CH 15 HW

11/20/2014
Ch 15 HW
Ch15HW
Due:11:59pmonTuesday,November25,2014
Youwillreceivenocreditforitemsyoucompleteaftertheassignmentisdue.GradingPolicy
Exercise15.4
Ultrasoundisthenamegiventofrequenciesabovethehumanrangeofhearing,whichisabout20000Hz.Wavesabove
thisfrequencycanbeusedtopenetratethebodyandtoproduceimagesbyreflectingfromsurfaces.Inatypical
ultrasoundscan,thewavestravelwithaspeedof1500m/s.Foragooddetailedimage,thewavelengthshouldbeno
morethan1.0mm.
PartA
Whatfrequencyisrequired?
ANSWER:
f
= 1.50106 Hz
Correct
AntonaTightrope
AlargeantisstandingonthemiddleofacircustightropethatisstretchedwithtensionTs .Theropehasmassperunit
length.Wantingtoshaketheantofftherope,atightropewalkermovesherfootupanddownneartheendofthe
tightrope,generatingasinusoidaltransversewaveofwavelengthandamplitudeA.Assumethatthemagnitudeofthe
accelerationduetogravityisg.
PartA
WhatistheminimumwaveamplitudeAmin suchthattheantwillbecomemomentarily"weightless"atsomepointas
thewavepassesunderneathit?Assumethatthemassoftheantistoosmalltohaveanyeffectonthewave
propagation.
ExpresstheminimumwaveamplitudeintermsofTs ,,,andg.
Hint1.Weightandweightless
Weightisgenerallydefinedasbeingtheequalandoppositeforcetothenormalforce.Onaflatsurfaceina
staticsituation,theweightisequaltotheforceduetogravityactingonamass.
"Weightless"isamorecolloquialtermmeaningthatifyousteppedonascale(e.g.,inafallingelevator)it
wouldreadzero.Thinkaboutwhathappenstothenormalforceinthissituation.
Notethattheforceduetogravitydoesnotchangeandwouldstillbethesameaswhentheelevatorwas
static.
Hint2.Howtoapproachtheproblem
Inthecontextofthisproblem,whenwilltheantbecome"weightless"?
http://session.masteringphysics.com/myct/assignmentPrintView?displayMode=studentView&assignmentID=3094650
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Ch 15 HW
ANSWER:
Whenithasnonetforceactingonit
Whenthenormalforceofthestringequalsitsweight
Whenthenormalforceofthestringequalstwiceitsweight
Whenthestringhasadownwardaccelerationofmagnitudeg
Hint3.Findthemaximumaccelerationofthestring
Assumethatthewavepropagatesasy(x, t)
= A sin(t kx)
.Whatisthemaximumdownward
accelerationamax ofapointonthestring?
Expressthemaximumdownwardaccelerationintermsofandanyquantitiesgivenintheproblem
introduction.
Usetheformulagivenforthedisplacementofthestringtofindtheaccelerationofthestringasa
functionofpositionandtime.Thendeterminewhatthemaximumvalueofthisaccelerationis.(At
sometime,thebitofropeunderneaththeantwillhavethismaximumdownwardacceleration.)
Hint2.Accelerationofapointonthestring
Findtheverticalaccelerationay (x, t) ofanarbitrarypointonthestringasafunctionoftime.
ExpressyouranswerintermsofA,,t,k,andx.
Hint1.Howtofindtheacceleration
Differentiatetheexpressiongivenfory(x, t) ,thedisplacementofapointonthestring,twice.
Hint2.Thefirstderivative
Differentiatethegivenequationforthedisplacementofthestringy(x, t) tofindthevertical
velocityv y (x, t) oftherope.
ExpressyouranswerintermsofA,,t,k,andx.
ANSWER:
v y (x, t)
Acos(t kx)
ANSWER:
ay (x, t)
Asin(t kx)
Hint3.Findthemaximumdownwardacceleration
(x, t)
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Themaximumdownwardaccelerationamax isthemostnegativepossiblevalueofay (x, t) .Asthe

wavepassesbeneaththeant,atsometimeoranothertheantwillbeatapointwherethe
accelerationofthestringhasthismostnegativevalue.
Whatisamax ?
Expressyouranswerintermsofandquantitiesgivenintheproblemintroduction.
Hint1.Whenwilltheaccelerationreachitsmostnegativevalue?
Themostnegativeaccelerationoccurswhensin(t kx)
= 1
ANSWER:
amax
Hint4.Determineintermsofgivenquantities
Theangularfrequencyofthewaveinthestringwasnotgivenintheproblemintroduction.Tosolve
theproblem,findanexpressionforintermsofgivenquantities.
ExpresstheangularfrequencyintermsofTs ,,,and.
Hint1.Howtoapproachthisquestion
Combineageneralformulafor,arelationshipamongfrequency,wavelength,andvelocity,
andaformulaforthevelocityofawaveonastringtofindanexpressionforintermsof
quantitiesgivenintheproblemintroduction.
Hint2.Generalformulafor
Theangularfrequencyofawaveisequalto2timesthenormalfrequency:
= 2f
Hint3.Relationshipamongfrequency,wavelength,andvelocity
Thefrequency,wavelength,andvelocityofawavearerelatedbyv
= f
Hint4.Speedofawaveonastring
Whatisthespeedv ofanywaveonthestringdescribedintheproblemintroduction?
ANSWER:
Ts
ANSWER:
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Ch 15 HW
Ts
ANSWER:
amax
Ts
) A
Hint4.Puttingitalltogether
Onceyouhaveanexpressionforthemaximumaccelerationofapointonthestringamax ,determinewhat
amplitudeisrequiredsuchthatamax
= g
.ThiswillbetheminimumamplitudeAmin forwhichtheant
becomesweightless.
ANSWER:
A min
g
Ts
Correct
Exercise15.12
x
Theequationy(x, t) = A cos 2f ( v
t) maybewrittenasy(x, t) = A cos[
(x vt)].
PartA
Usethelastexpressionfory(x, t) tofindanexpressionforthetransversevelocityv y ofaparticleinthestringon
whichthewavetravels.
ExpressyouranswerintermsofthevariablesA,v ,,x,t,andappropriateconstants.
ANSWER:
vy
A(
)vsin((
)( x vt))
Correct
PartB
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Findthemaximumspeedofaparticleofthestring.
ExpressyouranswerintermsofthevariablesA,v ,,x,t,andappropriateconstants.
ANSWER:
v max
A(
)v
Correct
StandardExpressionforaTravelingWave
LearningGoal:
Tounderstandthestandardformulaforasinusoidaltravelingwave.
Oneformulaforawavewithaydisplacement(e.g.,ofastring)travelinginthexdirectionis
y(x, t) = A sin(kx t)
Allthequestionsinthisproblemrefertothisformulaandtothewaveitdescribes.
PartA
Whichofthefollowingareindependentvariables?
Hint1.Whatareindependentvariables?
Independentvariablesarethosethatarefreelyvariedtocontrolthevalueofthefunction.Theindependent
variablestypicallyappearonthehorizontalaxisofaplotofthefunction.
ANSWER:
x
t
only
only
only
only
only
andt
andt
andkand
Correct
PartB
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Whichofthefollowingareparametersthatdeterminethecharacteristicsofthewave?
Hint1.Whatareparameters?
Parametersareconstantsthatdeterminethecharacteristicsofaparticularfunction.Forawavetheseinclude
theamplitude,frequency,wavelength,andperiodofthewave.
ANSWER:
x
t
only
only
only
only
only
andt
andt
andkand
Correct
PartC
Whatisthephase(x, t)ofthewave?
Expressthephaseintermsofoneormoregivenvariables(A,k,x,t,and)andanyneededconstantslike
.
Hint1.Definitionofphase
Thephaseistheargumentofthetrigfunction,whichisexpressedinradians.
ANSWER:
=
(x, t)
kx t
Correct
PartD
Whatisthewavelengthofthewave?
Expressthewavelengthintermsofoneormoregivenvariables(A,k,x,t,and)andanyneeded
constantslike.
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Hint1.Findingthewavelength
Considertheformofthewaveattimet = 0.Thewavecrossestheyaxis,slopingupwardatx = 0.The
wavelengthisthexpositionatwhichthewavenextcrossestheyaxis,slopingupward(i.e.,thelengthofone
completecycleofoscillation).
ANSWER:
=
2
k
Correct
PartE
WhatistheperiodT ofthiswave?
Expresstheperiodintermsofoneormoregivenvariables(A,k,x,t,and)andanyneededconstants
like.
ANSWER:
T
Correct
PartF
Whatisthespeedofpropagationv ofthiswave?
Expressthespeedofpropagationintermsofoneormoregivenvariables(A,k,x,t,and)andany
neededconstantslike.
Hint1.Howtofindv
Ifyou'vedonethepreviouspartsofthisproblem,youhavefoundthewavelengthandtheperiodofthiswave.
Thespeedofpropagationisafunctionofthesetwoquantities:v = /T .
ANSWER:
v
Correct
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VideoTutor:OutofPhaseSpeakers
First,launchthevideobelow.Youwillbeaskedtouseyourknowledgeofphysicstopredicttheoutcomeofan
experiment.Then,closethevideowindowandanswerthequestionontheright.Youcanwatchthevideoagainatany
point.
PartA
Twospeakersfaceeachother,andtheyeachemitasoundofwavelength.Onespeakeris180 outofphasewith
respecttotheother.Ifweseparatethespeakersbyadistance1.5,howfarfromtheleftmostspeakershouldwe
placeamicrophoneinordertopickuptheloudestsound?Ignorereflectionsfromnearbysurfaces.Selectallthat
apply.
Hint1.Howtoapproachtheproblem.
Makeasketchofthesituation.Notethatthespeakersrepresentantinodesoftheresultingstandingwave
pattern.
Willthesoundbeloudestatthenodesorantinodesofthestandingwavepattern?Wherewilltherebeno
sound?
ANSWER:
3
4
1
4
1
2
Correct
Exercise15.20:WeightyRope
Oneendofanylonropeistiedtoastationarysupportatthetopofaverticalmineshaftofdepth81.0m.Theropeis
kg
kg
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stretchedtautbyaboxofmineralsampleswithmass20.0kgattachedatthelowerend.Themassoftheropeis1.90kg.
Thegeologistatthebottomoftheminesignalstohiscolleagueatthetopbyjerkingtheropesideways.(Donotneglect
theweightoftherope.)
PartA
Whatisthewavespeedatthebottomoftherope?
ANSWER:
v
= 91.4 m/s
Correct
PartB
Whatisthewavespeedatthemiddleoftherope?
ANSWER:
v
= 93.6 m/s
Correct
PartC
Whatisthewavespeedatthetopoftherope?
ANSWER:
v
= 95.7 m/s
Correct
Exercise15.25
Ajetplaneattakeoffcanproducesoundofintensity9.20W/m2 at30.4maway.Butyoupreferthetranquilsoundof
2
normalconversation,whichis1.0W/m .Assumethattheplanebehaveslikeapointsourceofsound.
PartA
Whatistheclosestdistanceyoushouldlivefromtheairportrunwaytopreserveyourpeaceofmind?
Expressyouranswerusingtwosignificantfigures.
ANSWER:
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= 92 km
Correct
PartB
Whatintensityfromthejetdoesyourfriendexperienceifshelivestwiceasfarfromtherunwayasyoudo?
ANSWER:
I
= 0.25 W/m2
Correct
PartC
Whatpowerofsounddoesthejetproduceattakeoff?
ANSWER:
P
= 1.1105 W
Correct
Exercise15.33
Twotriangularwavepulsesaretravelingtowardeachotheronastretchedstringasshowninthefigure.Eachpulseis
identicaltotheotherandtravelsat2.00cm/s.Theleadingedgesofthepulsesare1.00cmapartatt=0.Supposethat
therighttravelingpulseinthefigureisbelowtheleveloftheunstretchedstring.
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PartA
Sketchtheshapeofthestringatt=0.250s.
ANSWER:
Correct
PartB
t
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ANSWER:
Correct
PartC
ANSWER:
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Correct
PartD
ANSWER:
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Correct
PartE
ANSWER:
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Correct
CreatingaStandingWave
LearningGoal:
Toseehowtwotravelingwavesofthesamefrequencycreateastandingwave.
Consideratravelingwavedescribedbytheformula
y
(x, t) = A sin(kx t)
Thisfunctionmightrepresentthelateraldisplacementofastring,alocalelectricfield,thepositionofthesurfaceofa
bodyofwater,oranyofanumberofotherphysicalmanifestationsofwaves.
PartA
Whichoneofthefollowingstatementsaboutthewavedescribedintheproblemintroductioniscorrect?
ANSWER:
Thewaveistravelinginthe+xdirection.
Thewaveistravelinginthexdirection.
Thewaveisoscillatingbutnottraveling.
Thewaveistravelingbutnotoscillating.
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Correct
PartB
Whichoftheexpressionsgivenisamathematicalexpressionforawaveofthesameamplitudethatistravelingin
theoppositedirection?Attimet = 0thisnewwaveshouldhavethesamedisplacementasy 1 (x, t) ,thewave
describedintheproblemintroduction.
ANSWER:
A cos(kx t)
A cos(kx + t)
A sin(kx t)
A sin(kx + t)
Correct
Theprincipleofsuperpositionstatesthatiftwofunctionseachseparatelysatisfythewaveequation,thenthesum(or
difference)alsosatisfiesthewaveequation.Thisprinciplefollowsfromthefactthateveryterminthewaveequationis
linearintheamplitudeofthewave.
Considerthesumoftwowavesy 1 (x, t) + y 2 (x, t) ,wherey 1 (x, t) isthewavedescribedinPartAandy 2 (x, t) isthe
wavedescribedinPartB.Thesewaveshavebeenchosensothattheirsumcanbewrittenasfollows:
y (x, t) = y (x)y (t)
s
Thisformissignificantbecausey e (x) ,calledtheenvelope,dependsonlyonposition,andy t (t) dependsonlyontime.

Traditionally,thetimefunctionistakentobeatrigonometricfunctionwithunitamplitudethatis,theoverallamplitudeof
thewaveiswrittenaspartofy e (x) .
PartC
Findy e (x) andy t (t) .Keepinmindthaty t (t) shouldbeatrigonometricfunctionofunitamplitude.
ExpressyouranswersintermsofA,k,x,,andt.Separatethetwofunctionswithacomma.
Hint1.Ausefulidentity
Ausefultrigonometricidentityforthisproblemis
.
sin(A + B) = sin(A) cos(B) + cos(A) sin(B)
Hint2.Applyingtheidentity
Sinceyoureallyneedanidentityforsin(A B),simplyreplaceBbyBintheidentityfromHintC.1,
keepinginmindthatsin(x)
= sin(x)
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ANSWER:
y (x)
e
,y t (t) =
sin(kx), 2Acos(t)
Correct
PartD
Whichoneofthefollowingstatementsaboutthesuperpositionwavey s (x, t) iscorrect?
ANSWER:
Thiswaveistravelinginthe+xdirection.
Thiswaveistravelinginthexdirection.
Thiswaveisoscillatingbutnottraveling.
Thiswaveistravelingbutnotoscillating.
Correct
Awavethatoscillatesinplaceiscalledastandingwave.Becauseeachpartofthestringoscillateswiththe
samephase,thewavedoesnotappeartomoveleftorrightrather,itoscillatesupanddownonly.
PartE
Atthepositionx
= 0
,whatisthedisplacementofthestring(assumingthatthestandingwavey s (x, t) ispresent)?
Expressyouranswerintermsofparametersgivenintheproblemintroduction.
ANSWER:
y (x = 0, t)
s
= 0
Correct
Thiscouldbeausefulpropertyofthisstandingwave,sinceitcouldrepresentastringtiedtoapostor
otherwiseconstrainedatpositionx = 0.Suchsolutionswillbeimportantintreatingnormalmodesthatarise
whentherearetwosuchconstraints.
PartF
Atcertaintimes,thestringwillbeperfectlystraight.Findthefirsttimet1
> 0
whenthisistrue.
Expresst1 intermsof,k,andnecessaryconstants.
cos(t) = 0
(x, t) = 0
s
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Thestringcanbestraightonlywhencos(t)
= 0
,fortheny s (x, t)
= 0
also(forallx).Foranyothervalue
ofcos(t),y s (x, t) willbeasinusoidalfunctionofpositionx.
ANSWER:
t1
Correct
PartG
FromPartFweknowthatthestringisperfectlystraightattimet =
.Whichofthefollowingstatementsdoesthe
string'sbeingstraightimplyabouttheenergystoredinthestring?
a. Thereisnoenergystoredinthestring:Thestringwillremainstraightforallsubsequenttimes.
b. Energywillflowintothestring,causingthestandingwavetoformatalatertime.
c. Althoughthestringisstraightattimet =
,partsofthestringhavenonzerovelocity.Therefore,
2
thereisenergystoredinthestring.
d. Thetotalmechanicalenergyinthestringoscillatesbutisconstantifaveragedoveracompletecycle.
ANSWER:
a
b
c
d
Correct
WhytheHighestPianoNotesHaveShortStrings
Thesteelusedforpianowirehasabreaking(tensile)strengthpT ofabout3 109
3
7800 kg/m .
N/m
andadensityof
PartA
Whatisthespeedv ofawavetravelingdownsuchawireifthewireisstretchedtoitsbreakingpoint?
Expressthespeedofthewavenumerically,inmeterspersecond,tothenearestinteger.
Hint1.Findthetravelingwavespeedinastretchedstring
Whatisthespeedv ofatravelingwaveinastretchedstringwhichhasmassperunitlengthandisunder
tensionFT ?
FT
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AnswerintermsofFT and.
Hint1.Usedimensionalanalysis
Onewaytoanswerthistypeofproblemistoconsiderthedimensionsofthequantitiesinvolved.Force
andmassperunitlengthhavedimensionskg m s2 andkg m1 respectively,whereaswe
requireananswerwithdimensionsofm s1 .ThereisonlyonecombinationofFT andthathas
thecorrectdimensions.
Notethatthismethoddoesnotgivethevalueofanyconstantmultiplieroftherelation.
ANSWER:
FT
Hint2.Expressvintermsoftensilestrengthanddensity
Whatisthespeedv ofatravelingwaveintermsofthetensilestrengthpT andwiredensity?
AssumethatthewirehascrosssectionalareaA.ThetensilestrengthpT andtensioninthewireat
thebreakingpointFT arerelatedbypT A
relatedbyA
= FT
.Thedensityandmassperunitlengthare
.UsetheseexpressionstoconvertthespeedintermsofFT andintoaspeedin
termsofpT and.
ANSWER:
pT
ANSWER:
v
= 620 m/s
Correct
Thisismuchlessthanthespeedofsoundinsteel(5941
m/s
)because,unlikesteel,inpianowirethetensile
strengthismuchlessthantheYoung'smodulus.
PartB
ImaginethatthewiredescribedintheproblemintroductionisusedforthehighestConapiano(C8
4000 Hz
).If
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thewireisintunewhenstretchedtoitsbreakingpoint,whatmustthevibratinglengthofthewirebe?
Expressthelengthnumerically,incentimeters,usingthreesignificantfigures.
Hint1.Alengthconstraintgivenf andv
InPartAofthisproblem,youfoundthespeedoftravelingwavesinthewire(v ).Youalsoknowthe
frequencyofoscillation(f ).Whatisthewavelength()ofthewaveintheC8pianostring,intermsofthese
quantities?
Expressyouranswerintermsofv andf .
ANSWER:
v
Hint2.Relationshipbetweenwavelengthandstringlength
Considertheboundaryconditionsforastretchedpianowire:Bothendsarefixed.Ifsuchawireisoscillating
atitsfundamentalfrequency(itsfirstnormalmode),thewavelengthwillnotbeequaltothewirelength.What
isthewavelengthofthefirstnormalmodeofastringoflengthLthatisfixedatbothends?
ExpressthewavelengthintermsofL.
ANSWER:
=
2L
ANSWER:
L
= 7.75 cm
Correct
NormalModesandResonanceFrequencies
LearningGoal:
Tounderstandtheconceptofnormalmodesofoscillationandtoderivesomepropertiesofnormalmodesofwavesona
string.
Anormalmodeofaclosedsystemisanoscillationofthesysteminwhichallpartsoscillateatasinglefrequency.In
generalthereareaninfinitenumberofsuchmodes,eachonewithadistinctivefrequencyf i andassociatedpatternof
oscillation.
Consideranexampleofasystemwithnormalmodes:astringoflengthLheldfixedatbothends,locatedatx = 0and
x = L .Assumethatwavesonthisstringpropagatewithspeedv .Thestringextendsinthexdirection,andthewaves
aretransversewithdisplacementalongtheydirection.
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Inthisproblem,youwillinvestigatetheshapeofthenormalmodesandthentheirfrequency.
Thenormalmodesofthissystemareproductsoftrigonometricfunctions.(Forlinearsystems,thetimedependanceofa
normalmodeisalwayssinusoidal,butthespatialdependenceneednotbe.)Specifically,forthissystemanormalmode
isdescribedby
y (x, t) = Ai sin(2
i
x
i
) sin( 2 f i t) .
PartA
Thestringdescribedintheproblemintroductionisoscillatinginoneofitsnormalmodes.Whichofthefollowing
statementsaboutthewaveinthestringiscorrect?
Hint1.Normalmodeconstraints
Thekeyconstraintwithnormalmodesisthattherearetwospatialboundaryconditions,y i (0, t)
y (L, t) = 0
i
= 0
and
,whichcorrespondtothestringbeingfixedatitstwoends.
ANSWER:
Thewaveistravelinginthe+xdirection.
Thewaveistravelinginthexdirection.
Thewavewillsatisfythegivenboundaryconditionsforanyarbitrarywavelengthi .
Thewavelengthi canhaveonlycertainspecificvaluesiftheboundaryconditionsaretobesatisfied.
Thewavedoesnotsatisfytheboundaryconditiony i (0; t)
= 0
Correct
PartB
Whichofthefollowingstatementsaretrue?
ANSWER:
Thesystemcanresonateatonlycertainresonancefrequenciesf i andthewavelengthi mustbesuch
thaty i (0; t) = y i (L; t) = 0 .
Ai
mustbechosensothatthewavefitsexactlyonthestring.
AnyoneofAi ori orf i canbechosentomakethesolutionanormalmode.
Correct
Thekeyfactorproducingthenormalmodesisthattherearetwospatialboundaryconditions,y i (0, t)
y (L, t) = 0
i
= 0
and
,thataresatisfiedonlyforparticularvaluesofi .
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PartC
Findthethreelongestwavelengths(callthem1 ,2 ,and3 )that"fit"onthestring,thatis,thosethatsatisfythe
boundaryconditionsatx
= 0
andx
= L
.Theselongestwavelengthshavethelowestfrequencies.
ExpressthethreewavelengthsintermsofL.Listthemindecreasingorderoflength,separatedby
commas.
Thenodesofthewaveoccurwhere
sin(2
x
i
) = 0.
Thisequationistriviallysatisfiedatoneendofthestring(withx
= 0
),sincesin(0)
= 0
Thethreelargestwavelengthsthatsatisfythisequationattheotherendofthestring(withx = L)aregiven
by2L/i = zi ,wherethezi arethethreesmallest,nonzerovaluesofzthatsatisfytheequation
sin(z) = 0
Hint2.Valuesofz thatsatisfysin(z)
= 0
Thespatialpartofthenormalmodesolutionisasinewave.Findthethreesmallest(nonzero)valuesofz
(callthemz1 ,z2 ,andz3 )thatsatisfysin(z) = 0.
Expressthethreenonzerovaluesofzasmultiplesof.Listtheminincreasingorder,separatedby
commas.
ANSWER:
z1
,z2 ,z3 = 1,2,3
Hint3.Pictureofthenormalmodes
Considerthelowestfourmodesofthestringasshowninthefigure.
TheletterNiswrittenovereachofthenodesdefinedasplaceswherethestringdoesnotmove.(Notethat
therearenodesinadditiontothoseattheendofthestring.)TheletterAiswrittenovertheantinodes,which
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arewheretheoscillationamplitudeismaximum.
ANSWER:
1
,2 ,3 =
2L, L,
2
3
Correct
Theproceduredescribedherecontainsthesamemathematicsthatleadstoquantizationinquantum
mechanics.
PartD
Thefrequencyofeachnormalmodedependsonthespatialpartofthewavefunction,whichischaracterizedbyits
wavelengthi .
Findthefrequencyf i oftheithnormalmode.
Expressf i intermsofitsparticularwavelengthi andthespeedofpropagationofthewavev .
Hint1.Propagationspeedforstandingwaves
Yourexpressionwillinvolvev ,thespeedofpropagationofawaveonthestring.Ofcourse,thenormal
modesarestandingwavesanddonottravelalongthestringthewaythattravelingwavesdo.Nevertheless,
thespeedofwavepropagationisaphysicalpropertythathasawelldefinedvaluethathappenstoappearin
therelationshipbetweenfrequencyandwavelengthofnormalmodes.
Hint2.Usewhatyouknowabouttravelingwaves
Therelationshipbetweenthewavelengthandthefrequencyforstandingwavesisthesameasthatfor
travelingwavesandinvolvesthespeedofpropagationv .
ANSWER:
f
v
i
Correct
Thefrequenciesf i aretheonlyfrequenciesatwhichthesystemcanoscillate.Ifthestringisexcitedatoneof
theseresonancefrequenciesitwillrespondbyoscillatinginthepatterngivenbyy i (x, t) ,thatis,with
wavelengthi associatedwiththef i atwhichitisexcited.Inquantummechanicsthesefrequenciesarecalled
theeigenfrequencies,whichareequaltotheenergyofthatmodedividedbyPlanck'sconstanth.InSIunits,
Planck'sconstanthasthevalueh = 6.63 1034 J s.
PartE
f
f
f
1
2
3
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Findthethreelowestnormalmodefrequenciesf 1 ,f 2 ,andf 3 .
ExpressthefrequenciesintermsofL,v ,andanyconstants.Listtheminincreasingorder,separatedby
commas.
ANSWER:
1
f1
,f 2 ,f 3 =
v
L
Correct
Notethat,forthestring,thesefrequenciesaremultiplesofthelowestfrequency.Forthisreasonthelowest
frequencyiscalledthefundamentalandthehigherfrequenciesarecalledharmonicsofthefundamental.When
otherphysicalapproximations(forexample,thestiffnessofthestring)arenotvalid,thenormalmode
frequenciesarenotexactlyharmonic,andtheyarecalledpartials.Inanacousticpiano,thehighestaudible
normalfrequenciesforagivenstringcanbeasignificantfractionofasemitonesharperthanasimpleinteger
multipleofthefundamental.Consequently,thefundamentalfrequenciesofthelowernotesaredeliberately
tunedabitflatsothattheirhigherpartialsarecloserinfrequencytothehighernotes.
Problem15.77
Auniformcylindricalsteelwire,54.0cmlongand1.20mmindiameter,isfixedatbothends.
PartA
Towhattensionmustitbeadjustedsothat,whenvibratinginitsfirstovertone,itproducesthenoteD# of
frequency311Hz?Assumethatitstretchesaninsignificantamount.(Hint:Thedensityofthesteelis7800kg/m3 .)
ANSWER:
T
= 250 N
Correct
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Yourscoreonthisassignmentis99.8%.
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11/20/2014

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