asp.net mvc 4 and the web api pdf free download : All jpg to one pdf converter application Library utility html asp.net windows visual studio mechanics4-part1311

1|Introduction
37
byanelectriceldbehaves sodierentlyfromtheeectby agravitationaleld? ? Therearelogically
twodierenttypesofmass.Oneisinertialmass,describingtheresistancetoacceleration;theotheris
gravitationalmass,describingthecouplingtothegravitationaleld.
~a
=
~
F
total
=m
i
;
and
~
F
gravitational
=
m
g
~g
Whosaysthatthesetwomassesarethesame?
Itis an experimental question. . Are e they? ? The e question was rstasked by Newton, but the
real creditgoes toEotvos. . In n thelate1800’s and continuingintothe rstdecade ofthe1900’s,he
carriedoutsomemiraculouslypreciseexperimentstomeasuretheratioofthetwomasses,
m
i
=m
g
. Is
itthesameforallmaterials? His s conclusionwasyes,toaprecisionofaboutonepartin10
8
. Later
experiments by Dicke improved this toone part in n 10
11
and more recently Adelbergeretal.added
anotherfactorofabout100. Atthispointitseemssafetosaythatthetwomassesarethesame.
1.3ConservationLaws
Classical mechanics is not just about forces s and accelerations. . Even n if f the e problems you examine
aretraditionaloneswithmassesinteracting|gravity,friction,magnetism,etc.Thereareotherways
to approach those e problems, , and d the conservation laws s are at t least as s fundamental l as
~
F
=
m~a
.
Conservationofenergy,ofmomentum,ofangularmomentumcanallbederivedfromthatequation,
butdid you knowthatyou can derive
~
F
=
m~a
from energy? ? If f so,then which of the two is more
fundamental.
Work,Energy
Inonedimensionwithpointmasses,thework-energytheoremisverysimpleandisnotatalldicult
toderivefromNewton’sequations. Thereareacoupleofwaystodoso,oneinvolvesthefullpowerof
calculusmanipulationsandafewlinesofalgebra;theotherismoreintuitive,butmoretedious.
Startwiththerstway(andifyouhaven’treviewedsection0.5thengobackanddoitnow).
F
x
=
ma
x
=
m
dv
x
dt
=
m
dv
x
dx
dx
dt
=
mv
x
dv
x
dx
(1
:
1)
Thisusedthechainrulefordierentiationandthen itused the denition ofvelocity. . Nowintegrate
thisequationwithrespectto
x
betweensomespeciedinitialandnallimits.
W
=
Z
x
f
x
i
F
x
dx
=
Z
x
f
x
i
mv
x
dv
x
dx
dx
=
Z
x
=
x
f
x
=
x
i
mv
x
dv
x
=
m
2
v
2
x
x
=
x
f
x
=
x
i
=
m
2
v
2
f
m
2
v
2
i
=
K
(1
:
2)
Striptheinterveningmaterialfromthisandyouhavethework-energytheorem:
W
=
Z
x
f
x
i
F
x
dx
=
m
2
v
2
f
m
2
v
2
i
=
K
(1
:
3)
This took two o (long) lines, but you need to have a a complete understanding of f the e chain rule e for
dierentiation and d of f the methods s to o change variables s in n an integral. . There e are two parts to this
derivation:First,knowinghowtodothemanipulations.Second,understandingwhatthemanipulations
meanandwhytheyarevalid.
Asecondwaytoreachthisresultislongerbutsimpler. Itdoesn’tinvolveany y specialcalculus
tricks,onlythedenitionofanintegral. Itstartswiththespecialcaseofaconstantforce,thensince
a
x
=
F
x
=m
theaccelerationisaconstant. That’safamiliarcase:
a
x
=constant  !
v
x
=
a
x
t
+
C;
then
x
=
1
2
a
x
t
2
+
Ct
+
D
All jpg to one pdf converter - Merge, append PDF files in C#.net, ASP.NET, MVC, Ajax, WinForms, WPF
Provide C# Demo Codes for Merging and Appending PDF Document
append pdf files reader; pdf merger
All jpg to one pdf converter - 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
pdf merge comments; pdf split and merge
1|Introduction
38
Usetheinitialconditionsthatat
t
=0positionis
x
=
x
0
andvelocityis
v
x
=
v
0
.
v
0
=
a
x
.
0+
C;
x
0
=
1
2
a
x
.
0
2
+
C
.
0+
D
so
v
x
=
a
x
t
+
v
0
;
and
x
=
1
2
a
x
t
2
+
v
0
t
+
x
0
Eliminatethevariable
t
betweenthelasttwoequations
t
=
v
x
v
0
=a
x
;
so
x
=
1
2
a
x
v
x
v
0
2
=a
2
x
+
v
0
v
x
v
0
=a
x
+
x
0
Rearrangethelastresult,simplifyingthealgebratoget
x
=
1
2
v
2
x
v
2
0
=a
x
+
x
0
or
ma
x
(
x
x
0
)=
1
2
mv
2
x
1
2
mv
2
0
Allofthisrearrangementwasjustelementaryalgebra,andthereasonforthenalmanipulation,mul-
tiplyingby
m
wastogetthecombination
ma
x
intherstterm. Thatbecomes
F
x
.
(
x
x
0
)=
1
2
mv
2
x
1
2
mv
2
0
(1
:
4)
Nowfortherealcase. Constantforcesaretextbookidealizationsoftherealworld. Atbestyou
canapproximatesomeforcetobecloseenoughtoconstantformostpurposes. Whataboutthemore
commoncaseforwhichtheforceisnotatallconstant? Answer:usesuccessiveapproximationsasyou
doincalculusandthensneakupontheresult. Approximateaposition-dependentforceasasequence
ofsteps.
F
x
(
x
)=
8
>
>
>
>
>
<
>
>
>
>
>
:
F
1
(
x
0
<x<x
1
)
F
2
(
x
1
<x<x
2
)
F
3
(
x
2
<x<x
3
)
F
4
(
x
3
<x<x
4
)
:::
F
N
(
x
N
1
<x<x
N
)
x
0
x
1
x
2

Fig.1.2
Ineachoftheseintervals,theequation(1.4)applies.Ateachpoint
x
0
,
x
1
,
:::
thespeedis
v
0
,
v
1
,etc.
F
1
(
x
1
x
0
)=
1
2
mv
2
1
1
2
mv
2
0
F
2
(
x
2
x
1
)=
1
2
mv
2
2
1
2
mv
2
1
F
3
(
x
3
x
2
)=
1
2
mv
2
3
1
2
mv
2
2
F
4
(
x
4
x
3
)=
1
2
mv
2
4
1
2
mv
2
3
F
5
(
x
5
x
4
)=
1
2
mv
2
5
1
2
mv
2
4

F
N
(
x
N
x
N
1
)=
1
2
mv
2
N
1
2
mv
2
N
1
Thesumofalltheseequationsis
F
1
(
x
1
x
0
)+
F
2
(
x
2
x
1
)+
F
3
(
x
3
x
2
)++
F
N
(
x
N
x
N
1
)=
1
2
mv
2
N
1
2
mv
2
0
becausethetermsontherighttelescope: Allthetermsexcepttherstandthelastcancelinpairs. In
conventionalnotationthisis
N
X
k
=1
F
k
x
k
=
N
X
k
=1
F
x
(
x
k
)
x
k
=
1
2
mv
2
f
1
2
mv
2
i
where
x
k
=
x
k
x
k
1
x
N
=
x
f
x
0
=
x
i
C# PDF Convert to Jpeg SDK: Convert PDF to JPEG images in C#.net
Turn multiple pages PDF into single jpg files respectively Support of converting from any single one PDF page and may customize the names of all converted JPEG
merge pdf; c# merge pdf files
VB.NET PDF Convert to Jpeg SDK: Convert PDF to JPEG images in vb.
Turn multiple pages PDF into multiple jpg files in VB Support of converting from any single one PDF page and And converted JPEG image preserves all the content
attach pdf to mail merge; best pdf merger
1|Introduction
39
Thelimitofthisequationas
x
k
!0isjustthedenitionoftheword\integral",sothisreproduces
Eq.(1.3).
These twoderivations of the sameequation,
W
= 
K
, lookcompletely y dierent, , but they
aren’t. When n you lookclosely y at t them and compare them step by step they are much h more e alike
thanyourstthink. Thesecond d derivationtellsyouthatthecombinationofforcetimesdistanceis
important|especiallyforce timesalittlebit ofdistance:
F
x
dx
. Writedownthis s combinationand
manipulateit.
F
x
dx
=
ma
x
dx
=
m
dv
x
dt
dx
=
m
dv
x
dt
dx
=
mdv
x
dx
dt
=
mv
x
dv
x
Howeveryoudoit,yougetthework-energyrelationstatingthattheworkonapointmass(
R
F
x
dx
)
equalsthechangeinitskineticenergy.
Whathappensifyoudon’thaveapointmass? Whatifyouaren’toperatinginonedimension?
Whatiftheforceisnotafunctionofpositionalone?Allgoodquestionsandallwillhavetobeaddressed
inchapters8,4,and2respectively.
MechanicalEnergy
Forthissamesimplecaseofapointmassinonedimension,andif theforceisafunctionofposition
only,youcanrearrangethework-energytheoremtogetaconservationlaw.
Theworkisanintegralof
F
x
(
x
)
dx
. Whenyouevaluateintegrals,themostcommonwayyou
useistond an anti-derivative and then toevaluateitatthe endpointsofthe integrationinterval,
thefundamentaltheoremofcalculus: If
f
hasanantiderivativeandif
f
isintegrablethenthetwoare
related.
If
f
(
x
)=
dF
(
x
)
=dx
then
Z
b
a
f
(
x
)
dx
=
F
(
b
F
(
a
)
(1
:
5)
In the present case the function you’re integrating is
F
x
(
x
) and Iwill denote its anti-derivative by
U
(
x
).Thatis,
F
x
(
x
)= 
dU
(
x
)
dx
;
then
Z
x
f
x
i
F
x
(
x
)
dx
U
(
x
f
)+
U
(
x
i
)
(1
:
6)
Nowapplythistothework-energytheorem,Eq.(1.3).
W
=
K
is
Z
x
f
x
i
F
x
(
x
)
dx
U
(
x
f
)+
U
(
x
i
)=
m
2
v
2
f
m
2
v
2
i
whichis
m
2
v
2
i
+
U
(
x
i
)=
m
2
v
2
f
+
U
(
x
f
)
(1
:
7)
Thisis the reasonfortheminussignin the denition of
U
inEq.(1.6). Ifyou u don’tputtheminus
signthereyouwouldnotgetaplussignhere,inthisconservationofenergyequation.
U
iscalledthepotentialenergycorrespondingtotheforce
F
x
(
x
),andthisequationsays that
thesumofthekineticandthepotentialenergieshasthesamevaluesthroughoutthemotion;thesum
isconserved.
Can’tyoualwaysdothismanipulationtogetconservationofmechanicalenergyoutofthework-
energy theorem? ? Can’tyou u always nd an anti-derivative (maybe in abig table of integrals)? ? For
example,whataboutcommondryfriction?Perhapsyourememberanequationsuchas
F
fr
k
F
N
.
Thefrictional forceasanobjectslidesoverasurfaceisthecoecientof(kinetic)friction timesthe
normalcomponentoftheforceon the surface. . It’sjustaconstant,soyou u cancertainly integratea
constantandgetapotentialenergy
U
andthengetconservationofmechanicalenergy.
Online Convert Jpeg to PDF file. Best free online export Jpg image
Easy converting! Drag and drop your JPG file, choose all the conversion settings you like, and download it with one click.
pdf mail merge plug in; append pdf
C# Create PDF from images Library to convert Jpeg, png images to
C#.NET Example: Convert One Image to PDF in Visual C# .NET 1.bmp")); images.Add(new Bitmap(Program.RootPath + "\\" 1.jpg")); images.Add All Rights Reserved.
pdf mail merge; build pdf from multiple files
1|Introduction
40
No.
Inthisexampleofdryfriction,thefrictionalforceisvelocitydependenteventhoughitdoesn’t
looklike it|there’snovelocityinthe expression  
k
F
N
. True,but t that’s becausethisexpression
is wrong. . Thefrictional l force is velocity dependentbecause itisalways opposite the velocity. . This
commonlyusedexpressionfortheforceissimplynotright. Amorecorrectwaytowriteitis
~
F
fr
k
F
N
^
v
(1
:
8)
where^
v
=
~v=v
istheunitvectorinthedirectionofthevelocityoftheslidingmass. Ifitslidesinthe
reversedirection,theforcefromfrictionisreversedandyoucanhavetwooppositevaluesoftheforce
atthesamevalueoftheposition. Itisnotafunctionof
x
aloneandno
U
exists. Whydon’tpeople
(includingme)writethisthecorrectway?Answer: It’sawkwardandI’mlazy.
Notice that I I keep using the clumsy y phrase \conservation n of mechanical energy" instead d of
\conservation of energy". . Why? ? That’s s tied in to the factthat conservation of energy was one of
themostdicultlaws to sortout. . Newtonmay y have written hisbasicequations in thelate1600s,
butconservationofenergydidnotbecomeanacceptedlawofphysicsuntilthemiddle1800s. These
two expressions,kineticenergy and potential energy,seem easy to derive now, buthistorically even
thesewerediscovered through tortuousand tortious routes.* * Theproblemisthatenergy is s notjust
mechanicalenergy.
Energy is s an n abstraction, a a bookkeeping device much h like money. . It t is s aprescription n saying
thatcertainmathematicallydescribedpropertiesofasystemaretobeaddedtogetherandifyoudoit
right,youcancomebacklater,redothesumandyou’regoingtogetthesameanswer. Thehistorical
confusionaroseintryingtogureoutwhattermstoincludeinthis sum. . Itwas s resolvedwhenheat
wasseentobeanotherformofenergy(aterminthesum). Thenlight,sound,chemicalenergy,and
eventuallymasswereaddedtothemix.Energyisnotathing,eventhoughyoucangetthatimpression
ifyoureadnewspaperarticlesonthesubject.
DidIjustsaythatmoneyisanabstraction? Isn’titjustthebillsandcoinsthatyoucarrywith
you?No.Ifyoubuyacar,doyoupayforitwithhundreddollarbills?Mostpeopledon’t,andbillsand
coinsarenowherenearconserved. Ifyouincludewhat’sinyourcheckingaccount,that’snotbillsand
coinsstoredinavault. Itisasetofbitsstoredonacomputerdisk. Asavingsaccountismoreofthe
same. Thenthere e are whatevermysteriousmanipulationsthe FederalReserveBoardcanimplement.
The total \money"isacombination ofmany dierentthings,ofwhich justatinyamountisinany
senseconcrete. Ifyouaskveeconomiststodenemoneyyouwilllikely y getveanswers(
M
1
,
:::
,
M
5
).Thedierencewiththeabstractioncalledenergyisthatwethinkweknowwhatgoesintoit.Or
dowe?Cosmologistsweresurprisedbythediscoverythattheexpansionoftheuniverseisaccelerating,
causedbysomethingcalled\darkenergy"forlackofabettername. Nooneknowswhatitis.
Derive
F
=
ma
AfewparagraphsbackIsaidthatyoucanderive
F
=
ma
fromenergy. How? Takeconservationof
mechanicalenergyanddierentiateit:
\
E
=
1
2
mv
2
x
+
U
(
x
) isconserved"
means
dE
dt
=0
d
dt
1
2
mv
2
x
+
U
(
x
)
=
1
2
m
2
v
x
dv
x
dt
+
dU
dx
dx
dt
=
v
x
m
dv
x
dt
+
dU
dx
=0
or,
m
dv
x
dt
dU
dx
* butnottorturous
VB.NET Create PDF from images Library to convert Jpeg, png images
take Gif image file as an example for converting one image to 1.bmp")) images.Add( New REImage(Program.RootPath + "\\" 1.jpg")) images.Add All Rights Reserved
pdf mail merge; pdf split and merge
C# PDF Image Extract Library: Select, copy, paste PDF images in C#
Get JPG, JPEG and other high quality image files from PDF document. Able to extract vector images from PDF. C#: Select All Images from One PDF Page.
pdf mail merge plug in; .net merge pdf files
1|Introduction
41
andthatis
F
x
=
ma
x
. This s isnotjustaprettytheorem,itis sometimesausefulwaytoderivethe
equationsofmotioninspeciccases.Oftenitinvolvesmucheasiermanipulationstoreachtheanswer,
andI’llregularlyusethismethodlaterinthetext.
Momentum
ThereareotherwaystomanipulateNewton’sequation. Therstandleastinterestingwayistostart
fromNewton’sequationforasinglepointmassandintegrateitwithrespecttotime.
~
F
=
m~a
!
Z
t
2
t
1
~
Fdt
=
Z
t
2
t
1
m~adt
=
m~v
(
t
2
m~v
(
t
1
)
Whenyouhavetwoormoreinteractingmasses,yougetsomethingmoreuseful
~
F
on1by2
=
m
1
~a
1
;
~
F
on2by1
=
m
2
~a
2
(1
:
9)
Addthese,andthetwoforcescancelbecauseofNewton’sthirdlaw,leaving
0=
m
1
d~v
1
dt
+
m
2
d~v
2
dt
=
d
dt
m
1
~v
1
+
m
2
~v
2
(1
:
10)
Thisisconservationofmomentum.Itdoesn’tmatterhowcomplicatedtotwoforcesareaslongasthe
thirdlawissatised.
Ifyouhavethreeorthreemillionparticlestheresultisthesame,onlythenotationchanges.Put
indices
i
and
j
onthemassesinsteadofsimple1and2. Thetotalforceonparticleiisthesumofthe
forcesfromalltheother particles
m
i
~a
i
=
X
j
6=
i
~
F
on
i
by
j
Nowaddthisequationoverallvaluesoftheindex
i
.
X
i
m
i
~a
i
=
X
i
X
j
6=
i
~
F
on
i
by
j
=0
Writethisoutforthreemasses! Really
(1
:
11)
Becauseaccelerationisthetime-derivativeofvelocity,thisequationsaysthat
d
dt
X
i
m
i
~v
i
=0
;
or
X
i
m
i
~v
i
=aconstant
(1
:
12)
CanyoureallygofromEq.(1.9)to(1.10)?Whatifmassisn’tconstant?Then
~
F
6=
m~a
anyway
andyoushouldhavebeenusing
~
F
=
d~p=dt
. Thatalsomakesthewholeprocessmuchmorenatural.
(Gobackanddoitthatway. It’seasy.)
AngularMomentum
Anothermanipulationof Newton’sequationusesthecrossproduct. . Pickan n originandlet
~r
bethe
coordinatevectorofasinglepointmassfromthatorigin.Noticethedierencehere:Thisresultdepends
onhavingchosenanorigin.Itdoesn’tmatterwhichoriginyoupick,butyouhavetopickone.
~r
~
F
=
~r
d~p
dt
=
~r
d~p
dt
+
d~r
dt
~p
=
d
dt
~r
~p
(1
:
13)
C# Image Convert: How to Convert MS PowerPoint to Jpeg, Png, Bmp
C:\output\"; // Convert PowerPoint to jpg and show This demo code convert PowerPoint file all pages to Bmp The last one is for rendering PowerPoint file to
add pdf pages together; scan multiple pages into one pdf
C# Image Convert: How to Convert Tiff Image to Jpeg, Png, Bmp, &
In general, conversion from one image to another should always retain the original size This demo code convert TIFF file all pages to jpg images.
add two pdf files together; acrobat combine pdf
1|Introduction
42
Whatwasthetrickthatappearsafterthesecondequalssign? ThetermthatIaddediszerobecause
itissimply
~v
~p
=
~v
m~v
=0. Thenaltermistheordinaryproductruleforderivatives,andthe
termsinthisequationgetnames,torqueandangularmomentum.
~
=
d
~
L
dt
is
torque=time-derivativeofangularmomentum
(1
:
14)
~
L
=
~r
1
~p
1
+
~r
2
~p
2
+
(1
:
15)
Again, when n you u have e many particles, , you u put t indices s on n the e masses and sum m over r all l the
particles,butunlikethecaseoflinearmomentumtheresultsaresucientlycomplexthattheydeserve
acompletechapter(eight)tothemselves.
Intheabsence ofexternalforces,you havethe conservationlaw:
~
L
total
=constant. Howto
derivethis? Theforceonasingleparticle,the
i
th
one,is
~
F
i
=
F
i;
external
+
X
j
6=
i
~
F
on
i
by
j
Thisrepresentsthetotalforceonasingleparticleas causedby everythingelse intheuniverse. . The
index
j
isforalltheotherparticlesinthebody.Inthepresentcase,thereisnoexternalforce,and
~
=
X
i
~r
i
~
F
i
=
X
i
~r
i
X
j
6=
i
~
F
on
i
by
j
Lookatoneparticularpairoftermsin thissum,theonesforwhich
i
=1
;j
=2and
i
=2
;j
=1.
Thoseare
~r
1
~
F
on1by2
+
~r
2
~
F
on2by1
Thesetwoforcesareopposite,sothisis
~r
1
~
F
on1by2
~r
2
~
F
on1by2
=
~r
1
~r
2
~
F
on1by2
Iftheforcesactingbetweenthesemassesactalongthelinebetweenthem thenthiscrossproductis
zero.Thesamewillapplytoallpairsofforces.Iftheforceisnotalignedthisway,thenthissumisnot
zero,andtheexpressionforangularmomentumpresentedhereisnotconserved. Seeproblem4.47for
anexampleofthis,thoughtheresolutionoftheproblemisleftforelsewhere.
Mass
Massisconserved. Iftwoobjectscollideandtheyhavemasses
m
1
and
m
2
,thenafterthecollisionthe
totalmasswillbethesame.
m
1
+
m
2
=
m
3
+
m
4
Thisdoesn’tmeanthat
m
3
=
m
1
and
m
4
=
m
2
. Masscanbemovedfromoneobjecttotheother,
perhapsbychippingoorbecausealoosepartofonemassbecomesattachedtotheother.
Youcanseewhythisistrueifyouassumethatmomentumconservationisvalid.Ifyoumeasure
thevelocityofsomemasstobe
~v
andafriendofyoursismovingbyatvelocity
~u
thenyourfriendwill
concludethatthevelocityofthemassis
~v
~u
. Ifmomentumconservationholdsforyouthenitshould
holdforyourfriend. Thetwoequationsare
m
1
~v
1
+
m
2
~v
2
=
m
3
~v
3
+
m
4
~v
4
and
m
1
(
~v
1
~u
)+
m
2
(
~v
2
~u
)=
m
3
(
~v
3
~u
)+
m
4
(
~v
4
~u
)
1|Introduction
43
Subtracttheseandyouhave(forall
~u
)
m
1
~u
+
m
2
~u
=
m
3
~u
+
m
4
~u
=)
m
1
+
m
2
=
m
3
+
m
4
(1
:
16)
Theideahereappearsagaininmuchmoredetailinchapternine,especiallysection9.11. There,inthe
contextofspecialrelativityyouwillseethatmassconservationisnotquiterightafterall.
Thisderivationshowedthatifyoustartfrommomentumconservationandlookatitfromanother
pointofview,yougetaveryinterestingresult. Whatifkineticenergyisconserved,acompletelyelastic
collision,whatdoesadierentpointofviewsayaboutthat?
1
2
m
1
v
2
1
+
1
2
m
2
v
2
2
=
1
2
m
3
v
2
3
+
1
2
m
4
v
2
4
=)
1
2
m
1
(
~v
1
~u
)
2
+
1
2
m
2
(
~v
2
~u
)
2
=
1
2
m
3
(
~v
3
~u
)
2
+
1
2
m
4
(
~v
4
~u
)
2
Ifthisistrueforallvaluesof
~u
,thenthecoecientsof
u
2
andof
~u
itselfmustagree.Theseare
m
1
+
m
2
=
m
3
+
m
4
and
m
1
~v
1
+
m
2
~v
2
=
m
3
~v
3
+
m
4
~v
4
Thissaysthatonceyou’veassumedthatyouhaveacompletelyelasticcollision,youautomaticallyget
conservationofmomentumandofmass. Thatis\automatically"ifyouassumethatdierentlymoving
observerswillallhave the samebasicequationsformechanics. . Galileowenttogreatpainstoargue
whythatistrue. Lookonpage294foraquotefromhistextonthesubject.
1.4TheTools
Ismechanicsacollectionoftrickstosolvevariouslycontrivedproblemsorisitasystematicapproach
to analyzingcomplex systems? ? I I think it is s thelatter,butit often appears more like the formerin
introductorytexts. Thereare ahandfulofrulesthat,systematicallyfollowed,willallowyoutosetup
evenvery complicated problems. . The e resulting equations can stillbe hard to solve, butthat’s only
becauseofthemathematics,notbecauseofthephysics.
Thesearemechanicalsystemsthatwe’redealingwithhere,andthatlimitsthesortofthingsyou
havetodealwith. Iwilllayoutasetofrulesandthenshowby y examplejustwhattheymean. . The
wholepointisthateveryproblemisattackedthesameway. Youdon’tlearnonemethodnowanda
dierentmethod later. . Youwillnotbelieveme e whenIsaythis,butfollowingthesesystematicrules
willsaveyouahugeamountoftime.
1.Drawasketchofthephysicalsystem.Itdoesn’thavetobepretty,butitshouldconvey
anideaofwhatishappening.
2. There e are typically several masses in eachproblem, so for r eachone e of them listin
ordinarylanguagethethingsthatareactingonit. Aforceisnotathing;anacceleration
isnotathing;evenan
m~a
isnotathing. Atableisathing;ahandisathing;aropeis
athing;theEarthisathing. Remember: Hereathingcanactonamassonlybyeither
beingincontactwiththemassorbyitsgravitationalpull.
3. Foreachmass,godownthelistofthings s acting onitand drawthatmasstogether
withthevectorsshowingthedirectionsoftheforcesexertedbythosethings. Also,draw
anaccelerationvectorforeachmass. Labeleachmassandeachvector.Usesymbolsthat
conveysomemeaning.
4.Pickabasisintermsofwhichyoucanwritethevectors.Notethatyouarenotrequired
tousethesamebasisasyouturnyourattentionfromonemasstoanother. Youcanthink
1|Introduction
44
ofthemasseparateproblems,andthemathematicswillunitethemforyou. Labelyour
coordinates.
5. Foreachmasswrite
~
F
=
m~a
in thebasischosen. . Or,
~
F
=
d~p=dt
if you need the
moregeneralform.
6. Breakthe e vectorequations intoequations fortheirvarious components,sothatyou
don’thavevectorstocarrythroughtherestofthealgebra.
7. DidyouneedtouseNewton’sthirdlaw? ? Isthemagnitudeofthisforceequaltothe
magnitudeofthatforce?
8.Countthenumberofequationsandthenumberofunknowns. Iftheymatch,you(may)
haveawinner.
9.Solve.
10. Analyzetheresults.
11. Themostpopularmethodtoturnaneasyproblemintoadicultoneistoskipsteps.
Example
Twomassesareincontactandaresittingonahorizontaltable. Pushtheleftonetowardtheright
andndtheircommonaccelerationandtheforcesbetweenthem.Assumezerofriction.
1.Sketch:
2.Actingontheleftmass:
1.you, 2.table e 3. . Earth(gravity) ) 4. . othermass
Actingontherightmass: 1.table, , 2. . Earth h 3. . othermass
Notice: Youarenotactingontherightmass.
3.
~
F
you
~
F
tabl
;
1
m
1
~g
~
F
contact
~
F
contact
~
F
tabl
;
2
m
2
~g
~a
(both)
4.
^
y
^
x
Thesamebasisforbothmasses
5.
~
F
1
=
F
you
^m
1
g^y
+
F
table
;
1
^F
contact
^x=m
1
a
x
^x
~
F
2
m
2
g
^
y
+
F
table
;
2
^
y
+
F
contact
^
x
=
m
2
a
x
^
x
6.
F
you
F
contact
=
m
1
a
x
F
contact
=
m
2
a
x
The
y
-componentsareoflittleuse.
7.Newton’sthirdlawwasusedinapplying
F
contact
tothetwomasses.
8.Theunknownsare
F
contact
and
a
x
,andtherearetwoequations.
9.
a
x
=
F
you
=
(
m
1
+
m
2
)
and
F
contact
=
F
you
m
2
=
(
m
1
+
m
2
)
10. Thedimensionsarecorrect,butyoudohavetolook.
If
m
1
m
2
thenthecontactforceisapproximately
F
you
.
If
m
1
m
2
thenthecontactforceismuchsmallerthen
F
you
.
F
you
m
1
m
2
Fig.1.3
Shouldyoubelievethis?Dotheexperiment: Useabookandapenor
abookandawadded-uppiece ofpaper. . Placethemincontactonatable
andpushononesideortheother. Howdoyoumeasurethecontactforce?
Putangerofyourotherhandbetweenthem. Thenreverseyourhandsand
1|Introduction
45
pushontheothermasstofeelthecontactforceagain.
Doyouhavetodothisevery timethatyousetupaproblem? ? Yes,untilthetimecomesthat
you nolongermakemistakes. . Then n youcan startskippingsteps. . (I’m still l waiting.) ) It’s s alsotrue
thatfollowingall these steps isagreattime-saver. . Youdon’tbelievethis. . Noonebelievesitatthe
beginning.
Example
Atwood’sMachine:Thisisaclassicexampleofamechanicalsystem,adevicethatappearsinevery
introductoryphysicsbook(probablybyan actofCongress). . You u canseepicturesofrealversionsof
thisapparatusatthiswebsiteofearlyscienticapparatus:
physics.kenyon.edu/EarlyApparatus/Mechanics/Atwoods
Machine/Atwoods
Machine.html
m
1
m
2
^
y
^
y
m
1
~g
T
1
m
2
~g
T
2
Fig.1.4
Mustyouexecutethissystematicprocedurebythenumbers?Maybenot.Hereitisanarration,
butallthestepsarepresent:
Hangtwomassesonstringsandrunthestringoveraxedpulley. Theheaviermasswillstarttodrop
andthelighteronetorise,andtheproblemistogureouttheaccelerationofeither. CanIneglect
themassofthepulley? Probablynot,butIwilldoitanywayfornow. CanIneglectthemassofthe
string? Again,maybenot,butdoitanywaythistime. Lateryoucanreturntothesubjectwhenyou
wishtoaddmorerealitytotheanalysis. Ichosethebasiswith ^
y
upforonemassandwithitdown
fortheother. Isthisnecessary? No,butit’saconvenientoptionbecauseyoucantheneasilyuseone
coordinate
y
forbothmasses,andaslongasthestringbehaves,
a
y
isthesameforboth.
The things s that t act on
m
1
are gravity and the string. . Acting g on
m
2
are again gravity y and
thestring. The e single coordinate
y
applies to themotion of eachmassifyou make the reasonable
assumptionthatthestringdoesn’tstretch(conservationofstring). ApplyNewton’slawtoeachmass,
treatingthetwomassesastwoseparateproblemshavingnothingtodowitheachother.
~
F
1
=
m
1
~g
+
~
T
1
=
m
1
~a
1
;
~
F
2
=
m
2
~g
+
~
T
2
=
m
2
~a
2
(1)
F
1
y
=
m
1
g
T
1
=
m
1
a
y
;
(2)
F
2
y
m
2
g
+
T
2
=
m
2
a
y
(1
:
17)
Thesearetwoequationsinthethreeunknowns(
a
y
,
T
1
,
T
2
),soyouneedanotherequation. Thatis
T
1
=
T
2
. Why y is this so? ? Itcomesfrom m atorqueequation. . Ifthemassofthepulley y is zero,any
non-zerotorque on it wouldgiveitinnite angularacceleration(Eq.(8.5)), and thatcan’thappen.
Withthesethreeequationsyoucansolveforeverything.
a
y
=
m
1
m
2
m
1
+
m
2
g;
T
1
=
T
2
=
2
m
1
m
2
m
1
+
m
2
g
(1
:
18)
Inanalyzingthissolutiontherearethreecasesthatpushittoitslimit:
m
1
=
m
2
,
m
1
m
2
,
and
m
1
m
2
.
First:
m
1
=
m
2
:
a
y
=0
; T
=
2
mm
m
+
m
g
=
mg
1|Introduction
46
Itbalances,givingzeroaccelerationandwithjustenoughtensioninthestringtomakethetotalforce
oneachmasszero.
Second:
m
1
m
2
:
a
y
m
1
m
1
g
=
g; T
2
m
1
m
2
m
1
g
=2
m
2
g
With
m
1
dominant,thatmassacceleratesat+
g
,causingtheothermasstoaccelerateupat+
g
. As
m
2
experiencesadownward force from gravityof
m
2
g
,thisrequiresthestringtopullupwith twice
thisforce,anditdoes.
Third:
m
1
m
2
:
a
y
 
m
2
m
2
g
g; T
2
m
1
m
2
m
2
g
=2
m
1
g
Now
m
2
acceleratestoward 
y
andpulls
m
1
upagainstgravity,requiringatensionof2
m
1
g
todoso.
Why did Atwood invent his machine and why is it important enough even to have aname?
Perhaps heinventeditas away toharassphysics students. . Perhaps s abetterreason is as adevice
toverify Newton’sLaws. . Alsoasadevicetomeasure
g
. In n 1780,electronictimers hadn’tyetbeen
invented,andthismachineslowedthemotionenoughtomakemeasurementsofaccelerationeasier. If
youlookattheKenyonhistoricallinktoearlyscienticapparatus,youwillseethatAtwood’soriginal
machinedoesnotlooklikethesimplepicturehere;ithascomplicationsdesignedtoreducetheeects
offrictionbecausegood,low-frictionbearingsdidn’texistthen.
Whatdoesthishavetodowithenergy?Inchaptertwoyouwillndmuchmoreonthesubject
of energy,but fornowI’ll assume that you have seen somethingofthesubjectbefore and thatthe
development of mechanical energy y in the e last t section is familiar. . You u remember r the e gravitational
potentialenergytobe
mgh
,andifyoudon’t,thentheequation(1.6)tellsyouthat
F
y
mg
dU
dy
;
so
U
(
y
)=
mgy
IntheAtwood machine asdrawn,Iwill assumethateverythingstartsatrestwhenthemasseshave
coordinates
y
=0. Takethezero-pointofpotentialenergy y tobezeroatthatpointtoo. . Laterthe
masseswillhavepickedupspeedandtheirpositionswillhavechanged.Writedownthetotalmechanical
energy.
E
=
1
2
m
1
v
2
y
+
1
2
m
2
v
2
y
+
m
2
gy
m
1
gy
Thetwokineticenergiesarepositiveand(forpositive
y
)the
m
2
getspositivepotentialenergyandthe
potentialenergyfor
m
1
isnegative.
Energyisconserved. Thatmeansthatthederivativeofthetotalenergywithrespecttotimeis
zero.
E
=
1
2
m
1
v
2
y
+
1
2
m
2
v
2
y
+
m
2
gy
m
1
gy
dE
dt
=0=
d
dt
1
2
m
1
v
2
y
+
1
2
m
2
v
2
y
+
m
2
gy
m
1
gy
=
m
1
v
y
dv
y
dt
+
m
2
v
y
dv
y
dt
+
m
2
g
dy
dt
m
1
g
dy
dt
0=(
m
1
+
m
2
)
dv
y
dt
+(
m
2
m
1
)
g
(1
:
19)
Thethirdlineusedthechainruleforderivativestodierentiate
v
2
y
withrespectto
t
. Thefourthline
used thedenitionof velocity,
v
y
=
dy=dt
,tocancelsomefactors. Solve e the nalequationforthe
acceleration
a
y
=
dv
y
=dt
andyouhavethe resultEq.(1.18). . Oratleastpartofit. Usingenergy
methodsisoftenaneasierwaytogettotheresultyouwant,butitdoesnotalwaysgiveyouall the
resultsyouwant. Inthiscaseitdoesn’tprovideanequationforthetensioninthestring. Ifyouneed
that,youcanuse
~
F
=
m~a
ononeofthetwomassesandgetthetensionfromthatsingleequation.
Documents you may be interested
Documents you may be interested