Textbook on Theoretical Mechanics.

354 Pages·2017·6.504 MB·English

by  Ракишева З.

AL-FARABI KAZAKH NATIONAL UNIVERSITY

Z. B. Rakisheva
A. S. Sukhenko

TEXTBOOK ON
THEORETICAL MECHANICS

Second edition

Almaty
«Qazaq university»
2017
UDC 531.011 (076)
LBC 22.21
R 17

Recommended for publication by the decision of the Academic Council
of the Faculty of Mathematics and Mechanics,
Editorial and Publishing Council of Al-Farabi Kazakh National University
(Protocol №1 dated 29.09.2017);
Educational and methodical association on groups of specialties
«Natural sciences», «Engineering and technology»
of Republican educational-methodical council on basis
Al-Farabi Kazakh National University
(Protocol №2 dated 29.06.2017)

Reviewers:
Doctor of Technical Sciences, Professor E.S. Temirbekov
Doctor of Technical Sciences, Professor Z.G. Ualiev
Doctor of Physical and Mathematical Sciences,
Associated Professor K.S. Zhilisbayeva

Authors:
Z.B. Rakisheva, candidate of physical
and mathematical sciences, associated professor – lecture course
A.S. Sukhenko, PhD, senior teacher – tasks and exercises

Rakisheva Z.B.
R 17 Textbook on Theoretical Mechanics / Z.B. Rakisheva,
A .S. Sukhenko. – 2nd ed. – Almaty: Qazaq university, 2017.
– 354 p.
ISBN 978-601-04-2895-9

Textbook was prepared on the base of a compulsorysubjectof theoretical mecha-
nics, read by the authors for the students of specialty «Mechanics». It contains a
lecture course, covering all topics of the Typical Curriculum of 2016, on the discipline
of «Theoretical Mechanics», and includes also thesections of rigid body dynamics and
analytical mechanics.After each studiedsection, the practical tasks, tasks for indepen-
dent work, control questions and assignments are given. In conclusion, tests on the
studied material with the keys of the correct answers are given. The textbook is
recommended for teaching students on the specialty «5В060300-Mechanics», as well
as for the students of all related specialties of science and technology, where the me-
chanics is studied.

Publishing in authorial release.
UDC 531.011 (076)
LBC 22.21

© Rakisheva Z.B., Sukhenko A.S., 2017
ISBN 978-601-04-2895-9 © Al-Farabi KazNU, 2017
Content

CONTENT

INTRODUCTION ..................................................................................... 7

1. KINEMATICS ....................................................................................... 9
1.1. The subject of mechanics. Kinematics of a point. Problems
of kinematic. Methods of the point’s motion setting. ................................. 9
Questions ................................................................................................... 13
1.2. The velocity and acceleration. The decomposition
of the velocity and acceleration. ................................................................ 13
Questions ................................................................................................... 18
Practice ...................................................................................................... 18
Self study of the student ............................................................................. 22
1.3. Mechanical system. Basic movements of a rigid body.
Translational motion of a rigid body. ......................................................... 23
Questions ................................................................................................... 25
1.4. Rotational motion of a rigid body around a fixed axis. ....................... 25
Questions ................................................................................................... 30
Practice ...................................................................................................... 30
Self study of the student ............................................................................. 33
1.5. Plane-parallel motion of a rigid body. Velocities of the points
of a plane figure. Instantaneous center of velocity. .................................... 35
Questions ................................................................................................... 40
1.6. Acceleration of the points of a plane figure. Instantaneous
center of acceleration. ................................................................................ 40
Questions ................................................................................................... 43
Practice ...................................................................................................... 43
Self study of the student ............................................................................. 49
1.7. Compound motion of a point. Full and relative derivatives
of the vector. Addition of velocities. .......................................................... 52
Questions ................................................................................................... 55
1.8. Theorem on the addition of accelerations (Coriolis theorem) ............. 56
Questions ................................................................................................... 58
Practice ...................................................................................................... 58
Self study of the student ............................................................................. 65
1.9. Complex motion of a solid body. Addition of the translational
velocities. Addition of the instant angular velocities. Addition
of the instant angular and translational velocities. ..................................... 68
Questions ................................................................................................... 71

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Textbook on Theoretical Mechanics
1.10. Reduction of the system of sliding vectors. Principle vector
and principle moment. Invariants of reduction .......................................... 71
Questions ................................................................................................... 73

2. STATICS ............................................................................................... 74
2.1. Statics. Basic definitions and axioms of statics. Constraints.
Constraint reactions. Axiom of constraints. ............................................... 74
Questions ................................................................................................... 78
2.2. System of forces. Convergent system of forces. Parallel forces.
Center of gravity. ....................................................................................... 79
Questions ................................................................................................... 87
Practice ...................................................................................................... 87
Self study of the student ............................................................................. 93
2.3. Moment of force relative to center and axis. Theory of couples. ........ 94
Questions ................................................................................................... 98
2.4. Arbitrary system of forces. Reduction of spatial system of forces.
Principle vector and principle moment. ..................................................... 98
Questions ................................................................................................... 101
2.5. Equilibrium conditions of arbitrary spatial system of forces.
Special cases of the equilibrium conditions ............................................... 102
Questions ................................................................................................... 104
Practice ...................................................................................................... 104
Self study of the student ............................................................................. 120
2.6. Conditions of equlibrium of a constrained solid body.
Friction and constraints with friction ......................................................... 122
Questions ................................................................................................... 127

3. DYNAMICS OF THE MASS POINT AND THE SYSTEM ................ 128
3.1. The laws of Newton. Direct and inverse problems of dynamics.
Motion equations ....................................................................................... 128
Questions ................................................................................................... 133
Practice ...................................................................................................... 133
3.2. Basic dynamic variables. Properties of internal forces
of the system. ............................................................................................. 137
Questions ................................................................................................... 146
3.3. Theorem of change of linear momentum of a mass point
and mechanical system. Theorem on the motion of the center of mass ..... 147
Questions ................................................................................................... 151
Self study of the student ............................................................................. 151
3.4. Angular momentum theorem of a mass point and mechanical
system ........................................................................................................ 154
Questions ................................................................................................... 160
Practice ...................................................................................................... 160
Self study of the student ............................................................................. 165
3.5. Work of force. Work of potential force. .............................................. 169

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Content
Questions ................................................................................................... 175
3.6. Work-Energy theorem for the mass point and mechanical system.
Energy integral........................................................................................... 175
Questions ................................................................................................... 179
Practice ...................................................................................................... 179
Self study of the student ............................................................................. 185
3.7. Rectilinear motion of a mass point. Harmonic oscillations
of the mass point. Parameters of oscillations. Oscillations in a resistant
medium. ..................................................................................................... 190
Questions ................................................................................................... 200
3.8. Forced oscillations in a medium without resistance and in
a resistant medium. Resonance. ................................................................. 200
Questions ................................................................................................... 205
Practice ...................................................................................................... 206
3.9. Motion of mass point under the action of central forces. Law
of areas. Binet formulas. ............................................................................ 210
Questions ................................................................................................... 215
3.10. Planetary motion problem. Kepler's laws. Derivation
of Newton’s law of gravitaion from Kepler's laws..................................... 215
Questions ................................................................................................... 217
3.11. Motion of a non-free mass point. Concept of a constraint.
Motion of a mass point over given curve. Motion of a mass point
over given surface. Geodetic line. .............................................................. 218
Questions ................................................................................................... 225
3.12. Relative motion and equlibrium of a mass point. Equations
of relative motion. Inertial forces of transportation motion,
Coriolis inertial force ................................................................................. 225
Questions ................................................................................................... 229
3.13. Apparent weight of a body. Deflection of bodies falling
on the Earth from the vertical. Work-energy theorem for relative
motion. Relative motion and equilibrium .................................................. 230
Questions ................................................................................................... 233

4. DYNAMICS OF THE SOLID BODY .................................................. 234
4.1. Mass geometry. Inertia moment. ......................................................... 234
Questions ................................................................................................... 238
4.2. Theorem of Guigens-Shteiner. Inertia moments relative
to the axes of the beam, coming from this point. ....................................... 239
Questions ................................................................................................... 241
Practice ...................................................................................................... 241
Self study of the student ............................................................................. 245
4.3. Rotation of solid body around fixed axis. Differential equations
of motion. Axle pressure. Motion of absolutely rigid body with one
fixed point. ................................................................................................. 248
Questions ................................................................................................... 253

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Textbook on Theoretical Mechanics
Practice ...................................................................................................... 254
Self study of the student ............................................................................. 262
4.4. Motion of absolutely rigid body with one fixed point.
Euler kinematic and dynamic equations. ................................................... 264
Questions ................................................................................................... 267
Practice ...................................................................................................... 267
4.5. General formulation of the problem of heavy solid body
with one fixed point. .................................................................................. 269
Questions ................................................................................................... 275
4.6. Special cases of integration and its geometrical interpretation:
the case of Euler-Puanso, the case of Lagrange-Puasson, the case of
Kovalevskaya. ............................................................................................ 276
Questions ................................................................................................... 277

5. ANALYTICAL MECHANICS ............................................................. 278
5.1. The notion of holonomic and nonholonomic systems.
Actual and virtual displacement of the point. ............................................ 278
Questions ................................................................................................... 281
Practice ...................................................................................................... 281
Self study of the student ............................................................................. 289
5.2. Virtual work of forces. Ideal constraints. Principle of virtual
work. d'Alambert's principle for the point and system. General dynamic
equation. .................................................................................................... 293
Questions ................................................................................................... 296
Practice ...................................................................................................... 296
Self study of the student ............................................................................. 300
5.3. Generalized coordinates. Generalized forces. Equations
of motion in generalized coordinates (Lagrange equations of 2-nd kind). . 304
Questions ................................................................................................... 307
5.4. Expression of kinetic energy in generalized coordinates. Lagrange
equations of 2-nd type for the system under action of potential force. ...... 307
Questions ................................................................................................... 309
Practice ...................................................................................................... 309
Self study of the student ............................................................................. 313

TESTS ....................................................................................................... 318

KEYS TO THE TEST ............................................................................... 351

CONCLUSION ......................................................................................... 352

REFERENCES .......................................................................................... 353

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Content

INTRODUCTION

Theoretical mechanics is the science about general laws of
mechanical interactions between material bodies, as well as general
laws of motion of bodies relative to each other.
Mechanical interaction between material bodies is the simplest
and at the same time common type of interaction between physical
objects. Mechanical motion being the simplest type of motion is a
fundamental property of matter.
Theoretical mechanics taught in the university contains four
sections: kinematics, statics, dynamics and analytical mechanics.
Kinematics is a part of mechanics in which the dependences
between quantities characterizing the state of systems motion are
studied, however the causes changing the state of the motion are not
considered.
Statics is a study of equilibrium of bodies set of some reference
system.
Dynamics is the part of mechanics that considers an influence of
forces to the state of motion of material objects system.
Analytical mechanics is the part of theoretical mechanics and
theoretical physics in which general principles of mechanics
(differential and integral) are formulated and used, on this basis main
differential equations of motion are derived and equations and
methods of their integration are investigated.
Purpose of studying a discipline «Theoretical mechanics» is a
formation of a necessary base of knowledge to study other technical
disciplines on a profile of future professional activity as a strength of
materials and the theory of mechanisms and machines.
Tasks of the discipline «Theoretical mechanics» are:
– development of practical skills in solving problems of me-
chanics by studying methods and algorithms of constructing
mathematical models of motion or state of the considering
mechanical systems, also research methods of these mathe-
matical models;
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Textbook on Theoretical Mechanics
– education of the natural-science worldview on the basis of
studying the basic laws of nature and mechanics.
Textbook gives the basic materials on each of main sections of
theoretical mechanics including the summary of lectures, tasks and
examples of solution, tasks for independent work, and literature on
theoretical mechanics.
Work with this textbook allows students studying the main
concepts, laws and theorems of theoretical mechanics, obtaining the
skills of mathematical modeling of physical processes, using the
obtained knowledge for the solution of practical tasks, analyzing the
solutions of the tasks and making conclusions.
This textbook was written on the base of the course of lectures,
which are read by prof. Zaure Rakisheva during last 20 years for the
students of the mechanical and mathematical faculty of al-Farabi
Kazakh National University.
8
1. Kinematics

1

KINEMATICS

1.1. The subject of mechanics. Kinematics of a point.
Problems of kinematic. Methods of the point’s motion setting

Theoretical mechanics is the science of the general laws of
mechanical motion and interaction of material bodies.
Motion is the mode of existence of matter. The simplest form of
matter in motion is a mechanical movement. The mechanical motion
is that the body changes its position in space in relation to other
objects in the course of time.
The notion of force is used in classical mechanics to account the
mechanical interaction that occurs between the bodies. However, the
nature of movement of the body depends on the forces as well as on
inertia of the body. The measure of inertia of a body is its mass,
which depends on the amount of the body.
Thus, the basic concepts of classical mechanics are: moving
matter (physical bodies), the space and time as forms of existence of
matter in motion, mass, as a measure of the inertia of material
bodies, the force as a measure of the mechanical interaction between
the bodies.
The relationships between the basic concepts of mechanics is
defined by axioms or basic laws of motion, which were given by
Newton.
First law (the law of inertia). Every body stays in its state of rest
or uniform rectilinear motion as long as it isn’t forced by the applied
forces to change this state.
Second law (the basic law of dynamics). Change of linear mo-
mentum is proportional to the applied driving force and occurs in the
direction of the straight line which the force acts along:

9
Textbook on Theoretical Mechanics

d(mv) 
F
dt

Third law (the law of action and reaction). Every action has an
equal and opposite reaction. In other words the interaction of two
bodies on each other are equal and opposite in direction:

 
F F
A B

Theoretical mechanics, as any science that uses mathematical
methods, does not deal with the real material objects, but with their
models. Such models are mass points, the system of mass points,
perfectly rigid bodies, deformable continuum.

Kinematics of the point

Basic concepts. Kinematics studies the motion of bodies from a
geometric point of view, excluding the causes of variation of this
motion (forces).
Mass point is the body, the size of which can be ignored when
studying its motion. The mass of mass point is not taken into account
in kinematics.
Position of the body in space can be defined only in relation to
an arbitrarily chosen another body, called the body or frame of
reference. If the position of the body relative to the chosen reference
frame doesn’t change with time, the body stays in rest relative to this
reference frame. But if the position of the body is changed, this body
moves relative to a given reference frame. Thus, the motion and the
rest are relative concepts, and make sense only in relation to a
particular reference frame.
Form of trajectory depends on the reference frame.
The motion of the body relative to the chosen reference frame
will be known, if at any arbitrary moment of time its position can be
determined (relative to this reference frame). The position of the
point is determined by the appropriate parameters (coordinates), and

10