As part of any educational course, the study of physics begins with mechanics. Not from theoretical, not from applied or computational, but from good old classical mechanics. This mechanics is also called Newtonian mechanics. According to legend, a scientist was walking in the garden and saw an apple falling, and it was this phenomenon that prompted him to discover the law of universal gravitation. Of course, the law has always existed, and Newton only gave it a form understandable to people, but his merit is priceless. In this article we will not describe the laws of Newtonian mechanics in as much detail as possible, but we will outline the fundamentals, basic knowledge, definitions and formulas that can always play into your hands.

Mechanics is a branch of physics, a science that studies the movement of material bodies and the interactions between them.

The word itself is of Greek origin and is translated as “the art of building machines.” But before we build machines, we are still like the Moon, so let’s follow in the footsteps of our ancestors and study the movement of stones thrown at an angle to the horizon, and apples falling on our heads from a height h.

Why does the study of physics begin with mechanics? Because this is completely natural, shouldn’t we start with thermodynamic equilibrium?!

Mechanics is one of the oldest sciences, and historically the study of physics began precisely with the foundations of mechanics. Placed within the framework of time and space, people, in fact, could not start with something else, no matter how much they wanted. Moving bodies are the first thing we pay attention to.

What is movement?

Mechanical motion is a change in the position of bodies in space relative to each other over time.

It is after this definition that we quite naturally come to the concept of a frame of reference. Changing the position of bodies in space relative to each other. Key words here: relative to each other . After all, a passenger in a car moves relative to the person standing on the side of the road at a certain speed, and is at rest relative to his neighbor in the seat next to him, and moves at some other speed relative to the passenger in the car that is overtaking them.

That is why, in order to normally measure the parameters of moving objects and not get confused, we need reference system - rigidly interconnected reference body, coordinate system and clock. For example, the earth moves around the sun in a heliocentric frame of reference. In everyday life, we carry out almost all our measurements in a geocentric reference system associated with the Earth. The earth is a body of reference relative to which cars, planes, people, and animals move.

Mechanics, as a science, has its own task. The task of mechanics is to know the position of a body in space at any time. In other words, mechanics builds a mathematical description of motion and finds connections between the physical quantities that characterize it.

In order to move further, we need the concept “ material point " They say that physics is an exact science, but physicists know how many approximations and assumptions have to be made in order to agree on this very accuracy. No one has ever seen a material point or smelled an ideal gas, but they exist! They are simply much easier to live with.

A material point is a body whose size and shape can be neglected in the context of this problem.

Sections of classical mechanics

Mechanics consists of several sections

• Kinematics
• Dynamics
• Statics

Kinematics from a physical point of view, it studies exactly how a body moves. In other words, this section deals with the quantitative characteristics of movement. Find speed, path - typical kinematics problems

Dynamics solves the question of why it moves the way it does. That is, it considers the forces acting on the body.

Statics studies the balance of bodies under the influence of forces, that is, answers the question: why doesn’t it fall at all?

Limits of applicability of classical mechanics

Classical mechanics no longer claims to be a science that explains everything (at the beginning of the last century everything was completely different), and has a clear framework of applicability. In general, the laws of classical mechanics are valid in the world we are accustomed to in size (macroworld). They stop working in the case of the particle world, when quantum mechanics replaces classical mechanics. Also, classical mechanics is not applicable to cases when the movement of bodies occurs at a speed close to the speed of light. In such cases, relativistic effects become pronounced. Roughly speaking, within the framework of quantum and relativistic mechanics - classical mechanics, this is a special case when the dimensions of the body are large and the speed is small.

Generally speaking, quantum and relativistic effects never go away; they also occur during the ordinary motion of macroscopic bodies at a speed much lower than the speed of light. Another thing is that the effect of these effects is so small that it does not go beyond the most accurate measurements. Classical mechanics will thus never lose its fundamental importance.

We will continue to study the physical foundations of mechanics in future articles. For a better understanding of the mechanics, you can always refer to to our authors, which will individually shed light on the dark spot of the most difficult task.

Statics is a branch of theoretical mechanics that studies the conditions of equilibrium of material bodies under the influence of forces, as well as methods for converting forces into equivalent systems.

In statics, a state of equilibrium is understood as a state in which all parts of a mechanical system are at rest relative to some inertial coordinate system. One of the basic objects of statics is forces and their points of application.

The force acting on a material point with a radius vector from other points is a measure of the influence of other points on the point under consideration, as a result of which it receives acceleration relative to the inertial reference system. Magnitude strength determined by the formula:
,
where m is the mass of the point - a quantity that depends on the properties of the point itself. This formula is called Newton's second law.

Application of statics in dynamics

An important feature of the equations of motion of an absolutely rigid body is that forces can be converted into equivalent systems. With this transformation, the equations of motion retain their form, but the system of forces acting on the body can be transformed into a simpler system. Thus, the point of application of force can be moved along the line of its action; forces can be expanded according to the parallelogram rule; forces applied at one point can be replaced by their geometric sum.

An example of such transformations is gravity. It acts on all points of a solid body. But the law of body motion will not change if the force of gravity distributed over all points is replaced by one vector applied at the center of mass of the body.

It turns out that if we add an equivalent system to the main system of forces acting on the body, in which the directions of the forces are changed to the opposite, then the body, under the influence of these systems, will be in equilibrium. Thus, the task of determining equivalent systems of forces is reduced to an equilibrium problem, that is, to a statics problem.

The main task of statics is the establishment of laws for transforming a system of forces into equivalent systems. Thus, statics methods are used not only in the study of bodies in equilibrium, but also in the dynamics of a rigid body, when transforming forces into simpler equivalent systems.

Statics of a material point

Let us consider a material point that is in equilibrium. And let n forces act on it, k = 1, 2, ..., n.

If a material point is in equilibrium, then the vector sum of the forces acting on it is equal to zero:
(1) .

In equilibrium, the geometric sum of the forces acting on a point is zero.

Geometric interpretation. If you place the beginning of the second vector at the end of the first vector, and place the beginning of the third at the end of the second vector, and then continue this process, then the end of the last, nth vector will be aligned with the beginning of the first vector. That is, we get a closed geometric figure, the lengths of the sides are equal to the modules of the vectors. If all vectors lie in the same plane, then we get a closed polygon.

It is often convenient to choose rectangular coordinate system Oxyz. Then the sums of the projections of all force vectors on the coordinate axes are equal to zero:

If you choose any direction specified by some vector, then the sum of the projections of the force vectors onto this direction is equal to zero:
.
Let's multiply equation (1) scalarly by the vector:
.
Here is the scalar product of the vectors and .
Note that the projection of the vector onto the direction of the vector is determined by the formula:
.

Rigid body statics

Moment of force about a point

Determination of moment of force

A moment of power, applied to the body at point A, relative to the fixed center O, is called a vector equal to the vector product of vectors and:
(2) .

Geometric interpretation

The moment of force is equal to the product of force F and arm OH.

Let the vectors and be located in the drawing plane. According to the property of the vector product, the vector is perpendicular to the vectors and, that is, perpendicular to the plane of the drawing. Its direction is determined by the right screw rule. In the figure, the torque vector is directed towards us. Absolute torque value:
.
Since then
(3) .

Using geometry, we can give a different interpretation of the moment of force. To do this, draw a straight line AH through the force vector. From the center O we lower the perpendicular OH to this straight line. The length of this perpendicular is called shoulder of strength. Then
(4) .
Since , then formulas (3) and (4) are equivalent.

Thus, absolute value of the moment of force relative to the center O is equal to product of force per shoulder this force relative to the selected center O.

When calculating torque, it is often convenient to decompose the force into two components:
,
Where . The force passes through point O. Therefore its moment is zero. Then
.
Absolute torque value:
.

Moment components in a rectangular coordinate system

If we choose a rectangular coordinate system Oxyz with a center at point O, then the moment of force will have the following components:
(5.1) ;
(5.2) ;
(5.3) .
Here are the coordinates of point A in the selected coordinate system:
.
The components represent the values ​​of the moment of force about the axes, respectively.

Properties of the moment of force relative to the center

The moment about the center O, due to the force passing through this center, is equal to zero.

If the point of application of the force is moved along a line passing through the force vector, then the moment, with such movement, will not change.

The moment from the vector sum of forces applied to one point of the body is equal to the vector sum of moments from each of the forces applied to the same point:
.

The same applies to forces whose continuation lines intersect at one point.

If the vector sum of forces is zero:
,
then the sum of the moments from these forces does not depend on the position of the center relative to which the moments are calculated:
.

Couple of forces

Couple of forces- these are two forces, equal in absolute magnitude and having opposite directions, applied to different points of the body.

A pair of forces is characterized by the moment they create. Since the vector sum of the forces entering the pair is zero, the moment created by the pair does not depend on the point relative to which the moment is calculated. From the point of view of static equilibrium, the nature of the forces involved in the pair does not matter. A couple of forces is used to indicate that a moment of force of a certain value acts on a body.

Moment of force about a given axis

There are often cases when we do not need to know all the components of the moment of a force about a selected point, but only need to know the moment of a force about a selected axis.

The moment of force about an axis passing through point O is the projection of the vector of the moment of force, relative to point O, onto the direction of the axis.

Properties of the moment of force about the axis

The moment about the axis due to the force passing through this axis is equal to zero.

The moment about an axis due to a force parallel to this axis is equal to zero.

Calculation of the moment of force about an axis

Let a force act on the body at point A. Let's find the moment of this force relative to the O′O′′ axis.

Let's construct a rectangular coordinate system. Let the Oz axis coincide with O′O′′. From point A we lower the perpendicular OH to O′O′′. Through points O and A we draw the Ox axis. We draw the Oy axis perpendicular to Ox and Oz. Let us decompose the force into components along the axes of the coordinate system:
.
The force intersects the O′O′′ axis. Therefore its moment is zero. The force is parallel to the O′O′′ axis. Therefore, its moment is also zero. Using formula (5.3) we find:
.

Note that the component is directed tangentially to the circle whose center is point O. The direction of the vector is determined by the right screw rule.

Conditions for the equilibrium of a rigid body

In equilibrium, the vector sum of all forces acting on the body is equal to zero and the vector sum of the moments of these forces relative to an arbitrary fixed center is equal to zero:
(6.1) ;
(6.2) .

We emphasize that the center O, relative to which the moments of forces are calculated, can be chosen arbitrarily. Point O can either belong to the body or be located outside it. Usually the center O is chosen to make calculations simpler.

The equilibrium conditions can be formulated in another way.

In equilibrium, the sum of the projections of forces on any direction specified by an arbitrary vector is equal to zero:
.
The sum of the moments of forces relative to an arbitrary axis O′O′′ is also equal to zero:
.

Sometimes such conditions turn out to be more convenient. There are cases when, by selecting axes, calculations can be made simpler.

Body center of gravity

Let's consider one of the most important forces - gravity. Here the forces are not applied at certain points of the body, but are continuously distributed throughout its volume. For every area of ​​the body with an infinitesimal volume ΔV, the force of gravity acts. Here ρ is the density of the body’s substance, and is the acceleration of gravity.

Let be the mass of an infinitely small part of the body. And let point A k determine the position of this section. Let us find the quantities related to gravity that are included in the equilibrium equations (6).

Let us find the sum of gravity forces formed by all parts of the body:
,
where is body mass. Thus, the sum of the gravitational forces of individual infinitesimal parts of the body can be replaced by one vector of the gravitational force of the entire body:
.

Let us find the sum of the moments of gravity, in a relatively arbitrary way for the selected center O:

.
Here we have introduced point C, which is called center of gravity bodies. The position of the center of gravity, in a coordinate system centered at point O, is determined by the formula:
(7) .

So, when determining static equilibrium, the sum of the gravity forces of individual parts of the body can be replaced by the resultant
,
applied to the center of mass of the body C, the position of which is determined by formula (7).

The position of the center of gravity for various geometric figures can be found in the corresponding reference books. If a body has an axis or plane of symmetry, then the center of gravity is located on this axis or plane. Thus, the centers of gravity of a sphere, circle or circle are located at the centers of the circles of these figures. The centers of gravity of a rectangular parallelepiped, rectangle or square are also located at their centers - at the points of intersection of the diagonals.

Uniformly (A) and linearly (B) distributed load.

There are also cases similar to gravity, when forces are not applied at certain points of the body, but are continuously distributed over its surface or volume. Such forces are called distributed forces or .

(Figure A). Also, as in the case of gravity, it can be replaced by a resultant force of magnitude , applied at the center of gravity of the diagram. Since the diagram in Figure A is a rectangle, the center of gravity of the diagram is located at its center - point C: | AC| = | CB|.

(Figure B). It can also be replaced by the resultant. The magnitude of the resultant is equal to the area of ​​the diagram:
.
The application point is at the center of gravity of the diagram. The center of gravity of a triangle, height h, is located at a distance from the base. That's why .

Friction forces

Sliding friction. Let the body be on a flat surface. And let be the force perpendicular to the surface with which the surface acts on the body (pressure force). Then the sliding friction force is parallel to the surface and directed to the side, preventing the movement of the body. Its greatest value is:
,
where f is the friction coefficient. The friction coefficient is a dimensionless quantity.

Rolling friction. Let a round shaped body roll or be able to roll on the surface. And let be the pressure force perpendicular to the surface from which the surface acts on the body. Then a moment of friction forces acts on the body, at the point of contact with the surface, preventing the movement of the body. The greatest value of the friction moment is equal to:
,
where δ is the rolling friction coefficient. It has the dimension of length.

References:
S. M. Targ, Short course in theoretical mechanics, “Higher School”, 2010.

Kinematics of a point.

1. Subject of theoretical mechanics. Basic abstractions.

Theoretical mechanics- is a science in which the general laws of mechanical motion and mechanical interaction of material bodies are studied

Mechanical movementis the movement of a body in relation to another body, occurring in space and time.

Mechanical interaction is the interaction of material bodies that changes the nature of their mechanical movement.

Statics is a branch of theoretical mechanics in which methods of transforming systems of forces into equivalent systems are studied and conditions for the equilibrium of forces applied to a solid body are established.

Kinematics - is a branch of theoretical mechanics that studies the movement of material bodies in space from a geometric point of view, regardless of the forces acting on them.

Dynamics is a branch of mechanics that studies the movement of material bodies in space depending on the forces acting on them.

Objects of study in theoretical mechanics:

material point,

system of material points,

Absolutely solid body.

Absolute space and absolute time are independent of one another. Absolute space - three-dimensional, homogeneous, motionless Euclidean space. Absolute time - flows from the past to the future continuously, it is homogeneous, the same at all points in space and does not depend on the movement of matter.

2. Subject of kinematics.

Kinematics - this is a branch of mechanics in which the geometric properties of the motion of bodies are studied without taking into account their inertia (i.e. mass) and the forces acting on them

To determine the position of a moving body (or point) with the body in relation to which the movement of this body is being studied, some coordinate system is rigidly associated, which together with the body forms reference system.

The main task of kinematics is to, knowing the law of motion of a given body (point), determine all the kinematic quantities that characterize its movement (speed and acceleration).

3. Methods for specifying the movement of a point

· The natural way

It should be known:

The trajectory of the point;

Origin and direction of reference;

The law of motion of a point along a given trajectory in the form (1.1)

· Coordinate method

Equations (1.2) are the equations of motion of point M.

The equation for the trajectory of point M can be obtained by eliminating the time parameter « t » from equations (1.2)

· Vector method

(1.3) Relationship between coordinate and vector methods of specifying the movement of a point (1.4)

Relationship between coordinate and natural methods of specifying the movement of a point

Determine the trajectory of the point by eliminating time from equations (1.2);

-- find the law of motion of a point along a trajectory (use the expression for the differential of the arc)

After integration, we obtain the law of motion of a point along a given trajectory:

The connection between the coordinate and vector methods of specifying the motion of a point is determined by equation (1.4)

4. Determining the speed of a point using the vector method of specifying motion.

Let at a moment in timetthe position of the point is determined by the radius vector, and at the moment of timet 1 – radius vector, then for a period of time the point will move.

(1.5) average point speed, the direction of the vector is the same as that of the vector

Speed ​​of a point at a given time

To obtain the speed of a point at a given time, it is necessary to make a passage to the limit

(1.6)

(1.7)

Velocity vector of a point at a given time equal to the first derivative of the radius vector with respect to time and directed tangentially to the trajectory at a given point.

(unit¾ m/s, km/h)

Average acceleration vector has the same direction as the vectorΔ v , that is, directed towards the concavity of the trajectory.

Acceleration vector of a point at a given time equal to the first derivative of the velocity vector or the second derivative of the radius vector of the point with respect to time.

(unit - )

How is the vector located in relation to the trajectory of the point?

In rectilinear motion, the vector is directed along the straight line along which the point moves. If the trajectory of a point is a flat curve, then the acceleration vector , as well as the vector ср, lies in the plane of this curve and is directed towards its concavity. If the trajectory is not a plane curve, then the vector ср will be directed towards the concavity of the trajectory and will lie in the plane passing through the tangent to the trajectory at the pointM and a line parallel to the tangent at an adjacent pointM 1 . IN limit when pointM 1 strives for M this plane occupies the position of the so-called osculating plane. Therefore, in the general case, the acceleration vector lies in the contacting plane and is directed towards the concavity of the curve.

20th ed. - M.: 2010.- 416 p.

The book outlines the fundamentals of the mechanics of a material point, a system of material points and a rigid body in a volume corresponding to the programs of technical universities. Many examples and problems are given, the solutions of which are accompanied by appropriate methodological instructions. For full-time and part-time students of technical universities.

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TABLE OF CONTENTS
Preface to the Thirteenth Edition 3
Introduction 5
SECTION ONE STATICS OF A SOLID BODY
Chapter I. Basic concepts and initial provisions of Articles 9
41. Absolutely rigid body; force. Statics problems 9
12. Initial provisions of statics » 11
$ 3. Connections and their reactions 15
Chapter II. Addition of forces. Converging Force System 18
§4. Geometrically! Method of adding forces. Resultant of converging forces, expansion of forces 18
f 5. Projections of force onto an axis and onto a plane, Analytical method of specifying and adding forces 20
16. Equilibrium of a system of converging forces_. . . 23
17. Solving statics problems. 25
Chapter III. Moment of force about the center. Power pair 31
i 8. Moment of force relative to the center (or point) 31
| 9. Couple of forces. Couple moment 33
f 10*. Theorems on equivalence and addition of pairs 35
Chapter IV. Bringing the system of forces to the center. Equilibrium conditions... 37
f 11. Theorem on parallel transfer of force 37
112. Bringing a system of forces to a given center - . , 38
§ 13. Conditions for equilibrium of a system of forces. Theorem about the moment of the resultant 40
Chapter V. Flat system of forces 41
§ 14. Algebraic moments of force and pairs 41
115. Reducing a plane system of forces to its simplest form.... 44
§ 16. Equilibrium of a plane system of forces. The case of parallel forces. 46
§ 17. Solving problems 48
118. Equilibrium of systems of bodies 63
§ 19*. Statically determinate and statically indeterminate systems of bodies (structures) 56"
f 20*. Definition of internal efforts. 57
§ 21*. Distributed forces 58
E22*. Calculation of flat trusses 61
Chapter VI. Friction 64
! 23. Laws of sliding friction 64
: 24. Reactions of rough bonds. Friction angle 66
: 25. Equilibrium in the presence of friction 66
(26*. Friction of thread on cylindrical surface 69
1 27*. Rolling friction 71
Chapter VII. Spatial force system 72
§28. Moment of force about the axis. Principal vector calculation
and the main moment of the force system 72
§ 29*. Bringing the spatial system of forces to its simplest form 77
§thirty. Equilibrium of an arbitrary spatial system of forces. Case of parallel forces
Chapter VIII. Center of gravity 86
§31. Center of Parallel Forces 86
§ 32. Force field. Center of gravity of a rigid body 88
§ 33. Coordinates of the centers of gravity of homogeneous bodies 89
§ 34. Methods for determining the coordinates of the centers of gravity of bodies. 90
§ 35. Centers of gravity of some homogeneous bodies 93
SECTION TWO KINEMATICS OF A POINT AND A RIGID BODY
Chapter IX. Kinematics of point 95
§ 36. Introduction to kinematics 95
§ 37. Methods for specifying the movement of a point. . 96
§38. Point velocity vector. 99
§ 39. Vector of the “torque of point 100”
§40. Determining the speed and acceleration of a point using the coordinate method of specifying motion 102
§41. Solving point kinematics problems 103
§ 42. Axes of a natural trihedron. Speed ​​numeric value 107
§ 43. Tangent and normal acceleration of a point 108
§44. Some special cases of motion of a point PO
§45. Graphs of motion, speed and acceleration of a point 112
§ 46. Solving problems< 114
§47*. Speed ​​and acceleration of a point in polar coordinates 116
Chapter X. Translational and rotational motions of a rigid body. . 117
§48. Forward movement 117
§ 49. Rotational motion of a rigid body around an axis. Angular velocity and angular acceleration 119
§50. Uniform and uniform rotation 121
§51. Velocities and accelerations of points of a rotating body 122
Chapter XI. Plane-parallel motion of a rigid body 127
§52. Equations of plane-parallel motion (movement of a plane figure). Decomposition of motion into translational and rotational 127
§53*. Determining the trajectories of points of a plane figure 129
§54. Determining the velocities of points on a plane figure 130
§ 55. Theorem on the projections of velocities of two points on a body 131
§ 56. Determination of the velocities of points of a plane figure using the instantaneous center of velocities. The concept of centroids 132
§57. Problem solving 136
§58*. Determination of accelerations of points of a plane figure 140
§59*. Instant acceleration center "*"*
Chapter XII*. The motion of a rigid body around a fixed point and the motion of a free rigid body 147
§ 60. Motion of a rigid body having one fixed point. 147
§61. Euler's kinematic equations 149
§62. Velocities and accelerations of body points 150
§ 63. General case of motion of a free rigid body 153
Chapter XIII. Complex point movement 155
§ 64. Relative, portable and absolute movements 155
§ 65, Theorem on the addition of velocities » 156
§66. Theorem on the addition of accelerations (Coriolns theorem) 160
§67. Problem solving 16*
Chapter XIV*. Complex motion of a rigid body 169
§68. Addition of translational movements 169
§69. Addition of rotations around two parallel axes 169
§70. Spur gears 172
§ 71. Addition of rotations around intersecting axes 174
§72. Addition of translational and rotational movements. Screw movement 176
SECTION THREE DYNAMICS OF A POINT
Chapter XV: Introduction to Dynamics. Laws of dynamics 180
§ 73. Basic concepts and definitions 180
§ 74. Laws of dynamics. Problems of the dynamics of a material point 181
§ 75. Systems of units 183
§76. Main types of forces 184
Chapter XVI. Differential equations of motion of a point. Solving point dynamics problems 186
§ 77. Differential equations, motion of a material point No. 6
§ 78. Solution of the first problem of dynamics (determination of forces from a given movement) 187
§ 79. Solution of the main problem of dynamics for rectilinear motion of a point 189
§ 80. Examples of solving problems 191
§81*. Fall of a body in a resisting medium (in the air) 196
§82. Solution of the main problem of dynamics, with the curvilinear movement of a point 197
Chapter XVII. General theorems of point dynamics 201
§83. The amount of movement of a point. Force impulse 201
§ S4. Theorem on the change in momentum of a point 202
§ 85. Theorem on the change in angular momentum of a point (theorem of moments) " 204
§86*. Movement under the influence of a central force. Law of areas... 266
§ 8-7. Work of force. Power 208
§88. Examples of calculating work 210
§89. Theorem on the change in kinetic energy of a point. "... 213J
Chapter XVIII. Not free and relative to the movement of the point 219
§90. Non-free movement of the point. 219
§91. Relative motion of a point 223
§ 92. The influence of the Earth’s rotation on the balance and movement of bodies... 227
§ 93*. Deviation of the falling point from the vertical due to the rotation of the Earth "230
Chapter XIX. Rectilinear oscillations of a point. . . 232
§ 94. Free vibrations without taking into account resistance forces 232
§ 95. Free oscillations with viscous resistance (damped oscillations) 238
§96. Forced vibrations. Rezonayas 241
Chapter XX*. Movement of a body in the field of gravity 250
§ 97. Motion of a thrown body in the gravitational field of the Earth "250
§98. Artificial Earth satellites. Elliptical trajectories. 254
§ 99. The concept of weightlessness."Local frames of reference 257
SECTION FOUR DYNAMICS OF THE SYSTEM AND SOLID BODY
G i a v a XXI. Introduction to system dynamics. Moments of inertia. 263
§ 100. Mechanical system. External and internal forces 263
§ 101. Mass of the system. Center of mass 264
§ 102. Moment of inertia of a body relative to an axis. Radius of inertia. . 265
$ 103. Moments of inertia of a body about parallel axes. Huygens' theorem 268
§ 104*. Centrifugal moments of inertia. Concepts about the main axes of inertia of a body 269
$105*. The moment of inertia of a body about an arbitrary axis. 271
Chapter XXII. Theorem on the motion of the center of mass of the system 273
$ 106. Differential equations of motion of a system 273
§ 107. Theorem on the motion of the center of mass 274
$ 108. Law of conservation of motion of the center of mass 276
§ 109. Solving problems 277
Chapter XXIII. Theorem on the change in the quantity of a movable system. . 280
$ BUT. System movement quantity 280
§111. Theorem on the change in momentum 281
§ 112. Law of conservation of momentum 282
$113*. Application of the theorem to the movement of liquid (gas) 284
§ 114*. Body of variable mass. Rocket movement 287
Gdava XXIV. Theorem on changing the angular momentum of a system 290
§ 115. Main moment of momentum of the system 290
$ 116. Theorem on changes in the principal moment of the system’s quantities of motion (theorem of moments) 292
$117. Law of conservation of principal angular momentum. . 294
$118. Problem solving 295
$119*. Application of the theorem of moments to the movement of liquid (gas) 298
§ 120. Equilibrium conditions for a mechanical system 300
Chapter XXV. Theorem on the change in kinetic energy of a system. . 301.
§ 121. Kinetic energy of the system 301
$122. Some cases of calculating work 305
$ 123. Theorem on the change in kinetic energy of a system 307
$124. Solving problems 310
$125*. Mixed problems "314
$126. Potential force field and force function 317
$127, Potential Energy. Law of conservation of mechanical energy 320
Chapter XXVI. "Application of general theorems to rigid body dynamics 323
$12&. Rotational motion of a rigid body around a fixed axis ". 323"
$129. Physical pendulum. Experimental determination of moments of inertia. 326
$130. Plane-parallel motion of a rigid body 328
$131*. Elementary theory of the gyroscope 334
$132*. The motion of a rigid body around a fixed point and the motion of a free rigid body 340
Chapter XXVII. D'Alembert's principle 344
$ 133. D'Alembert's principle for a point and a mechanical system. . 344
$ 134. Main vector and main moment of inertia 346
$135. Solving problems 348
$136*, Didemical reactions acting on the axis of a rotating body. Balancing rotating bodies 352
Chapter XXVIII. The principle of possible displacements and the general equation of dynamics 357
§ 137. Classification of connections 357
§ 138. Possible movements of the system. Number of degrees of freedom. . 358
§ 139. The principle of possible movements 360
§ 140. Solving problems 362
§ 141. General equation of dynamics 367
Chapter XXIX. Equilibrium conditions and equations of motion of a system in generalized coordinates 369
§ 142. Generalized coordinates and generalized velocities. . . 369
§ 143. Generalized forces 371
§ 144. Conditions for equilibrium of a system in generalized coordinates 375
§ 145. Lagrange equations 376
§ 146. Solving problems 379
Chapter XXX*. Small oscillations of the system around the position of stable equilibrium 387
§ 147. The concept of stability of equilibrium 387
§ 148. Small free oscillations of a system with one degree of freedom 389
§ 149. Small damped and forced oscillations of a system with one degree of freedom 392
§ 150. Small combined oscillations of a system with two degrees of freedom 394
Chapter XXXI. Elementary Impact Theory 396
§ 151. Basic equation of impact theory 396
§ 152. General theorems of impact theory 397
§ 153. Impact recovery coefficient 399
§ 154. Impact of a body on a stationary obstacle 400
§ 155. Direct central impact of two bodies (impact of balls) 401
§ 156. Loss of kinetic energy during an inelastic collision of two bodies. Carnot's theorem 403
§ 157*. Hitting a rotating body. Impact Center 405
Subject index 409

General theorems on the dynamics of a system of bodies. Theorems on the movement of the center of mass, on the change in momentum, on the change in the main angular momentum, on the change in kinetic energy. D'Alembert's principles and possible movements. General equation of dynamics. Lagrange equations.

Content

The work done by the force, is equal to the scalar product of the force vectors and the infinitesimal displacement of the point of its application:
,
that is, the product of the absolute values ​​of the vectors F and ds by the cosine of the angle between them.

The work done by the moment of force, is equal to the scalar product of the torque vectors and the infinitesimal angle of rotation:
.

d'Alembert's principle

The essence of d'Alembert's principle is to reduce problems of dynamics to problems of statics. To do this, it is assumed (or it is known in advance) that the bodies of the system have certain (angular) accelerations. Next, inertial forces and (or) moments of inertial forces are introduced, which are equal in magnitude and opposite in direction to the forces and moments of forces that, according to the laws of mechanics, would create given accelerations or angular accelerations

Let's look at an example. The body undergoes translational motion and is acted upon by external forces. We further assume that these forces create an acceleration of the system's center of mass. According to the theorem on the motion of the center of mass, the center of mass of a body would have the same acceleration if a force acted on the body. Next we introduce the force of inertia:
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After this, the dynamics problem:
.
;
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For rotational motion proceed in the same way. Let the body rotate around the z axis and be acted upon by external moments of force M e zk . We assume that these moments create an angular acceleration ε z. Next, we introduce the moment of inertia forces M И = - J z ε z. After this, the dynamics problem:
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Turns into a statics problem:
;
.

The principle of possible movements

The principle of possible displacements is used to solve statics problems. In some problems, it gives a shorter solution than composing equilibrium equations. This is especially true for systems with connections (for example, systems of bodies connected by threads and blocks) consisting of many bodies

The principle of possible movements.
For the equilibrium of a mechanical system with ideal connections, it is necessary and sufficient that the sum of the elementary works of all active forces acting on it for any possible movement of the system is equal to zero.

Possible system relocation- this is a small movement in which the connections imposed on the system are not broken.

Ideal connections- these are connections that do not perform work when the system moves. More precisely, the amount of work performed by the connections themselves when moving the system is zero.

General equation of dynamics (D'Alembert - Lagrange principle)

The D'Alembert-Lagrange principle is a combination of the D'Alembert principle with the principle of possible movements. That is, when solving a dynamic problem, we introduce inertial forces and reduce the problem to a static problem, which we solve using the principle of possible displacements.

D'Alembert-Lagrange principle.
When a mechanical system with ideal connections moves, at each moment of time the sum of the elementary works of all applied active forces and all inertial forces on any possible movement of the system is zero:
.
This equation is called general equation of dynamics.

Lagrange equations

Generalized q coordinates 1 , q 2 , ..., q n is a set of n quantities that uniquely determine the position of the system.

The number of generalized coordinates n coincides with the number of degrees of freedom of the system.

Generalized speeds are derivatives of generalized coordinates with respect to time t.

Generalized forces Q 1 , Q 2 , ..., Q n .
Let us consider a possible movement of the system, at which the coordinate q k will receive a movement δq k. The remaining coordinates remain unchanged. Let δA k be the work done by external forces during such a movement. Then
δA k = Q k δq k , or
.

If, with a possible movement of the system, all coordinates change, then the work done by external forces during such movement has the form:
δA = Q 1 δq 1 + Q 2 δq 2 + ... + Q n δq n.
Then the generalized forces are partial derivatives of the work on displacements:
.

For potential forces with potential Π,
.

Lagrange equations are the equations of motion of a mechanical system in generalized coordinates:

Here T is kinetic energy. It is a function of generalized coordinates, velocities and, possibly, time. Therefore, its partial derivative is also a function of generalized coordinates, velocities and time. Next, you need to take into account that coordinates and velocities are functions of time. Therefore, to find the total derivative with respect to time, you need to apply the rule of differentiation of a complex function:
.

References:
S. M. Targ, Short course in theoretical mechanics, “Higher School”, 2010.