Conceptual evolution of Newtonian and relativistic mechanics

134 Pages·2018·1.555 MB·English

by  Ghosh, Amitabha

Undergraduate Lecture Notes in Physics
Amitabha Ghosh
Conceptual Evolution
of Newtonian and
Relativistic Mechanics
Undergraduate Lecture Notes in Physics
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Amitabha Ghosh
Conceptual Evolution
of Newtonian and Relativistic
Mechanics
123
AmitabhaGhosh
Department ofAerospace Engineering
andAppliedMechanics
IIESTShibpur
Howrah, West Bengal
India
ISSN 2192-4791 ISSN 2192-4805 (electronic)
Undergraduate Lecture Notesin Physics
ISBN978-981-10-6252-0 ISBN978-981-10-6253-7 (eBook)
https://doi.org/10.1007/978-981-10-6253-7
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To
dear
Meena
Preface
ThisbookistheoutcomeofascienceelectivecourseofferedatIndianInstituteofTechnology
Kanpurby me along with anothercolleague ofmine. The course usedto cover bothclassical
and quantum mechanics, but in this volume, only the classical part is being covered and the
otherpartisexpectedtobepenneddownbymycolleague.However,themotivationtodesign
and offer a course on ‘Conceptual Evolution of Mechanics’ germinated from an interesting
episodeintheearlylifeoftheauthorofthisvolume.Itmaynotbeoutofplacetogiveabrief
account of that here.
Iwasveryfondofeggcurryfrommychildhood,butmymothernevergavememorethan
one eggat atime fearing it could cause stomach problems forme.As a young boy,I usedto
thinkthatwhenIgrowupandbecomeindependent,IwouldtakeasmanyeggsatatimeasI
pleased. Long after those childhood days, suddenly an opportunity came. I had a combined
handwhousedtodoallcookingandotherhouseholdchoresformeduringthelastyearsofthe
1960s when I was a young faculty at Bengal Engineering College, Shibpur, Howrah. In
December1970,whenIdecidedtochangeovertoIITKanpur,Ihadsentmycombinedhand
tomynativehome260kmawaytotakehouseholdbelongingsthere.Mywifewasalsothere
asmysonwastooyoungtoundergotheproblemsoftransferringresidencetoafarawaycity.
Iwasalone,andsuddenlyitcomestomymindtheolddesireofconsumingasmanyeggsasI
wanted. I had never learned cooking but prepared an egg curry with three eggs following
whatever steps came to my mind. To my utter surprise, I could not eat the curry as it tasted
horribly awesome. I realized that though I ate and digested (and enjoyed too) egg curry so
much during the previous 25 years of my life, I did not know how to cook egg curry. This
event gave me a realization that we teach our students only cooked science. As a result, it
becomesdifficultforthemtocreatenewscience.Fromthattime,Iplannedtodesignacourse
inwhichthestudentsofmechanicscanbecomefamiliarwiththeevolutionaryprocessthrough
which the science of motion developed and achieved maturity. Mechanics being a very basic
subject and fundamental to many branches of physical science and engineering, I considered
this subject to be the most suited for my experiment.
Itshouldbenotedthatthe‘conceptualevolution’issomewhatdifferentfromthe‘history’.
There are excellent books on history of mechanics. It is also not a textbook on mechanics.
I have tried to emphasize the process through which the basic concepts evolved, transformed
andledtotheconsolidationofthescientificprinciplesinvolved.ThecourseatIITKanpurwas
offered with a hope to give the students some taste of the process through which science is
created. It was hoped that the effort would be somewhat useful in enabling the students to
create new science when the occasion arises. As a secondary outcome, the course helped to
remove many incorrect impressions about some major scientific discoveries in the field of
mechanics.
Quite naturally, the major emphasis has been given on the development of Newtonian
mechanics as that isconsidered as one of the starting points of modern science. The chapters
on relativistic mechanics are much shorter as the evolutionary processes for the two theories
were confined to a much smaller extent of ‘space–time’, to use a relativity terminology. The
vii
viii Preface
periodoftheirdevelopmentwasonlyacoupleofdecadesandinvolvedamuchsmallergroup,
Einstein occupying the predominant place.
While leaving IIT Kanpur, I was requested by many of my colleagues to compile a book
using the material used by me in the part of the course I developed for the use by younger
faculty members desirous of offering similar courses. Realizing the desirability of their sug-
gestion, I planned this book. It is only the students of the subject can decide if I have been
successful(atleastpartially)inmyoriginalendeavour.Ifthisbookisusedforofferingsimilar
courses, it will be my greatest satisfaction. It goes without saying that there are many short
coming and mistakes in this book and I will remain perpetually grateful for suggestions and
corrections.
It has taken a long time in writing this book, and I gratefully acknowledge the help and
suggestions I received from my students and faculty colleagues of IIT Kanpur. Professors
Ashok Kumar Mallik, Pinaki Guptabhaya, Raminder Singh, H.S. Mani and Late Himanshu
Hatwalarethe mostprominent among them.I also gratefullyacknowledge thekind help and
encouragement received from Professor E.C.G. Sudarshan of University of Texas at Austin.
IalsoacknowledgethehelpreceivedfromthePhysicsDepartmentofUTAustinbygivingme
free access to the library. It would not have been possible to complete the book without the
activesupportfrommywifeMeenawhotookthewholeburdenofrunningthehouseholdwith
negative help from my side. I also gratefully acknowledge the help from my sister-in-law
SabitaGhoshofAsansolforprovidingmethenecessaryrefugeatherhomethatgavemefree
and undisturbed time required to finish the work. The financial support from Indian National
Science Academy, New Delhi, and National Academy of Sciences, Allahabad, India, during
the period when the book was being written is thankfully acknowledged. The enthusiasm
shownbyMs.SwatiMehershiofSpringerinpublishingthemanuscriptandthecarefultyping
of the handwritten manuscript by Mr. Sourav Kundu also deserve my sincere thanks.
The preparation of the manuscript had many interruptions, and it has taken almost nine
years to complete the book. I will remain grateful to the readers for their suggestions for
further improvement of the book.
Shibpur, Howrah, India Amitabha Ghosh
April 2017 1 Boisakh, 1424 (Bengali New Year)
Contents
1 Evolution of Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Early Concepts and Aristotelian Physics. . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The Earth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Earth–Moon Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Measuring the Sun’s Distance. . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.4 Size of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Role of Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Difficulties in Discovering the Laws of Motion. . . . . . . . . . . . . . . . . . . . 5
1.4 Pre-copernican Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 Hipparchus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.2 The Epicycle–Deferent Model and Ptolemy . . . . . . . . . . . . . . . . . 6
1.4.3 Problems with Explaining the Observations
with Ptolemaic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.4 Progress During the Period Between Ptolemy and Copernicus. . . . . 11
1.5 Copernican Model: Rediscovery of the Heliocentric Theory . . . . . . . . . . . 12
1.6 Tycho Brahe: Improvement in Accuracy for Naked-Eye Astronomy. . . . . . 14
1.7 Kepler: Beginning of Modern Astronomy and Foundation
of Science of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7.1 Discovery of the Laws of Planetary Motion . . . . . . . . . . . . . . . . . 15
1.7.2 Transition from Geometric to Physical Model. . . . . . . . . . . . . . . . 19
1.7.3 Early Concept of Action-at-a-Distance and Gravitation,
and the Concept of Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.8 Galileo: Naked Eye to Telescopic Astronomy . . . . . . . . . . . . . . . . . . . . . 20
1.8.1 Observation of the Moon and Discarding the Concept
of Fifth Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.8.2 Discovery of Jupiter’s Moons and its Implications. . . . . . . . . . . . . 21
1.8.3 Discovery of the Phases of Venus: A Further Proof
of Heliocentric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.9 Galileo: Experimental Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.9.1 Early Works on Accelerated Change: Merton School. . . . . . . . . . . 23
1.9.2 Galileo’s Work on Free Fall and Uniformly
Accelerated Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.9.3 Discovery of the Law of Inertia of Motion in Its
Primitive Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.9.4 Laws of Compound Motion: Projectiles. . . . . . . . . . . . . . . . . . . . 25
1.9.5 Galilean Relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.10 Collapse of the Old Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
ix
x Contents
1.11 Descartes: Beginning of Inertial Science. . . . . . . . . . . . . . . . . . . . . . . . . 28
1.11.1 Law of Inertia of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.11.2 Collision Problems and Early Concept of Momentum
Conservation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.11.3 Descartes’ Concept of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.12 Huygens: Breakthrough in the Discovery of Dynamics. . . . . . . . . . . . . . . 30
1.12.1 Theory of Collision and Conservation of Momentum. . . . . . . . . . . 30
1.12.2 Kinematics of Circular Motion and ‘Centrifugal Force’ . . . . . . . . . 32
1.12.3 Modern Concept of Force and the ‘Force–Acceleration’
Relation: Second Law of Motion in Primitive Form. . . . . . . . . . . . 34
1.12.4 Early Concept of the Principle of Equivalence . . . . . . . . . . . . . . . 35
1.13 Halley, Wren and Hooke: Rudiments of Gravitation. . . . . . . . . . . . . . . . . 36
1.14 Newton and the Final Synthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.14.1 Concepts of Mass, Momentum, Force and the Second Law
of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.14.2 Collision Problem and the Discovery of the Third Law . . . . . . . . . 38
1.14.3 Law of Universal Gravitation and Planetary Motion. . . . . . . . . . . . 39
1.14.4 Universality of Gravitational Force . . . . . . . . . . . . . . . . . . . . . . . 41
1.14.5 Orbit for Inverse Square Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.15 Newtonian Dynamics in Matured State. . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.15.1 Concept of Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.15.2 Principia and Subsequent Development . . . . . . . . . . . . . . . . . . . . 45
2 Some Basic Concepts in Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . 47
2.1 Nature of Motion and Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.1.1 Newton’s Concept of Absolute Space and Time . . . . . . . . . . . . . . 47
2.1.2 Newton’s Bucket Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.1.3 Newton’s Bucket Experiment Follow up . . . . . . . . . . . . . . . . . . . 49
2.1.4 Ernst Mach and Mach’s Principle . . . . . . . . . . . . . . . . . . . . . . . . 50
2.1.5 Quantification of Mach’s Principle—Concept of Inertial
Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.1.6 Origin of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2 Relative–Absolute Duality of Nature of Motion. . . . . . . . . . . . . . . . . . . . 53
2.2.1 The Nature of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2.2 Absolute Motion in Terms of Relative Motion . . . . . . . . . . . . . . . 53
2.3 Inertial and Gravitational Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.1 Inertial Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3.2 Gravitational Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3.3 Equivalence of Inertial and Gravitational Mass . . . . . . . . . . . . . . . 57
2.4 Space–Time and Symmetry in Newtonian Mechanics. . . . . . . . . . . . . . . . 58
2.5 Early Concept of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.6 The Principle of Relativity and Galilean Transformation. . . . . . . . . . . . . . 60
2.6.1 The Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6.2 Symmetry and Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.6.3 Form Invariance of Physical Laws. . . . . . . . . . . . . . . . . . . . . . . . 63
2.6.4 Energy and Energy Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.6.5 Energy Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.7 Laws of Motion and the Properties of Space and Time. . . . . . . . . . . . . . . 66
2.7.1 The Second Law of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.7.2 The Third Law of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68