PHYS1121 Course Notes

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

CompiledbyJohnTrieu2009 Page1


ResolutionandUnitVectorsVectorsareresolvedonthebasisoftheirdirectionfromtheorigin.Aguideforthenotationisprovidedbelow.

Unitvectorsallhaveamagnitudeof1.Theyarealwaysunderlinedandindicatedbya^signabovethewordedvector(eg.)PolarCoordinateSystemsThisisbase ontheprinciplethatanypointrofcoordinates(x,y)willconsistof2dcomponent indicatingitslocation.Thesecomponentsare:s

VectorAddition



VectorSubtraction:v=vuToworkoutvectorvuwemustreverseuandaddtov.Letusimaginethatacarismovingat20m/sNanditturnsacornersoitisnowmoving20m/sW.Itispossibletouseavectordiagramtodeterminethechangeinvelocity.

Step1:Draworiginalvectordiagram

Step2:Reverseu

Step3:Solveforx. 800Therefore 202Or28.28ms1SW(2DecimalPlaces)

EquatingVectorsTheprocessofequatingvectorsisachievedbyensuringallcomponentsareequalandthenensuringthatallforcesareequal(asforcesarevectorsinthemselves).Assuchthemagnitudeanddirectionofthevectorsshouldbeequal.EquationsofMotioninVectorForm



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.



CentripetalAccelerationandPeriodofRotationThisreferstoanaccelerationwhichiscentreseeking,andiscalculatedaccordingtotheequation:

Notethatcentripetalaccelerationisneverconstantowingtothepersistentchangingofvelocity.Theperiodofrotationcanbecalculatedaccordingtotheequation:

2

ExampleAplanetravelsinahorizontalcircle,speedvandradiusr.Foragivenv,whatistherforwhichthenormalforceexertedbytheplaneonthepilotistwiceherweight.Whatisthedirectionofthisforce?

CentripetalForce:

VerticalForces:

Byeliminating,

30,TangentialandRadialAccelerationTangentialaccelerationisthecomponentwhichwillcauseachangeinthevelocityoftheparticle.Ithasparallelswithinstantaneousvelocityandisgivenby:

Radialaccelerationisderivedfromthechangesindirectionofthevelocity,andisgivenby:

Becauseradialandtangentialaccelerationareperpendicularcomponentvectorsofacceleration,themagnitudeoftotalaccelerationcanbegivenby:



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.



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.



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



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,

.



Thiscanbeappliedtotheproblembelow:

ariableForcesV

HookesLawHookesLawdealswiththebehaviourofproductswhichdisplaylinearelasticity.Itisbasedontheprinciplethattheforceappliedtoanobjectwillbeproportionaltoanydeformation. However, helawhasamajorflawwithit;itonlyappliestoasmallportionofthe

of

tive

hbelow:

tgraphofforcevsdeformation(orintermolecularseparation,asthisisanindicatordeformation).Indeformationbybending,someseparationsarestretched,whileothersarecompressed.Therealsoexistsaneutralpositionwherethereisnorelastretchingorcompression.Whenthisstretchingorcompressionexceedsthelimitationsoftheobject,thelawfails.Assuch,itcanonlybeusedasanapproximation.Observethegrap



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



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:



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:



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:




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.


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



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



AngularMomentum




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.



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.



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

.



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.



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.



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.



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.



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.




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 .



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.



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



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



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.



WaveMotionropagationofaDisturbanceP

Therearetwomajortypesofwaves.Theyaremechanicalwaves(wheresomemediummustbedisturbedandthewavepropagatesthroughthismedium)andelectromagneticwaves.Theenergyofanydisturbanceistransferredoverdistance,butnotthematter(theparticlesmovebackandforthorupanddown). ApulseonaropecanbecreatedbyflickingaropeundertensionindirectThispulsecanthentravelthroughtherope,causingthepulsetohaveadefiniteheightandspeedofpropagation(whichistypicallyuniqueforeachmedium).

ion.

Iftheopeis

Longitudinalwavesarethosewherethetravellingwavecausestheparticlestomoveparalleltothedirectionofthewave.

Complexwavescanalsobeformed,suchaswaterwaves.Thesecanexhibitacombinationofthepropertiesoftransverseandlongitudinalwaves.

r continuouslyflicked,aperiodicdisturbancecanbeformedasawave. Wavescantraveleitherparallelorperpendiculartothemotionofthedisturbance.Transversewavesarecomparabletoasinecurve,andtheparticlesaredisplacedperpendiculartothedirectionofpropagation.



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



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.



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.



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


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.



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



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



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.



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.



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:




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

PHYS1121 Course Notes

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