asp.net mvc 4 and the web api pdf free download : Combine pdf files application software cloud windows html azure class mechanics10-part1280

3|SimpleHarmonicMotion
97
3.7StableMotion
Whendoyouencounterharmonicoscillators?Notjustspringsandpendulumsorevenelectriccircuits.
Ifyouhaveasysteminequilibriumandyoudisturbitalittle,whathappens? Ifit’sastableequilibrium
thenitispushedbacktowarditsoriginalconguration. Ifit’sunstablethenitispushedfartheraway.
Ifyou’reskateboardinginavalley oron topofahillyouhavedierentconcerns. . Ifanatomis s ina
moleculeanditgetsbumpedtothesidethentheotheratomsinthemoleculewillpushitbackinto
place.
Everyoneoftheseexamplesleadstoaharmonicoscillator,andyoucanseewhybylookingat
thepotentialenergy,
E
=
mv
2
=
2+
U
(
x
). Theconditionforequilibriumisthat
F
x
dU=dx
=0,
andwhetherthisisstableornotdependsonwhetherthepotentialenergyfunctioncurvesupordown.
U
/
x
2
x
U
x
2
x
U
/
x
3
x
Fig.3.8
Therstgraphleadstostableoscillationsandthesecondgraphprovidesaforcethatpushesthe
massawayfromequilibrium.
F
x
dU=dx
,soapositiveslopefor
U
impliesanegative
F
x
andvice
versa. Intherstgraphthatforceistowardtheoriginandinthesecondgraphthatisawayfromit.
The thirdgraph(
U
=
Ax
3
)isneither,butitneverhappensinpractice*becauseeventheslightest
changeinthesystemchangesthegraphintosomethingquitedierent.Ifyouadd10
20
x
tothis
U
thenitsbehaviorchanges.Seeproblem3.19.
Youcangettheequationofmotiondirectlyfromenergyconservationbysayingthat
dE=dt
=0.
d
dt
1
2
mv
2
+
U
(
x
)
=0=
mv
dv
dt
+
dU
dx
dx
dt
=
v
m
dv
dt
+
dU
dx
(3
:
46)
Cancelthe
v
andyouseethatusingenergymethodsisoftenafasterwaytotheequationsofmotion.
Whatifthere’sfriction? Seeproblem3.22.
Thepowerseriesexpansionof
U
allowsyoutoanalyzethemotionquantitatively.
U
(
x
)=
U
(
x
0
)+(
x
x
0
)
U
0
(
x
0
)+
1
2
(
x
x
0
)
2
U
00
(
x
0
)+
1
3!
(
x
x
0
)
3
U
000
(
x
0
)+
If
x
0
isanequilibriumpointofthepotentialenergy
U
0
(
x
0
)=0,thenthedominanttermafterthatis
thequadratic,
U
00
(
x
0
)(
x
x
0
)
2
=
2. If
U
00
(
x
0
)
>
0thenthislooksliketherstpictureaboveandyou
havestableequilibrium
m
x
=
F
x
dU=dx
U
00
(
x
0
)(
x
x
0
)
Thisis astandard harmonic oscillatorwith centerat
x
0
. Take e aspecicexample and do it several
dierentways.
U
(
x
)= 
A
sech(
x
)= 
A
cosh
x
Fig.3.9
* Thisshapeiscalledgenericallyunstable.Alittlechangemakeaqualitativedierence,notsimply
quantitative.
Combine pdf files - Merge, append PDF files in C#.net, ASP.NET, MVC, Ajax, WinForms, WPF
Provide C# Demo Codes for Merging and Appending PDF Document
merge pdf; pdf split and merge
Combine pdf files - VB.NET PDF File Merge Library: Merge, append PDF files in vb.net, ASP.NET, MVC, Ajax, WinForms, WPF
VB.NET Guide and Sample Codes to Merge PDF Documents in .NET Project
c# combine pdf; pdf combine files online
3|SimpleHarmonicMotion
98
Therstmethod(andusuallythehardest)istoplugintotheTaylorseriesformulatogettheseries
representationfor
U
outtothequadraticterm.
U
(0)= 
A
,and
U
0
(
x
)=
A
sech
x
tanh
x
!
U
0
(0)=0
U
00
(
x
)=
2
A
sech
2
x
tanh
2
x
+sech
3
x
!
U
00
(0)=
2
A
U
(
x
)= 
A
+
1
2
2
Ax
2
+
Theequationofmotionis
m
x
=
F
x
dU=dx
2
Ax
(3
:
47)
andthisisthesimpleharmonicoscillatorwithfrequency
!
0
=
p
A=m
. If
issmall,thepotential
welliswideandhasaweakrestoringforce;thecorrespondingfrequencyissmall.
Thesecondmethodtoobtaintheapproximateequationofmotionfromthepotentialistostop
afterdoingtherstderivative.
m
x
=
F
x
dU
dx
A
sech
x
tanh
x
(3
:
48)
Forsmall
x
,thesechis essentially 1,and tanh
x
x
sothisequationreproducedEq.(3.47)for
smalloscillations.
Thethirdmethodis touseseriesexpansionsonthepotentialenergyandtodierentiatethat.
Eq.(0.1)includesthecoshandthebinomialexpansions.
U
(
x
)= 
A
cosh
x
A
1+
1
2
2
x
2+
A
1
2
2
x
2
+
andtheforcecomesfromthederivativeofthesecondterm,givingthesameequationasbefore. Once
youhavemorepracticewithseries,thisiscommonlytheeasiestchoice.
Whichever way y you arrive at it(only occasionally the rst method), , the equation to solve is
Eq.(3.47),andthatsolution isthestandard
x
(
t
)=
C
cos(
!
0
t
+
) with
!
2
0
=
2
A=m
. Isthisan
exactsolutionforEq.(3.48)?. No,butjustbecausetheapproximationthat
j
x
j1isrequired.This
ishoweverexactly thesamedicultythatoccurredinEq.(3.1)foramassattachedtoaspring.Isaid
thattheforcebythespringisrepresentedby
F
x
kx
,butthat’sanapproximationtoo. Itisvalid
onlyifj
x
jissmallenoughinsomeunspeciedway. ThedierenceinthepresentexampleisthatIcan
specifyjustwhatImeanby\smallenough",andforthespringIcouldn’t.
Thissortofexpansionistypicalofwhytheharmonicoscillatorappearssooften,evenincontexts
forwhichyoudon’texpectit.Itwillevenshowupinthestudyofplanetaryorbits,section6.5.
Example
Whatisthebehaviorofapieceofwood oatinginthewater?Ifyouleaveitalone,nothing. Ifyou
disturbitthenitwilloscillateupanddownaccordingto
~
F
=
m~a
.Makeitarectangularblock,sothe
geometryiseasy,andlet
y
bethedistancethebottomisbelowthesurface(dashedline)ofthewater.
Theforcesonitcomefromgravityandthesurroundingwater,andthelatteriscomputedeasilybythe
rulethatArchimedesfoundforbuoyantforces.
F
y
=+
mg
w
gAy
=
m
y
h
A
y
Fig.3.10
Online Merge PDF files. Best free online merge PDF tool.
RasterEdge C#.NET PDF document merging toolkit (XDoc.PDF) is designed to help .NET developers combine PDF document files created by different users to one PDF
acrobat reader merge pdf files; pdf merge documents
C# Word - Merge Word Documents in C#.NET
RasterEdge C#.NET Word document merging toolkit (XDoc.Word) is designed to help .NET developers combine Word document files created by different users to one
pdf merger online; combine pdf files
3|SimpleHarmonicMotion
99
The mass density of wateris
w
,and
m
is themass of thebox.
A
isthe areaof the top. . This s is
aninhomogeneous,linear,constantcoecientdierentialequation. Asolutionfortheinhomogeneous
partisaconstant,
y
inh
=
m=
w
A
,theequilibriumposition. Afterthatyouhaveaharmonicoscillator
withsolution
y
hom
(
t
)=
C
cos(
!
0
t
+
)and
!
2
0
=
w
gA=m
.
Inthisexamplethedierentialequationisexactlyaharmonicoscillator,withnoapproximations
needed. Orisit? Waterhasto owaroundtheboxas s theboxmoves,and  uid owisnotoriously
complicatedtogureout.Therewillattheleastbe uidfriction(viscosity),andprobablymuchmore.
Butthatisanotherbook.
Iftheboxisnotrectangulartheprocedureisthesame,butthevolumesubmergedwillnotbe
assimpletocalculateas
Ay
,andtheresultingdierentialequationwillusuallyhavetobeexpandedin
apowerseriesabouttheequilibriumpoint.
3.8UnstableMotion
Balanceapencilonitspoint. Itiseasytowritetheequationsforitsmotion;justwritethetotalenergy.
E
=
1
2
I!
2
+
Mg‘
cos
(3
:
49)
I
isthemomentofinertiaaboutthetipand
isthedistancefromthetiptothecenterofmass. The
angle
ismeasuredfromthetop. Simplerconceptually,butcompletelyequivalentmathematically:Put
apointmass
M
onthetopofamasslessrodoflength
andbalancetherodontheotherend.Then
E
=
1
2
Mv
2
+
Mgh
=
1
2
M‘
2
_
2
+
Mg‘
cos
(3
:
50)
Fortheequationofmotion,simplydierentiatethiswithrespecttotime.
dE
dt
=0=
M‘
2
_
Mg‘
sin
_
Cancelthe
_
factorandyouhave
g
sin
=0
Justaswiththestandardpendulum,thisinvertedpendulumsimpliesenormouslyifyoumakethesmall
angleapproximation,then
g
=0
(3
:
51)
andthislinearequationishandledthesamewayasalltheotherlinearequationsinthischapter. Assume
anexponential.
=
Ae
t
!
g
=
‘A
2
e
t
gAe
t
=0  !
‘
2
g
=0  !
=
q
g=‘
(3
:
52)
Thegeneralsolutiontothedierentialequationisnow
(
t
)=
Ae
!t
+
Be
!t
;
with
!
=
q
g=‘
Thisislikeusingthecomplexexponentialsolutionsforthesimpleharmonicoscillator. Itisoften
moreconvenienttousesinesandcosinesdirectlyinthatcase,andhereitisoftenmoreconvenientto
usehyperbolicsinesandcosinesasinsection0.2.
(
t
)=
Ae
!t
+
Be
!t
=
C
cosh
!t
+
D
sinh
!t
(3
:
53)
C# PowerPoint - Merge PowerPoint Documents in C#.NET
RasterEdge C#.NET PowerPoint document merging toolkit (XDoc.PowerPoint) is designed to help .NET developers combine PowerPoint document files created by
pdf mail merge plug in; build pdf from multiple files
C# PDF: C#.NET PDF Document Merging & Splitting Control SDK
C#.NET PDF Merger to Combine PDF Files. Using following C#.NET PDF document merging APIs, you can easily merge two or more independent PDF files to create a
add two pdf files together; pdf combine
3|SimpleHarmonicMotion
100
Atypicalinitialconditionistostartatrestwithinitialanglefromthevertical
0
,then
(0)=
C
=
0
and
_
(0)=
!D
=0
(
t
)=
0
cosh
!t;
andif
=20cm=0
:
2m
then
!
=
q
10
=
0
:
2=7
:
07s
1
(3
:
54)
Howmuchtimewouldthistaketohitthetablestartingwithaninitialangleofonedegree?
1
cosh
!t
=90
!
t
=
!
1
cosh
1
90=5
:
193
=
p
50=0
:
73seconds
Howmuchmoretimewould ittaketofallthis fariftheinitialangle issmaller? ? Perhapsstartingat
0
:
1
or0
:
01
?Plugtheminandndoutjusthowlittledierencethischangemakes: respectively1.09
and1.39seconds.
MaybethelinearapproximationmadeinEq.(3.51)isnotgoodenough. Howmuchdierence
wouldtherebeinthetimeittakestohitthetable? YouhavetogobacktoEq.(3.50)andseparate
variables.
E
=
1
2
M‘
2
_
2
+
Mg‘
cos
!
dt
=
d
p
2(
E
Mg‘
cos
)
=M‘
2
(3
:
55)
Integratethisfrom
0
to
=
2togetthetimetofall. Whatis
E
?Assumingthateverythingstartsfrom
restthatis
E
=
Mg‘
cos
0
,makingthis
Z
dt
=
Z
=
2
0
d
!
p
2(cos
0
cos
)
(3
:
56)
This is notan integralthat you’re likely to have seen before, , and youcannot do it in terms s of the
functions(polynomials,roots,trigfunctions,exponentials)thatyouarefamiliarwith. Itisanelliptic
integral,andtherewillbeabriefintroductiontothesubjectinsection4.6. Forthethreeanglesused
one paragraph back, the results using this s integral l are respectively y 0.742s, , 1.069s, 1.379s. . See
problem3.70.
Theyareremarkableclosetothenumbersyougetbyusingthelinearapproximationinthesecond
paragraphback,sothatinvitesthequestion: Why? Thelinearapproximationtothesinefunctionis
goodevenat30
,butat90
,it’sterrible. Thetimespredictedforthefallhavenorighttobeasgood
astheyare. Ordothey?TheexplanationofwhyEq.(3.54)givessuchgoodresultsisthatasthemass
fallsitspendsmostofitstimemovingslowly,whenitisnearthetop. Mostofitstimeisspentwhere
thesmallangleapproximationisgood,andtheniszipsthroughthelastpartofitsfall.
3.9CoupledOscillations
Oscillatorsdon’talwayscomesingly.Yougetasingleoscillatorwhenyouhaveastableequilibriumand
youthenexaminesmalloscillationsaboutthatequilibriumpoint.Amoleculeconsistsofseveralatoms,
andeachisinsomesortofequilibrium. Iftheyaredisturbedinanywaytheywilloscillateaboutthat
point,butnoweachoscillatorcanaectalltheotheroscillators.That’sthesubjectnow.
Asaprototypeforalloscillators,I’lluseamassonaspringagain. Rather,twomassesonthree
springs. Whenyouunderstandthisexampleyoucangraduatetothevibrationalpropertiesofwaterand
thentothoseofhemoglobin.
x
1
x
2
k
1
m
1
k
2
m
2
k
3
Fig.3.11
C# PDF File Split Library: Split, seperate PDF into multiple files
Split PDF document by PDF bookmark and outlines. Also able to combine generated split PDF document files with other PDF files to form a new PDF file.
attach pdf to mail merge in word; break a pdf into multiple files
VB.NET Word: Merge Multiple Word Files & Split Word Document
As List(Of DOCXDocument), destnPath As [String]) DOCXDocument.Combine(docList, destnPath) End imaging controls, PDF document, image to pdf files and components
c# merge pdf; c# merge pdf files
3|SimpleHarmonicMotion
101
The twocoordinates,
x
1
and
x
2
aremeasuredfrom the equilibrium positionoftherespective
masses,sothetotalforcesarezeroat
x
1
=0and
x
2
=0. Afterthat,whenthecoordinateschange
thentheadditionalforcesthatarisewillappearbecauseofthechangeinlengthoftheattachedsprings.
Forthe rstmass,theforcefromtheleftspringis  
k
1
x
1
. That’seasy. Theforcefromthemiddle
springisproportionaltothecompressionofthatspring,thatisto(
x
1
x
2
). Nail
x
2
downandlet
x
1
alonevary,thenitiseasiertoseethattheforcethisspringappliesis 
k
2
(
x
1
x
2
).
m
1
d
2
x
1
dt
2
k
1
x
1
k
2
(
x
1
x
2
)
and
m
2
d
2
x
2
dt
2
k
3
x
2
k
2
(
x
2
x
1
)
(3
:
57)
Thereasoningforthesecondmassrepeatsthatfortherst. Noticethatthetwoforcesfromthemiddle
springobeyNewton’sthirdlaw,andthattheforceonthemiddlespringiszero. That’srequiredbecause
oftheapproximationthatthespringshavezeromass. (OrdidIforgettosaythat? Seeproblem3.44
forsomethingaboutthis.)
Theseequationsarenotasformidable asyoumay expect,because the are the familiarlinear,
homogeneous,constantcoecientdierentialequationsthatthiswholechapterisdevotedto.It’sjust
thattheyaresimultaneousdierentialequations,soyouhavetolearnacoupleofnewtechniques.One
ofthekeypropertiesoftheseequations,onethatI’vealreadyusedseveraltimesisthatthesumoftwo
solutionsisasolution. Thatmeansthatifthepairoffunctions
x
1
A
(
t
)and
x
2
A
(
t
)areasolutionand
if
x
1
B
(
t
)and
x
2
B
(
t
)are,thensoistheirsum.
m
1
d
2
(
x
1
A
+
x
1
B
)
dt
2
k
1
(
x
1
A
+
x
1
B
k
2
(
x
1
A
+
x
1
B
) (
x
2
A
+
x
2
B
)
and
m
1
d
2
(
x
2
A
+
x
2
B
)
dt
2
k
3
(
x
2
A
+
x
2
B
k
2
(
x
2
A
+
x
2
B
) (
x
1
A
+
x
1
B
)
Justlookatthetermswithsubscript
A
andtheyalreadyagree. Thesamewiththe
B
-terms. Thisis
likesayingthatifcos
!
0
t
andsin
!
0
t
satisfyEq.(3.1)thensodoescos
!
0
t
+sin
!
0
t
.
ADigression
Thereisalittlebitofelementaryalgebrathatyouprobablyalreadyknow,butI’mgoing
todoitagainanyway.It’simportant.
Solve
ax
+
by
=0
cx
+
dy
=0
for
x
and
y
.
(3
:
58)
Tosolvefor
x
,multiplytherstequationby
d
andthesecondby
b
,thensubtract. Now
startoverandeliminate
x
bymultiplyingrespectivelyby
c
and
a
andsubtracting.
dax
+
dby
=0
bcx
+
bdy
=0
)
adx
bcx
=0
;
cax
+
cby
=0
acx
+
ady
=0
)
bcy
ady
=0
(3
:
59)
Thesearebothin the form(
ad
bc
)(
x
or
y
)=0. Iftheproductoftwonumbers s is
zerothenoneorbothofthenumbersiszero,andthisimpliesthatif
ad
bc
6=0then
both
x
and
y
arezero. The e onlyway tohaveanon-zerosolutionfor
x
or
y
istohave
ad
bc
=0.Thequantity
ad
bc
isthedeterminantofthesesimultaneousequations.
Afurtherpoint:If
ad
=
bc
thenthetwoequations(3.58)arethesameequation.Atleast
oneofthenumbers
a
and
b
isnon-zero. (Otherwisewhybother?)Say
a
6=0,thenmultiply
thesecondofthepairofequationsby
a
toget
acx
+
ady
=0=
acx
+
bcy
=
c
(
ax
+
by
).
Thesecondequationisamultipleoftherst. (If
c
=0,lookatthatcaseforyourselfand
showthattheresultstillholds.)
VB.NET TIFF: Merge and Split TIFF Documents with RasterEdge .NET
String], docList As [String]()) TIFFDocument.Combine(filePath, docList powerful & profession imaging controls, PDF document, tiff files and components
apple merge pdf; how to combine pdf files
VB.NET TIFF: .NET TIFF Merger SDK to Combine TIFF Files
VB.NET TIFF merging API only allows developers to combine two source powerful & profession imaging controls, PDF document, image to pdf files and components
reader merge pdf; c# merge pdf pages
3|SimpleHarmonicMotion
102
NowreturntoEqs.(3.57)andsolvethem. Theseequationsarelinear,homogeneous,constant
coecientequations,andanexponentialsolutionworksherejustasitdidinthepreviouscases. Now
howevertherearetwofunctions,withtwocoecients.
x
1
(
t
)=
Ae
t
;
x
2
(
t
)=
Be
t
(3
:
60)
Plugin.
m
1
2
Ae
t
k
1
Ae
t
k
2
(
Ae
t
Be
t
)
m
2
2
Be
t
k
3
Be
t
k
2
(
Be
t
Ae
t
)
(3
:
61)
e
t
isacommonfactor;that’swhythismethodworks. Cancelitandrearrange.
m
1
2
+
k
1
+
k
2
A
k
2
B
=0
k
2
A
+
m
2
2
+
k
3
+
k
2
B
=0
(3
:
62)
TheseequationsareinpreciselytheformofEq.(3.58). Ifthereistobeanon-zerosolutionfor
A
and
B
thenthedeterminantofthecoecientsmustvanish. Thatdetermines
.
Insteadofgrindingthroughtherathermessyalgebraofthisgeneralproblem,takeaspecialand
moresymmetriccase. Letthetwomassesbeequalandletthetwospringsontheendsbeequal.
m
1
=
m
2
=
m
and
k
1
=
k
3
=
k
=)
m
2
+
k
+
k
2
A
k
2
B
=0
k
2
A
+
m
2
+
k
+
k
2
B
=0
(3
:
63)
Thedeterminantoftheseequationsmustbezero.
m
2
+
k
+
k
2
2
k
2
2
=0
Thisisthedierenceofsquares,soitfactors.
m
2
+
k
+
k
2
+
k
2
 
m
2
+
k
+
k
2
k
2
=0=
m
2
+
k
+2
k
2
 
m
2
+
k
andthefourrootsfor
arenow
i
r
k
m
=
i!
and
i
r
k
+2
k
2
m
=
i!
0
(3
:
64)
You’renotdone. Thistellsonlythefrequenciesofoscillation,nowyouneed
A
and
B
inordertoget
thefunctions
x
1
and
x
2
.
In the equations (3.58),onceyou knowthatthe determinantis zero, there is really just one
equation,aseachisamultipleoftheother. Pickone,whicheverismoreconvenient,andsolveforthe
ratioof
x
to
y
.Geometricallythatdeterminesastraightlinethroughtheorigininthe
x
-
y
plane.
Forthepresentapplication,pickoneofthealphas,say
2
k=m
andsubstituteitintothe
rstofEqs.(3.63).
m
k
m
+
k
+
k
2
A
k
2
B
=0
;
or
A
=
B
Theotherchoiceforalphawiththesameequationgives
m
k
+2
k
2
m
+
k
+
k
2
A
k
2
B
=0
or
A
B
3|SimpleHarmonicMotion
103
Younowhavefoursolutionstotheoriginalequations,andthegeneralsolutionisalinearcombination
ofallofthem.
!
=
q
k=m;
then
x
1
(
t
)=
A
1
e
i!t
+
A
2
e
i!t
;
with
x
2
(
t
)=
x
1
(
t
)
!
0
=
q
k
+2
k
2
=m;
then
x
1
(
t
)=
A
3
e
i!0t
+
A
4
e
i!0t
;
with
x
2
(
t
)= 
x
1
(
t
)
Thetotal,generalsolutionisnowthesumofthese.
x
1
(
t
)=
A
1
e
i!t
+
A
2
e
i!t
+
A
3
e
i!0t
+
A
4
e
i!0t
x
2
(
t
)=
A
1
e
i!t
+
A
2
e
i!t
A
3
e
i!0t
A
4
e
i!0t
OR,ofteneasiertodealwith,
x
1
(
t
)=
C
1
cos
!t
+
C
2
sin
!t
+
C
3
cos
!
0
t
+
C
4
sin
!
0
t
x
2
(
t
)=
C
1
cos
!t
+
C
2
sin
!t
C
3
cos
!
0
t
C
4
sin
!
0
t
(3
:
65)
Whatdothesesolutionslooklike?Pickone:
C
1
6=0,
C
2
;
3
;
4
=0.
x
1
(
t
)=
C
1
cos
!t;
x
2
(
t
)=
C
1
cos
!t
(3
:
66)
Thetwomassesaremovingtogether,soasonemovestotherightsodoestheother. Thelengthof
themiddlespringdoes notchange,andthe (
x
1
x
2
)termsinEq.(3.57)areidenticallyzero. The
solutionbehavesasifthemiddlespringisnotthere(leftpicture).
!
!
0
Fig.3.12
Fortheotherformofsolution,
C
3
6=0,
C
1
;
2
;
4
=0.
x
1
(
t
)=
C
3
cos
!
0
t;
x
2
(
t
)= 
C
3
cos
!
0
t
(3
:
67)
Herethetwomassesareoscillatinginoppositedirections,andthemiddlespringissqueezedfromboth
sides. That’swhythefrequency
!
0
involvesthecombination
k
+2
k
3
(rightpicture). Astheleftmass
movesby
x
1
totheright,the
k
3
springissqueezedby
x
1
x
2
=2
x
1
. Thisfrequencyishigherthen
therstonebecausethere’snowalargerrestoringforceappliedtoeachmass. Thesesolutionsarethe
twomodesofoscillationofthesystem.*
Example
Takeasinitialconditionsthatyoupushthecoordinate
x
1
tothevalue
x
0
whileholding
x
2
inplace
atzero. Thenreleaseeverythingfromrest.
x
1
(0)=
x
0
=
C
1
+
C
3
;
x
2
(0)=0=
C
1
C
3
_
x
1
(0)=0=
!C
2
+
!
0
C
4
;
_
x
2
(0)=0=
!C
2
!
0
C
4
* Ifyou u wanttoseethis normalmodeproblemfullyworkedoutwithout makingassumptionsof
symmetry,sothat
k
1
6=
k
3
and
m
1
6=
m
2
,lookupchapterfourofthetextMechanicsbyKeithSymon.
Itisnodierentconceptuallybutitinvolvesmuchmorealgebra.
3|SimpleHarmonicMotion
104
Thesolutionsare
C
2
=
C
4
=0and
C
1
=
C
3
=
x
0
=
2.
x
1
(
t
)=
x
0
2
cos
!t
+cos
!
0
t
x
2
(
t
)=
x
0
2
cos
!t
cos
!
0
t
Youwillhaveseenequationslikethesebeforeifyou’vestudiedthesubjectofbeatsinsound.Tointerpret
them,itismucheasierifyourewritethemusingtwonot-so-well-knowntrigonometricidentitiesforthe
sumanddierenceoftwocosines. Recallproblem0.55.
x
1
(
t
)=
x
0
cos
!
+
!
0
2
t
cos
!
!
0
2
t
; x
2
(
t
)=
x
0
sin
!
+
!
0
2
t
sin
!
0
!
2
t
(3
:
68)
!
0
islargerthan
!
becauseofitsextra2
k
2
,representingtheextraforcefromthemiddlespringpushing
themassesapart. Drawgraphsofthemotionforthecasethat
k
2
isnotverybig,making
!
0
only a
littlelargerthan
!
. Thatmakesthetermsinvolving
!
0
!
inEq.(3.68)oscillatemuchmoreslowly
thantheothers. Agraphof
x
2
(
t
)andthetwofactorsthatcomposeitis
x
2
Fig.3.13
Thisgraphshowsthefunctionsfromthesecondoftheequations(3.68). Thetwosinefactors
thatcompose
x
2
aredrawnlightly,andtheproductisdrawnastheheavygraph. Whenyoudrawthe
samesortofgraphfor
x
1
youwillseethattheenergythatwasgiventomass#1graduallyshiftsover
to#2andthenbackagainto#1. Thisisthesamesortofphenomenonyouhearwhenyoulistento
twomusicalnotes thathaveslightlydierentfrequenciesandthatareplayedtogether. . Theperiodic
pulsationsinintensityarecalledbeatsinthatcontext,andthemathematicsdescribingthemisthesame
thatyouseehere,withtheamplitudeofoscillation ofmass#1slowlygoingfromzerotomaximum
andbackagain.
Tondalife-savingapplicationofcoupledoscillators,gotosection10.5.Thereyoucanseehow
atunedmassdamper,whichisaspecialkindofcoupledoscillator,canbeusedtomaketallbuildings
moreresistanttodamagefromearthquakes. Youmaychoosetoskipthedetailsthere,butifnothing
else,lookattheinterpretationaccompanyingFigure10.6.
3.10NormalModes
Thetwomodesofoscillationrepresentedbytheequations(3.66)and(3.67)arecallednormalmodes
ofoscillation.Thereasonforthisisnotthatthewordnormalmeans\typical"oranythinglikethat. It
isusedinthesenseof\perpendicular"aswhentwolinesintersectatrightangles.
You canthinkofthe components of amodeofoscillation as the coecients in theequation
(3.66),sothattherst(
!
)modeisrepresentedbythecolumnmatrix
P
=
C
1
C
1
SimilarlyEq.(3.67)statesthecomponentsforthesecond(
!
0
)modeas
Q
=
C
3
C
3
3|SimpleHarmonicMotion
105
Theseareorthogonaltoeachotherinthesenseofthescalarproduct
~
P
m
1
0
0
m
2
Q
=(
C
1
C
1
)
m
1
0
0
m
2

C
3
C
3
=
m
1
C
1
C
3
m
2
C
1
C
3
=0
(3
:
69)
because
m
1
=
m
2
,andwherethetildeoverthe
P
meanstranspose.Thisislikethescalarproduct
~
A
.
~
B
=
AB
cos
=
A
x
^
x
+
A
y
^
y
.
B
x
^
x
+
B
y
^
y
=
A
x
B
x
+
A
y
B
y
andthevectors
~
A
and
~
B
areperpendicularif
~
A
.
~
B
=0. Thisexamplecouldbemisleadingbecause
theequalmasseshere make the examplesospecial,andyoumaythinkthat you couldsimply omit
themass matrixinthemiddleofEq.(3.69). . StartingwithEq.(3.63) themasses
m
1
and
m
2
were
setequal,andifyougobacktotheequationprecedingit(3.62),carryingoutallthecalculations,you
wouldndthattheproductinEq.(3.69)wouldstillbezeroprovidedthatyouhavethetwomasses
m
1
and
m
2
intheproduct. Themodesthemselveswillbemorecomplicated,buttheproduct
(
C
1
C
0
1
)
m
1
0
0
m
2

C
3
C
0
3
willstillbezero.
(3
:
70)
Thisredenitionofthescalarproductmaybeunfamiliar,butitispartofalargerpicturethatisexplored
inchapterten,anditisaspecialcaseofEq.(10.22). Itcanwaituntilthen. Thecomplexconjugates
weren’tneededfortheprecedingexamples,butwhenyouwritethesolutionsintermsof
e
i!t
instead
ofsinesandcosinestheybecomenecessary.Thecoecientswillbecomplextoointhatcase.Youcan
refertothesolutionmentionedinthefootnoteonpage103toseethatmoregeneralsolution,andthen
youcanndhowthemassesenterintotheorthogonality.
Youcanusethisorthogonalityrelationasaguidetodeterminetheshapeofthemodesinmore
complexcases. Seeforexampleproblem3.80.
3.11Green’sFunctions
Isinspiredguessworktheonlywaytosolvefortheinhomogeneoussolutionto(3.28)? No. Thereisa
systematicmethoddevelopedbyGreen,amethodthatturnsouttobeofastoundinglygreatutility. It
iscentralinthesubjectofscatteringtheoryinquantummechanicsandquantumeldtheory.*
Takethespecicproblemoftheundampedharmonicoscillatorwithaforcingfunction.
m
d
2
x
dt
2
+
kx
=
F
ext
(
t
)
(3
:
71)
The ideabehind this method is totreatthe addedforce as asum of impulses, and because of the
atomicnatureofmatter,thisisnotsofarfromthetruthanyway. Ifthemassisatrestandyoukick
it,yougiveitaninitialvelocityandafterthatitobeysthehomogeneousequation
mx
+
kx
=0. An
initialvelocity of
v
0
attime
t
=0produces
x
(
t
)=(
v
0
=!
0
)sin
!
0
t
for
t>
0. Ifyouadministerthe
kickattime
t
0
insteadofzero,thatsimplyshiftstheorigin,so
x
(
t
)=
v
0
!
0
sin
!
0
(
t
t
0
)
(
t>t
0
)
(3
:
72)
Ispecifyexplicitlythatthesolution=0for
t<t
0
. Thissaysthatit’stheexternalforcethatiscausing
themotion,notsomedistantinitialconditions.
* GeorgeGreen’slifeasascientististhestuofction. . Hestartedlifeinthefamilybusiness,as
amiller,andwaslargelyself-taughtinmathematicswhenhepublishedhismostfamousworks.
3|SimpleHarmonicMotion
106
Howisthis
v
0
relatedtotheappliedforce? Aforce
F
1
appliedfora(short)timeinterval
t
1
changesthemomentumby
Z
dp
=
Z
Fdt
!
F
1
t
1
=
mv
1
andthis
v
1
isthe
v
0
oftheprecedingequation.Ifyoulatergiveasecondkick
F
2
t
2
youdon’thaveto
knowtheresultoftherstkicktondtheresultofthesecondkick.Thekeyfactabouttheequation
thatyou’retryingtosolveisthatitislinear.
m
d
2
x
1
dt
2
+
kx
1
=
F
1
(
t
)
and
m
d
2
x
2
dt
2
+
kx
2
=
F
2
(
t
)
then
m
d
2
(
x
1
+
x
2
)
dt
2
+
k
(
x
1
+
x
2
)=
F
1
(
t
)+
F
2
(
t
)
x
1
x
2
x
1
+
x
2
+
=
Fig.3.14
In these graphs you apply an impulseat one time (
t
1
) to get therst graph. . Separately y apply an
impulseatalatertime(
t
2
)togetthesecondgraph. Thethirdgraphisthesumofthersttwo,and
itisthesolutiontotheoriginaldierentialequationwhenbothimpulsesareapplied.
In the example that started from Eq. (3.32), the process required you to solve the problem
completelyuptothesecondtime(
T
inthatcase)andthentousetheterminalconditionsbefore
T
as
theinitialconditionsafter
T
,leadingtothesolution(3.35). Inthecurrentproblem,withtwokicks,a
similarsolutionwouldusetheterminalconditionsjustbeforethesecondkickastheinitialconditionsat
thesecondkick. Green’sinsightwasthatthelinearityofthisdierentialequationallowsyoutoavoid
thatcomplexityandtohandlethetwoimpulsesindependently.
Butwhataboutanothersortofforcingfunction,one thatis notjust the sum of acouple of
impulses? Another r insight: every function n is a a sum m of impulses, , orat least itis s a a limitof such h a
sum. Justaswhenyoudeneanintegral,youdividetheindependentvariableintosmallpiecesand
approximatethefunctionbyasequenceofsteps.
k
=0
k
=1
k
=2
=
+
+
+
+
Fig.3.15
The
k
th
intervalhaswidth
t
k
=
t
k
t
k
1
,and theforcetherehasheight
F
(
t
k
). Onesuch
boxisanimpulse,oratleastitwillbewheneventually
t
k
!0. Takeatypicaloneoftheseboxes
and compute its eectontheharmonicoscillator,assumingthat
t
k
issmall. Theequation(3.72)
becomes
F
(
t
k
)
t
k
=
mv
init
so
x
k
(
t
)=
F
(
t
k
)
t
k
m!
0
sin
!
0
(
t
t
k
)
;
(
t>t
k
)
Documents you may be interested
Documents you may be interested