Statics and Dynamics

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Introduction to
S
TATICS
and
D
YNAMICS
Filename:Saskyalaunch3517
Andy Ruina and Rudra Pratap
OxfordUniversityPress(Preprint)
MostrecentmodificationsonJanuary20,2015.
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
iv Chapter0.
(cid:13)c Rudra Pratap and Andy Ruina, 1994-2014. 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,Hye
Yeon Choe, Jason Cortell, Gabor Domokos, Max Donelan, Thu Dong, Gail Fish, Mike Fox, John Gibson, Robert Ghrist,
Vivek Gupta, Saptarsi Haldar, Dave Heimstra, Theresa Howley, Herbert Hui, Dirk Martin (Mark) Luchtenburg, Michael
Marder,ElainaMcCartney,SaskyavanNouhuys,HorstNowacki,JimPapadopoulos,KalpanaPratap,DaneQuinn,Richard
Rand, C.V. Radakrishnan, Nidhi Rathi, Phoebus Rosakis, Les Schaffer, Ishan Sharma, David Shipman, Jill Startzell, Brett
Tallman,TianTang,KimTurnerandBillZobrist.Ouron-againoff-againeditorPeterGordonhasbeensupportivethroughout.
Manyotherfriends,colleagues,relatives,students,andanonymousreviewershavealsomadehelpfulsuggestions.
WecertifyArthurOgawa,IvanDobrianov,andStephenHicksasTeXgeniuses.
MikeColemanworkedextensivelyonthetext,wrotemanyoftheexamplesandhomeworkproblemsandmademanyfigures.
DavidHo,R.Manjula,AbhayandMiekeRuinadreworimprovedmostofthedrawings. Creditforsomeofthehomework
problems retrieved from Cornell archives is due to various Theoretical and Applied Mechanics faculty. Harry Soodak and
MartinTierstenprovidedsomeproblemsfromtheirincompletebook.
SoftwarewehaveusedtopreparethisbookincludesTEXshop(forLATEX)withmanycustomfeaturesimplementedbyStephen
Hicks,AdobeIllustrator,GraphicsConverterandMATLAB.
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2014.
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2014. 1
2 Chapter0.
Brief Contents
Fronttables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
BriefContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
DetailedContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Tothestudent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Part I: Basics for Mechanics
26
1 Whatismechanics? . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Vectors: position,forceandmoment . . . . . . . . . . . . . . . . 42
3 FBDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Part II: Statics
190
4 Staticsofoneobject. . . . . . . . . . . . . . . . . . . . . . . . . 190
5 Trussesandframes . . . . . . . . . . . . . . . . . . . . . . . . . 264
6 Transmissionsandmechanisms. . . . . . . . . . . . . . . . . . . 330
7 Tension,shearandbendingmoment . . . . . . . . . . . . . . . . 384
8 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
Part III: Dynamics
426
9 Dynamicsin1D . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
10 Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
11 Particlesinspace . . . . . . . . . . . . . . . . . . . . . . . . . . 552
12 Manyparticlesinspace . . . . . . . . . . . . . . . . . . . . . . . 600
13 Straightlinemotion . . . . . . . . . . . . . . . . . . . . . . . . . 626
14 Circularmotionofaparticle . . . . . . . . . . . . . . . . . . . . 666
15 Circularmotionofarigidobject . . . . . . . . . . . . . . . . . . 698
16 Planarmotionofanobject . . . . . . . . . . . . . . . . . . . . . 778
17 Time-varyingbasisvectors . . . . . . . . . . . . . . . . . . . . . 864
18 Constrainedparticlesandrigidobjects . . . . . . . . . . . . . . . 934
Appendices
1004
A Unitsanddimensions . . . . . . . . . . . . . . . . . . . . . . . .1004
B Friction: perspectivesonfrictionlaws . . . . . . . . . . . . . . .1016
C ThesimplestODEsandtheirsolutions . . . . . . . . . . . . . . .1026
D TheoremsforSystems . . . . . . . . . . . . . . . . . . . . . . .1030
Answerstosomehomeworkproblems . . . . . . . . . . . . . . . . .1040
Backtables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1049
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2014.
Detailed Contents
Fronttables i
Summaryofmechanics . . . . . . . . . . . . . . . . . . . i
Somebasicdefinitions . . . . . . . . . . . . . . . . . . . . ii
BriefContents 2
DetailedContents 3
Preface 12
General issues about content, level, organization, style and motivation.
Studyadvicestartsonpage ??.
Tothestudent 16
Howtostudy. Theuseofcomputers.
0.1 Anoteoncomputation . . . . . . . . . . . . . . . . . . . . . 21
Box: Informalcomputercommands . . . . . . . . . . . . . 24
Part I: Basics for Mechanics
26
1 Whatismechanics? 26
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 . . . . . . . . . . . . . . . . . . . . . . . . 27
1.2 Mechanicsiswrong,whystudyit? . . . . . . . . . . . . . . 33
1.3 Thehierarchyofmodels . . . . . . . . . . . . . . . . . . . . 35
2 Vectors: position,forceandmoment 42
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
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2014. 3
4 Chapter0.DetailedContents DetailedContents
theformulawhichcorrectlycalculatestheheuristicallymotivatedquan-
titiesofmomentandmomentaboutanaxis.
2.1 Notationandaddition . . . . . . . . . . . . . . . . . . . . . 44
Box2.1Thescalarsinmechanics . . . . . . . . . . . . . . 45
Box2.2TheVectorsinMechanics . . . . . . . . . . . . . 46
2.2 Thedotproductoftwovectors . . . . . . . . . . . . . . . . 62
Box2.3Basicfeaturesofthevectordotproduct. . . . . . . 62
Box2.4abcos(cid:18) a b a b a b . . . . . . . 67
x x y y z z
) C C
2.3 Vectorcrossproduct . . . . . . . . . . . . . . . . . . . . . . 71
Box2.5Usesofthecrossproduct . . . . . . . . . . . . . . 72
Box2.6Crossproductasamatrixmultiply . . . . . . . . . 78
Box2.7Thecrossproduct: fromgeometrytocomponents . 79
2.4 Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.5 Solvingvectorequations . . . . . . . . . . . . . . . . . . . . 97
Box2.8Therulesofvectoralgebra. . . . . . . . . . . . . 98
Box2.9Vectortrianglesandthelawsofsinesandcosines . 100
Box2.10Existence,uniqueness,andgeometry . . . . . . . 112
ProblemsforChapter2 . . . . . . . . . . . . . . . . . . . . . . . 117
3 FBDs 126
Afree-bodydiagramisasketchofthesystemtowhichyouwillapplythe
lawsofmechanics. Thediagramshowsallofthenon-negligibleexternal
forcesandcoupleswhichactonthesystem. Thediagramtellswhatma-
terial is in the system and also what is known, and what is not known,
about the forces. Mechanics reasoning depends on free-body diagrams
so we give tips about how to avoid common mistakes. On a free-body
diagram systems of forces are often replaced with ‘equivalent’ forces, a
specialcaseofwhichisaweightforceatthecenterofgravity.
3.1 Equivalentforcesystems . . . . . . . . . . . . . . . . . . . . 129
Box3.1 meansadd . . . . . . . . . . . . . . . . . . . . 131
Box3.2Equivalentatonepoint equivalentatallpoints 132
P )
Box3.3A“wrench”canrepresentanyforcesystem . . . . 133
3.2 Centerofmassandgravity. . . . . . . . . . . . . . . . . . . 138
Box3.4Like ,thesymbol alsomeansadd . . . . . . . 139
Box3.5Eachsubsystemislikeaparticle . . . . . . . . . . 143
P R
Box3.6TheCOMofatriangleisath=3 . . . . . . . . . . 147
3.3 Interactions,forces&partialFBDs . . . . . . . . . . . . . . 153
VectornotationforFBDs . . . . . . . . . . . . . . . . . . 156
Box3.7Free-bodydiagramfirst,mechanicsreasoningafter 163
Box3.8ActionandreactiononpartialFBD’s . . . . . . . 165
3.4 Contact: Sliding,friction,androlling . . . . . . . . . . . . . 168
ProblemsforChapter3 . . . . . . . . . . . . . . . . . . . . . . . 182
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2014.
Chapter0.DetailedContents DetailedContents 5
Part II: Statics
190
4 Staticsofoneobject 190
One object is in equilibrium if the forces and moments balance. For a
particle,forcebalancetellsall. Butforanextendedobject,momentbal-
ance is also essential. There are special shortcuts for an objects that
hasexactlytwoorexactlythreeforcesactingonit. Iffrictionforcesare
relevant the possibility of motion needs to be taken into account. Many
real-world problems are not statically determinate and thus yield either
only partial solutions, or yield full solutions after you have made extra
assumptions.
4.1 Staticequilibriumofaparticle . . . . . . . . . . . . . . . . . 192
Box4.1Existenceanduniqueness . . . . . . . . . . . . . 196
Box4.2Thesimplificationofdynamicstostatics . . . . . . 199
4.2 Equilibriumofoneobject . . . . . . . . . . . . . . . . . . . 206
Box4.3Two-forcebodies . . . . . . . . . . . . . . . . . . 211
Box4.4Three-forcebodies . . . . . . . . . . . . . . . . . 212
Box4.5Momentbalanceabout3pointsissufficientin2D . 213
Box4.6Howtoholdsomethinginplacestaticallydetermi-
nately? . . . . . . . . . . . . . . . . . . . . . . . . . 214
4.3 Equilibriumwithfrictionalcontact . . . . . . . . . . . . . . 219
Box4.7Undrivenwheelsandtwo-forcebodies . . . . . . 223
4.4 Internalforces . . . . . . . . . . . . . . . . . . . . . . . . . 233
4.5 3Dstatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Box4.8Staticallydeterminatewaystoholdanobjectin3D 245
ProblemsforChapter4 . . . . . . . . . . . . . . . . . . . . . . . 249
5 Trussesandframes 264
Hereweconsidercollectionsofpartsdesignedtoholdsomethinguporin
place. Emphasisisontrusses, assembliesofstraightbarsconnectedby
pinsattheirends. Trussesareanalyzedbydrawingfree-bodydiagrams
of the pins (method of joints) or of bigger parts of the truss (method of
sections). Frameworks, built with other than two-force bodies are also
analyzed by drawing free-body diagrams of parts. Trusses and frames
can be rigid or not and redundant or not, as can be determined by the
equilibriumequations.
5.1 Methodofjoints . . . . . . . . . . . . . . . . . . . . . . . . 266
5.2 Themethodofsections . . . . . . . . . . . . . . . . . . . . 283
5.3 Solvingtrussesonacomputer . . . . . . . . . . . . . . . . . 290
5.4 Framesandstructures . . . . . . . . . . . . . . . . . . . . . 301
Box5.1The‘methodofbarsandpins’fortrusses . . . . . 304
5.5 Advancedtrussconcepts: determinacy . . . . . . . . . . . . 311
Box5.2Structuralrigidityandgeometriccongruence . . . 316
Box5.3Rigidity,redundancy,linearalgebraandmaps . . 317
ProblemsforChapter5 . . . . . . . . . . . . . . . . . . . . . . . 322
6 Transmissionsandmechanisms 330
Sometimessolidpartsareassembledtocauseforceortorqueinoneplace
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2014.
6 Chapter0.DetailedContents DetailedContents
whenadifferentforceortorqueisappliedatanotherplace. Suchassem-
blies include levers, gear boxes, presses, pliers, clippers, chain drives,
and crank-drives. Besides solid parts connected by pins, a few special-
purposepartsarecommonlyused, includingsprings, stringsandgears.
Tricks for amplifying force are usually based on principles idealized by
pulleys,levers,wedgesandtoggles. Force-analysisoftransmissionsand
mechanisms is done by drawing free-body diagrams of the parts, writ-
ingequilibriumequationsforthese,andsolvingtheequationsfordesired
unknowns.
6.1 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
Box6.1‘Zero-length’springs . . . . . . . . . . . . . . . . 333
Box6.2Howstiffaspringisasolidrod . . . . . . . . . . 340
Box6.3Stifferbutweaker . . . . . . . . . . . . . . . . . . 340
Box6.4Apuzzlewithtwospringsandthreeropes. . . . . . 341
Box6.52Dgeometryofspringstretch . . . . . . . . . . . 344
6.2 Forceamplification . . . . . . . . . . . . . . . . . . . . . . 353
6.3 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 364
Box6.6Shearswithgears . . . . . . . . . . . . . . . . . . 368
ProblemsforChapter6 . . . . . . . . . . . . . . . . . . . . . . . 375
7 Tension,shearandbendingmoment 384
The‘internalforces’(tension,shearandbendingmoment)canvaryfrom
point to point in long narrow objects. Here we introduce the notion of
graphing this variation and noting the features of these graphs. This
graphingisafavoritechoreofcivilengineers.
7.1 Arbitrarycuts . . . . . . . . . . . . . . . . . . . . . . . . . 385
7.2 Singularityfunctions . . . . . . . . . . . . . . . . . . . . . . 401
ProblemsforChapter7 . . . . . . . . . . . . . . . . . . . . . . . 406
8 Hydrostatics 408
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 somewhere between 1/2 and 2/3 of the way down. The net
force acting on a totally submerged object in a constant density fluid is
thedisplacedfluid’sweight,actingatthecentroid.
8.1 Fluidpressure . . . . . . . . . . . . . . . . . . . . . . . . . 409
Box8.1AddingforcestoderiveArchimedes’principle . . . 412
Box8.2Pressuredependsonpositionbutnotonorientation 413
ProblemsforChapter8 . . . . . . . . . . . . . . . . . . . . . . . 422
Part III: Dynamics
426
9 Dynamicsin1D 426
ThescalarequationF Dmaintroducestheconceptsofmotionandtime
derivatives to mechanics. In particular the equations of dynamics are
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2014.