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Theoretical and analytical mechanics. Solving problems in theoretical mechanics

In all its beauty and elegance. With its help, Newton once derived his law of universal gravitation based on Kepler's three empirical laws. The subject, in general, is not that complicated and is relatively easy to understand. But passing is difficult, since teachers are often terribly picky (like Pavlova, for example). When solving problems, you need to be able to solve diffuses and calculate integrals.

Key Ideas

In essence, theoretical mechanics in this course is the application of the variational principle to calculate the “motion” of various physical systems. Calculus of variations is covered briefly in the course Integral Equations and Calculus of Variations. Lagrange's equations are Euler's equations, which are the solution to a problem with fixed ends.

One problem can usually be solved by 3 different methods at once:

Lagrange method (Lagrange function, Lagrange equations)
Hamilton method (Hamilton function, Hamilton equations)
Hamilton-Jacobi method (Hamilton-Jacobi equation)

It is important to choose the simplest one for a specific task.

Materials

First semester (test)

Basic formulas

View in large size!

Theory

Videos

Lectures by V.R. Khalilova - Attention! Not all lectures are recorded

Second semester (exam)

We need to start with the fact that different groups take the exam differently. Usually Examination ticket consists of 2 theoretical questions and 1 task. Questions are required for everyone, but you can either get rid of a task (for excellent work in the semester + written tests) or grab an extra one (and more than one). Here you will be told about the rules of the game at seminars. In the groups of Pavlova and Pimenov, theormin is practiced, which is a kind of admission to the exam. It follows that this theory must be known perfectly.

Exam in Pavlova groups goes something like this: First, a ticket with 2 term questions. There is little time to write, and the key here is to write it absolutely perfectly. Then Olga Serafimovna will be kind to you and the rest of the exam will go very pleasantly. Next is a ticket with 2 theory questions + n problems (depending on your work in the semester). Theory in theory can be written off. Solve problems. Having a lot of problems in an exam is not the end if you know how to solve them perfectly. This can be turned into an advantage - for each exam point you get a +, +-, -+ or -. The rating is given “based on the overall impression” => if in theory everything is not perfect for you, but then you get 3+ for the tasks, then the overall impression is good. But if you had no problems in the exam and the theory is not ideal, then there is nothing to smooth it out.

Theory

Julia. Lecture notes (2014, pdf) - both semesters, 2nd stream
Second stream tickets part 1 (lecture notes and part for tickets) (pdf)
Second stream tickets and table of contents for all these parts (pdf)
Answers to tickets for the 1st stream (2016, pdf) - in printed form, very convenient
Recognized theory for the exam for Pimenov groups (2016, pdf) - both semesters
Answers to theorymin for Pimenov groups (2016, pdf) - neat and seemingly error-free

Tasks

Pavlova's seminars 2nd semester (2015, pdf) - neat, beautifully and clearly written
Problems that may be on the exam (jpg) - once in some shaggy year they were in the 2nd stream, may also be relevant for V.R. groups. Khalilov (he gives similar problems in kr)
Problems for tickets (pdf)- for both streams (on the 2nd stream these tasks were in A.B. Pimenov’s groups)

Examples of solving problems in theoretical mechanics

Statics

Problem conditions

Kinematics

Kinematics of a material point

The task

Determining the speed and acceleration of a point using the given equations of its motion.
Using the given equations of motion of a point, establish the type of its trajectory and for the moment of time t = 1 s find the position of the point on the trajectory, its speed, total, tangential and normal acceleration, as well as the radius of curvature of the trajectory.
Equations of motion of a point:
x = 12 sin(πt/6), cm;
y = 6 cos 2 (πt/6), cm.

Kinematic analysis of a flat mechanism

The task

The flat mechanism consists of rods 1, 2, 3, 4 and a slider E. The rods are connected to each other, to the sliders and fixed supports using cylindrical hinges. Point D is located in the middle of rod AB. The lengths of the rods are equal, respectively
l 1 = 0.4 m; l 2 = 1.2 m; l 3 = 1.6 m; l 4 = 0.6 m.

The relative arrangement of the mechanism elements in a specific version of the problem is determined by the angles α, β, γ, φ, ϑ. Rod 1 (rod O 1 A) rotates around a fixed point O 1 counterclockwise with a constant angular velocity ω 1.

For a given position of the mechanism it is necessary to determine:

linear velocities V A, V B, V D and V E of points A, B, D, E;
angular velocities ω 2, ω 3 and ω 4 of links 2, 3 and 4;
linear acceleration a B of point B;
angular acceleration ε AB of link AB;
positions of instantaneous speed centers C 2 and C 3 of links 2 and 3 of the mechanism.

Determination of absolute speed and absolute acceleration of a point

The task

The diagram below considers the motion of point M in the trough of a rotating body. Using the given equations of portable motion φ = φ(t) and relative motion OM = OM(t), determine the absolute speed and absolute acceleration of a point at a given point in time.

Download the solution to the problem >>>

Dynamics

Integration of differential equations of motion of a material point under the influence of variable forces

The task

Load D with mass m, having received at point A initial speed V 0 moves in a curved pipe ABC located in a vertical plane. In a section AB, the length of which is l, the load is acted upon by a constant force T (its direction is shown in the figure) and a force R of the medium resistance (the modulus of this force R = μV 2, the vector R is directed opposite to the speed V of the load).

The load, having finished moving in section AB, at point B of the pipe, without changing the value of its speed module, moves to section BC. In section BC, the load is acted upon by a variable force F, the projection F x of which on the x axis is given.

Considering the load to be a material point, find the law of its motion in section BC, i.e. x = f(t), where x = BD. Neglect the friction of the load on the pipe.

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Theorem on the change in kinetic energy of a mechanical system

The task

The mechanical system consists of weights 1 and 2, a cylindrical roller 3, two-stage pulleys 4 and 5. The bodies of the system are connected by threads wound on the pulleys; sections of threads are parallel to the corresponding planes. The roller (a solid homogeneous cylinder) rolls along the supporting plane without sliding. The radii of the stages of pulleys 4 and 5 are respectively equal to R 4 = 0.3 m, r 4 = 0.1 m, R 5 = 0.2 m, r 5 = 0.1 m. The mass of each pulley is considered to be uniformly distributed along its outer rim . The supporting planes of loads 1 and 2 are rough, the sliding friction coefficient for each load is f = 0.1.

Under the action of a force F, the modulus of which changes according to the law F = F(s), where s is the displacement of the point of its application, the system begins to move from a state of rest. When the system moves, pulley 5 is acted upon by resistance forces, the moment of which relative to the axis of rotation is constant and equal to M 5 .

Determine the value of the angular velocity of pulley 4 at the moment in time when the displacement s of the point of application of force F becomes equal to s 1 = 1.2 m.

Download the solution to the problem >>>

Application of the general equation of dynamics to the study of the motion of a mechanical system

The task

For a mechanical system, determine the linear acceleration a 1 . Assume that the masses of blocks and rollers are distributed along the outer radius. Cables and belts should be considered weightless and inextensible; there is no slippage. Neglect rolling and sliding friction.

Download the solution to the problem >>>

Application of d'Alembert's principle to determining the reactions of the supports of a rotating body

The task

The vertical shaft AK, rotating uniformly with an angular velocity ω = 10 s -1, is fixed by a thrust bearing at point A and a cylindrical bearing at point D.

Rigidly attached to the shaft are a weightless rod 1 with a length of l 1 = 0.3 m, at the free end of which there is a load with a mass of m 1 = 4 kg, and a homogeneous rod 2 with a length of l 2 = 0.6 m, having a mass of m 2 = 8 kg. Both rods lie in the same vertical plane. The points of attachment of the rods to the shaft, as well as the angles α and β are indicated in the table. Dimensions AB=BD=DE=EK=b, where b = 0.4 m. Take the load as a material point.

Neglecting the mass of the shaft, determine the reactions of the thrust bearing and the bearing.

Theoretical mechanics

Theoretical mechanics- the science of general laws mechanical movement and interaction of material bodies. Being essentially one of the branches of physics, theoretical mechanics, having absorbed a fundamental basis in the form of axiomatics, became an independent science and was widely developed due to its extensive and important applications in natural science and technology, of which it is one of the foundations.

In physics

In physics, theoretical mechanics refers to the part of theoretical physics that studies mathematical methods of classical mechanics that are alternative to the direct application of Newton's laws (so-called analytical mechanics). This includes, in particular, methods based on Lagrange equations, principles of least action, Hamilton-Jacobi equation, etc.

It should be emphasized that analytical mechanics can be either non-relativistic - then it intersects with classical mechanics, or relativistic. The principles of analytical mechanics are so general that its relativization does not lead to fundamental difficulties.

In technical sciences

In technical sciences, theoretical mechanics means a set of physical and mathematical methods that facilitate calculations of mechanisms, structures, aircraft etc. (so-called applied mechanics or engineering mechanics). Almost always, these methods are derived from the laws of classical mechanics - mainly from Newton's laws, although in some technical problems Some of the methods of analytical mechanics turn out to be useful.

Theoretical mechanics is based on a certain number of laws established in experimental mechanics, accepted as truths that do not require proof - axioms. These axioms replace the inductive truths of experimental mechanics. Theoretical mechanics is deductive in nature. Relying on axioms as a foundation known and tested by practice and experiment, theoretical mechanics erects its edifice with the help of strict mathematical deductions.

Theoretical mechanics, as a part of natural science that uses mathematical methods, deals not with the actual material objects themselves, but with their models. Such models studied in theoretical mechanics are

material points and systems of material points,
absolutely rigid bodies and systems of rigid bodies,
deformable continuous media.

Usually in theoretical mechanics there are such sections as

Methods are widely used in theoretical mechanics

vector calculus and differential geometry,

Theoretical mechanics was the basis for the creation of many applied areas that have received great development. These are fluid and gas mechanics, mechanics of deformable solids, theory of oscillations, dynamics and strength of machines, gyroscopy, control theory, flight theory, navigation, etc.

In higher education

Theoretical mechanics is one of the fundamental mechanical disciplines in the mechanics and mathematics faculties of Russian universities. In this discipline, annual All-Russian, national and regional student Olympiads are held, as well as International Olympiad.

Notes

Literature

theoretical mechanics- general mechanics A section of mechanics that sets out the basic laws and principles of this science and studies general properties movement of mechanical systems. [Collection of recommended terms. Issue 102. Theoretical mechanics. Academy of Sciences of the USSR. Committee… …

See MECHANICS Dictionary foreign words, included in the Russian language. Pavlenkov F., 1907 ...

theoretical mechanics- theoretical mechanics; general mechanics A branch of mechanics that sets out the basic laws and principles of this science and studies the general properties of the motion of mechanical systems... Polytechnic terminological explanatory dictionary

Noun, number of synonyms: 1 theoretical mechanics (2) Dictionary of synonyms ASIS. V.N. Trishin. 2013… Synonym dictionary

theoretical mechanics- teorinė mechanika statusas T sritis fizika atitikmenys: engl. theoretical mechanics vok. theoretische Mechanik, f rus. theoretical mechanics, f pranc. mécanique rationnelle, f … Fizikos terminų žodynas

- (Greek mechanike, from mechane machine). Part of applied mathematics, the science of force and resistance in machines; the art of applying force to action and building machines. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. MECHANICS... ... Dictionary of foreign words of the Russian language

Mechanics- Science of mechanical movement and mechanical interaction of material bodies. [Collection of recommended terms. Issue 102. Theoretical mechanics. Academy of Sciences of the USSR. Committee of Scientific and Technical Terminology. 1984] Topics theoretical... ... Technical Translator's Guide

- (from the Greek mechanike (techne) the science of machines, the art of building machines), the science of mechanics. movement mater. bodies and the interactions occurring between them. Under mechanical movement is understood as a change in the relative position of bodies over time or ... Physical encyclopedia

Theoretical physics is a branch of physics in which the main way of understanding nature is to create mathematical models of phenomena and compare them with reality. In this formulation, theoretical physics is... ... Wikipedia

- (Greek: μηχανική art of building machines) area of physics that studies the movement of material bodies and the interaction between them. Movement in mechanics is the change in time of the relative position of bodies or their parts in space.... ... Wikipedia

V. I. Dront, V. V. Dubinin, M. M. Ilyin and others; Under general ed. K. S. Kolesnikova “Course of Theoretical Mechanics: Textbook for Universities” Publishing House of Moscow State Technical University named after. N. E. Bauman, 2005, 736 pp. (7.17 mb. djvu)

The textbook presents such sections as: kinematics, statics, dynamics of a point, a rigid body and a mechanical system. As well as analytical mechanics, theory of vibrations, theory of impact, introduction to the dynamics of bodies variable mass, fundamentals of celestial mechanics. All sections are accompanied by examples of problem solving. The course of the textbook is presented according to the course of lectures and in accordance with the program read by the authors at MSTU. N. E. Bauman.

The book can be used as tutorial for students of mechanical engineering universities and technical universities. Will assist graduate students and teachers in preparing and conducting lectures and classes. As well as specialists working in the field of applied statics and dynamics of mechanical systems, mechanical and instrument engineering.
ISBN 5-7038-1695-5 (Vol. 1)
ISBN 5-7038-1371-9

Preface.

The textbook is the result of many years of teaching activity of the authors at MSTU. N. E. Bauman, which produces design engineers and researchers who specialize in the field of mechanical and instrument engineering. It was preceded by textbooks also written by university teachers V.V. Dobronravov, A.L. Dvornikov, K.N. Nikitin, which were republished several times and played a big role in teaching students.

The transition to university engineering education required an expansion of the course content, a more complete physical interpretation of a number of issues and a natural complication of the mathematical apparatus used. For this purpose, in the “Kinematics” section, the chapter “ General case motion of a rigid body."

Statics is presented as an independent section, since subjects such as strength of materials, theory of mechanisms and mechanics of machines, machine parts, engineering design subjects, require the student to have a clear understanding of the methods of transformation and transmission of force interactions in machine mechanisms.

Significant additions have been made in the “Dynamics” section. Here integral variational principles and elements of celestial mechanics are introduced; the theory of oscillations, the theory of impact and some other issues are more fully presented.

Some information from vector theory 9
B. 1. Scalar and vector quantities. Unit vectors 9
AT 2. Projections of a vector onto the axis and plane 11
V.Z. Vector coordinates. Analytical assignment of a vector. Radius vector point 12
AT 4. Addition and subtraction of vectors 14
AT 5. Vector multiplication 16
AT 6. Vectors and matrices 24
AT 7. Relationship between vector projections on the axes of two rectangular coordinate systems 29
AT 8. Vector function. Vector hodograph. Differentiation of a vector by a scalar argument 32

Section 1. KINEMATICS

Chapter I. Kinematics of a point 39
1.1. Point speed 39
1.2. Acceleration point 41
1.3. Vector method of specifying the movement of a point 44
1.4. Coordinate method of specifying the movement of a point 44
1.5. The natural way specifying the movement of point 61

Chapter 2. The simplest motions of a rigid body 70
2.1. Degrees of freedom and the velocity projection theorem 70
2.2. Translational motion of a rigid body 73
2.3. Rotation of a rigid body around a fixed axis 73

Chapter 3. Plane motion of a rigid body 85
3.1. Decomposition of the plane motion of a rigid body into translational and rotational motions 85
3.2. Equations of motion, angular velocity and angular acceleration of a rigid body in plane motion 87
3.3. Velocities of body points during plane motion 89
3.4. Instant speed center 90
3.5. Instant center of rotation. Centroids 94
3.6. Calculation of the angular velocity of a rigid body in plane motion
3.7. Accelerations of body points during plane motion 98
3.8. Instant Acceleration Center 102
3.9. Methods for calculating the angular acceleration of a body in plane motion 106

Chapter 4. Rotation of a rigid body around a fixed point 110
4.1. Number of degrees of freedom. Euler angles. Rotation equations 110
4.2. Direction cosine matrix. Body point trajectory 114
4.3. Instantaneous rotation axis. Axoids 116
4.4. Instantaneous angular velocity and angular acceleration 119
4.5. Velocities of body points. Euler's kinematic equations 122
4.6. Accelerations of body points 128
4.7. angular acceleration of the body 130

Chapter 5. General case of rigid body motion 134
5.1. Number of degrees of freedom. Generalized coordinates. Equations of motion 134
5.2. Trajectory of an arbitrary point of the body 139
5.3. Speed of an arbitrary body point 140
5.4. Acceleration of an arbitrary point of a body 141

Chapter 6. Complex point movement 143
6.1. Relative, portable and absolute movements of a point 143
6.2. Absolute and relative derivatives of a vector. Formula Borax 145
6.3. Velocity addition theorem 148
6.4. The theorem on the addition of accelerations, or the Coriolis kinematic theorem. Coriolis acceleration 150
6.5. Addition of accelerations in special cases of portable motion 153

Chapter 7. Complex rigid body motion 162
7.1. Theorem on the addition of angular velocities in complex motion of a rigid body 162
7.2. Addition of rotations around intersecting axes 164
7.3. Addition of rotations around parallel axes. Pararotations 165
7.4. Addition of translational movements 168
7.5. Addition of translational and rotational motions 169

Section 2. STATICS

Chapter 8. Axioms and fundamental principles of statics 173
8.1. Axioms of statics 174
8.2. Main types of bonds and their reactions 177
83. System of converging forces 181
8.4. Moment of force relative to a point and relative to an axis 189
8.5. Addition of parallel forces. Pair of forces 196
8.6. Reducing the system of forces to the simplest system 204

Chapter 9 Balance of bodies 214
9.1. Conditions for equilibrium of a system of forces 214
9.2. Equilibrium of a system of bodies 222
9.3. Definition of internal forces 225
9.4. Statically definable and statically indeterminate systems of bodies 227
9.5. Calculation of flat trusses 228
9.6. Distributed forces 229

Chapter 10. Friction 236
10.1. Laws of sliding friction 236
10.2. Reactions of rough surfaces. Friction angle 237
10.3. Rolling coupling reaction 238
10.4. Equilibrium of a body in the presence of friction. Friction cone 239

Chapter 11. Center of gravity 248
11.1. Center of parallel force system 248
11.2. Center of gravity of a rigid body 251
11.3. Methods for determining the coordinates of the centers of gravity of bodies 253

Chapter 12. Equilibrium of flexible and inextensible thread 260
12.1. Differential equations balance thread 260
12.2. Special cases external forces 263
12.3. Chain line 265

Section 3. DYNAMICS

Chapter 13. Dynamics of a material point 271
13.1. Axioms of dynamics 271
13.2. Differential equations of motion of a material point 273
13.3. Two main problems of the dynamics of a material point 275
13.4. Movement of a non-free material point 280
13.5. Dynamics of relative motion 288
13.6. Equilibrium and motion of a material point relative to the Earth 293

Chapter 14. Geometry of masses 298
14.1. Center of mass of mechanical system 298
14.2. Moments of inertia 301
14.3. Dependence of moments of inertia relative to parallel axes (Huygens-Steiner theorem) 304
14.4. Moments of inertia of homogeneous bodies 305
14.5. Moments of inertia of homogeneous bodies of rotation 310
14.6. Moment of inertia about an axis passing through given point 315
14.7. Ellipsoid of inertia. Main axes of inertia 318
14.8. Properties of the main axes of inertia of a body 321
14.9. Determining the direction of the main axes of inertia 326

Chapter 13. General theorems of dynamics 331
13.1. Mechanical system. External and internal forces 331
15.2. Differential equations of motion of a mechanical system 334
15.3. Theorem on the motion of the center of mass of a mechanical system 335
15.4. Theorem on the change in momentum 342
15.5. Theorem on the change in angular momentum of a material point. Theorem on the change in the main moment of momentum of a mechanical system 353
15.6. Theorem on the change in kinetic energy 382
15.7. Potential force field 400
15.8. Examples of using general theorems of dynamics 412

Chapter 16. Rigid body dynamics 424
16.1. Translational motion of a rigid body. Rotation of a rigid body around a fixed axis. Plane motion of a rigid body 424
16.2. Spherical motion of a rigid body 436
16.3. General case of rigid body motion 465

Chapter 17. D'Alembert's principle. Dynamic reactions of connections 469
17.1. D'Alembert's principle. Inertia force 469
17.2. D'Alembert's principle for a mechanical system 471
17.3. Main vector and main point inertia forces 473
17.4. Dynamic reactions of supports 475
17.5. Static and dynamic balance of a rigid body rotating around a fixed axis 482
17.6. Balancing rotors 487

Chapter 18. Fundamentals of Analytical Mechanics 493
18.1. Basic concepts 493
18.2. Possible work of force. Perfect connections 504
18.3. Generalized forces 507
18.4. Differential principles of analytical mechanics 513
18.5. Lagrange equation of the second kind 527
18.6. Integral variational principles of mechanics 536

Chapter 19. Oscillation theory 555
19.1. Stability of the equilibrium position of a mechanical system 555
19.2. Differential equations of small oscillations of a linear system with one degree of freedom 559
19.3. Free motions of a linear system with one degree of freedom 568
19.4. Forced oscillations of a linear system with one degree of freedom 582
19.5. Fundamentals of the theory of recording instruments 607
19.6. Vibration Protection Basics 612
19.7. Differential equations of small oscillations of a linear system with a finite number of degrees of freedom 615
19.8. Free vibrations of a linear conservative system with two degrees of freedom 625
19.9. Forced oscillations of a linear system with two degrees of freedom under harmonic excitation.
Dynamic vibration damper 637
19.10. Oscillations linear systems with a finite number of degrees of freedom 645

Chapter 20. Impact theory 653
20.1. Basic concepts and assumptions. Impact model 653
20.2. Theorems on the change in momentum and on the movement of the center of mass of the system upon impact 658
20.3. Theorem on the change in the main moment of momentum of the system upon impact 660
20.4. Recovery factor 662
20.5. Theorem on the change in the kinetic energy of a system upon impact. Carnot's theorem 664
20.6. A blow to a body rotating around a fixed axis. Impact Center 672
20.7. Impact on a rigid body with a fixed point. Impact center. Impact on a free rigid body 677
20.8.0 connections upon impact. General equation mechanics 679
20.9 Lagrange equation of the second kind for impact in mechanical system 682
20.10. Impact of two bodies during translational motion. Energy ratios 684
20.11. Impact of a material point on a stationary rough surface 691
20.12. Hit two balls. Hertz Model 699

Chapter 21. Introduction to the dynamics of bodies of variable mass 705
21.1. Basic concepts and assumptions 705
21.2. Generalized Meshchersky equation, reactive forces 707
21.3. Special cases of the Meshchersky equation 709
21.4. Some classical problems of the dynamics of a point of variable mass 712

Chapter 22. Fundamentals of Celestial Mechanics 717
22.1. Binet's formulas 717.
22.2. The law of universal gravitation. Kepler's Laws 720
22.3. Energy classification of orbits 723
22.4. Motion of a point in orbit 725
22.5. Two body problem 727
22.6.0 n-body problem and other problems of celestial mechanics 729

Download the book for free 7.17 MB. djvu

Introduction

Theoretical mechanics is one of the most important fundamental general scientific disciplines. It plays a significant role in the training of engineers of any specialization. General engineering disciplines are based on the results of theoretical mechanics: strength of materials, machine parts, theory of mechanisms and machines, and others.

The main task of theoretical mechanics is the study of the movement of material bodies under the influence of forces. An important particular task is the study of the equilibrium of bodies under the influence of forces.

Lecture course. Theoretical mechanics

The structure of theoretical mechanics. Basics of statics

Equilibrium conditions for an arbitrary system of forces.

Equilibrium equations for a rigid body.

Flat system of forces.

Special cases of rigid body equilibrium.

Balance problem for a beam.

Determination of internal forces in rod structures.

Fundamentals of point kinematics.

Natural coordinates.

Euler's formula.

Distribution of accelerations of points of a rigid body.

Translational and rotational movements.

Plane-parallel motion.

Complex point movement.

Basics of point dynamics.

Differential equations of motion of a point.

Particular types of force fields.

Fundamentals of the dynamics of a system of points.

General theorems on the dynamics of a system of points.

Dynamics of rotational motion of the body.

Dobronravov V.V., Nikitin N.N. Course of theoretical mechanics. M., graduate School, 1983.

Butenin N.V., Lunts Ya.L., Merkin D.R. Course of theoretical mechanics, parts 1 and 2. M., Higher School, 1971.

Petkevich V.V. Theoretical mechanics. M., Nauka, 1981.

Collection of tasks for coursework in theoretical mechanics. Ed. A.A. Yablonsky. M., Higher School, 1985.

Lecture 1. The structure of theoretical mechanics. Basics of statics

In theoretical mechanics, the motion of bodies relative to other bodies, which are physical reference systems, is studied.

Mechanics allows not only to describe, but also to predict the movement of bodies, establishing causal relationships in a certain, very wide range of phenomena.

Basic abstract models of real bodies:

material point – has mass, but no size;

absolutely solid – a volume of finite dimensions, completely filled with a substance, and the distances between any two points of the medium filling the volume do not change during movement;

continuous deformable medium – fills a finite volume or unlimited space; the distances between points in such a medium can vary.

Of these, systems:

System of free material points;

Connected systems;

An absolutely solid body with a cavity filled with liquid, etc.

"Degenerate" models:

Infinitely thin rods;

Infinitely thin plates;

Weightless rods and threads connecting each other material points, etc.

From experience: mechanical phenomena occur differently in different places physical reference system. This property is the heterogeneity of space, determined by the physical reference system. Here, heterogeneity is understood as the dependence of the nature of the occurrence of a phenomenon on the place in which we observe this phenomenon.

Another property is anisotropy (non-isotropy), the movement of a body relative to a physical reference system can be different depending on the direction. Examples: river flow along the meridian (from north to south - Volga); projectile flight, Foucault pendulum.

The properties of the reference system (inhomogeneity and anisotropy) make it difficult to observe the movement of a body.

Practically free from this - geocentric system: the center of the system is in the center of the Earth and the system does not rotate relative to the “fixed” stars). The geocentric system is convenient for calculating movements on Earth.

For celestial mechanics(for solar system bodies): heliocentric frame of reference, which moves with the center of mass solar system and does not rotate relative to the “fixed” stars. For this system not yet discovered heterogeneity and anisotropy of space

in relation to mechanical phenomena.

So, the abstract is introduced inertial frame of reference for which space is homogeneous and isotropic in relation to mechanical phenomena.

Inertial reference frame- one whose own motion cannot be detected by any mechanical experiment. Thought experiment: “a point alone in the whole world” (isolated) is either at rest or moving in a straight line and uniformly.

All reference systems moving relative to the original one rectilinearly and uniformly will be inertial. This allows the introduction of a unified Cartesian coordinate system. Such a space is called Euclidean.

Conventional agreement - take the right coordinate system (Fig. 1).

IN time– in classical (non-relativistic) mechanics absolutely, the same for all reference systems, that is, the initial moment is arbitrary. In contrast to relativistic mechanics, where the principle of relativity is applied.

The state of motion of the system at time t is determined by the coordinates and velocities of the points at this moment.

Real bodies interact and forces arise that change the state of motion of the system. This is the essence of theoretical mechanics.

How is theoretical mechanics studied?

The doctrine of the equilibrium of a set of bodies of a certain frame of reference - section statics.

Chapter kinematics: part of mechanics in which dependencies between quantities characterizing the state of motion of systems are studied, but the reasons causing a change in the state of motion are not considered.

After this, we will consider the influence of forces [MAIN PART].

Chapter dynamics: part of mechanics that deals with the influence of forces on the state of motion of systems of material objects.

Principles for constructing the main course - dynamics:

1) based on a system of axioms (based on experience, observations);

Constantly - ruthless control of practice. Sign of exact science – presence of internal logic (without it - a set of unrelated recipes)!

Static is called that part of mechanics where the conditions are studied that the forces acting on a system of material points must satisfy in order for the system to be in equilibrium, and the conditions for the equivalence of systems of forces.

Equilibrium problems in elementary statics will be considered using exclusively geometric methods based on the properties of vectors. This approach is used in geometric statics(in contrast to analytical statics, which is not considered here).

The positions of various material bodies will be related to the coordinate system, which we will take as stationary.

Ideal models of material bodies:

1) material point – a geometric point with mass.

2) an absolutely rigid body is a collection of material points, the distances between which cannot be changed by any actions.

By forces we will call objective causes that are the result of the interaction of material objects, capable of causing the movement of bodies from a state of rest or changing the existing movement of the latter.

Since force is determined by the movement it causes, it also has a relative nature, depending on the choice of reference system.

The question of the nature of forces is considered in physics.

A system of material points is in equilibrium if, being at rest, it does not receive any movement from the forces acting on it.

From everyday experience: forces have a vector nature, that is, magnitude, direction, line of action, point of application. The condition for equilibrium of forces acting on a rigid body is reduced to the properties of vector systems.

Summarizing the experience of studying the physical laws of nature, Galileo and Newton formulated the basic laws of mechanics, which can be considered as axioms of mechanics, since they have are based on experimental facts.

Axiom 1. The action of several forces on a point of a rigid body is equivalent to the action of one resultant force constructed according to the rule of vector addition (Fig. 2).

Consequence. The forces applied to a point on a rigid body add up according to the parallelogram rule.

Axiom 2. Two forces applied to a rigid body mutually balanced if and only if they are equal in size, directed in opposite directions and lie on the same straight line.

Axiom 3. The action of a system of forces on a rigid body will not change if add to this system or discard from it two forces of equal magnitude directed towards opposite sides and lying on the same straight line.

Consequence. The force acting on a point of a rigid body can be transferred along the line of action of the force without changing the equilibrium (that is, the force is a sliding vector, Fig. 3)

1) Active - create or are capable of creating the movement of a rigid body. For example, weight force.

2) Passive - do not create movement, but limit the movement of a solid body, preventing movement. For example, the tension force of an inextensible thread (Fig. 4).

Axiom 4. The action of one body on a second is equal and opposite to the action of this second body on the first ( action equals reaction).

We will call the geometric conditions limiting the movement of points connections.

Terms of communication: for example,

- rod of indirect length l.

- flexible non-stretchable thread of length l.

Forces caused by connections and preventing movement are called forces of reactions.

Axiom 5. The connections imposed on a system of material points can be replaced by reaction forces, the action of which is equivalent to the action of the connections.

When passive forces cannot balance the action of active forces, movement begins.

Two particular problems of statics

1. System of converging forces acting on a rigid body

A system of converging forces This is called a system of forces whose lines of action intersect at one point, which can always be taken as the origin of coordinates (Fig. 5).

Projections of the resultant:

;

If , then the force causes the motion of the rigid body.

Equilibrium condition for a converging system of forces:

2. Balance of three forces

If three forces act on a rigid body, and the lines of action of the two forces intersect at some point A, equilibrium is possible if and only if the line of action of the third force also passes through point A, and the force itself is equal in magnitude and opposite in direction to the sum (Fig. 6).

Examples:

Moment of force about point O let's define it as a vector, in size equal to twice the area of a triangle, the base of which is the force vector with the vertex at a given point O; direction– orthogonal to the plane of the triangle in question in the direction from which the rotation produced by the force around point O is visible counterclockwise. is the moment of the sliding vector and is free vector(Fig.9).

So: or

Where ;;.

Where F is the force modulus, h is the shoulder (the distance from the point to the direction of the force).

Moment of force about the axis is the algebraic value of the projection onto this axis of the vector of the moment of force relative to an arbitrary point O taken on the axis (Fig. 10).

This is a scalar independent of the choice of point. Indeed, let us expand :|| and in the plane.

About moments: let O 1 be the point of intersection with the plane. Then:

a) from - moment => projection = 0.

b) from - moment along => is a projection.

So, moment about an axis is the moment of the force component in a plane perpendicular to the axis relative to the point of intersection of the plane and the axis.

Varignon's theorem for a system of converging forces:

Moment of resultant force for a system of converging forces relative to an arbitrary point A is equal to the sum of the moments of all component forces relative to the same point A (Fig. 11).

Proof in the theory of convergent vectors.

Explanation: addition of forces according to the parallelogram rule => the resulting force gives a total moment.

Control questions:

1. Name the main models of real bodies in theoretical mechanics.

2. Formulate the axioms of statics.

3. What is called the moment of force about a point?

Lecture 2. Equilibrium conditions for an arbitrary system of forces

From the basic axioms of statics, elementary operations on forces follow:

1) force can be transferred along the line of action;

2) forces whose lines of action intersect can be added according to the parallelogram rule (according to the rule of vector addition);

3) to the system of forces acting on a rigid body, you can always add two forces, equal in magnitude, lying on the same straight line and directed in opposite directions.

Elementary operations do not change the mechanical state of the system.

Let's call two systems of forces equivalent, if one from the other can be obtained using elementary operations (as in the theory of sliding vectors).

A system of two parallel forces, equal in magnitude and directed in opposite directions, is called a couple of forces(Fig. 12).

Moment of a couple of forces- a vector equal in size to the area of the parallelogram built on the vectors of the pair, and directed orthogonally to the plane of the pair in the direction from where the rotation imparted by the vectors of the pair is seen to occur counterclockwise.

, that is, the moment of force relative to point B.

A pair of forces is completely characterized by its moment.

A pair of forces can be transferred by elementary operations to any plane parallel to the plane of the pair; change the magnitude of the forces of the pair in inverse proportion to the shoulders of the pair.

Pairs of forces can be added, and the moments of pairs of forces are added according to the rule of addition of (free) vectors.

Bringing a system of forces acting on a rigid body to an arbitrary point (center of reduction)- means replacing the current system with a simpler one: system of three forces, one of which passes through a predetermined point, and the other two represent a pair.

It can be proven using elementary operations (Fig. 13).

A system of converging forces and a system of pairs of forces.

- resultant force.

Resulting pair.

That's what needed to be shown.

Two systems of forces will equivalent if and only if both systems are reduced to one resultant force and one resultant pair, that is, when the conditions are met:

General case of equilibrium of a system of forces acting on a rigid body

Let us reduce the system of forces to (Fig. 14):

Resultant force through the origin;

The resulting pair, moreover, through point O.

That is, they led to and - two forces, one of which passes through a given point O.

Equilibrium, if the two on the same straight line are equal and opposite in direction (axiom 2).

Then it passes through point O, that is.

So, General terms equilibrium of a rigid body:

These conditions are valid for an arbitrary point in space.

Control questions:

1. List the elementary operations on forces.

2. What systems of forces are called equivalent?

3. Write the general conditions for the equilibrium of a rigid body.

Lecture 3. Equilibrium equations for a rigid body

Let O be the origin of coordinates; – resultant force; – moment of the resultant pair. Let point O1 be new center casts (Fig. 15).

New power system:

When the reduction point changes, => only changes (in one direction with one sign, in the other direction with another). That is, the point: the lines match

Analytically: (colinearity of vectors)

; coordinates of point O1.

This is the equation of a straight line, for all points of which the direction of the resulting vector coincides with the direction of the moment of the resulting pair - the straight line is called dynamo.

If the dynamism => on the axis, then the system is equivalent to one resultant force, which is called resultant force of the system. At the same time, always, that is.

Four cases of bringing forces:

1.) ;- dynamism.

2.) ;- resultant.

3.) ;- pair.

4.) ;- balance.

Two vector equilibrium equations: the main vector and the main moment are equal to zero,.

Or six scalar equations in projections onto Cartesian coordinate axes:

Here:

The complexity of the type of equations depends on the choice of the reduction point => the skill of the calculator.

Finding the equilibrium conditions for a system of solid bodies in interaction<=>the problem of the equilibrium of each body separately, and the body is acted upon by external forces and internal forces (the interaction of bodies at points of contact with equal and oppositely directed forces - axiom IV, Fig. 17).

Let us choose for all bodies of the system one adduction center. Then for each body with the equilibrium condition number:

, , (= 1, 2, …, k)

where , is the resulting force and moment of the resulting pair of all forces, except internal reactions.

The resulting force and moment of the resulting pair of forces of internal reactions.

Formally summing by and taking into account the IV axiom

we get necessary conditions for the equilibrium of a solid body:

Example.

Equilibrium: = ?

Control questions:

1. Name all cases of bringing a system of forces to one point.

2. What is dynamism?

3. Formulate the necessary conditions for equilibrium of a system of solid bodies.

Lecture 4. Flat force system

A special case of the general delivery of the problem.

Let all active forces lie in the same plane - for example, a sheet. Let us choose point O as the reduction center - in the same plane. We obtain the resulting force and the resulting steam in the same plane, that is (Fig. 19)