The collection is a detailed selection of problems on the dynamics of the motion of a material point acted on by a central gravitational force, in particular, the dynamics of space flight. As an exception, the book presents several problems on the motion of a point acted on central non-gravitational forces. The book is written mainly for correspondence students. Topics covered include Kepler’s laws, the integral of areas, Binet’s formulas for central forces, the energy balance and velocity along a space trajectory, time of motion along a space trajectory, conditions for the existence of elliptical trajectories, transfer from orbit to orbit, sphere of action, third escape velocity problems, two-body problem, and the generalized third law of Kepler, along with miscellaneous problems.

This collection is a textbook for the course of theoretical mechanics (“Point Dynamics” section). students in correspondence departments of Leningrad State University and other higher educational institutions used in part by students in the day and evening departments. Moreover, the problem book may prove useful for beginning instructors in providing practical exercises mechanics, particularly, when they prepare modifications of test problems.

Most of the collection is a detailed sampling of problems on the dynamics of a material point acted on by gravitational force, in particular, problems on the elementary dynamics of space flight. Several problems on the motion of a point acted on by central non-gravitational forces are presented. Altogether, the collection includes about 200 problems of varying degrees of difficulties, with solutions.

Translation of *Sbornik zadach po dinamike tochki v pole tsentral’nykh sil,* Leningrad, Leningrad University Press, 1974, pp. 1-145. The book was published by NASA in May 1975 under its technical translation programme (NASA TT F-16,263).

Foreword iv

Table of Contents vi

Chapter One. Central Forces. Force of Gravity and Its Dynamic Characteristics 1

Chapter Two. Kepler’s Laws 11

Chapter Three. Integral of Areas 19

Chapter Four. Binet’s Formulas for Central Forces 26

Chapter Five. Energy Balance and Velocity Along a Space Trajectory 36

Chapter Six. Time of Motion in a Space Trajectory 63

Chapter Seven. Conditions for the Existence of Elliptical Trajectories 77

Chapter Eight. Transfer from Orbit to Orbit 90

Chapter Nine. Sphere of Action. Problems of Third Escape Velocity 109

Chapter Ten. Two-Body Problem. Third Kepler’s Law Generalized 118

Chapter Eleven. Miscellaneous Problems 127

References 163

N. V. Efimov, E. R. Rozendorn

About the book

This book was conceived as a text combining the course of linear algebra and analytic geometry. It originated as a course of lectures delivered by N. V. Efimov at Moscow State University (mechanics and mathematics department) in 1964-1966. However, the material of these lectures has been completely reworked and substantially expanded. We have tried to bear in mind the requirements of other mathematical disciplines and also of mechanics and physics. We hope that all parts of the text will be useful. The only preparation required for this text can be given an a first- semester course of analytic geometry and algebra at the most elementary level. All that is needed is a firm grasp of the elements of these subjects. For Chapter XII the student should be acquainted with projective transformations and the projective properties of figures in the plane. Also, in Chapter X the reader may simplify his task by skipping Subsections 13 to 23 (Section 3) and Subsection 10 of Section 7. What is left of Chapter X can serve as a minimal algebraic basis for the theory of multidimensional integration.

It may be noted in conclusion that the first five chapters already contain material with broad applications in mathematics, mechanics, and physics. These chapters, supplemented with some of the material of subsequent chapters, can be utilized in higher technical schools with a more advanced mathematics curriculum.

The book was translated from the Russian by George Yankovsky and published by Mir in 1975.

Many thanks to shankar.leo for providing the scans and the pdf.

Note: I tried to optimise the file for size, but somehow the archive kept on rejecting for some error in the pdf. I could have tried a few more things, but didn’t get time for that, hence the delay in post. For now, hence this large file (~120M) was uploaded. This file is OCRed but without bookmarks and pagination.

I will try to update a smaller file in the future, or if someone can add a link to a smaller file, it would be great.

Preface 9

Introduction 11

Chapter I. Linear Spaces

1. Axioms in linear space 15

2. Examples Of linear spaces 17

3. Elementary corrolaries to the axioms of a linear space 23

4. Linear combinations. Linear dependence.25

5. Lemma on the basis minor 27

6. Basic lemma on two systems of vectors 30

7. The rank of a matrix 32

8. Finite-dimensional and infinite-dimensional spaces. Bases 34

9. Linear operations in components 36

10. Isomorphism between linear spaces 38

11. Correspondence between complex and real spaces 40

12. Linear subspace 42

13. Linear hull 44

14. Sum of subspaces. Direct sum 47

Chapter II. Linear Transformations of Variables. Transformations of Coordinates

1. Abbreviated notation for summation 53

2. Linear transformation of variables. The product of linear

transformations of variables and matrix products 56

3. Square matrices and nonsingular transformations 60

4. The rank of a product of matrices 64

5. Transformation of coordinates in a change of basis 66

Chapter III. Systems of Linear Equations. Planes In Affine Space

1. Affine space 70

2. Affine coordinates 71

3. Planes 73

4. Systems of first-degree equations 77

5. Homogeneous systems 81

6. Nonhomogeneous systems 88

7. Mutual positions of planes 91

8. Systems of linear inequalities and convex polyhedrons 98

Chapter IV. Linear, Bilinear and Quadratic Forms

1. Linear forms 108

2. Bilinear forms 112

3. The matrix of a bilinear form 116

4. Quadratic forms 118

5. Reducing a quadratic form to canonical form by Lagrange’s method 121

6. The normal form of a quadratic form 124

7. The law of inertia of quadratic forms 125

8. Reducing a quadratic form to canonical form by Jacobi’s method 127

9. Positive definite and negative definite quadratic forms 129

10. Gram’s determinant. The Cauchy-Bunyakovsky inequality 132

11. Zero subspaces of a bilinear and a quadratic form 134

12. The zero cone of a quadratic form.137

13. Elementary examples of zero cones of quadratic forms 139

Chapter V. Tensor Algebra

1. Reciprocal bases. Contravariant and covariant vectors 142

2. Tensor product of linear spaces 149

3. Basis in a tensor product. Components of a tensor 153

4. Tensors of bilinear forms 159

5. Multiple-order tensors. Tensor product 162

6. Components of multiple-order tensors 166

7. Multilinear forms and their tensors 168

8. Symmetrization and antisymmetrization (alternation). Skewsymmetric forms 170

9. An alternative description of the tensor product of two linear spaces 174

Chapter VI. Groups and Some Applications

1. Groups and subgroups. Distribution of bases into classes

with respect to a given subgroup of matrices. Orientation 180

2. Transformation groups. Isomorphism and homomorphism of groups 186

3. Invariants. Axial invariants. Pseudoinvariants 191

4. Tensor quantities 197

5. The oriented volume of a parallelepiped. The discriminant tensor 201

Chapter VII. Linear Transformations of Linear Spaces

1. Generalities 207

2. A linear transformation as a tensor 210

3. The geometrical meaning of the rank and determinant of a linear transformation. The group of nonsingular linear transformations.213

4. Invariant subspaces 216

5. Examples of linear transformations 218

6. Eigenvectors and the characteristic polynomial of a transformation 224

7. Basic theorems on the characteristic polynomial and eigenvectors 227

8 Nilpolent transformations. The general structure of singular

transformations 229

9. The canonical basis of a nilpotent transformation 233

10. Reducing a transformation matrix to the Jordan normal form 242

11. Transformations of a simple structure 248

12. Equivalence of matrices 250

13. The Hamilton-Cayley formula 252

Chapter VIII. Spaces with Quadratic Metric

1. Scalar products 254

2. The norm of a vector 256

3. Orthonormal bases 258

4. Orthogonal projection. Orthogonalization 259

5. Metric isomorphism 265

6. ^-orthogonal matrices and ^ orthogonal groups 266

7. The group of Euclidean rotations 270

8. The group of hyperbolic rotations 278

9. Tensor algebra in quadratic-metric spaces 287

10. The equation of a hyperplane in quadratic-metric space 295

11. Euclidean space. Orthogonal matrices. Orthogonal group 297

12. The normal equation of a hyperplane in Euclidean space 302

13. The volume of a parallelepiped in Euclidean space. The discriminant tensor. Vector product 304

Chapter IX. Linear Transformations of Euclidean Space

1. Adjoint of a transformation 308

2. Lemma on the characteristic roots of a symmetric matrix 310

3. Self-adjoint transformations 311

4. Reducing a quadratic form to canonical form in an orthonormal basis 317

5. The joint reduction to canonical form of two quadratic forms 319

6. Skew-adjoint transformations 322

7. Isometric transformations 325

8. The canonical form of an isometric transformation 330

9. The motion of a rigid body with one fixed point 335

10. The curvature and torsion of a space curve 338

11. The decomposition of an arbitrary linear transformation into the product of a self-adjoint and an isometric transformation 340

12. Applications to the theory of elasticity. The strain tensor and the stress tensor 343

Chapter X. Multivectors and Outer Forms

1. Alternation 346

2. Multivectors. Outer product 351

3. Bivectors 357

4. Simple multivectors 366

5. Vector product 370

6. Outer forms and operations on them 376

7. Outer forms and covariant multivectors 379

8. Outer forms in three-dimensional Euclidean space 386

Chapter XI. Quadric Hypersurfaces

1. The general equation of a quadric hypersurface 391

2. Changes in the left member of the equation under translation of the origin 392

3. Changes in the left member of the equation for a change in the orthonormal basis 395

4. The centre of a quadric hypersurface 397

5. Reducing to canonical form the general equation of a quadric hypersurface in Euclidean space 399

6. Classification of quadric hypersurfaces in Euclidean space 402

7. Affine transformations. 410

8. Affine classification of quadric hypersurfaces 414

9. The intersection of a straight line with a quadric hypersurface. Asymptotic directions 415

10. Conjugate directions.418

Chapter XII. Projective Space

1. Homogeneous coordinates in affine space. Points at infinity 422

2. The concept of a projective space 425

3. A bundle of planes in affine space 435

4. Central projection 443

5. Projective equivalence of figures 446

6. Projective classification of quadric hypersurfaces 453

7. The intersection of a quadric hypersurface and a straight line. Polars 459

Appendix 1. Proof of the theorem on the classification of linear quantities 467

Appendix 2. Hermitian forms. Unitary space.471

Bibliography. 484

Index. 486

]]>About the book

At the Crossroads of Infinities is a story about the struggle of ideas out of which the modern physical picture of the world was born. Can anything move faster than light? Is the universe finite or infinite? Is time reversible? What lies at the basis of the realities which we perceive as space, time or matter? These are the questions taken up in this book. And more, for it also tells of the roads of knowledge, of the way man has probed the mysteries of the infinitely large and infinitely small, yet at root integral world. (From the Front Jacket)

The book is a tour of the development of ideas of modern physics at the beginning of the 20th century. The book covers historical as well as philosophical issues which are at the core of modern physics. The struggle of ideas which the creators and developers of quantum mechanics and special relativity had to deal with is described very well.

The book was translated from the Russian by Vladimir Talmy and was published by Mir in 1971.

Contents

FACE TO FACE WITH THE UNIVERSE 7

PART I. LOOKING BACK 11

PART II. BUILDING BLOCKS OF THE UNIVERSE 86

PART III. AT THE THRESHOLD OF A UNIFIED THEORY 200

PART IV. SPACE. TIME. VACUUM 252

PART V. THE MEGAWORLD 297

PART VI. THE UNIVERSE AND INFINITY 361

]]>

This is a book in the Science for Everyone Series. With this post, we have almost completed this series. As of now, we have updated all the dead links with Internet Archive ones. Next, we will update the Little Mathematic Library.

Now only **ONE** book in this series remains (Earth, Sweet Earth by Ekaterina Radkevich (1990)). Many people contributed to this collection becoming almost complete, a big thanks to all those who have contributed to making this possible.

One of the most important problems in modern science is the origin of the Earth and formation of its shell. Modern views on the chemical composition of meteorites, planets and other bodies of the Solar System are presented in this book. On the basis of recent achievements in cosmochemistry, the author, who is one of the leading scientists in the field of geochemistry, geophysics, describes the most probable processes that determined the chemical composition of the Earth in the remote past. Prof. George Voitkevich is also a winner of the Karpinsky Prize, which is given to a scientist for outstanding work in geology by the USSR Academy of Sciences. (From the back cover)

In this book the author attempts to describe in popular form some problems of the Earth’s origin and its chemical changes over its long geological history on the basis of cosmochemical, geochemical and geophysical data obtained in recent years. The chemical evolution of the Earth is part of the chemical evolution of Space. Modern cosmochemical and geochemical data reveal that the chemical history of the Earth as well as of other bodies of the Solar System is associated not only with preserved stable and nonstable isotopes but also with extinct radioactive isotopes, including the isotopes of transuranium elements. (From the Introduction)

The book was translated from the Russian by V. F. Agranat and V. F. Pominov and was first published by Mir in 1988.

Preface 5

The Distribution of Elements in the Solar System and Their Geochemical Properties 9

Evidence of the Early History of the Solar System 31

The Nature and Chemical Composition of Planets 45

Composition and Constitution of the Earth 71

The Present and Past Radioactivity of the Earth 87

The Birth of Atoms in Space 97

Chemical Evolution of the Protoplanetary Material 112

Formation of the Earth’s Principal Shells 136

Origin and Evolution of the Ocean and Atmosphere 152

Principal Trends in the Chemical Changes in the Earth’s Crust and Biosphere 181

The Chemical Evolution of the Earth’s Crust 202

Conclusion 223

Bibliography 232

]]>Paradoxes in the evolution of technological systems. Who designed the global telephone and communication network? The whole is the sum of the parts plus their interactions. Is a simple data register really simple? Automata exploring unknown worlds. Optimum or expediency? Random interactions and information exchange. Daydreaming and cynical automata.

“I thought he thought I thought he. .. .” Heterogeneity in a group of automata is a key to success. Are we haunted by the ghost of Erehwon City? Puppets without strings: who is the puppeteer? Boundaries of centralization. How do systems evolve?

A fresh look at design. Evolution goes on. (from the Back Cover)

We seek to present a popular account of the control problems that arise in complex systems which are more generally called large-scale systems in control theory. In systems of this kind, centralized control often gives way to decentralized control, the transition being a penalty for the system’s complexity. This is because the system’s complexity makes centralized control either inefficient or impracticable. How do large-scale systems arise, and is it possible that the category of large-scale system is merely a far-fetched nothing? We have tried to show in this book that large-scale man-made systems which surround us are steadily becoming more numerous and still more complex. The evolution of man-made systems out of the already existing ones goes on in much the same way as living organisms evolve. Decentralized control is but a natural product of this evolution. We hope to convince our readers that it is just so. (from Instead of a Preface)

The book was translated from the Russian by A. Kandaurov and was first published by Mir in 1988.

Instead of a Preface 8

**Chapter 1. Decentralized Control: the Whys and Wherefores 11**

1.1. The Man-Made World 11

1.2. Systems That Have Never Been Designed as a Whole 17

1.3. A Few Instructive Examples 23

1.4. Analysis of the Examples 28

1.5. Why Decentralization? 33

**Chapter 2. Is It Easy to Exist in a Contradictory World? 40**

2.1. The Pros and Cons of Common Sense 40

2.2. A “Small Animal” 45

2.3. Reaping the Fruits of Linear Tactics 49

2.4. Smart Machines: Reckless and Cautious 55

2.5. How to Live in a Transient World 60

2.6. Hungry Bats and Aerobatics 76

2.7. Put Your Heads Together 81

**Chapter 3. “How Comes This Gentle Concord in the World? 86
**

3.1. The Sukharev Tower Pact 86

3.2. When Everybody Is Alike 103

3.3. Distribution of Limited Resources 119

3.4. What Shall ’ We Do with Random Interactions? 127

3.5. He Thought I Thought He 141

3.6. Optimists and Pessimists in the World of Automata 150

3.7. Three More Simple Models 161

**Chapter 4. Jump the Queue and Call It Fair! 170**

4.1. Where Do All the Queues Come From? 170

4.2. Barbers, Clients, and Priorities 178

4.3. How to Learn to Be a Foreman 187

4.4. One Circus Ring Is Not Enough 194

4.5. Problem Faced by Housing Board and Similar Problems 200

4.6. “Stubborn” Automata and Voting 212

**Chapter 5. Stringless Puppets Make a Show 223**

5.1. Wait and See Them Fire 223

5.2. Have Them Fire All at Once 227

5.3. Marching and Wandering Automata 236

5.4. Praise Be to Homogeneous Structures 241

5.5. Why Yoga Is Not Our Way? 255

**Chapter 6. Dialectics of the Simple and the Complex 262**

6.1. Synthesogenesis and Integration of Efforts 262

6.2. Segregatiogenesis and Its Effects 272

6.3. Evolution in the Erehwon City 282

6.4. Instead of a Conclusion. Evolution Goes on 287

About the book

This book is ideal for those who wish to start learning about the intriguing modem concepts of astrophysics; the birth and evolution of the universe, its large-scale structure, galaxies and their clusters, and stars. The book provides

just enough text to give a feeling of what it is all about, such as the non* Euclidean geometry of the universe, while the main point is to cover all the recent major discoveries and novel hypotheses and theories in astrophysics, from quasars and relict radiation to black holes and the neutrino rest mass, the latter having a profound impact on the entire philosophy of the universe. The Russian edition of this book ran into 100,000 copies and is sold out. (from the back cover)

This book covers the cosmogony of stars and galaxies, a new field in astrophysics. The modern development of cosmogony is related to the astronomic discoveries of the past two decades, starting from the discovery of quasars in 1963 and microwave background radiation in 1965. The researchers in this area proceed from the achievements of cosmology, a science dealing with the universe as a whole, and make use of data from different branches of astronomy, physics, and mathematics

This book deals with the latest achievements of cosmogony, its problems and prospects, using the simple language of school physics and astronomy. In fact, the basic ideas and hypotheses allow a demonstrative presentation without mathematical formulas, and we hope that the general reader, keen on science news, will be eagerly interested.

This popular-science book follows our monograph Introduction to Cosmogony (Nauka, Moscow, 1978, in Russian), which reviewed and analyzed the investigations of the authors and their colleagues and gave a general presentation of the modern science of cosmogony. The principal ideas of the monograph are reflected in this book as well; furthermore, we added new material on recent star formation, the final stages of their evolution, and the role of neutrinos in cosmogony.

The book was translated from the Russian by Michael Burov and was published by Mir in 1987.

Though not the copy in the scan, I have had a copy of this book for a long time. This book is one of my first Mir books that I had purchased in the mid-90s, when you could still get these off-the-shelf. This is one of the few books that I have perhaps read more than once. The conceptual clarity is great and helps you understand the main physics behind phenomena.

This current scan copy was purchased from Blossoms in Bangalore, just for Rs. 40, a few years back. (The label you see above was removed during the cleaning.) Blossoms has a dedicated section for Mir books (and fantastic other sections too). If you are in Bangalore, please do visit and get your hands full for all kinds of books. You can spend hours at this book shop, get exhausted physically, cognitively, and economically yet there are more books than you can imagine or carry back with you. A true bibliophile’s candy shop. Thanks to @DhwaniB for the photos

Preface 5

**Chapter 1. The Universe 9
**

Stars and Galaxies 9

Cosmological Expansion 16

The Geometry of the Universe 21

The Horizon 22

Relict Radiation 25

**Chapter 2. The Origin of the Large-Scale Structure of the Universe 32
**

Stars, Galaxies, and Cosmological Expansion 32

Gravitational Instability 34

Pregalactic Structure 41

Entropy Perturbations and the Relict Background 51

The Formation of Clusters of Galaxies 54

The Large-Scale Structure of the Universe 59

**Chapter 3. Stellar Eddies 63
**

Rotation of Galaxies 67

Eddy Cosmogony 69

Protogalactic Turbulence 71

Primordial Eddies? 77

Tidal Torques? 83

The Birth of Eddies 84

A Protocluster as a Turbulent Layer 89

The Spiral Structure 101

**Chapter 4. The Birth and Evolution of Stars 109
**

The Sun and Stars 109

Gravitational Condensation 111

Cascade Fragmentation 113

The Interstellar Medium 118

Young Stars 126

Instabilities and Clouds 134

The Life of a Star 138

Close Binary Stars 145

**Chapter 5. The Evolution of Stellar Systems 156
**

From a Protogalaxy to a Stellar System 156

The Motion of Stars in Galaxies 161

Violent Relaxation 162

The Evolution of Star Clusters 164

**Chapter 6. “Hidden Masses”, Neutrinos, and Einstein’s Vacuum 168**

“Hidden Masses” 169

The Neutrino Rest Mass 178

Neutrino Coronas 179

A Closed Universe? 184

Einstein’s Vacuum 186

**Conclusion 192**

**Recommended Literature 200**

The principal advances in geoscience are discussed in this book. Prof. N. Yasamanov tells us about the role that geology plays in human society, about the Earth’s structure, origin and history. He describes the beginnings of life on Earth, the distribution of natural resources over the globe and the problems connected with environmental protection.The book is intended for senior schoolchildren, teachers and all who are interested in Earth sciences; it would also be helpful for students of geological exploration in technical schools.

The geological science is versatile. One book cannot show all of its specifics, are search techniques and major advances in all of its areas. We hence merely mention here such principal avenues as mineralogy and petrology, crystallography and petrography, geochemistry and geophysics, engineering geology and hydrogeology. This book will help a school teacher to unfold more openly the essence of one of the fundamental sciences and will ease the correct and duly professional orientation of a youngster.

The book was translated from the Russian by V. F. Agranat and was published by Mir in 1990. This might perhaps be one of the last books published by Mir and other Soviet-era publishers.

Many, many thanks to the Russian book lover **Angelika** for the raw scans of the book.

Preface 5 Introduction 9

Geology As the Fundamental Science of the Earth 15

Geology and Humans 15

Geological Processes 19

Geology and Cities 23

The Planet Earth 27

The Shape and Size of the Earth 33

Shells of the Earth 36

The Internal Structure of the Earth 44

The Origin of the Earth and Evolution of Its Interior 52

The Birth of the Earth 52

Gravitational Differentiation 56

The Origin of the Earth’s Crust 60

The Time Scale and History of the Earth 64

The Age of Rocks and Geological Time 64

Geological Time Scale 70

Principal Stages in the Formation of the Earth’s Crust 76

This Variable Face of the Earth 84

Weathering and Soils 86

Surface and Subsurface Waters 90 Glaciers 98

Wind Action 104

Geological Activity of Seas 108

Volcanoes and Earthquakes 111

Present-day Volcanoes 112

Volcanic Activity 118

Causes and Distribution of Earthquakes 123

Earthquake Studies and Prediction 128

A Biography of Life on the Earth 134

Origin of Organisms 134

The Appearance of Skeletal Faunas 138

The Conquest of Land 144

The Time of Dinosaurs and Mammals 147

The Life of Microorganisms 156

History of the Earth’s Climate and Atmosphere 164

Origin of the Atmosphere 165

Climatic Variations in the Geological Past 168

Climate and the Evolution of Organisms 174

Climate in the Future 179

Marine Geology 182

Origin and Evolution of Waters of the World Ocean 183

Why Is the Sea Salty? 186

The Structure and Geology of the Ocean Floor 191

Marine Research Laboratories 201

Motions of Continents 21

A. Wegener’s Hypothesis 211

Paleomagnetism and Neomobilism 216

The Tectonics of the Lithospheric Plates 220

The Mechanism of Motion of Lithospheric Plates 224

Global Reconstructions 227

Geosynclines as Folded Mountain Systems 236

The History of the Mediterranean Sea 241

Earth’s Natural Resources and Environmental Protection 246

Energy Resources 247

Mineral Resources 251

On the Protection of the Earth’s Interior and the Environment 259

Conclusions 275

Index 278

This book deals with the mysteries of geology. The basic concepts about solid mineral materials, symmetry and its elements, single crystals, intergrowths and twins, and habitual and simple crystallographic forms, which manifest themselves both in the outward appearance and in the external atomic structure of minerals are presented in this book.

This book is the result of many years of the author’s studies on an interesting mineral assemblage in sulfate-carbonate rocks. The problems, connected with the specifics of calcium sulfate conversion to sulfur and calcite, are discussed in the book. (from the back cover)

Mineralogy is a fundamental science; concepts about the origin of mineral deposits can be based solely on the information provided by this science. This book is the result of many years of the author’s studies on an interesting mineral assemblage in sulfate- carbonate rocks. It is intended for geologists, students, and lovers of stones. The basic concepts about solid mineral materials, symmetry and its elements, single crystals, intergrowths and twins, and habitual and simple crystallographic forms, which manifest themselves both in the outward appearance and in the internal atomic structure of minerals, are presented in this book.

The reader is acquainted with the marble onyx secret and with the secrecy of Egyptian pyramids long- living. It is intended for geologists, students, and lovers of stones. (from the Preface)

The book was translated from the Russian by V. F. Agranat and was published by Mir in 1989.

Many, many thanks to **Angelika** (a Russian booklover!) for providing the raw scans of this and Modern Geology (which we will see in the next post) in SFE series.

Preface 5

Why Are the Egyptian Pyramids So Long-Lived? 10

Gypsum and Anhydrite 10

Barite 25

Resistance of Stone to Weather 37

The World of Crystals 45

The Mystery of Onyx Marble 57

Calcite 57

Ornamental Calcite 72

A Mineral of Vital Importance 85

Salt Domes 91

A Mineral of the Future 111

Cryptocrystalline Sulphur 126

Coarse-Crystalline Sulphur 133

Microbiological Sulphur Accumulation 164

Contaminating Elements in Sulphur 167

Sulphur Isotopy 170

Sulphur As a Mercury Mineral Settler 173

Sulphur Caves 179

Ubiquitous and Diverse 180

Bluish Quartz 185

Chalcedony 186

Quartzine 190

Precious Opal 192

The Mystery of Melanophlogite 196

The World of Minute Minerals 204

Clay Minerals from Sulphur Deposits 214

Clay Minerals Derived from Underwater Weathering 225

Conclusion 230

Bibliography 233

Subject Index 236

]]>This Short Course of Theoretical Mechanics is designed for students of higher and secondary technical schools. It treats of the basic methods of theoretical mechanics and spheres of their application along with some topics which are of such importance todays that no course of mechanics, even a short one, can neglect them altogether.

In preparing the original Russian edition for translation the text has been substantially revised, with additions, changes and corrections in practically all the chapters.

Most of the additions are new sections containing supplementary information on the motion of a rigid body about a fixed point (the kinematic and dynamic Euler equations) and chapters setting forth the fundamentals of the method of generalized coordinates (the Lagrange equations), since the demands to the course of theoretical mechanics in training engineers of different specialities makes it necessary to devote some space to this subject even in a short course.

Also the book presents an essential minimum on the elementary theory of the gyroscope and such highly relevant topics as motion in gravitational fields (elliptical paths and space flights) and the motion of a body of variable mass (rocket motion); a new section discusses weightlessness.

The structure of this book is based on the profound conviction, born out by many years of experience, that the best way of presenting study material, especially when it is contained in a short course, is to proceed from the particular to the general. Accordingly, in this book, plane statics comes before three-dimensional statics, particle dynamics before system dynamics, rectilinear motion before curvilinear motion, etc. Such an arrangement helps the student to understand and digest the material better and faster and the teaching process itself is made more graphic and consistent.

Alongside with the geometrical and analytical methods of mechanics the book makes wide use of the vector method as one of the main generally accepted methods, which, furthermore, possesses a number of indisputable advantages. As a rule, however, only those vector operations are used which are similar to corresponding operations with scalar quantities and which do not require an acquaintance with many new concepts.

Considerable space—more than one-third of the book—is devoted to examples and worked problems. They were chosen with an eve to ensure a clear comprehension of the relevant mechanical phenomena and cover all the main types of problems solved by the methods described. There are 176 such examples (besides worked problems); their solutions contain instructions designed to assist the student in his independent work on the course. In this respect the book should prove useful to all students of engineering, notably those studying by correspondence or on their own.

The book was translated from the Russian by V. Talmy.

There are several editions and reprints of the book. First, it was published under **Foreign Languages Publishing House** in the 1950s and 1960s, later under **Mir**, with the last reprint in 1988.

This post has copies from both Mir (Link 1) and FLPH (Links 2 and 3).

Link 1 (Mir 1988 reprint, credits to the original uploader, converted djvu to pdf [I am not a big fan of djvu format], added pagination, bookmarks, OCR and cover)

Link 2 (FLPH 1960s print, cleaned, bookmarked, paginated copy of the link below. Note that this is not a hi-resolution scan, though the OCR has worked well for most of the words. A better copy could be suggested.)

Link 3 (from Public Resource collection, original for the cleaned copy above)

Note: The contents of the FLPH and Mir editions are a bit different. There is slight reorganisation of the topics and a few new topics in the Mir edition and it has 32 chapters, one more than the FLPH edition and more pages 528 as compared to 427 in FLPH edition.

Preface to the English Edition 5

Introduction 15

**Part 1. STATICS OF RIGID BODIES**

Chapter 1. Basic Concepts and Principles

1. The subject of statics 19

2. Force 21

3. Fundamental principles 22

4. Constraints and their reactions 26

5. Axiom of constraints 28

Chapter 2. Composition of Forces. Concurrent Force Systems

6. Geometrical method of composition of forces. Resultant of con current forces 30

7. Resolution of forces 32

8. Projection of a force on an axis and on a plane 36

9. Analytical method of defining a force 37

10. Analytical method of composition of forces 38

11. Equilibrium of a system of concurrent forces 40

12. Problems statically determinate and statically indeterminate 42

13. Solution of problems of statics 43

14. Moment of force about an axis (or a point) 53

15. Varignon’s theorem of the moment of a resultant 54

16*. Equations of moments of concurrent forces 55

Chapter 3. Parallel Forces and Force Couples in a Plane 64

17. Composition and resolution of parallel forces 58

18. A force couple. Moment of a couple 60

19. Equivalent couples 62

20. Composition of coplanar couples. Conditions for the equilibrium of couples 64

Chapter 4. General Case of Forces in a Plane

21. Theorem of translation of a force 67

22. Reduction of a coplanar force system to a givencentre 68

23. Reduction of a coplanar force system to the simplestpossible form 71

24. Conditions for the equilibrium of a coplanar force system. The case of parallel forces 73

25. Solution of problems 75

26. Equilibrium of systems of bodies 84

27*. Determination of internal forces (stresses) 88

28*. Distributed forces 89

Chapter 5. Elements of Graphical Statics

29. Force and string polygons. Reduction of a coplanar force system to two forces 93

30. Graphical determination of a resultant 95

31. Graphical determination of a resultant couple 96

32. Graphical conditions of equilibrium of a coplanarforce system 96

33. Determination of the reactions of constraints 97

Chapter 6. Solution of Trusses

34. Trusses. Analytical analysis of plane trusses 99

35*. Graphical analysis of plane trusses 103

36*. The Maxwell-Cremona diagram 104

Chapter 7. Friction

37. Laws of static friction 107

38. Reactions of rough constraints. Angle offriction 109

39. Equilibrium with friction 110

40*. Belt friction 114

41*. Rolling friction and pivot friction 116

Chapter 8. Couples and Forces in Space

42. Moment of a force about a point as a vector 118

43. Moment of a force with respect to an axis 120

44. Relation between the moments of a force about a point and an axis 123

45. Vector expression of the moment of a couple 124

46*. Composition of couples in space. Conditions of equilibrium of couples 125

47. Reduction of a force system in space to a given centre 128

48*. Reduction of a force system in space to the simplest possible form 130

49. Conditions of equilibrium of an arbitrary force system in space.

The case of^ parallel forces 132

50. Varignon’s theorem of the moment of a resultant with respect to

an axis 134

51. Problems on equilibrium of bodies subjected to action of force systems in space 134

52*. Conditions of equilibrium of a constrained rigid body. Concept of stability of equilibrium 144

Chapter 9. Centre of Gravity

53. Centre of parallel forces 146

54. Centre of gravity of a rigid body 148

55. Coordinates of centres of gravity of homogeneous bodies 149

56. Methods of determining the coordinates of the centre of gravity of bodies 150

57. Centres of gravity of some homogeneous bodies 153

**Part 2 KINEMATICS OF A PARTICLE AND A RIGID BODY**

Chapter 10. Kinematics of a Particle

58. Introduction to kinematics 156

59. Methods of describing motion of a particle. Path 158

60*. Conversion from coordinate to natural method of describing its motion is described by the coordinate method 161

61. Velocity vector of a particle 163

62. Acceleration vector of a particle 164

63. Theorem of the projection of the derivativeof a vector 166

64. Determination of the velocity and acceleration of a particle when its motion is described by coordinate method 167

65. Solution of problems of particle kinematics 168

66. Determination of the velocity of a particle when its motion is described by the natural method 173

67. Tangential and normal accelerations of a particle 174

68. Some special cases of particle motion 178

69. Graphs of displacement, velocity and acceleration of a particle 180

70. Solution of problems 182

71*. Velocity in polar coordinates 185

72*. Graphical analysis of particle motion 186

Chapter 11. Translational and Rotational Motion of a Rigid Body

73. Translational motion 191

74. Rotational motion of a rigid body. Angular velocity and angular acceleration 193

75. Uniform and uniformly variable rotations 195

76. Velocities and accelerations of the points of a rotating body 196

Chapter 12. Plane Motion of a Rigid Body

77. Equations of plane motion. Resolution of motion into translation and rotation 201

78. Determination of the path of a point of a body 203

79. Determination of the velocity of a point of a body 204

80. Theorem of the projections of the velocities of two points of a body 206

81. Determination of the velocity of a point of a body using the instantaneous centre of zero velocity. Centrodes 207

82. Solution of problems 212

83*. Velocity diagram 217

84. Determination of the acceleration of a point of a body 219

85*. Instantaneous centre of zero acceleration 227

Chapter 13. Motion of a Rigid Body Having One Fixed Point and Motion of a Free Rigid Body

86. Motion of a rigid body having one fixed point 231

87*. Velocity and acceleration of a point of a body 233

88. The general motion of a free rigid body 236

Chapter 14. Resultant Motion of a Particle

89. Relative, transport, and absolute motion 239

90. Composition of velocities 241

91*. Composition of accelerations 245

92. Solution of problems 249

Chapter 15. Resultant Motion of a Rigid Body

93. Composition of translational motions 257

94. Composition of rotations about two parallel axes 257

95*. Toothed spur gearing 260

96*. Composition of rotations about intersecting axes 264

97*. Euler kinematic equations 266

98*. Composition of a translation and a rotation. Screwmotion 268

**Part 3 PARTICLE DYNAMICS**

Chapter 16. Introduction of Dynamics. Laws of Dynamics

99. Basic concepts and definitions 271

100. The laws of dynamics 273

101. Systems of units 275

102. The problems of dynamics for a free and a constrained particle 275

103. Solution of the first problem of dynamics (determination of the forces if the motion is known) 276

Chapter 17. Differential Equations of Motion for a Particle and Their Integration

104. Rectilinear motion of a particle 279

105. Solution of problems 282

106*. Body falling in a resisting medium (in air) 288

107. Curvilinear motion of a particle 291

108. Motion of a particle thrown at an angle to the horizon in a uniform gravitational field 292

Chapter 18. General Theorems of Particle Dynamics

109. Momentum and kinetic energy of a particle 295

110. Impulse of a force 296

111. Theorem of the change in the momentum of a particle 297

112. Work done by a force. Power 298

113. Examples of calculation of work 302

114. Theorem of the change in the kinetic energy of a particle 306

115. Solution of problems 307

116. Theorem of the change in the angular momentum of a particle

(the principle of moments) 315

117*. Motion under the action of a central force. Law of areas 317

Chapter 19. Constrained Motion of a Particle

§ 118. Equations of motion of a particle along a given fixed curve 319 § 119. Determination of the reactions of constraints 322

Chapter 20. Relative Motion of a Particle

120. Equations of relative motion and rest of a particle 325

121. Effect of the rotation of the earth on the equilibrium and motion of bodies 328

122*.Deflection of a falling particle from the vertical by the earth’s rotation 331

Chapter 21. Rectilinear Vibration of a Particle

123. Free vibrations neglecting resisting forces 335

124. Free vibration with a resisting force proportional to velocity (damped vibration) 341

125. Forced vibration. Resonance 343

Chapter 22*. Motion of a Body in the Earth’s Gravitational Field

126. Motion of a particle thrown at an angle to the horizon in the earth’s gravitational field 353

127. Artificial earth satellites. Elliptical paths 357

128. Weightlessness 360

**Part 4 DYNAMICS OF A SYSTEM AND A RIGID BODY**

Chapter 23. Introduction to the Dynamics of a System. Moments of Inertia of Rigid Bodies

129. Mechanical systems. External and internal forces 366

130. Mass of a system. Centre of mass 367

131. Moment of inertia of a body about an axis. Radius of gyration 368

132. Moments of inertia of a body about parallel axes. The parallel axis (Huygens’) theorem 372

133*. Product of inertia. Principal axes of inertia of a body 374

Chapter 24. Theorem of the Motion of the Centre of Mass of a System

134. The differential equations of motion of a system 378

135. Theorem of motion of centre of mass 379

136. The law of conservation of motion of centre of mass 380

137. Solution of problems 382

Chapter 25. Theorem of the Change in the Linear Momentum of a System

138. Linear momentum of a system 387

139. Theorem of the change in linear momentum 388

140. The law of conservation of linear momentum 389

141. Solution of problems 391

142*. Bodies having variable mass. Motion of a rocket 393

Chapter 26. Theorem of the Change in the Angular Momentum of a System

143. Total angular momentum of a system 397

144. Theorem of the change in the total angular momentum of a system (the principle of moments) 399

145. The law of conservation of the total angular momentum 401

146. Solution of problems 403

Chapter 27. Theorem of the Change in the Kinetic Energy of a System

147. Kinetic energy of a system 407

148. Some cases of computation of work 411

149. Theorem of the change in the kinetic energy of a system 414

150. Solution of problems 416

151. Conservative force field and force function 422

152. Potential energy 426

153. The law of conservation of mechanical energy 427

Chapter 28. Applications of the General Theorems to Rigid-body Dynamics

154. Rotation of a rigid body 429

155. The compound pendulum 432

156. Plane motion of a rigid body 435

157*. Approximate theory of gyroscopic action 443

158*. Motion of a rigid body about a fixed point and motion of a free rigid body 448

Chapter 29. Applications of the General Theorems to the Theory of Impact

159. The fundamental equation of the theory of impact 454

160. General theorems of the theory of impact 455

161. Coefficient of restitution 457

162. Impact of a body against a fixed obstacle 458

163. Direct central impact of two bodies (impact of spheres) 460

164. Loss of kinetic energy in perfectly inelastic impact. Carnot’s theorem 462 165*. Impact with a rotating body 464

Chapter 30. D’Alembert’s Principle. Forces Acting on the Axis of a Rotating Body

166. D’Alembert’s principle 469

167. The principal vector and the principal moment of the inertia forces of a rigid body 472

168. Solution of problems 473

169*. Dynamic reactions on the axis of a rotating body. Dynamic balancing of masses 479

Chapter 31. The Principle of Virtual Displacements and the General Equation of Dynamics 485

170. Virtual displacements of a system. Degrees of freedom 485

171. The principle of virtual displacements 486

172. Solution of problems 488

173. The general equation of dynamics 494

Chapter 32*. Equilibrium Conditions and Equations of Motion of a System in Generalised Coordinates 499

174. Generalised coordinates and generalised velocities 499

175. Generalised forces 501

176. Equilibrium conditions for a system in generalised coordinates 505

177. Lagrange’s equations 507

178. Solution of problems 510

Index 520

Preface 9

Introduction 10

**PART 1. STATICS OF RIGID BODIES**

Chapter 1. Basic Concepts and Principles

1. The Subject of Statics 13

2. Force 14

3. Fundamental Principles 16

4. Constraints and Their Reactions 19

5. Axiom of Constraints 22

Chapter 2. Concurrent Force Systems

6. Geometrical method of Composition of forces. Concurrent Forces 23

7. Resolution of Forces 25

8. Projection of a Force on an Axis and on a Plane 28

9. Analytical Method ol Defining a Force 30

10. Analytical Method for (he Composition of Forces 31

11. Equilibrium of a System of Concurrent Forces 32

12. Problems Statically Determinate and Statically Indeterminate 34

13. Solution of Problems of Statics 35

14. Moment of a Force About an Axis (or a Point) 43

15. Varignon’s Theorem of the moment of a Resultant 45

16. Equations of Moments of Concurrent Forces 46

Chapter 3. Parallel Forces and Couples in a Plane

17. Composition and Resolution of Parallel Forces 47

18. Force Couples. Moment of a Couple 50

19. Equivalent Couples 51

20. Coplanar Couples. Conditions for the Equilibrium of couples 53

Chapter 4. General Case of Forces in a Plane

21. Theorem of the Translation of a Force to a Parallel Position 55

22. Reduction of a Coplanar Force System to a Given Centre 56

23. Reduction of a Coplanar Force System to the Simplest Possible Form 58

24. Conditions for the Equilibrium of a Coplanar Force System 61

25. Equilibrium of a Coplanar System of Parallel Forces 63

26. Solution of Problems 63

27. Equilibrium of Systems of Bodies 70

28. Distributed Forces 74

Chapter 5. Elements of Graphical Statics

29. Force and String Polygons. Reduction of a Coplanar Force Systems to Two Forces 78

30. Graphical Determination of a Resultant 80

31. Graphical Determination of a Resultant Couple 80

32. Graphical Conditions of Equilibrium of a Coplanar Force System 81

33. Determination of the Reactions of Constraints 81

34. Graphical Analysis of Plane Trusses 82

35. The Maxwell Diagram 85

Chapter 6. Friclion

36. Laws of Static Friction 86

37. Reactions of Rough Constraints. Angle of Friction 88

38. Equilibrium with Friction 89

39. Belt Friction 92

40. Rolling Friction and Pivot Friction 94

Chapter 7. Couples and Forces in Space

41. Moment of a Force About a Point as a Vector 95

42. Moment of a Force with Respect to an Axis 97

43. Relation Between the Moments of a Force about a Point and an Axis 100

44. Vector Expression of the Moment of a Couple 101

45. Composition of Couples in Space. Conditions of Equilibrium of Couples 101

46. Reduction of a Force System in Space to a Given Centre 104

47. Reduction of a Force System in Space to the Simplest Possible Form 106

48. Condition of Equilibrium of an Arbitrary Force System in Space. The Case of Parallel Forces 108

49. Varignon’s Theorem of the Moment of a Resultant with Respect to an Axis 109

50. Problems on the Equilibrium of Bodies Subjected to the Action of Force Systems in Space 110

51. Conditions of Equilibrium of a Constrained Rigid Body. Concept of Stability of Equilibrium 117

Chapter 8. Centre of Gravity

52. Centre of Parallel Forces 118

53. Centre of Gravity of a Rigid Body 120

54. Coordinates of Centres of Gravity of Homogeneous Bodies 122

55. Methods of Determining the Coordinates of the Centre of Gravity of Bodies 122

56. Centre of Gravity of Some Homogeneous Bodies 125

**PART 2. KINEMATICS OF A PARTICLE AND A RIGID BODY**

Chapter 9. Rectilinear Motion of a Particle

57. Introduction to Kinematics 128

58. Equation of Rectilinear Motion 129

59. Velocity and Acceleration of a Particle in Rectilinear Motion 130

60. Some Examples of Rectilinear Motion of a Particle 132

61. Graphs of Displacement, Velocity and Acceleration of a Particle 134

62. Solution of Problems 138

Chapter 10. Curvilinear Motion of a Particle

63. Vector Method of Describing Motion of a Particle 137

64. Velocity Vector of a Particle 138

65. Acceleration Vector of a Particle 130

66. Theorem of the Projection of the Derivative of a Vector 141

67. Coordinate Method of Describing Motion. Determination of the Path, Velocity and Acceleration of a Particle 142

68. Natural Method of Describing Motion. Determination of the Velocity of a Particle 147

69. Tangential and Normal Acceleration of a Particle 148

70. Some Special Cases of Particle Motion 151

71. Velocity in Polar Coordinates 156

72. Graphical Analysis of Particle Motion 156

Chapter 11. Translatory and Rotational Motion of a Rigid Body

73. Motion of Translation 160

74. Rolatiotral Atotion of a Rigid Body. Angular Velocity and Angular Acceleration 162

75. Uniform and Uniformly Variable Rotation 164

76. Velocities and Accelerations of llie Points of a Rotating Body 166

Chapter 12. Plane Motion of a Rigid Body

77. Equations of Plane Motion. Resolution of Motion into Translation and Rotation 170

78. Determination of the Paths of the Points of a Body 172

79. Determination of the Velocity of Any Point of a Body 173

80. Theorem of the Projections of the Velocities of Two Points of a Body 174

81. Determination of the Velocity of Any Point of a Body Using the

Instantaneous Centre of Zero Velocity 175

82. Solution of Problems 178

83. Velocity Diagram 182

84. Determination of the Acceleration of Any Point of a Body 184

85. Instantaneous Centre of Zero Acceleration 191

Chapter 13. Motion of a Rigid Body Having One Fixed Point and Motion of a Free Rigid Body

86. Motion of a Rigid Body Having One Fixed Point 193

87. Acceleration of Any Point of a Body 195

88. The Most General Motion of a Free Rigid Body 196

Chapter 14. Resultant Motion of a Particle

89. Relative, Transport, and Absolute Motion 198

90. Composition of Velocities 200

91. Composition of Accelerations. Coriolis Theorem 203

92. Calculation of Coriolis Acceleration 207

93. Solution of Problems 207

Chapter 15. Resultant Motion of a Rigid Body

94. Composition of Translatory Motions 215

95. Composition of Rotations About Two Parallel Axes 215

96. Toothed Spur Gearing 218

97. Composition of Rotations About Two Intersecting Axes 221

98. Composition of a Translation and a Rotation. Screw Motion. 223

**PART 3. PARTICLE DYNAMICS**

Chapter 16. Introduction to Dynamics, Laws of Dynamics

99. Basic Concepts and Definitions 226

100. The Laws of Dynamics 227

101. Systems of Units 230

102. The Problems of Dynamics for a Free and a Constrained Particle 230

103. Solution of the First Problem of Dynamics 231

Chapter 17. Differential Equations of Motion for a Particle and Their Integration

104. Rectilinear Motion of a Particle 233

105. Solution of Problems 236

106. Body Falling in a Resisting Medium (in Air) 241

107. Curvilinear Motion of a Particle 244

108. Motion of a Particle Thrown at an Angle to the Horizon in a Uniform Gravitational Field 245

Chapter 18. General Theorems of Particle Dynamics

109. Momentum and Kinetic Energy of a Particle 248

110. Impulse of a Force 249

111. Theory m of the Change in the Momentum of a Parlicie 250

112. Work Done by a Force. Power 251

113. Examples of Calculation of Work 254

114. Theorem of the Change in the Kinetic Energy of a Particle 256

115. Solution of Problems 258

116. Theorem of the Change in the Angular Momentum of a Particle (the Principle of Moments) 264

Chapter 19. Constrained Motion of a Particle and D’Alembert’s Principle

117. Equations of Motion of a Particle Along a Given Fixed Curve 268

118. Determination of the Reactions of Constraints 270

119. D’Alembert’s Principle 272

Chapter 20. Relative Motion of a Particle

120. Equations of Relative Motion and Rest of a Parlicie 275

121. Effect of the Rotation of the Earth on the Equilibrium and Motion of bodies 278

122. Deflection of a Falling Particle from the Vertical by the Earth’s

Rotation 281

Chapter 21. Vibration of a Particle

123. Free Harmonic Motion 284

124. The Simple Pendulum 288

125. Damped Vibrations

126. Forced Vibrations. Resonance 291

Chapter 22. Motion of a Body In the Earth’s Gravitational Field

127. Motion of a Particle Thrown at an Angle to the Horizon In the Earth’s Gravitational Field 299

128. Artificial Earth Satellites. Elliptical Paths 304

**PART 4. DYNAMICS OF A SYSTEM AND A RIGID BODY**

Chapter 23. Introduction to the Dynamics of a System. Moments of Inertia of Rigid Bodies

129. Mechanical Systems. External and Internal Forces 308

130. Mass of a System. Centre of Mass 309

131. Moment of inertia of a Body About an Axis. Radius of Gyration 310

132. Moments of Inertia of Some Homogeneous Bodies 311

133. Moments of Inertia of a Body About Parallel Axes. The Parallel-Axis (Huygens’) Theorem 313

Chapter 24. Theorem of the Motion of the Centre of Mass of a System

131. The Differential Equations of Motion of a System 315

135. Theorem of the Motion of Centre of Mass 316

136. The Law of Conservation of Motion of Centre of Mass 317

137. Solution of Problems 319

Chapter 25. Theorem of the Change in the Linear Momentum of a System

138. Linear Momentum of a System 323

139. Theorem of the Change in Linear Momentum 324

140. The Law of Conservation of Linear Momentum 325

141. Solution of Problems 326

142. Bodies Having Variable Mass. Motion of a Rocket 329

Chapter 26. Theorem of the Change In the Angular Momentum of a System

143. Total Angular Momentum of a System 332

144. Theorem of the Change in the Angular Momentum of a System

(Ihe Principle of Moments) 333

145. The Law of Conservation of the Total Angular Momentum 334

146. Solution of Problems 337

Chapter 27. Theorem of the Change In the Kinetic Energy of a System

147. Kinetic Energy of a System 339

148. Theorem of Ihe Change in the Kinetic Energy of a System 344

149. Some Cases of Computation of Work 316

150. Solution of Problems 318

151. Field of Force. Potential Energy 353

152. The Law of Conservation ol Mechanical Energy 355

Chapter 28. Some Cases of Rigid-Body Motion

153. Rotation of a Rigid Body 355

154. The Compound Pendulum 359

155. Determination of Moments ol Inertia by Experiment 361

156. Plane Motion of a Rigid Body 361

157. Approximate Theory of Gyroscopic Action 368

Chapter 29. D’Alembert’s Principle. Forces Acting on the Axis ol a – y r Rotating Body

158. D’Alembert’s Principle for a System 373

159. The Principal Vector and the Principal Moment of the Inertia

Forces of a Rigid Body 374

160. Solution of Problems 376

161. Dynamical Pressures on the Axis til a Rotating Body 380

162. The Principal Axes of Inertia of a Body. Dynamic Balancing of Masses 382

Chapter 30. The Principle of Virtual Work and Ihe General Equation of Dynamics

163. Virtual Displacements of a System. Degrees of Freedom 386

164. Idea] Constraints 388

165. The Principle of Virtual Work 368

166. Solution of Problems 390

167. The General Equation of Dynamics 395

Chapter 31. The Theory of Impact

168. The Fundamental Equation of Ihe Theory of Impact 398

169. General Theorems of Ihe Theory of Impact 400

170. Coefficient of Restitution 401

171. Impact of a Body Against a Fixed Obstacle 403

172. Direct Central Impact of Two Bodies (Impact of Spheres) 405

173. Loss of Kinetic Energy in Perfectly Inelastic Impact. Carnot’s Theorem 407

174. Impact with a Rotating Body 409

Name Index 414

Subject Index 414

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Studies of the asteroids and their relationship to meteroids and comets are discussed, and details are given on the more interesting asteroids. Direct astronomical observation and laboratory data on meteorites, singly and in combination, are used to study the asteroids. The real and apparent motion and orbits are described, along with asteroid family identification and the three-body problem. The characteristics are given of the Trojan group, Eros, Ganymede and its group, Hidalgo, Icarus, and the Amor, Apollo, Adonis, and Hermes group. the small bodies in the solar system are identified as asteroids, comets, and products of their destruction (meteorites, meteoric bodies, and cosmic dust), and their physical characteristics and composition are outlined. The composition of tektites and locations of discovery are discussed, and it is felt that they are most likely to be glassy meteorites. The hypothetical planet Phaeton is also considered and it is pointed out that the problem of the origin of the asteroid belt is still unsolved.

Where did that group of minor planets come from, revolving around the Sun between the orbits of Mars and Jupiter? Are the minor planets related to the meteorites that strike the Earth? What role could the minor planets play in the plans for the conquest of space? These are a few of the questions discussed in this book by F. Yu. Zigel.

The reader will also learn about the history of the study of asteroids, modern methods of investigating them, and about some of the interesting minor planets— Icarus, Hermes, Eros and others.

The book was translated to English from the Russian “Malyye Planety” Nauka Press, Moscow, 1969 by NASA in 1972.

Note: There are many translations of space-technology and astronomy/astrophysics related Russian books done under Technical Translations of NASA Technical Documents. Do explore this collection.

**Bonus: **These two articles may be of interest in the same subject area

Meteors – Development of Meteoric Astronomy in the USSR by S. Levin

Sputniks and Meteors by B. Levin

The above two articles are part of Defense Technical Information Archive which also has loads of translations from Russian sources.

ASTEROIDS – THEIR SIGNIFICANCE TODAY 1

SOME HISTORY 4

METHODS OF STUDYING THE MINOR PLANETS 15

THE MOTION AND ORBITS OF THE ASTEROIDS 23

IMPORTANT ASTEROIDS 29

THE PHYSICAL NATURE OF THE MINOR PLANETS 38

SMALL BODIES IN THE SOLAR SYSTEM 49

ASTEROIDS IN THE LABORATORY 64

THE ENIGMA OF TEKTITES 77

WAS THERE EVER A PLANET PHAETON? 83

ASTEROIDS AND ASTRONAUTICS 99

REFERENCES 102

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