90.PendulumsThemotionofasimplependulumintheverticalplaneisdrivenbygravitationalforce.Thismotionisverysimilartothatofparticlesundergoingsimpleharmonic
retheangleofthependulumissmall.
motion,whe
Fromthediagramabove,theforcesactingonthebobaretension(T)andweightforce(mg).Thetangentialcomponentofgravitationalforceisarestoringforce,givenby sin .Thearclengthi givenbys .
Hence
and
Inthetangentialdirection, sin
ThelengthLofthependulumisconstant,henceforsmallvaluesof,
sin
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Hencethemotionissimpleharmonicwith
Thefunction as:canbewritten cos
Theangularfrequencyisgivenby:
Theperiodisgivenby:
2
islongerorisinafieldoflowergravitationalplete1period.
DampedOscillations
Basedontheabove,ifthependulumforce,itwilltakelongertocom
Inmanyrealsystems,nonconservativeforcesarepresent,suchasfriction,airresistanceandviscosity.Insuchcases,themechanicalenergyofthesystemwilldiminishovertime,andhencethemotionisdescribedasbeingdamped.Belowis
thegraphofadampedoscillation:
Basedonthisdiagram,theamplitudeoftheoscillationdecreasewithtime,wherethebluelineisrepresentativeoftheenvelopeofthemotion. Anexampleofdampedmotioniswhenanobjectisattachedtoaspringandsubmergedinaviscousliquid.Theretardingforcecanbeexpressedas ,wherebisapositiveconstant,knownasthedampingconstant.FromNewtonssecondlaw, Therearethreetypesofdamping,shownbythegraphbelowofpositionversustime:
.
(a) Isanunderdampedoscillator(b) Isacriticallydampedoscillator(c) Isanoverdampedoscillator
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ForcedOscillationsBecauseoftheconceptofdampedoscillations,theprocessofforcedoscillationswasdeveloped.Thisiswherelossofenergyinadampedsystemiscompensatedforbyapplyinganexternalforce,forinstanceperiodicallypushingapendulum.Theamplitudeofthemotionremainsconstantiftheenergyinputpercycleexactlyequalsthemechanicalenergylostbythesystemineachcyclefromresistiveforces.Generallyspeaking: sin sin W is eherextisthetransientforce,andxss th steadystateforce.
Whilethemotionofanobjectundergoingsimpleharmonicmotionisgivenb am fy . cos ,the plitudeo thedrivenoscillationisgivenby:
Whereisthenaturalfrequencyoftheundampedoscillator,givenby
.
Quiteclearly,theamplitude fthemotionisdependentonthefrequency.oResonanceWhenth frequencyofthedrivingforceisnearthenaturalfrequency,(oe r ),therewillbeanincreaseinamplitude.Thisdramaticincreaseisreferredtoasresonance,whilethenaturalfrequencyofthesystemisalsoreferredtoastheresonancefrequencyofthesystem.
Themaximumpeakofresonanceoccurswhenthedrivingfrequencyequalsthenaturalfrequency.Theamplitudethenincreaseswithdecreaseddamping.Consequently,thecurve(shownbelow)willbroadenasdampingincreases.Theshapeo
fthiscurveishencedependentonb.
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WaveMotionropagationofaDisturbanceP
Therearetwomajortypesofwaves.Theyaremechanicalwaves(wheresomemediummustbedisturbedandthewavepropagatesthroughthismedium)andelectromagneticwaves.Theenergyofanydisturbanceistransferredoverdistance,butnotthematter(theparticlesmovebackandforthorupanddown). ApulseonaropecanbecreatedbyflickingaropeundertensionindirectThispulsecanthentravelthroughtherope,causingthepulsetohaveadefiniteheightandspeedofpropagation(whichistypicallyuniqueforeachmedium).
ion.
Iftheopeis
Longitudinalwavesarethosewherethetravellingwavecausestheparticlestomoveparalleltothedirectionofthewave.
Complexwavescanalsobeformed,suchaswaterwaves.Thesecanexhibitacombinationofthepropertiesoftransverseandlongitudinalwaves.
r continuouslyflicked,aperiodicdisturbancecanbeformedasawave. Wavescantraveleitherparallelorperpendiculartothemotionofthedisturbance.Transversewavesarecomparabletoasinecurve,andtheparticlesaredisplacedperpendiculartothedirectionofpropagation.
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TravellingWavesAtravellingwaveorpulsecanberepresentedbyanequationwhichgivesthedisturbanceasafunctionofdisplacementandtime.
w:
Inthediagrambelo
Evidently,theshapeofthepulseatt=0canberepresentedby, 0 .Thisdescribesthetransverseposition,y,oftheelementofthestringlocatedateachvalueofxatt=0.Thespeed ofthepulseisvandintimet,willtraveladistanceofvt.Attime=t,theshapeisrepresentedby .Thisisb seecausethepulmovesdowntheropeandisntalwayssinusoidal.Hence,toachievetheoriginalcurve,thedisplacementmustbesubtracted. Consequently,foracurvetravel t,lingtotherigh , .Meanwhile,foracurvetravellingtotheleft,, .Assuch,y(x,t)isthewavefunction,representingtheycoordina
teofanyelementlocatedatposition
x ytimet.Forafixedt,itiscalledthewaveform,as aatan itdefinesthecurve tanyspecifictime.PropertiesofWavesTheamplitudeofawaveisthemaximumdisplacementfromtheequilibriumposition.Thewavelengthisthedistancebetweenany2identicalpointsonadjacentwaves.
ncyisthenumberofwavespersecond.Theperiodisthetimetakenforonewavelength.ThefrequeWaveFunction
Thisisgivenby, . sin .Itdescribesthemotionofawave
movinginthepositivexdirectionwithaspeedofv.Conversely,
, . sin describesmotioninthenegativexdirection.
Inperiodicformitisgivenby:
, . sin 2
. sin
HencethewavenumberIsgivenby
andtheangularfrequencyby
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WaveSpeedWhilethetransversespeedofawave givenbycanbe
, . . cos ,this wave
itself,givenby:
isdifferenttothespeedofthe
Thespeedofawaveonthestringcanalsobegivenby:
/
PowerandIntensityinWaveMotion
Thekineticenergyofa veisgivenbywa ,where .
Thetotalkineticenergyinawaveisgivenby
.Thissameequation
canalsobeusedtodefinepotentialener ence,the rgygyinawave.H totalene ofa
waveisgivenby
.
Thepowerassociatedwithawaveisgivenby:
red,amplitudesquaredandwavespeed.PrincipleofSuperposition
y
Hencepowerisproportionaltofrequencysqua
Theprincipleofsuperpositioninvolvestheenergyofaseriesofwavesaddingatangivenpoint;hencetheircombinationisthealgebraicsumoftheirvalues.However,theprincipleofsuperpositioncanonlyapplytolinearwaves,wheretheamplitudeissmallerthanthewavelength. Theprinciplestatesthattravellingwavescanpassthroughoneanotherwithoutbeingdestroyedoraltered.Thisresultsintheircombinationinaresultantwaveknownasinterference.InterferenceofWavesTheretwotypesofinterference:constructiveanddestructive.Constructiveinterferenceiswherethedisplacementscausedbythetwopulsesareinthesamedirectionandhencetheamplitudeoftheresultingwaveisgreaterthaneitherwave.Destructiveinterferenceiswherethedisplacementsareinoppositedirectionsandhencetheamplitudeoftheresultingwaveislessthaneitherwave.Ineffectthewavescanceleachotherouttoacertainextent.
Mathematically:1 . sin 2.sin 21 2.cos
sin
Hencetheresultingwaveissinusoidal,withthesamefrequencyandwavelength,but
withanamplitudeof2. cos andaphaseof
.Henceduringperfectconstructive
ndthewa
interference,thephaseiszero(wherebothwavesareinphaseeverywhere)andperfectdestructiveinterference,thephaseis
anoddmultipleof(a vescanceloneanotherout)resultinginanconsequently,theamplitudeis2A.In
amplitudeof0.Generallyspeaking,ifthephaseisbetween0andanoddmultipleoftheamplitudeisbetween0and2Aandthefunctionswillcontinuetoadd.
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StandingWavesIf2waveshavethesameamplitude,wavelengthandfrequency,andtravelintheoppositedirectiononeanotherwiththeequations: . sin . sin Wherethedifferentsignsindicatethedifferentdirections,theywillinterfereaccordingtothesuperpositionprincipledescribedintheequationbelow: 2. sin cos
Thisisthewavefunctionofastandingwave.Evidently,thereisno component,becausethewaveisnolongertravelling.Thisbecomesobviouswhenastandingwaveisobserved,wherethereisnosenseofmotionastheparticlesappeartooscillatearoundanode. Anodeoccursatapointofzeroamplitude.Inotherwords:
sin sin 0.Thiscorrespondstowhen
, .
Anantinodeoccursatapointofmaximumdisplacement(2A).Inotherwords:
sin 1 towhere .Thiscorresponds
,wherenisapositiveodd
number. Evidently,theamplitudeofanindividualwaveisA,whiletheamplitudeof
dergoingSHMis2. sin, theamplitudeofastandinganyparticleun while waveis2A.However,theamplitudeofanyparticleinastandingwaveisgivenbythesameequationasforSHM. Keytostandingwavesis tatpointsofmaximaldisplacement,thethefactthaparticlesaremomentarilystationary,whileatzerodisplacement,theparticleshavedifferinginstantaneousvelocitiesasssomeparticlesmoveupwhileothersmovedown. Forastandingwavetobeestablished,theendpointsmustbenodesabdfixed(hencehavezerodisplacement).Thiswillthenresultinasetofnormalmodes(oraseriesofantinodes),whereeachnormalmodereferstothenumberofantinodes.Therelationshipbetweennormalmodesandthewavelengthviewedis
givenby
,wherenisthenthnormalmodeofoscillation.Since ,the
naturalfrequencyisgivenby
.Ifthestringisundertension,
thiscanb by ,ifafreelyhangingweightisused.Inastandingwaveegiven ,thenumberofnodesisonegreaterthanthatofantinodesanditwillshowsymmetryaboutthemidpointofthestrinQuantisation
g.
Thisiswhereonlycertainfrequenciesofoscillationareallowed.Thisisparticularlycommonwhereboundaryconditionsmustbemet.
esHarmonicSeriThenaturalfrequencycorrespondsto 1,andisthelowestfrequency.Thefrequenciesofthereaminingnaturalmodesareintegermultiplesofthefunamentalfrequency,andwillformaharmonicseries.Thenormalmodesmaybecalledharmonics,orresonantmodes.
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SoundWavespeedofSoundhespeedofsoundisdependentuponthepropertiesofthemediumitistravellingST
through.Itcanbesummedupbytheequation
.Inaliquidora
gas,theBulkmodulus,orB,isgivenby
.Thedensityofthematerialis
givenby,whilethespeedofsoundinaliquidorgasisgivenby .
Thespeedofsoundinasolidisdeterminedviadifferentmeans.Itutilises
Youngsmodulus:
.Again,the
d isgivenby.Hencethevelocityofsoundinasolidisgivenbyensity .
Thespeedof fthemediumsoundisalsodependentuponthetemperatureo ,
especiallyingases.Forair,thisrelationshipisgivenby 3311
,
whereuponthefrequency.
mpressionsandrarefactionsinthemedium
TcisthetemperatureinCelsius.Unliketransversewaves,thevelocitydoesnotdepend
TravellingLongitudinalWavesSoundwavesarecausedbyaseriesofcoitistravellingthrough.Theseregionsmovewiththesamespeedassound.Theperceptionofsoundcanbeviathechangeinpressure(causedbythedifferencesinpressureofcompressionsandrarefactions),orbytheinterpretationofthepulses. EachelementofthemediummoveswithSHMparalleltothedirectionofpropagation.Consequently,thedisplacementofanyelementisgivenby:, cos ,wheresmaxisthemaximumpositionfromtheequilibriumposition,andisoftencalledthedisplacementamplitudeofthewave.
Thevariationingaspressure,orisalsoperiodicandcanbegivenbytheequation
Physics1ATopic3Waves
PowerandIntensityofSoundWavesIfapistonweretocompressagas,itwouldtransferenergytotheelementsofairinthetube.Thisenergywouldthenpropagateawayfromthepistonasasoundwave.
kineticandpotentialenergy,togivethetotalUsingthis,itispossibletocalculatetheenergyofthewave, cos , . sin
12
1
2. . . sin
Byusing sin
14.
mentsundergoSHM,thepotentialenergyofonewavelengthisthesameasitskineticenergy.Hence, hetotalmechanicalenergyisgivenby:Astheele
t
12.
enby:
Therateofenergytransferisthepowerofthewaveandisgiv
12. .
inceS .Thisgivestheenergythatpassesagivenpointduringoneperiodof
oscillation. Theintensityofawaveisdefinedasthepowerperunitarea.Itistherateatwhich nergytransportedbythewavepassesthroughaunitarea;thee
In aseoftheir,thisisgivenby:thec . .Intermsofthepressure
amplitude: . . .
.
Apointsourcewillemitsoun alldirections.Consequentdwavesequallyin ly,thepowerwillbeevendistributedinthisspherearoundthesource.
.Evidently,thislawobey inversesquarelaw.sthe
Theintensityofsoundwavesismeasuredindecibels.Whiletherangeofintensitiesdetectablebythehumanearareratherlarge,theyaredetectedaccordingtoalogarithmicsca isinfact10le,suchthatasoundperceivedtobetwiceasloudti sloud.Theintensitycanbedeterminedvia:mesa
10 log
I0isconsideredthereferenceintensity,andistakentobethethresholdofhearing.Itisequivalenttothefaintestdiscernableintensityofsoundinasilentroom.Itisequalto1 10/ .Atthislevel,theintensityis0dB.0dBdoesnotnecessarilymean
Theloudnesherethe
ncesound(oftentakentobe
thereisnosound;justnosoundwhichcanbedetectedbythehumanear.Consequently,anegativevalueofdBispossible. sofasoundisoftenrelatedtoaphysicalmeasurementofthestrengthofasound.However,apsychologicalassessmentcanalsooccur,wbodycalibratesasoundbycomparingittoarefere1000Hz,whichisthethresholdofhearing).Generally,doublingtheloudnesswillcauseanincreaseof10dB.
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Physics1ATopic3Waves
DopplerEffectThisistheapparentchangeinfrequency(orwavelength)ofawave,whichoccursbecauseofrelativemotionbetweenthesourceandtheobserver.Whentherelativespeedisgreaterthanthewavespeed,thefrequencyappearstoincrease.Conversely,whentherelativespeedislowerthanwavespeed,thefrequencyappearstodecrease. Ifthesourceisstationaryandtheobservermovestowardsthesource,thenthespeedofthewavesrelativetotheobserverwillbe .Consequently,theobservedfrequencywillbe:
Wherefistheobservedfrequencyandvisthevelocityofthewavesandfisthefrequencyofthewaves.Consequently,ifthesourceisstationaryandtheobservermovesawayfromthesource,theobservedfrequencywillbe:
Ifthesourceisinmotionandtheobserverisatrest,thenthedistancebetweenwavefrontswillchangeby
.Hencethedistancebetweenwavefronts
w omeillbec
.Therefore:
"
"
Mostimportantlyistha thewavespeedremainsunchanged.Wavespeedistpurelydependentuponthemediumandisnotaffectedbyanyrelativemotionofthesource. Whenthesourceismovingtowardstheobserver:
"
Whenthesourceismovingawayfromtheobserver:
"
Ifboththesourceandobserverismoving,then:
eyto forachangeinfrequencytooccur.Ifatrainapproachesatconstantvelocity,thedistancebetweenwavefrontswillessentiallybethesameandtherefore,therewillbenochangeinfrequency.
K theDopplerEffectisthepresenceofacceleration
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Physics1ATopic3Waves
ShockWavesItispossibleforthespeedofthesourcetoexceedthespeedofthewave.Theresultisanenvelopeofthesewavefrontsintheformofacone,wheretheapexhalfangleisgivenbysin
.ThisanglemayalsobecalledtheMachangle.
TheMachnumberistheratioofthesourcespeedtothewavespeed.Itisgivenby
.TherelationshipbetweentheMachnumberandMachangleisgiven
by:sin
.Whenthemachnumberisgreaterthan1,ashockwave
willform hespeedisconsideredsupersonic.Theshockwavecarriesalargeandtamountofenergy,concentratedinthesurfaceofthecone.Consequently,thepres rdingtolocation.surevariesgreatlyacco Whenanaircraftflieswithconstantvelocityfromcoldairtowarmair,theMachnumberwilldecrease.ResonanceA miscapableofoscillatinginoneormorenormalmodes.Consequently,ifasysteperiodicforceisappliedtoasystem,theresultingmotionisgreatestwhenthefrequencyoftheappliedmotionisequaltooneofthenaturalfrequenciesofthesystem.Thesenaturalfrequenciesarereferredtoasresonancefrequencies,andissymbolisedby .Insuchsystems,themaximumamplitudeisonlylimitedbythefrictioninthesystem.StandingLongitudinalWavesinAirColumnsSuchwavescan esetupinaircolumnsasaresultofinterferencebetweenblongitudinalsoundwavestravellingintheoppositedirectiontoeachother.Thephaserelationshipbetweentheincidentandreflectedwavesdependonwhethertheendofthepipeisopenorclosed. Iftheendofthepipeisclosed,thenadisplacementnodeisformedattheendofthepipe(andsincedisplacementandpressureare90outofphase,itisapressureantinode).Thisisbecausethewallwillnotallowanyfurtherlongitudinalm intheair.Consequently,thereflectedwaveis180outofphasewiththeotionincidentwave,creatingastandingwave.Itisworthnotingthattheopenendisadisplacementantinode,whiletheclosedendremainsadisplacementnode.Thefirst
resonancewillgivenby ,whilethefirstfundamentalfrequencyisgivenby
.Consequentlyfrequenciesofhighermodesaregivenby:
4 2 1 2 1
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Physics1ATopic3Waves
Iftheendofthepipeisopen,thenattheendofthepipewillbeadisplacementantinode(orpressurenode)becausethecompressedairisfreetoexpandintotheatmosphere(andhencethereisnopressurevariation).Itisworthnotingthatbothendsofthetubearedisplacementantinodes.Thefirstresonance
w ofillbe ,whilethefundamentalfrequencywillbe
.Higher
resonanceswillbeequalto . .
.
Inpractiseanantinodeformingattheopenendofatubewillbeslightlybeyondtheendofthetube.Thisadditionallengthmustbeaccountedforwhenconsidering
resonance.Forthetubebelow,thefirstresonancewillbegivenby 2 .
Anexampleofresonanceinaircolumnscanbegivenbytubespartiallyfilledwithwater.Whenatuningforkisbroughtnearthetopofthetube,andthelengthfromthetoptothewatercorrespondstoaresonancefrequencyofthepipe,thesoundwillbelouder.Usingtheselengths,itis ossibletocalculatethelengthswherepresonanceoccurs.
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Physics1ATopic3Waves
StandingLongitudinalWavesinRodsIfarodisclampedinthemiddle,andstrokesareappliedtherod(andinthesamedirectionastherod),longitudinalwaveswillpassthroughit,causingtherodtooscillate.Theclamphoweverwillforcetheappearanceofadisplacementnode.Theendsoftherod,however,willbefreetovibrateandhencewillformdisplacementantinodes.
Iftherodisclampedatapointotherthanthemiddle,othernormalmodesofoscillationcanbeproduced.Iftherodisclampedadistanceof
fromoneendofthe
rod(where
),thenthesecondnormalmodewillbeproduced.Thisconceptis
utilisedonmusicalinstrumentssuchasxylophonesandchimes.StandingWaveinMembranes
circularface resultingsoundwillnotbeharmonic,sincethestandingwaveshavefrequencieswhicharenotintegermultiples.Thefundament
Twodimensionaloscillationscanbesetupinaflexiblemembranestretchedovera.The
alfrequencywill
SpatialandTemporalInterference
containonenodalcurve.
Spatialinterferenceiswhentheamplitudeoftheoscillationinamediumvarieswiththepositioninspaceoftheelement,suchaswithstandingwaves.Temporalinterferenceiswhenthewavesareperiodicallyinandoutofphase.Consequently,
eenconstructiveanddestructiveinterference,thereisatemporalalternationbetwsuchasinbeats.
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Physics1ATopic3Waves
Beatstemporalinterferencewilloccurwhentheinterferingwavehaveslightlydifferentequencies.Beatingistheperiodicvariationinamplitudeatagivenpointduetotheuperpositionoftwowaveshavingslightlydifferentfrequencies.
Afrs . cos . cos2 . cos . cos2
2 cos 2 2
cos 2 2
Consequently,thewavehasameanfrequencyof:
andismodulatedbythe
timevaryingamplitudeof2 cos 2 .
Thebeatfrequencyisthenumberofamplitudemaximapersecond.Itoccurs
whencos 2 1.Consequently,itisthe ifferencebetweenthed
frequenciesoftwosources,givenby | |,althoughthehumanearcano tectabeatfrequencyofupto20beatspersecond.nlyde
Thewavepatternsproducedbymusicalinstrumentsaretheresultofthesuperpositioningofvariousharmonics.Thehumanperceptiveresponseassociatedwiththesemixturesisthequalityortimbreofthesound.Forinstance,atuningforkproducesasinusoidalpattern:
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Thesamenoteonaflutesoundsdifferently,giventhesecondharmonicisverystrongandthefourthissimilarinstrengthtothefirst:
Whenawavepatternisperiodic,itcanbecloselyapproximatedbyacombinationofsinusoidalwaveswhichformaharmonicseries.ThistechniqueisdescribedbyFouriersTheorem,andutilisestheFourierSeries.heseriesismadeupofoddnumberedharmonicsandisgivenby:T
sin2 cos2
Where
and n n
andA andB aretheamplitudesofthewaves.
PHYS1121_Topic 1_MechanicsPHYS1121_Topic 2_Thermal PhysicsPHYS1121_Topic 3_Waves