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Online electronic version
May not be emailed or posted ANYWHERE
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Introduction to
S
TATICS
and
D
YNAMICS
Andy Ruina and Rudra Pratap
OxfordUniversityPress(Preprint)
MostrecentmodificationsonFebruary6,2010.
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
(cid:13)c Rudra Pratap and Andy Ruina, 1994-2009. All rights reserved. No part of this book may be reproduced, stored in a
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,LesSchaeffer,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
22
1 Whatismechanics? . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 FBDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Part II: Statics
180
4 Staticsofoneobject. . . . . . . . . . . . . . . . . . . . . . . . . 180
5 Trussesandframes . . . . . . . . . . . . . . . . . . . . . . . . . 248
6 Transmissionsandmechanisms. . . . . . . . . . . . . . . . . . . 310
7 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
8 Tension,shearandbendingmoment . . . . . . . . . . . . . . . . 380
Part III: Dynamics
400
9 Dynamicsin1D . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
10 Particlesinspace . . . . . . . . . . . . . . . . . . . . . . . . . . 518
11 Manyparticlesinspace . . . . . . . . . . . . . . . . . . . . . . . 566
12 Straightlinemotion . . . . . . . . . . . . . . . . . . . . . . . . . 592
13 Circularmotion . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
14 Planarmotionofanobject . . . . . . . . . . . . . . . . . . . . . 740
15 Time-varyingbasisvectors . . . . . . . . . . . . . . . . . . . . . 826
16 Constrainedparticlesandrigidobjects . . . . . . . . . . . . . . . 894
Appendices
962
A Units&Centerofmasstheorems . . . . . . . . . . . . . . . . . . 962
Answerstosomehomeworkproblems . . . . . . . . . . . . . . . . . 983
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991
Backtables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1992-2009. 1
Detailed Contents
Fronttables i
Summaryofmechanics . . . . . . . . . . . . . . . . . . . i
Somebasicdefinitions . . . . . . . . . . . . . . . . . . . . ii
BriefContents 1
DetailedContents 2
Preface 10
Generalissuesaboutcontent,level,organization,styleandmotivation.
Tothestudent 14
Howtostudy. Theuseofcomputers.
0.1 Anoteoncomputation . . . . . . . . . . . . . . . . . . . . . 18
Box: Informalcomputercommands . . . . . . . . . . . . . 21
Part I: Basics for Mechanics
22
1 Whatismechanics? 22
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 . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2 Mechanicsiswrong,whystudyit? . . . . . . . . . . . . . . 29
1.3 Theheirarchyofmodels . . . . . . . . . . . . . . . . . . . . 31
2 Vectors 36
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
theformulawhichcorrectlycalculatestheheuristicallymotivatedquan-
titiesofmomentandmomentaboutanaxis.
2 IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1992-2009.
Chapter0.DetailedContents DetailedContents 3
2.1 Notationandaddition . . . . . . . . . . . . . . . . . . . . . 38
Box2.1Thescalarsinmechanics . . . . . . . . . . . . . . 39
Box2.2TheVectorsinMechanics . . . . . . . . . . . . . 40
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 Crossproductandmoment . . . . . . . . . . . . . . . . . . 67
Box2.5Crossproductasamatrixmultiply . . . . . . . . . 77
Box2.6Thecrossproduct: fromgeometrytocomponents . 78
2.4 Solvingvectorequations . . . . . . . . . . . . . . . . . . . . 87
Box2.7Vectortrianglesandthelawsofsinesandcosines . 90
Box2.8Existence,uniqueness,andgeometry . . . . . . . 102
2.5 Equivalentforcesystems . . . . . . . . . . . . . . . . . . . . 107
Box2.9 meansadd . . . . . . . . . . . . . . . . . . . . 109
Box2.10Equivalentatonepoint equivalentatallpoints 110
P )
Box2.11A“wrench”canrepresentanyforcesystem . . . 111
2.6 Centerofmassandgravity. . . . . . . . . . . . . . . . . . . 116
Box2.12Like ,thesymbol alsomeansadd . . . . . . 117
Box2.13Eachsubsystemislikeaparticle . . . . . . . . . 121
P R
Box2.14TheCOMofatriangleisath=3 . . . . . . . . . 125
ProblemsforChapter2 . . . . . . . . . . . . . . . . . . . . . . . 131
3 FBDs 142
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 . . . . . . . . . . . . . . 144
VectornotationforFBDs . . . . . . . . . . . . . . . . . . 147
Box3.1Freebodydiagramfirst,mechanicsreasoningafter 154
Box3.2ActionandreactiononpartialFBD’s . . . . . . . 156
3.2 Contact: Sliding,friction,androlling . . . . . . . . . . . . . 163
Box3.3Aproblemwiththeconceptofstaticfriction . . . . 167
Box3.4AcritiqueofCoulombfriction . . . . . . . . . . . 172
ProblemsforChapter3 . . . . . . . . . . . . . . . . . . . . . . . 176
Part II: Statics
180
4 Staticsofoneobject 180
Equilibrium of one object is defined by the balance of forces and mo-
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1992-2009.
4 Chapter0.DetailedContents DetailedContents
ments. Force balance tells all fora particle. Foran extended body mo-
ment balance is also used. There are special shortcuts for bodies with
exactly two or exactly three forces acting. If friction forces are relevant
thepossibilityofmotionneedstobetakenintoaccount. Manyreal-world
problemsarenotstaticallydeterminateandthusonlyyieldpartialsolu-
tions,orfullsolutionswithextraassumptions.
4.1 Staticequilibriumofaparticle . . . . . . . . . . . . . . . . . 182
Box4.1Existenceanduniqueness . . . . . . . . . . . . . 186
Box4.2Thesimplificationofdynamicstostatics . . . . . . 188
4.2 Equilibriumofoneobject . . . . . . . . . . . . . . . . . . . 194
Box4.3Two-forcebodies . . . . . . . . . . . . . . . . . . 199
Box4.4Three-forcebodies . . . . . . . . . . . . . . . . . 200
Box4.5Momentbalanceabout3pointsissufficientin2D . 201
4.3 Equilibriumwithfrictionalcontact . . . . . . . . . . . . . . 206
Box4.6Wheelsandtwoforcebodies . . . . . . . . . . . . 210
4.4 Internalforces . . . . . . . . . . . . . . . . . . . . . . . . . 220
4.5 3Dstaticsofonepart . . . . . . . . . . . . . . . . . . . . . 226
ProblemsforChapter4 . . . . . . . . . . . . . . . . . . . . . . . 234
5 Trussesandframes 248
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 . . . . . . . . . . . . . . . . . . . . . . . . 250
5.2 Themethodofsections . . . . . . . . . . . . . . . . . . . . 262
5.3 Solvingtrussesonacomputer . . . . . . . . . . . . . . . . . 269
5.4 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Box5.1The‘methodofbarsandpins’fortrusses . . . . . 282
5.5 3Dtrussesandadvancedtrussconcepts . . . . . . . . . . . . 289
Box5.2Stucturalrigidityandgeometriccongruence . . . 295
Box5.3Rigidity,redundancy,linearalgebraandmaps . . 296
ProblemsforChapter5 . . . . . . . . . . . . . . . . . . . . . . . 303
6 Transmissionsandmechanisms 310
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
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.
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1992-2009.
Chapter0.DetailedContents DetailedContents 5
6.1 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
Box6.1‘Zero-length’springs . . . . . . . . . . . . . . . . 313
Box6.2Apuzzlewithtwospringsandthreeropes. . . . . . 320
Box6.3Howstiffaspringisasolidrod . . . . . . . . . . 321
Box6.4Stifferbutweaker . . . . . . . . . . . . . . . . . . 321
Box6.52Dgeometryofspringstretch . . . . . . . . . . . 324
6.2 Forceamplification . . . . . . . . . . . . . . . . . . . . . . 333
6.3 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 343
Box6.6Shearswithgears . . . . . . . . . . . . . . . . . . 347
ProblemsforChapter6 . . . . . . . . . . . . . . . . . . . . . . . 354
7 Hydrostatics 364
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.
7.1 Fluidpressure . . . . . . . . . . . . . . . . . . . . . . . . . 365
Box7.1AddingforcestoderiveArchimedes’principle . . . 368
Box7.2Pressuredependsonpositionbutnotonorientation 369
ProblemsforChapter7 . . . . . . . . . . . . . . . . . . . . . . . 377
8 Tension,shearandbendingmoment 380
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.
8.1 Arbitrarycuts . . . . . . . . . . . . . . . . . . . . . . . . . 381
ProblemsforChapter8 . . . . . . . . . . . . . . . . . . . . . . . 397
Part III: Dynamics
400
9 Dynamicsin1D 400
ThescalarequationF Dmaintroducestheconceptsofmotionandtime
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.
9.1 Forceandmotionin1D . . . . . . . . . . . . . . . . . . . . 402
Box9.1WhatdothetermsinF mamean? . . . . . . . 407
D
Box9.2SolutionsofthesimplestODEs . . . . . . . . . . . 412
Box9.3D’Alembert’smechanics: beginnersbeware . . . . 415
9.2 Energymethodsin1D . . . . . . . . . . . . . . . . . . . . . 422
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1992-2009.
6 Chapter0.DetailedContents DetailedContents
Box9.4Particlemodelsfortheenergeticsoflocomotion. . 433
9.3 Vibrations: mass,springanddashpot . . . . . . . . . . . . . 440
Box9.5Acos.(cid:21)t/ Bsin.(cid:21)t/ Rcos.(cid:21)t (cid:30)/ . . . . . 444
C D (cid:0)
Box9.6Solutionofthedamped-oscillatorequations . . . . 450
9.4 Coupledmotionsin1D . . . . . . . . . . . . . . . . . . . . 465
Box9.7Normalmodes: themathandtherecipe . . . . . . 472
9.5 Collisionsin1D . . . . . . . . . . . . . . . . . . . . . . . . 480
Box9.8Whenequalrodscollidethevibrationsdisappear . 485
9.6 Advanced: forcing&resonance . . . . . . . . . . . . . . . . 489
Box9.9ALoudspeakerconeisaforcedoscillator. . . . . . 494
Box9.10Solutionoftheforcedoscillatorequation . . . . . 496
Box9.11Thevocabularyofforcedoscillations . . . . . . . 497
ProblemsforChapter9 . . . . . . . . . . . . . . . . . . . . . . . 505
10 Particlesinspace 518
This chapter is about the vector equation F* D m*a for one particle.
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 . . 520
10.1 Dynamicsofaparticleinspace . . . . . . . . . . . . . . . . 521
Box10.2Thederivativeofavectordependsonframe . . . 528
10.2 Momentumandenergy . . . . . . . . . . . . . . . . . . . . 537
Box10.3Conservativeforcesandnon-conservativeforces 543
Box10.4Particletheoremsformomentaandenergy . . . . 545
10.3 Central-forcemotionandcelestialmechanics . . . . . . . . . 549
ProblemsforChapter10. . . . . . . . . . . . . . . . . . . . . . . 559
11 Manyparticlesinspace 566
Thismoreadvancedchapterconcernsthemotionoftwoormoreparticles
inspace. WewilluseF*D m*aforeachparticle. WewilluseCartesian
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 . . . . . . . . . . . . . . . . . . . . 568
11.2 particlecollisions . . . . . . . . . . . . . . . . . . . . . . . 576
Box11.1Effectivemass . . . . . . . . . . . . . . . . . . . 578
Box11.2Energeticsofcollisions . . . . . . . . . . . . . . 580
Box11.3Coefficientofgeneration . . . . . . . . . . . . . 583
Box11.4Aparticlecollisionmodelofrunning . . . . . . . 584
ProblemsforChapter11. . . . . . . . . . . . . . . . . . . . . . . 589
12 Straightlinemotion 592
Here is an introduction to kinematic constraint in its simplest context,
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1992-2009.