Classical Mechanics Lecture 23 - Simon Fraser University · Types of Waves Movie (lwave) Movie...
Transcript of Classical Mechanics Lecture 23 - Simon Fraser University · Types of Waves Movie (lwave) Movie...
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Classical MechanicsLecture 23
Today’sConcept:
HarmonicWaves
MechanicsLecture23,Slide1
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Your comments are very important to us...
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Your comments are still very important to us...
So,thetensioninaropeisrepresentedwiththesymbol"T",insteadofaforcewithasubscriptlike"F_t".Wellthatmakessense,becausethere'snootherquanEtysymbolizedwith"T"thatcomesupalltheEmewhendealingwithwaves...OHWAIT,YESTHEREIS,IT'SPERIOD!!!Seriously,whoismakingthesedecisions?!IsitthesameguywhodecidedtouseomegaforangularfrequencyANDangularvelocity?Thatguyisoverdueforagoodlynching...
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InformationUnit14and15AcEvityGuideswillnotbegraded
Pleaseturnin:! Unit14WriXenHomeworkonDec7� TheMini-labbookonyourSHMorKarateProject,Dec9
FinalExamEme:Wed.,Dec.9,3:30pm
Finalexamroom:2740
SmartPhysicsUnit22(SuperposiEon)isopEonal! WavesandsuperposiEon! Prelecture,CheckpointandHomeworkallopEonal! Willnotbeonexam! DoitforextrasmartPhysicsmarks! ReviewonMonday,noPrelecturediscussionunlessasked
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What is a Wave?
AwaveisatravelingdisturbancethattransportsenergybutnotmaXer.
Examples:!Soundwaves(airmovesback&forth)!Stadiumwaves(peoplemoveup&down)!Waterwaves(watermovesup&down)!Lightwaves(whatmoves?)
MechanicsLecture23,Slide2
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Types of Waves
Movie(lwave)
Movie(twave)
Longitudinal:ThemediumoscillatesinthesamedirecEonasthewaveismoving.
MechanicsLecture23,Slide3
Transverse:ThemediumoscillatesperpendiculartothedirecEonthewaveismoving.ThisiswhatwestudyinPhysics140.
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MechanicsLecture23,Slide4
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Amplitude:ThemaximumdisplacementA ofapointonthewave.
A
Wavelength:Thedistanceλ betweenidenCcalpointsonthewave. Movie(tspeed)
Period:TheCmeP ittakesforanelementofthemediumtomakeonecompleteoscillaCon.
Wave Properties
MechanicsLecture23,Slide5
Wavelengthλ
String:
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WavesAandBshownabovearepropagaCnginthesamemedium.Howdotheirfrequenciescompare?
A)ωA > ωB
B)ωA < ωB
C)ωA = ωB
A
B
x
y
x
y
CheckPoint
MechanicsLecture23,Slide6
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WavesAandBshownabovearepropagaCnginthesamemedium.Howdotheirfrequenciescompare?B)ωA < ωB C)ωA = ωB A)ωA > ωB
B)AstheybothhavethesamevelociCes,andthewavelengthofAislarger,thefrequencymustbesmallerthanthatofB.C)Thetensionandmassdensityarethesameinbothstringssotheyhavethesamefrequency.
A)It'speriodisshorter.
A
B
x
y
x
y
MechanicsLecture23,Slide7
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Thespeedofsoundinairisabitover300 m/s,andthespeedoflightinairisabout300,000,000 m/s.
Supposewemakeasoundwaveandalightwavethatbothhaveawavelengthof3 meters.WhatistheraEoofthefrequencyofthelightwavetothatofthesoundwave?
A)About1,000,000B)About0.000001C)About1000
Clicker Question
MechanicsLecture23,Slide8
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How to make a Function Move
SupposewehavesomefuncEony = f (x):
x
y
0
MechanicsLecture23,Slide9
f (x − a) isjustthesameshapemovedadistanceatotheright:
x = ax
y
0
Leta = vt Then
f (x − vt) willdescribethesameshapemovingtotherightwithspeedv.
x = vt
v
x
y
0
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y(x) = A cos
2⇡
�x
!
A
λ
x
y
Harmonic Wave
Considerawavethatisharmonicinxandhasawavelengthofλ.
MechanicsLecture23,Slide10
Iftheamplitudeismaximumatx = 0 thishasthefuncEonalform:
y(x) = A cos
2⇡
�(x � vt)
!
Now,ifthisismovingtotherightwithspeedvitwillbedescribedby:
x
y
v
k =2⇡�
! =2⇡v�
“wavenumber”
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1 cycle per second
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2 cycles per second
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IfafuncEonmovingtotherightwithspeedvisdescribedbyf (x − vt) thenwhatdescribesthesamefuncEonmovingtothelehwithspeedv?
v
x
y
0
y = f (x − vt)
v A) y = − f (x − vt)
B)y = f (x + vt)
C)y = f (−x + vt)
Clicker Question
MechanicsLecture23,Slide11
x
y
0 x
y
0 x
y
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SupposethefuncConhasitsmaximumatf (0).
A)y = − f (x − vt)
B)y = f (x + vt)
C)y = f (−x + vt)
x – vt = 0
x = vty = f (x − vt)
x – vt = 0
x = vt
x + vt = 0
x = −vt
−x + vt = 0
x = vt
v
v
v
v
MovestotherightwhenthesignsinfrontofthexandttermsaredifferentMovestotheleYwhenthesignsinfrontofthexandttermsarethesame
MechanicsLecture23,Slide12
v
x
y
0 x
y
0 x
y
v
0 x
y
x
y
0
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WehaveshownthatthefuncConalformy(x,t) = Acos(kx − ωt) representsawavemovinginthe+x direcCon.
Whichofthefollowingrepresentsawavemovinginthe–xdirecCon?
A)y(x,t) = Acos(ωt − kx)
B)y(x,t) = Asin(kx - ωt)
C)y(x,t) = Acos(kx + ωt)
CheckPoint
MechanicsLecture23,Slide13
Ax
y
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MechanicsLecture23,Slide14
WehaveshownthatthefuncConalformy(x,t) = Acos(kx − ωt) representsawavemovinginthe+x direcCon.
Ax
y
CheckPoint
Whichofthefollowingrepresentsawavemovinginthe –x direcCon?
A)y(x,t) = Acos(ωt − kx)CosineisanoddfuncCon,soreversingkxandwtwillreversethesignofthefuncConfromposiCvetonegaCve
C)y(x,t) = Acos(kx + ωt)Withparabolas,(x-a),whereaisanynumber,resultsinagraphshiYedtotheright/orposiCvexdirecCon.Therefore,theopposit[e](x+a)shi[f]tsittotheleY.
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Energy
MechanicsLecture23,Slide15
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WavesAandB shownabovearepropagaEnginthesamemediumwiththesameamplitude.Whichonecarriesthemostenergyperunitlength?
A)A
B)B
C)Theycarrythesameenergyperunitlength
CheckPoint
MechanicsLecture23,Slide16
A
B
x
y
x
y
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A
B
x
y
x
y
Whichonecarriesthemostenergyperunitlength?
A)AB)B C)Same
A)SinceAhasalargerwavelength,itcarriesmoreenergyperunit.
B)kineCcenergyisdirectlyproporConalto(ωA2),sincetheiramplitudearethesameandBhasalargerω,Bhasmoreenergy
C)AssumingthatCwassupposedtobeA=B,theirenergiesareequalsincetheiramplitudesareequal.
MechanicsLecture23,Slide17
CheckPoint
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MechanicsLecture23,Slide18
Homework Problem
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MechanicsLecture23,Slide19
Homework Problem
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MechanicsLecture23,Slide20
Homework Problem
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Distancetraveledbyapieceofropeduringoneperiodis4A
Averagespeed=distancetraveled/Cmetaken
MechanicsLecture23,Slide21
Homework Problem