You Can Purchase a Copy of This Book Here
Overview
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 today 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 specialties 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 rectilinear 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.
Download the Book Here
Contents
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
Chet Aero Marine 