how to open pdf file in new tab in mvc : Convert pdf to text control software system web page windows wpf console mechanics0-part692

Mechanics
(draft,2013)
byJamesNearing
PhysicsDepartment
UniversityofMiami
jnearing@miami.edu
Copyright2008,JamesNearing
Permissiontocopyfor
individualorclassroom
useisgranted.
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Contents
Bibliography
iii
0
MathematicalPrependix
1
Series
HyperbolicFunctions
CoordinateSystems
Vectors
Dierentiation
Velocity,Acceleration
ComplexAlgebra
Separationofvariables
ConstantCoecientODEs
Matrices
IterativeSolutions
1
Introduction
34
DimensionsandUnits
TypesofMass
ConservationLaws
TheTools
CheckingSolutions
2
OneDimensionalMotion
52
SolvingF=ma:F(t)
SolvingF=ma:F(v)
SolvingF=ma:F(x)
Fallingwithresistance
Equilibrium
ConservationofEnergy
3
SimpleHarmonicMotion
77
SimplestCase
ComplexExponentials
DampedOscillators
OtherOscillators
ForcedOscillations
HarmonicForcing
StableMotion
UnstableMotion
CoupledOscillations
NormalModes
Green’sFunctions
AnharmonicExample
4
ThreeDimensionalMotion
120
ProjectileMotion
GeneralResults
EandBelds
MagneticMirrors
ApproximateSolutions
Pendulum,largeangles
5
Non-InertialSystems
152
GalileanTransformation
RotatingSystem
CoriolisForce
CentrifugalForce
ShapeoftheEarth
Tides
FoucaultPendulum
6
Orbits
183
HarmonicOscillator
PlanetaryOrbits
KeplerProblem
Insolation
ApproximateSolutions
SphericalPendulum
CenterofMassTransformation
ExtrasolarPlanets
AnotherOrbit
HyperbolicOrbits
TimeDependence
PerihelionofMercury
EllipticIntegrals
7
Waves
227
AString
Staticcase
TheWaveEquation
EnergyandPower
ComplexForm
ThreeDimensionalWaves
Re ections
StandingWaves
AnAlgebraicAside
PerturbationTheory
Stiness
OtherWaves
OtherVelocities
WavesandTides
8
RigidBodyMotion
260
CenterofMass
AngularMomentum
TensorComponents
i
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PrincipalAxes
PropertiesofEigenvectors
Dynamics
PerturbationTheory
9
SpecialRelativity
294
TimeDilation,LengthContraction
Space-TimeDiagrams
RelativeVelocity
Space-TimeIntervals
SuperluminalSpeeds
Acceleration
Rapidity
EnergyandMomentum
Applications
YarkovskyEect
ConservationLaws
Energy-Momentumtransformations
Poynting-RobertsonEect
10
CoupledOscillators
333
Energy
NormalModes
ScalarProducts
InitialValues
TunedMassDampers
ChainofMasses
PerturbationTheory
11
NonlinearOscillations
356
ForcedPendulum,Qualitatively
AMethodthatFails
AKludge
Workable,butSpecialMethod
GeneralApproach
12
StaticsandBifurcations
363
Bifurcations
Index
369
ii
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Bibliography
.
AnalyticalMechanicsbyGrantR.Fowles. Brooks-Cole.
AnalyticalMechanicsbyFowlesandCassiday. Asecondauthorwasadded. Iprefertheoriginal.
MechanicsbySymon. Addison-Wesley
Thesamesubjectasthistext,ataboutthesamelevel.It’s
beeninprintforalmost40years,soit’sgottobeprettygood.
IntroductiontoClassicalMechanicsbyArya. AllynandBacon
Ithinktherecenteditionisquite
good.
Special Relativity by A.P. French. . MITPress
I thinkthis s remains the bestintroduction to the
subject.
MathematicalMethodsforPhysicsandEngineeringbyRiley,Hobson,andBence.CambridgeUni-
versityPress Forthequantityofwell-writtenmaterialhere,itissurprisinglyinexpensiveinpaperback.
MathematicalMethodsinthePhysicalSciencesbyBoas. JohnWileyPub
Abouttherightlevel
andwithaveryusefulselectionoftopics.Ifyouknoweverythinginhere,you’llndallyourupperlevel
coursesmucheasier.
Street-FightingMathematicsby Mahajan. . MITPress
\In problemsolving,as instreetghting,
rulesareforfools: dowhateverworks{don’tjuststandthere!"Aspracticalasyoucanget.
MathematicalToolsforPhysicsbyNearing.DoverPuboranonlineversion Inmyunbiasedopinion,
it’sprettygood.
Schaum’s Outlines by various.
There are e many good d and inexpensive books in this series: for
example,\ComplexVariables",\AdvancedCalculus",\BookkeepingandAccounting",\AdvancedMath-
ematicsforEngineersandScientists". Amazonlistshundreds.
LinearDierentialOperatorsbyLanczos. Itexploitsthecloserelationshipbetweendierentialequa-
tionsandmatricestogaindeepinsights. ReadtheusercommentsonAmazon.
ABriefonTensorAnalysisbyJamesSimmonds. Springer r ThisistheonlytextontensorsthatI
willrecommend. Toanyone.Underanycircumstances.
LinearAlgebraDoneRightbyAxler. Springer r Don’tletthetitleturnyouaway. . It’sprettygood.
LinearAlgebraDoneWrongby Treil. anonlinetextasapdf.
Stillanotherviewofthesubject,
andmaybeevenbetter.(Viewsdier.)
iii
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MathematicalPrependix
.
Thispreliminarychaptercouldaseasilybeanappendixtothetext,butIprefertoputithere.
Itisacollectionoftopicsthatyouwillneedatmanyplaceslateron,butthatyoudon’thavetostudy
in detailnow. IneachsectionIwilltry y toindicatewherethematerialisused,andwhenyougetto
thechapterwhereyouneeditIwillindicatethereferencehere. Youshouldatleastskimthismaterial
now,sothatyouwillhaveseen whereitis. Ifsomethingcatches s youreyeandyouwanttostudy it
now,don’tletmestopyou.
0.1Series
Thissectionisusedinsomeformineverychapterinthetext.
Inniteseries is atoolthatyou seein anintroductory calculus course, and you may y notthenhave
realizedjusthowusefulitis.Especiallypowerseries.Thereareafewseriesthatshowupsooftenthat
youneedtohavetheminstantlyavailable.Binomial,trigonometric,exponential,geometric,occasionally
thelogarithm.
e
x
=1+
x
+
x
2
2!
+
x
3
3!

=
1
X
0
x
k
k
!
(a)
sin
x
=
x
x
3
3!
+
x
5
5!

=
X1
0
( 1)
k
x
2
k
+1
(2
k
+1)!
(b)
cos
x
=1 
x
2
2!
+
x
4
4!

=
X1
0
( 1)
k
x
2
k
(2
k
)!
(c)
ln(1+
x
)=
x
x
2
2
+
x
3
3

=
1
X
1
( 1)
k
+1
x
k
k
(j
x
j
<
1)
(0
:
1)
(1+
x
)
n
=1+
nx
+
n
(
n
1)
x
2
2!
+ =
X1
k
=0
n
(
n
1)(
n
k
+1)
k
!
x
k
(j
x
j
<
1)
(e)
sinh
x
=
x
+
x
3
3!
+
x
5
5!
+
=
X1
0
x
2
k
+1
(2
k
+1)!
(f)
cosh
x
=1+
x
2
2!
+
x
4
4!
+
=
X1
0
x
2
k
(2
k
)!
(g)
1
x
=1+
x
+
x
2
+
x
3
+
=
1
X
0
x
k
(j
x
j
<
1)
(h)
Thehyperbolicfunctions,sinhandcoshhavepowerseriessimilartothoseforsinandcosexcept
thatthesignsareallpositive(f,g)insteadofalternating(b,c). Dierentiatethepowerseriesforsine
andcosinetogetthefamiliardierentiationformulas. Nowdothesamethingforthehyperbolicsine
andcosine.Dierentiatetheseriesforthelogarithm(d),andrelateittothelastseries(h)onthelist,
1
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MathematicalPrependix
2
calledthe\geometricseries". Thisgeometricseriesisaspecialcaseofthebinomialseriesfor
n
= 1.
Whatis
p
2? Usethe5
th
series,thebinomialexpansion: (1+1)
1
=
2
=1+
1
2
=1
:
5. Notbad
fortwoterms,eventhoughitisattheedgeofitsdomainofvalidity.
Whatis1
=
0
:
9?Usethelastseries: 1
=
(1 0
:
1)=1
:
1.
Whatis0
:
99
1
=
10
?Usethebinomialseriesagain: (1 0
:
01)
1
=
10
=1 
:
001=0
:
999.
Evaluatethislimit. Thisexamplepullstogetherseveraltechniqueswithseries;itisveryworth
yourtimetobeabletoreproducethisexampleonyourown.
lim
x
!0
1
p
x
2
x
(0
:
2)
Usethebinomialexpansionafewtimes. Workrstonthecomplicatedfraction:
1
p
x
=
1
1
2
x
1
8
x

=
1
1
2
x
+
1
8
x
2+
=
1
1
2
x
1+
1
4
x
+

=
2
x
1
4
x
+
+
2

Putthisintotheoriginalexpression
2
x
1
2
2
x
1
2
Testthislimitbyanumericalexperimentonacalculator.Take
x
=0
:
5
;
0
:
1
;
0
:
01
Whatis(sin0
:
1)
=
(sinh0
:
1)?Use(b),(f),andthegeometric(h)series:
(0
:
1 0
:
001
=
6)
=
(0
:
1+0
:
001
=
6)=(1 
:
01
=
6)
=
(1+
:
01
=
6)
=(1 
:
01
=
6)(1 
:
01
=
6)=(1 
:
02
=
6)=0
:
996667.
Acalculatorgives0
:
99667221535
Whatis the behaviorof the function
x
(
t
) =
p
c
2
t
2
+
c
4
=a
2
c
2
=a
for small time? Use
thebinomialexpansion,butrstyoumust arrangethesquarerootas
p
1+small. Thissimply
involvesfactoringthelargertermoutofthesquareroot.
x
(
t
)=
c
2
a
r
1+
a
2
t
2
c
2
c
2
a
c
2
a
1+
a
2
t
2
2
c
2
c
2
a
=
1
2
at
2
(0
:
3)
Thisexpressionfor
x
(
t
)istherelativisticexpressionformotionwithconstant(proper)accelera-
tion,and
at
2
=
2isthenon-relativisticapproximationtoit. Itisderivedinsection9.6.
Suddenly apply a force to a mass that is attached to a spring. The result for
x
is (see
Eq.(3.33))
x
(
t
)=
F
0
k
1 cos
!
0
t
where
k
isthespringconstant,and
!
0
=
r
k
m
Whatisthebehaviorof
x
forsmalltime?Andrememberthatsmall isnotzero.
x
(
t
)=
F
0
k
1 (1 
!
0
t
2
=
2+)
=
F
0
k
!
2
0
t
2
=
2
=
F
0
k
k
m
t
2
2
=
F
0
m
t
2
2
andthatis
at
2
=
2. Atthestartofthemotionthespringhasn’tyetstretched,sotheonlyforce
istheonethatyouapply.
AlloftheseexpansionsarespecialcasesoftheTaylorseries.
MathematicalPrependix
3
f
(
x
)=
f
(
x
0
)+(
x
x
0
)
f
0
(
x
0
)+
1
2
(
x
x
0
)
2
f
00
(
x
0
)+
1
3!
(
x
x
0
)
3
f
000
(
x
0
)+
(0
:
4)
Wheredoesthisrepresentationcomefrom? Ifyouassumethatthereisanexpansionoftheform
f
(
x
)=
A
+
B
(
x
x
0
)+
C
(
x
x
0
)
2
+
D
(
x
x
0
)
3
+
(0
:
5)
then evaluate both sides at
x
=
x
0
and you immediately have
A
=
f
(
x
0
). Nowdierentiate e the
hypothesizedequation(0.5)for
f
f
0
(
x
)=
B
+2
C
(
x
x
0
)+3
D
(
x
x
0
)
2
+
andagainevaluateitat
x
=
x
0
.Thisgives
B
=
f
0
(
x
0
).Anotherderivativeandanevaluationandyou
have2
C
=
f
00
(
x
0
). Andagain,andagain,
:::
ThisiswhereallthecoecientsinEq.(0.4)comefrom,andeveryoneoftheseriesinEq.(0.1)
canbederivedthisway(with
x
0
=0inallcases).
Themanipulationsintheexampleofequation(0.3)aretypicalofthemostcommonway that
seriesareusedinthis text. Whenyouhave e acomplicatedmathematicalresultforthesolution toa
problem,themostimportantstepistounderstandthatresult. Seriesapproximationsareapowerfultool
todigsimpleresultsoutofcomplexmathematics.Therearetechnicaldetailstolearntoo,thoughthat
isnotthepointofthistext: Underwhatconditionsdotheseseriesconverge? Underwhatconditions
dotheyconvergetothefunctionstheysupposedlyrepresent?
Example
Whatisthepowerseriesexpansionofthefunction
f
(
x
)=
a
bx
+
cx
3
aboutitminimum,outto
secondorderterms? Take
a;b;c>
0.
Firstwhereistheminimum?
f
0
(
x
)= 
b
+2
cx
2
=0  !
x
min
=
q
b=
2
c
Whichifeitherofthesetwopointsis aminimum?Foranansweryoucaneithertakeasecondderivative
(doso)oryoucansketchagraph. Closetotheoriginthe 
bx
termin
f
has itgoingdowntoward
therightanduptowardtheleft,thenthe
cx
3
termeventuallybecomesverypositiveontherightand
verynegativeontheleft. Thatmeanstheminimumistheoneontherightandtheoneontheleftis
amaximum. Callit
x
0
.
x
0
=+
q
b=
2
c;
f
(
x
0
)=
a
b
b
2
c
1
=
2
+
c
b
2
c
3
=
2
=
a
b
3
=
2
2(2
c
)
1
=
2
f
0
(
x
0
)=0
;
f
00
(
x
0
)=6
cx
0
=3
p
2
bc
f
(
x
)=
f
(
x
0
)+(
x
x
0
)
.
0+
1
2
f
00
(
x
0
)(
x
x
0
)
2
+
=
a
b
3
=
2
2(2
c
)
1
=
2
+
3
2
p
2
bc
(
x
x
0
)
2
+
MathematicalPrependix
4
0.2HyperbolicFunctions
Thissectionappearsinchapters3,9,and10.
Thecirculartrigonometricfunctionssuchassineandcosinearefamiliar,butthehyperbolictrigonometric
functionsmaynotbe. Thesefunctionsaredenedintermsofexponentialsas
cosh
x
=
e
x
+
e
x
2
sinh
x
=
e
x
e
x
2
tanh
x
=
sinh
x
cosh
x
(0
:
6)
Their reciprocals are sech, csch, coth in analogy with h the denitions of the corresponding circular
functions.
Whyarethesehyperbolic?First,whyaretheotherscircular?Answer:thesineandcosinesatisfy
asimpleidentitythatallowsthemtodescribeacircle.
If
x
=cos
and
y
=sin
;
then
x
2
+
y
2
=cos
2
+sin
2
=1
Fig.0.1
Thereisasimilaridentityforhyperbolicfunctions,anditsderivationinvolvesnothingmorethanusing
thedenitions.
cosh
2
sinh
2
=
e
+
e
2
!
2
e
e
2
!
2
=
e
2
+2+
e
2
e
2
+2 
e
2
4
=1
Dividethisequationbycosh
2
toget1 tanh
2
=sech
2
.
If
x
=cosh
and
y
=sinh
;
then
x
2
y
2
=cosh
2
sinh
2
=1
(0
:
7)
Thecoordinates
x
and
y
describeahyperbola. Inthecircularcasethereisasimplegeometricinter-
pretationof
.Inthehyperboliccasethereisnot. It’snotthatthereisn’taninterpretationatall,just
thatitisnotveryuseful.
Whatarethederivativesof thesefunctions? Dierentiatetheequations s (0.6)ortheseries in
(0.1)forcoshandsinh,thentheproduct(orquotient)rulefortanh.
d
dx
cosh
x
=sinh
x
d
dx
sinh
x
=cosh
x
d
dx
tanh
x
=sech
2
x
(0
:
8)
Thehyperbolicfunctionsinvolveexponentials,soitshouldnotbetoosurprisingthattheinverse
hyperbolicfunctionsinvolvelogarithms.
y
=sinh
1
x
means
x
=sinh
y
=
1
2
e
y
e
y
(0
:
9)
MathematicalPrependix
5
Multiplyby2
e
y
andrearrange.Theresultisaquadraticequation.
e
2
y
2
xe
y
1=
e
y
2
2
xe
y
1=0
whichimplies
e
y
=
x
p
x
2+1
Theexponential
e
y
ispositive,sothatforcesthe+signontheright.Nowtakethelogarithm.
y
=sinh
1
x
=ln
x
+
p
x
2+1
similarly,
cosh
1
x
=ln
x
p
x
1
(
x
1)
(0
:
10)
Commoncirculartrigonometricidentitieshavetheirparallelhere.Forexample
cosh(
x
+
y
)=cosh
x
cosh
y
+sinh
x
sinh
y;
sinh(
x
+
y
)=sinh
x
cosh
y
+cosh
x
sinh
y
(0
:
11)
Use the series for the e circular and d the hyperbolic functions s to see the relations between the
twosetsoffunctions. Forexample,whatis s cos(
ix
)? Substitute
ix
intothethirdseriesofEq.(0.1).
Similarly,substitute
ix
intotheseriesforthesinetoseeitsrelationtothehyperbolicsine.
Fig.0.2
Thesearesixgraphsofthesix
hyperbolicfunctions. It’s
uptoyoutopuzzleoutwhich
curvesgowithwhichfunctions.
Togettheinversefunctions,
invertthesegraphsinthe
45
line
x
=
y
.
0.3CoordinateSystems
Thissectionappearsinchapters3,4,5,6,7,8(atleast).
Thereareafewcommoncoordinatesystemsthatyouwilluseallthetime.Intwodimensionsyouhave
rectangularandpolar.Inthreedimensionsthecommononesarerectangular,cylindrical,andspherical.
Whenyouneedparaboliccoordinatesortoroidalcoordinates,youcanlookthem upsomewhereelse,
butthevedrawnherearetheonesyouhavetomaster.
x
y
x
y
r
x
y
rectangular
polar
MathematicalPrependix
6
x
y
z
r
z
r
x
y
z
x
y
z
x
y
z
rectangular
cylindrical
spherical
Fig.0.3
All the coordinate lines simply represent the functions that say all the other coordinates are
constant. Inthecommonplanerectangularcoordinatesthelinesparalleltothe
x
-axisarethegraphs
oftheequations
y
=1,or
y
= 10,etc.Theequation
x
=5isalineparalleltothe
y
-axis,andthe
equationforthatlineis
x
=0.
Inplanepolarcoordinatesthecoordinatelinesare
r
=constant(circles)or
=constant(rays
thatstartfromtheoriginbecause
r
0).
Therelationbetweenthesetwocoordinatesystemsissimple. Whenyoudescribeasinglepoint
using(
x;y
)onetimeand(
r;
)anothertime,theequationsrelatingthesepairsare
x
y
r
r
=
p
x
2+
y
2
; 
=tan
1
(
y=x
)
x
=
r
cos
; y
=
r
sin
(0
:
12)
The only place tostumble inthis transformation is incomputing
from
y
and
x
. Youneed d both
numbers
y
and
x
tospecify the quadrantfor
becausetheinversetangentismultiplevalued. The
signs of
y
and
x
will however tell you that tan
1
(1
=
1) =
=
4 and that tan
1
( 1
=
1) = 3
=
4,
therebyremovingtheambiguity.
Inthreedimensions, rectangularcoordinates s (
x;y;z
) are much likethosein twodimensions,
exceptthatanequationsuchas
y
=1isnowaplaneperpendiculartothe
y
-axisas
x
and
z
varyfrom
minustoplusinnity. Togetalineyouneedtwoequations. Forexamplethepairf
x
=5and
z
=6g
describesalineparalleltothe
y
-axisandpuncturingthe
x
-
z
planeat(5
;
6).The
z
-axisitselfisdescribed
bythetwoequationsf
x
=0and
y
=0g.
Cylindricalcoordinates (
r;;z
)areanextensionof two-dimensionalpolarcoordinates,simply
stretchedparalleltothe
z
-axis. The
z
-coordinateisthesameastherectangular
z
-coordinate,andthe
(
r;
)arethesameasthetwodimensionalpolarcoordinates. Theequation
z
=constantisaplane
paralleltothe
x
-
y
planeasbefore,andtheequation
r
=constantisnowacylindercenteredalongthe
z
-axis. Thethirdcoordinate
=constantisahalf-planewithoneedgealongthe
z
-axis(0
<r<
1
and 1
<z<
1).
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