Table Of ContentOUPCORRECTEDPROOF – FINAL,8/7/2020,SPi
EXPLORING CLASSICAL MECHANICS
OUPCORRECTEDPROOF – FINAL,8/7/2020,SPi
Exploring Classical Mechanics
A Collection of 350+ Solved Problems for Students,
Lecturers, and Researchers
Gleb L. Kotkin
Valeriy G. Serbo
Novosibirsk State University,Russia
Second revised and enlarged English edition
1
OUPCORRECTEDPROOF – FINAL,8/7/2020,SPi
3
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ISBN978-0-19-885378-7(hbk.)
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DOI:10.1093/oso/9780198853787.001.0001
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OUPCORRECTEDPROOF – FINAL,8/7/2020,SPi
Preface to the second English edition
This book was written by the working physicists for students of physics faculties of
universities.
ThefirstEnglisheditionofthisbookunderthetitleCollectionofProblemsinClassical
Mechanics was published by Pergamon Press in 1971 with the invaluable help by the
translation editor D. ter Haar. This second English publication is based on the fourth
Russianeditionof2010andincludesnewproblemsfromamongthoseusedinteaching
at the physics faculty of Novosibirsk State University as well as the problems added in
the publications in Spanish and French. As a result, this book contains 357 problems
insteadofthe289problemsthatappearedinthefirstEnglishedition.
We are grateful to A.V.Mikhailov for useful discussions of some new problems,
to Z.K.Silagadze for numerous indications of misprints and inaccuracies in previous
editions,andtoO.V. Karpushinaforaninvaluablehelpinpreparationofthismanuscript.
Inthisedition,themainnotationsare:
m, e, r, p, and M=[r,p] – mass, charge, radius vector, momentum, and angular
momentumofaparticle,respectively;
L,H,E,and U – Lagrangian function,Hamiltonian function,energy,and potential
energyofasystem,respectively;
EandB–electricandmagneticfieldintensities,respectively;
ϕ andA–scalarandvectorpotentials,respectively,oftheelectromagneticfield;
c–velocityoflight;and
d(cid:3)–solidangleelement.
Forproblemsaboutthemotionofparticlesinelectromagneticfields,weuseGaussian
units,andinproblemsonelectricalcircuits,SIunits.
OUPCORRECTEDPROOF – FINAL,8/7/2020,SPi
From the Preface to the first English edition
This collection is meant for physics students. Its contents correspond roughly to the
mechanicscourseinthetextbooksbyLandauandLifshitz[1],Goldstein[4],orterHaar
[6]. We hope that the reading of this collection will give pleasure not only to students
studying mechanics, but also to people who already know it. We follow the order in
which the material is presented by Landau and Lifshitz,except that we start using the
Lagrangianequationsin§4.Theproblemsin§§1–3canbesolvedusingtheNewtonian
equationsofmotiontogetherwiththeenergy,linearmomentumandangularmomentum
conservationlaws.
As a rule, the solution of a problem is not finished with obtaining the required
formulae.Itisnecessarytoanalysetheresult,andthisisbynomeansthe“mechanical”
part of the solution.It is also very useful to investigate what happens if the conditions
oftheproblemarevaried.Wehave,therefore,suggestedfurtherproblemsattheendof
severalsolutions.
A large portion of the problems were chosen for the practical classes with students
fromthephysicsfacultyoftheNovosibirskStateUniversityforacourseontheoretical
mechanics given by Yu.I. Kulakov. We want especially to emphasize his role in the
choiceandcriticaldiscussionofalargenumberofproblems.WeoweagreatdebttoI.F.
Ginzburg for useful advice and hints which we took into account.We are very grateful
toV.D.Krivchenkovwhoseactiveinteresthelpedustopersevereuntiltheend.
WeareextremelygratefultoD.terHaarforhishelpinorganizinganEnglishedition
ofourbook.
OUPCORRECTEDPROOF – FINAL,8/7/2020,SPi
§1
Integration of one-dimensional
equations of motion
1.1. DescribethemotionofaparticleinthefollowingpotentialfieldsU(x):
a) U(x)=A(e−2αx−2e−αx)(Morsepotential,Fig.1a);
b) U(x)=− U0 (Fig.1b);
cosh2αx
c) U(x)=U tan2αx(Fig.1c).
0
(a) (b) (c) U
U
U
x x x
Figure1
1.2. Describe the motion of a particle in the field U
U(x)=−Ax4 forthecasewhenitsenergyisequaltozero.
E
1.3. Give an approximate description of the motion of a
particleinthefieldU(x)nearthetuningpointx=a(Fig.2). a x
Hint:UseaTaylorexpansionofU(x)nearthepointx=a.
ConsiderthecasesU(cid:2)(a)(cid:3)=0andU(cid:2)(a)=0,U(cid:2)(cid:2)(a)(cid:3)=0. Figure2
1.4. DeterminehowtheperiodofaparticlemovinginthefieldinFig.3tendstoinfinity
asitsenergyE approachesU .
m
ExploringClassicalMechanics:ACollectionof350+SolvedProblemsforStudents,Lecturers,andResearchers.FirstEdition.
GlebL.KotkinandValeriyG.Serbo,OxfordUniversityPress(2020).©GlebL.KotkinandValeriyG.Serbo2020.
DOI:10.1093/oso/9780198853787.001.0001
OUPCORRECTEDPROOF – FINAL,8/7/2020,SPi
4 ExploringClassicalMechanics [1.5
U
U
U E
m
U
E m
a x x1 b a c x2 x
Figure3 Figure4
1.5. a) Estimate the period of the particle motion in the field U(x) (Fig. 4), when its
energyisclosetoU (i.e.,E−U (cid:4)U −U ).
m m m min
b)Determineduringwhichpartoftheperiodtheparticleisintheintervalfromxto
x+dx.
c)Determineduringwhichpartoftheperiodtheparticlehasamomentummx˙inthe
intervalfromptop+dp.
d) In the plane x, p=mx˙ represent qualitatively lines E(x,p)=const for the cases
E<U ,E=U ,E>U .
m m m
1.6. Aparticleofmassmmovesalongacircleofradiusl inaverticalplaneunderthe
influence of the field of gravity (mathematical pendulum).Describe its motion for the
casewhenitskineticenergyE inthelowestpointisequalto2mgl.
Estimate the period of revolution of the pendulum for the case when
E−2mgl(cid:4)2mgl.
1.7. Describe the motion of a mathematical pendulum for an arbitrary value
oftheenergy.
Hint:Thetimedependenceoftheanglethependulummakeswiththeverticalcanbe
expressedintermsofellipticfunctions(e.g.see[1],§37).
1.8. Determinethechangeinthemotionofaparticlemovingalongasectionwhichdoes
notcontainturningpointswhenthefieldU(x)ischangedbyasmallamountδU(x).
Consider the applicability of the results obtained for the case of a section near the
turningpoint.
1.9. FindthechangeinthemotionofaparticlecausedbyasmallchangeδU(x)inthe
fieldU(x)inthefollowingcases:
a) U(x)= 1mω2x2, δU(x)= 1mαx3;
2 3
b) U(x)= 1mω2x2, δU(x)= 1mβx4.
2 4
1.10. Determine the change in the period of a finite orbit of a particle caused by the
changeinthefieldU(x)byasmallamountδU(x).
OUPCORRECTEDPROOF – FINAL,8/7/2020,SPi
1.12] §1. Integrationofone-dimensionalequationsofmotion 5
1.11. FindthechangeintheperiodofaparticlemovinginafieldU(x)causedbyadding
tothefieldU(x)asmalltermδU(x)inthefollowingcases:
a) U(x)= 1mω2x2 (aharmonicoscillator), δU(x)= 1mβx4;
2 4
b) U(x)= 1mω2x2, δU(x)= 1mαx3;
2 3
c) U(x)=A(e−2αx−2e−αx), δU(x)=−Veαx (V (cid:4)A).
1.12. TheparticlemovesinthefieldU(x)= U0 withtheenergyE>U .Findthe
cosh2αx 0
particledelaytimeatthemotionfromx=−∞tox=+∞incomparisonwiththefree
motiontimewiththesameenergy.
OUPCORRECTEDPROOF – FINAL,8/7/2020,SPi
§2
Motion of a particle in
three-dimensional fields
2.1. Describe qualitatively the motion of a particle in the field U(r)=−α − γ for
r r3
differentvaluesoftheangularmomentumandoftheenergy.
2.2. Findthetrajectoriesandthelawsofmotionofaparticleinthefield
(cid:2)
−V, whenr<R, U
U(r)=
0, whenr>R R
r
(Fig.5,“spherical rectangular potential well”) for different values −V
oftheangularmomentumandoftheenergy.
2.3. Determine the trajectory of a particle in the field U(r)= Figure5
α + β . Give an expression for the change in the direction of
r r2
velocity when the particle is scattered as a function of angular
momentumandenergy.
2.4. Determine the trajectory of a particle in the field U(r)= α − β .Find the time it
r r2
takestheparticletofalltothecentreofthefieldfromadistancer.Howmanyrevolutions
aroundthecentrewilltheparticlethenmake?
2.5. Determine the trajectory of a particle in the field U(r)=−α + β . Find the
r r2
angle(cid:5)ϕ betweenthedirectionofradiusvectorattwosuccessivepassagesthroughthe
pericentre(i.e.,whenr=r );alsofindtheperiodoftheradialoscillations,T .Under
min r
whatconditionswilltheorbitbeaclosedone?
2.6. Determine the trajectory of a particle in the field U(r)=−α − β .A field of this
r r2
kindarisesinthemotionofarelativisticparticleintheCoulombfieldinthespecialtheory
ofrelativity;see[7],§42.1fordetails.
ExploringClassicalMechanics:ACollectionof350+SolvedProblemsforStudents,Lecturers,andResearchers.FirstEdition.
GlebL.KotkinandValeriyG.Serbo,OxfordUniversityPress(2020).©GlebL.KotkinandValeriyG.Serbo2020.
DOI:10.1093/oso/9780198853787.001.0001
OUPCORRECTEDPROOF – FINAL,8/7/2020,SPi
2.16] §2. Motionofaparticleinthree-dimensionalfields 7
2.7. ForwhatvaluesoftheangularmomentumM isitpossibletohavefiniteorbitsin
thefieldU(r)forthefollowingcases:
a) U(r)=−αe−(cid:7)r; b)U(r)=−Ve−(cid:7)2r2.
r
2.8. AparticlefallsfromafinitedistancestowardsthecentreofthefieldU(r)=−αr−n.
Will it make a finite number of revolutions around the centre? Will it take a finite time
tofalltowardsthecentre?Findtheequationoftheorbitforsmallr.
2.9. AparticleinthefieldU(r)fliesofftoinfinityfromadistancer(cid:2)=0.Isthenumber
ofrevolutionsaroundthecentremadebytheparticlefiniteforthefollowingcases?
a) U(r)=αr−n b)U(r)=−αr−n
2.10. HowlongwillittakeaparticletofallfromadistanceRtothecentreofthefield
U(r)=−α/r.The initial velocity of the particle is zero.Treat the orbit as a degenerate
ellipse.
2.11. Oneparticleofmassmmovesalongthex-axisfromalongdistancewithvelocity
v towards the origin O of the coordinate system. Another particle of the same mass
movestowardstheoriginOalongthey-axisfromalongdistancewiththesamevelocity
magnitude. If the particles didn’t interact, the second would pass through point O in
timeτ afterthefirstone.However,theyrepulsefromeachother,andpotentialenergyof
interaction is U(r)=α/r,where r is the distance between particles.Find the minimum
distancebetweentheparticles.
2.12. Two particles with masses m and m move with velocities v and v from long
1 2 1 2−→
distances along the crossing lines, the distance between which is equal to AB=ρ. If
particlesdidn’tinteract,particle1wouldpassthepointofminimumdistanceAattime
τ earlierthanparticle2wouldpassthepointB.However,thereistheforceofattraction
betweentheparticles,whichisgivenbythepotentialenergyU(r)=−β/r2.
a) Atwhatrelationbetweenρ andτ willparticlescollide?
b) AtwhatdistancefromthepointAwillsuchacollisionoccur?
2.13. Determinetheminimaldistancebetweentheparticles,theoneapproachingfrom
infinitywithanimpactparameterρ andaninitialvelocityvandtheotheroneinitiallyat
rest.Themassesoftheparticlesarem andm ,andtheinteractionlawisU(r)=α/rn.
1 2
2.14. Determine in the centre of a mass system the finite orbits of two particles of
massesm andm ,andaninteractionlawU(r)=−α/r.
1 2
2.15. Determinethepositionofthefocusofabeamofparticlesclosetothebeamaxis,
whentheparticlesarescatteredinacentralfieldU(r)undertheassumptionthataparticle
flyingalongtheaxisisturnedback.
2.16. Findtheinaccessibleregionofspaceforabeamofparticlesflyingalongthez-axis
withavelocityvandbeingscatteredbyafieldU(r)=α/r.
