Statics and Dynamics - Andy Ruina

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Introduction to
S
TATICS
and
D
YNAMICS
Andy Ruina and Rudra Pratap
OxfordUniversityPress(Preprint)
MostrecentmodificationsonAugust21,2010.
Reference Tables: The front and back tables concisely summarize much of the text material.
Summary of Mechanics
0) Thelawsofmechanicsapplytoanycollectionofmaterialor‘body.’ Thisbodycouldbetheoverallsystemofstudy
oranypartofit. Intheequationsbelow,theforcesandmomentsarethosethatshowonafreebodydiagram. Interacting
bodiescauseequalandoppositeforcesandmomentsoneachother.
I) LinearMomentumBalance(LMB)/ForceBalance
EquationofMotion Fi L The total force on a body is equal (I)
to its rate of change of linear
momentum.
Impulse-momentum t2
(integratingintime) Fi·dt L Netimpulseisequaltothechangein (Ia)
t1 momentum.
Conservationofmomentum L=0
When there is no net force the linear (Ib)
(if Fi 0) L=L2 L1 0 momentum does not change.
Statics
Fi 0 If the inertial terms are zero the (Ic)
(ifLisnegligible)
net force on system is zero.
II) AngularMomentumBalance(AMB)/MomentBalance
Equationofmotion M H The sum of moments is equal to the (II)
C C
rateofchangeofangularmomentum.
Impulse-momentum(angular) t2
M dt H The net angular impulse is equal to (IIa)
(integratingintime) C C
t1 the change in angular momentum.
Conservationofangularmomentum H 0
C If there is no net moment about point (IIb)
(if M 0) H H H 0
C C C2 C1 C then the angular momentum about
point C does not change.
Statics
M 0 If the inertial terms are zero then the (IIc)
(ifH isnegligible) C
C total moment on the system is zero.
III) PowerBalance(1stlawofthermodynamics)
Equationofmotion Q P E E E Heat flow plus mechanical power (III)
K P int
into a system is equal to its change
E
in energy (kinetic + potential +
internal).
t2 t2
forfinitetime Qdt Pdt E Thenetenergyflowgoinginisequal (IIIa)
t1 t1 tothenetchangeinenergy.
ConservationofEnergy E 0
If no energyflows into a system, (IIIb)
(ifQ P 0) E E E 0
2 1 then its energydoesnotchange.
Statics
Q P E E If there is no change of kinetic energy (IIIc)
(ifE isnegligible) P int
K then the change of potential and
internal energy is due to mechanical
work and heat flow.
PureMechanics
(ifheatflowanddissipation P E E In a system well modeled as purely (IIId)
K P
arenegligible) mechanical the change of kinetic
and potential energy is due to mechanical
work on the system.
Filename:Summaryofmechanics
Some definitions
(Alsoseetheindexandbacktables)
*r or *x Position e.g.,*r *r is the position of a point i
i (cid:17) i=O
relativetotheorigin,O.
d*r
*v Velocity e.g.,*v *v is the velocity of a point i
(cid:17) dt i (cid:17) i=O
relativetoO,measuredinanon-rotatingref-
erenceframe.
d*v d2*r
*a Acceleration e.g.,*a *a istheaccelerationofapointi
(cid:17) dt D dt2 i (cid:17) i=O
relativetoO,measuredinaNewtonianframe.
*
F Force e.g.,theforceonAfromBisF .
AfromB
* * *
M or M M MomentorTorque e.g., the moment of a collection of forces
CD =C
aboutpointC.
*! Angularvelocity Ameasureofrotationalvelocityofarigidob-
ject.*! =angularvelocityofrigidobjectB.
B
*(cid:11) *! Angularacceleration Ameasureofrotationalaccelerationofarigid
(cid:17) P
object.
m*v discrete
* i i
L Linearmomentum Ameasureofasystem’snettranslationalrate
(cid:17) 8 P*vdm continuous (weightedbymass).
<
m R*v
D :tot cm
m*a discrete
* i i
LP Rate of change of linear momen- The aspect of motion that balances the net
(cid:17) 8 P*adm continuous tum forceonasystem.
<
m R*a
D :tot cm
*r m*v discrete
H* i=C(cid:2) i i AngularmomentumaboutpointC Ameasureoftherotationalrateofasystem
=C (cid:17) 8< P*r=C (cid:2)*vdm continuous about a point C (weighted by mass and dis-
tancefromC).
R
:
*r m*a discrete
H*P=C (cid:17) 8< P*r=iC=C(cid:2)(cid:2)*admi i continuous RtuamteaobfocuhtapnogientoCfangularmomen- TtohrequaesopnecatsoyfstmemotiaobnouthtaatpboainlatnCc.es the net
R
:
1 m v2 discrete
E 2 i i Kineticenergy Ascalarmeasureofnetsystemmotion.
K (cid:17) 8 1Pv2dm continuous
< 2
R
:
E (heat-liketerms) Internalenergy The non-kinetic non-potential part of a sys-
int D
tem’stotalenergy.
P F**v M* *! Powerofforcesandtorques The mechanical energy flow into a system.
(cid:17) i(cid:1) i C i(cid:1) i
Also,P W,rateofwork.
P P (cid:17) P
Icm Icm Icm
xx xy xz
(cid:140)Icm(cid:141) 2 Icm Icm Icm 3 Moment of inertia matrix about Ameasureofthemassdistributioninarigid
(cid:17) xy yy yz
centerofmass(cm) object.
66 Ixczm Iyczm Izczm 77
6 7
4 5
Introduction to
S
TATICS
and
D
YNAMICS
Andy Ruina and Rudra Pratap
c Rudra Pratap and Andy Ruina, 1994-2009. All rights reserved. No part of this book may be reproduced, stored in a
(cid:13)
retrievalsystem,ortransmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,orotherwise,without
priorwrittenpermissionoftheauthors.
Thisbookisapre-releaseversionofabookinprogressforOxfordUniversityPress.
Acknowledgements. Thefollowingareamongstthosewhohavehelpedwiththisbookaseditors,artists,texprogrammers,
advisors,criticsorsuggestersandcreatorsofcontent:WilliamAdams,AlexaBarnes,PranavBhounsule,JosephBurns,Jason
Cortell,GaborDomokos,MaxDonelan,ThuDong,GailFish,MikeFox,JohnGibson,RobertGhrist,SaptarsiHaldar,Dave
Heimstra, Theresa Howley, Herbert Hui, Michael Marder, Elaina McCartney, Horst Nowacki, Jim Papadopoulos, Kalpana
Pratap, DaneQuinn, RichardRand, C.V.Radakrishnan, NidhiRathi, PhoebusRosakis, LesSchaffer, IshanSharma, David
Shipman, JillStartzell, SaskyavanNouhuys, TianTang, KimTurnerandBillZobrist. Ouron-againoff-againeditorPeter
Gordonhasbeensupportivethroughout. Manyotherfriends,colleagues,relatives,students,andanonymousreviewershave
alsomadehelpfulsuggestions.
WecertifyArthurOgawa,IvanDobrianov,andStephenHicksasTeXgeniuses.
MikeColemanworkedextensivelyonthetext,wrotemanyoftheexamplesandhomeworkproblemsandmademanyfigures.
David Ho, R. Manjula and Abhay drew or improved most of the drawings. Credit for some of the homework problems
retrieved from Cornell archives is due to various Theoretical and Applied Mechanics faculty. Harry Soodak and Martin
Tierstenprovidedsomeproblemsfromtheirincompletebook.
SoftwarewehaveusedtopreparethisbookincludesTEXshop(forLATEX)withmanycustomfeaturesimplementedbyStephen
Hicks,AdobeIllustrator,GraphicsConverterandMATLAB.
Brief Contents
Fronttables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
BriefContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
DetailedContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Tothestudent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Part I: Basics for Mechanics
24
1 Whatismechanics? . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Vectors: position,forceandmoment . . . . . . . . . . . . . . . . 38
3 FBDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Part II: Statics
184
4 Staticsofoneobject. . . . . . . . . . . . . . . . . . . . . . . . . 184
5 Trussesandframes . . . . . . . . . . . . . . . . . . . . . . . . . 254
6 Transmissionsandmechanisms. . . . . . . . . . . . . . . . . . . 320
7 Tension,shearandbendingmoment . . . . . . . . . . . . . . . . 376
8 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
Part III: Dynamics
412
9 Dynamicsin1D . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
10 Particlesinspace . . . . . . . . . . . . . . . . . . . . . . . . . . 530
11 Manyparticlesinspace . . . . . . . . . . . . . . . . . . . . . . . 578
12 Straightlinemotion . . . . . . . . . . . . . . . . . . . . . . . . . 604
13 Circularmotion . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
14 Planarmotionofanobject . . . . . . . . . . . . . . . . . . . . . 752
15 Time-varyingbasisvectors . . . . . . . . . . . . . . . . . . . . . 838
16 Constrainedparticlesandrigidobjects . . . . . . . . . . . . . . . 906
Appendices
974
A Unitsanddimensions . . . . . . . . . . . . . . . . . . . . . . . . 974
B Friction: perspectivesonfrictionlaws . . . . . . . . . . . . . . . 986
C TheoremsforSystems . . . . . . . . . . . . . . . . . . . . . . . 996
Answerstosomehomeworkproblems . . . . . . . . . . . . . . . . .1006
Backtables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1015
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1992-2009. 1
Detailed Contents
Fronttables i
Summaryofmechanics . . . . . . . . . . . . . . . . . . . i
Somebasicdefinitions . . . . . . . . . . . . . . . . . . . . ii
BriefContents 1
DetailedContents 2
Preface 10
General issues about content, level, organization, style and motivation.
Studyadvicestartsonpage 14.
Tothestudent 14
Howtostudy. Theuseofcomputers.
0.1 Anoteoncomputation . . . . . . . . . . . . . . . . . . . . . 18
Box: Informalcomputercommands . . . . . . . . . . . . . 22
Part I: Basics for Mechanics
24
1 Whatismechanics? 24
Mechanicscanpredictforcesandmotionsbyusingthethreepillarsofthe
subject: I. models of physical behavior, II. geometry, and III. the basic
mechanicsbalancelaws. Thelawsofmechanicsareinformallysumma-
rized in this introductory chapter. The extreme accuracy of Newtonian
mechanicsisemphasized,despiterelativityandquantummechanicssup-
posedly having ‘overthrown’ seventeenth century physics. Various uses
oftheword‘model’aredescribed.
1.1 Thethreepillars . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2 Mechanicsiswrong,whystudyit? . . . . . . . . . . . . . . 31
1.3 Theheirarchyofmodels . . . . . . . . . . . . . . . . . . . . 33
2 Vectors: position,forceandmoment 38
The key vectors for statics, namely relative position, force, and mo-
ment, are used to develop vector skills. Notational clarity is empha-
sized because good vector calculation demands distinguishing vectors
fromscalars. Vectoradditionismotivatedbytheneedtoaddforcesand
relativepositions. Dotproductsaremotivatedasthetoolwhichreduces
vectorequationstoscalarequations.Andcrossproductsaremotivatedas
2 IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1992-2009.
Chapter0.DetailedContents DetailedContents 3
theformulawhichcorrectlycalculatestheheuristicallymotivatedquan-
titiesofmomentandmomentaboutanaxis.
2.1 Notationandaddition . . . . . . . . . . . . . . . . . . . . . 40
Box2.1Thescalarsinmechanics . . . . . . . . . . . . . . 41
Box2.2TheVectorsinMechanics . . . . . . . . . . . . . 42
2.2 Thedotproductoftwovectors . . . . . . . . . . . . . . . . 58
Box2.3Basicfeaturesofthevectordotproduct. . . . . . . 58
Box2.4abcos(cid:18) a b a b a b . . . . . . . 63
x x y y z z
) C C
2.3 Vectorcrossproduct . . . . . . . . . . . . . . . . . . . . . . 67
Box2.5Usesofthecrossproduct . . . . . . . . . . . . . . 68
Box2.6Crossproductasamatrixmultiply . . . . . . . . . 73
Box2.7Thecrossproduct: fromgeometrytocomponents . 74
2.4 Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.5 Solvingvectorequations . . . . . . . . . . . . . . . . . . . . 92
Box2.8Therulesofvectoralgebra. . . . . . . . . . . . . 93
Box2.9Vectortrianglesandthelawsofsinesandcosines . 95
Box2.10Existence,uniqueness,andgeometry . . . . . . . 107
2.6 Equivalentforcesystems . . . . . . . . . . . . . . . . . . . . 112
Box2.11 meansadd . . . . . . . . . . . . . . . . . . . 114
Box2.12Equivalentatonepoint equivalentatallpoints 115
P )
Box2.13A“wrench”canrepresentanyforcesystem . . . 116
2.7 Centerofmassandgravity. . . . . . . . . . . . . . . . . . . 121
Box2.14Like ,thesymbol alsomeansadd . . . . . . 122
Box2.15Eachsubsystemislikeaparticle . . . . . . . . . 126
P R
Box2.16TheCOMofatriangleisath=3 . . . . . . . . . 130
ProblemsforChapter2 . . . . . . . . . . . . . . . . . . . . . . . 136
3 FBDs 148
Afree-bodydiagramisasketchofthesystemtowhichyouwillapplythe
lawsofmechanics,andallthenon-negligibleexternalforcesandcouples
whichactonit.Thediagramindicateswhatmaterialisinthesystem.The
diagramshowswhatis,andwhatisnot,knownabouttheforces. Gener-
allythereisaforceormomentcomponentassociatedwithanyconnection
thatcausesorpreventsamotion.Conversely,thereisnoforceormoment
component associated with motions that are freely allowed. Mechanics
reasoning entirely rests on free body diagrams. Many student errors in
problem solving are due to problems with their free body diagrams, so
wegivetipsabouthowtoavoidvariouscommonfree-bodydiagrammis-
takes.
3.1 Interactions,forces&partialFBDs . . . . . . . . . . . . . . 150
VectornotationforFBDs . . . . . . . . . . . . . . . . . . 153
Box3.1Freebodydiagramfirst,mechanicsreasoningafter 161
Box3.2ActionandreactiononpartialFBD’s . . . . . . . 163
3.2 Contact: Sliding,friction,androlling . . . . . . . . . . . . . 170
ProblemsforChapter3 . . . . . . . . . . . . . . . . . . . . . . . 180
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1992-2009.
4 Chapter0.DetailedContents DetailedContents
Part II: Statics
184
4 Staticsofoneobject 184
Equilibrium of one object is defined by the balance of forces and mo-
ments. Foraparticle,forcebalancetellsall. Butforanextendedobject,
moment balance is also useful. There are special shortcuts for an ob-
jects thathas exactly twoor exactly threeforces acting onit. If friction
forces are relevant the possibility of motion needs to be taken into ac-
count. Manyreal-worldproblemsarenotstaticallydeterminateandthus
yield either only partial solutions, or yield full solutions after you have
madeextraassumptions.
4.1 Staticequilibriumofaparticle . . . . . . . . . . . . . . . . . 186
Box4.1Existenceanduniqueness . . . . . . . . . . . . . 190
Box4.2Thesimplificationofdynamicstostatics . . . . . . 193
4.2 Equilibriumofoneobject . . . . . . . . . . . . . . . . . . . 199
Box4.3Two-forcebodies . . . . . . . . . . . . . . . . . . 204
Box4.4Three-forcebodies . . . . . . . . . . . . . . . . . 205
Box4.5Momentbalanceabout3pointsissufficientin2D . 206
4.3 Equilibriumwithfrictionalcontact . . . . . . . . . . . . . . 211
Box4.6Undrivenwheelsandtwoforcebodies . . . . . . 215
4.4 Internalforces . . . . . . . . . . . . . . . . . . . . . . . . . 225
4.5 3Dstaticsofonepart . . . . . . . . . . . . . . . . . . . . . 231
ProblemsforChapter4 . . . . . . . . . . . . . . . . . . . . . . . 240
5 Trussesandframes 254
Hereweconsidercollectionsofpartsassembledsoastoholdsomething
up or hold something in place. Emphasis is on trusses, assemblies of
bars connected by pins at their ends. Trusses are analyzed by drawing
free body diagrams of the pins or of bigger parts of the truss (method
ofsections). Frameworksbuiltwithotherthantwo-forcebodiesarealso
analyzedbydrawingfreebodydiagramsofparts. Structurescanberigid
or not and redundant or not, as can be determined by the collection of
equilibriumequations.
5.1 Methodofjoints . . . . . . . . . . . . . . . . . . . . . . . . 256
5.2 Themethodofsections . . . . . . . . . . . . . . . . . . . . 273
5.3 Solvingtrussesonacomputer . . . . . . . . . . . . . . . . . 280
5.4 Framesandstructures . . . . . . . . . . . . . . . . . . . . . 291
Box5.1The‘methodofbarsandpins’fortrusses . . . . . 294
5.5 Advancedtrussconcepts: determinacy . . . . . . . . . . . . 301
Box5.2Stucturalrigidityandgeometriccongruence . . . 306
Box5.3Rigidity,redundancy,linearalgebraandmaps . . 307
ProblemsforChapter5 . . . . . . . . . . . . . . . . . . . . . . . 312
6 Transmissionsandmechanisms 320
Some collections of solid parts are assembled so as to cause force or
torque in one place given a different force or torque in another. These
include levers, gear boxes, presses, pliers, clippers, chain drives, and
crank-drives. Besides solid parts connected by pins, a few special-
purpose parts are commonly used, including springs and gears. Tricks
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1992-2009.
Chapter0.DetailedContents DetailedContents 5
for amplifying force are usually based on principals idealized by pul-
leys, levers, wedges and toggles. Force-analysis of transmissions and
mechanisms is done by drawing free body diagrams of the parts, writ-
ingequilibriumequationsforthese,andsolvingtheequationsfordesired
unknowns.
6.1 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
Box6.1‘Zero-length’springs . . . . . . . . . . . . . . . . 323
Box6.2Howstiffaspringisasolidrod . . . . . . . . . . 330
Box6.3Stifferbutweaker . . . . . . . . . . . . . . . . . . 330
Box6.4Apuzzlewithtwospringsandthreeropes. . . . . . 331
Box6.52Dgeometryofspringstretch . . . . . . . . . . . 334
6.2 Forceamplification . . . . . . . . . . . . . . . . . . . . . . 343
6.3 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 355
Box6.6Shearswithgears . . . . . . . . . . . . . . . . . . 359
ProblemsforChapter6 . . . . . . . . . . . . . . . . . . . . . . . 366
7 Tension,shearandbendingmoment 376
The ‘internal forces’ tension, shear and bending moment can vary from
point to point in long narrow objects. Here we introduce the notion of
graphingthisvariationandnotingthefeaturesofthesegraphs.
7.1 Arbitrarycuts . . . . . . . . . . . . . . . . . . . . . . . . . 377
ProblemsforChapter7 . . . . . . . . . . . . . . . . . . . . . . . 393
8 Hydrostatics 396
Hydrostaticsconcernstheequivalentforceandmomentduetodistributed
pressure on a surface from a still fluid. Pressure increases with depth.
With constant pressure the equivalent force has magnitude = pressure
times area, acting at the centroid. For linearly-varying pressure on a
rectangular plate the equivalent force is the average pressure times the
area acting 2/3 of the way down. The net force acting on a totally sub-
mergedobjectinaconstantdensityfluidisthedisplaceweightactingat
thecentroid.
8.1 Fluidpressure . . . . . . . . . . . . . . . . . . . . . . . . . 397
Box8.1AddingforcestoderiveArchimedes’principle . . . 400
Box8.2Pressuredependsonpositionbutnotonorientation 401
ProblemsforChapter8 . . . . . . . . . . . . . . . . . . . . . . . 409
Part III: Dynamics
412
9 Dynamicsin1D 412
ThescalarequationF maintroducestheconceptsofmotionandtime
D
derivatives to mechanics. In particular the equations of dynamics are
seen to reduce to ordinary differential equations, the simplest of which
have memorable analytic solutions. The harder differential equations
needbesolvedonacomputer. Weexplorevariousconceptsandapplica-
tions involving momentum, power, work, kinetic and potential energies,
oscillations,collisionsandmulti-particlesystems.
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1992-2009.
6 Chapter0.DetailedContents DetailedContents
9.1 Forceandmotionin1D . . . . . . . . . . . . . . . . . . . . 414
Box9.1WhatdothetermsinF mamean? . . . . . . . 419
D
Box9.2Theunitsofforce . . . . . . . . . . . . . . . . . . 421
Box9.3SolutionsofthesimplestODEs . . . . . . . . . . . 424
Box9.4D’Alembert’smechanics: beginnersbeware . . . . 427
9.2 Energymethodsin1D . . . . . . . . . . . . . . . . . . . . . 434
Box9.5Particlemodelsfortheenergeticsoflocomotion. . 445
9.3 Vibrations: mass,springanddashpot . . . . . . . . . . . . . 452
Box9.6Acos.(cid:21)t/ Bsin.(cid:21)t/ Rcos.(cid:21)t (cid:30)/ . . . . . 456
C D (cid:0)
Box9.7Solutionofthedamped-oscillatorequations . . . . 462
9.4 Coupledmotionsin1D . . . . . . . . . . . . . . . . . . . . 477
Box9.8Normalmodes: themathandtherecipe . . . . . . 484
9.5 Collisionsin1D . . . . . . . . . . . . . . . . . . . . . . . . 492
Box9.9Whenequalrodscollidethevibrationsdisappear . 497
9.6 Advanced: forcing&resonance . . . . . . . . . . . . . . . . 501
Box9.10ALoudspeakerconeisaforcedoscillator. . . . . 506
Box9.11Solutionoftheforcedoscillatorequation . . . . . 508
Box9.12Thevocabularyofforcedoscillations . . . . . . . 509
ProblemsforChapter9 . . . . . . . . . . . . . . . . . . . . . . . 517
10 Particlesinspace 530
This chapter is about the vector equation F* m*a for one particle.
D
Conceptsandapplicationsincludeballisticsandplanetarymotion. The
differential equations of motion are set-up in cartesian coordinates and
integratedeithernumerically,orforspecialsimplecases,byhand. Con-
straints,forcesfromropes,rods,chains,floors,railsandguidesthatcan
onlybefoundonceoneknowstheacceleration,arenotconsidered.
Box10.1Newton’slawsinNewtonianreferenceframes . . 532
10.1 Dynamicsofaparticleinspace . . . . . . . . . . . . . . . . 533
Box10.2Thederivativeofavectordependsonframe . . . 540
10.2 Momentumandenergy . . . . . . . . . . . . . . . . . . . . 549
Box10.3Conservativeforcesandnon-conservativeforces 555
Box10.4Particletheoremsformomentaandenergy . . . . 557
10.3 Central-forcemotionandcelestialmechanics . . . . . . . . . 561
ProblemsforChapter10. . . . . . . . . . . . . . . . . . . . . . . 571
11 Manyparticlesinspace 578
Thismoreadvancedchapterconcernsthemotionoftwoormoreparticles
inspace. WewilluseF* m*aforeachparticle. WewilluseCartesian
D
coordinates only. The start is the set up of “two-body” type problems
whichareeasilygeneralizedto3ormoreparticles. Thefirstsectioncon-
cerns smooth motions due to forces from gravity, springs, smoothly ap-
pliedforcesandfriction. Thesecondsectionconcernsthesuddenchange
invelocitieswhenimpulsiveforcesareapplied.
11.1 Coupledparticlemotion . . . . . . . . . . . . . . . . . . . . 580
11.2 particlecollisions . . . . . . . . . . . . . . . . . . . . . . . 588
Box11.1Effectivemass . . . . . . . . . . . . . . . . . . . 590
Box11.2Energeticsofcollisions . . . . . . . . . . . . . . 592
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1992-2009.