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Within any training 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, the science that studies movement. material bodies and 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. Keywords 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 heliocentric system countdown. 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 or smelled a material point 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 the classical one is replaced by quantum 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 physical foundations mechanics in the following 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.

The course covers: the kinematics of a point and a rigid body (and from different points of view it is proposed to consider the problem of the orientation of a rigid body), classical problems of the dynamics of mechanical systems and the dynamics of a rigid body, elements of celestial mechanics, the movement of systems of variable composition, impact theory, differential equations analytical dynamics.

The course presents all the traditional sections of theoretical mechanics, but special attention is paid to the consideration of the most meaningful and valuable sections of dynamics and methods of analytical mechanics for theory and applications; statics is studied as a section of dynamics, and in the section of kinematics the concepts and mathematical apparatus necessary for the section of dynamics are introduced in detail.

Informational resources

Gantmakher F.R. Lectures on analytical mechanics. – 3rd ed. – M.: Fizmatlit, 2001.
Zhuravlev V.F. Fundamentals of theoretical mechanics. – 2nd ed. – M.: Fizmatlit, 2001; 3rd ed. – M.: Fizmatlit, 2008.
Markeev A.P. Theoretical mechanics. – Moscow – Izhevsk: Research Center “Regular and Chaotic Dynamics”, 2007.

Requirements

The course is designed for students who are proficient in analytical geometry and linear algebra within the scope of the first-year program at a technical university.

Course program

1. Kinematics of a point
1.1. Kinematics problems. Cartesian coordinate system. Decomposition of a vector in an orthonormal basis. Radius vector and point coordinates. Speed ​​and acceleration of a point. Trajectory of movement.
1.2. Natural trihedron. Decomposition of velocity and acceleration in the axes of a natural trihedron (Huygens' theorem).
1.3. Curvilinear coordinates of a point, examples: polar, cylindrical and spherical coordinate systems. Components of velocity and projections of acceleration on the axis of a curvilinear coordinate system.

2. Methods for specifying the orientation of a rigid body
2.1. Solid. A fixed and body-related coordinate system.
2.2. Orthogonal rotation matrices and their properties. Euler's finite rotation theorem.
2.3. Active and passive points of view on orthogonal transformation. Addition of turns.
2.4. Angles of final rotation: Euler angles and "airplane" angles. Expressing an orthogonal matrix in terms of finite rotation angles.

3. Spatial motion of a rigid body
3.1. Progressive and rotational movement solid body. Angular velocity and angular acceleration.
3.2. Distribution of velocities (Euler's formula) and accelerations (Rivals' formula) of points of a rigid body.
3.3. Kinematic invariants. Kinematic screw. Instant screw axis.

4. Plane-parallel motion
4.1. The concept of plane-parallel motion of a body. Angular velocity and angular acceleration in the case of plane-parallel motion. Instantaneous velocity center.

5. Complex motion of a point and a rigid body
5.1. Fixed and moving coordinate systems. Absolute, relative and portable movements of a point.
5.2. Theorem on the addition of velocities during complex motion of a point, relative and portable velocities of a point. Coriolis theorem on the addition of accelerations during complex motion of a point, relative, transport and Coriolis accelerations of a point.
5.3. Absolute, relative and portable angular velocity and angular acceleration of a body.

6. Motion of a rigid body with a fixed point (quaternion presentation)
6.1. The concept of complex and hypercomplex numbers. Quaternion algebra. Quaternion product. Conjugate and inverse quaternion, norm and modulus.
6.2. Trigonometric representation of a unit quaternion. Quaternion method of specifying body rotation. Euler's finite rotation theorem.
6.3. Relationship between quaternion components in different bases. Addition of turns. Rodrigue-Hamilton parameters.

7. Examination paper

8. Basic concepts of dynamics.
8.1 Impulse, angular momentum (kinetic moment), kinetic energy.
8.2 Power of forces, work of forces, potential and total energy.
8.3 Center of mass (center of inertia) of the system. The moment of inertia of the system about the axis.
8.4 Moments of inertia about parallel axes; Huygens–Steiner theorem.
8.5 Tensor and ellipsoid of inertia. Main axes of inertia. Properties of axial moments of inertia.
8.6 Calculation of angular momentum and kinetic energy of a body using the inertia tensor.

9. Basic theorems of dynamics in inertial and non-inertial reference systems.
9.1 Theorem on the change in momentum of a system in an inertial reference frame. Theorem on the motion of the center of mass.
9.2 Theorem on the change in angular momentum of a system in an inertial reference frame.
9.3 Theorem on the change in the kinetic energy of a system in an inertial reference frame.
9.4 Potential, gyroscopic and dissipative forces.
9.5 Basic theorems of dynamics in non-inertial reference systems.

10. Motion of a rigid body with a fixed point by inertia.
10.1 Dynamic Euler equations.
10.2 Euler's case, first integrals of dynamic equations; permanent rotations.
10.3 Interpretations of Poinsot and McCullagh.
10.4 Regular precession in the case of dynamic symmetry of the body.

11. Motion of a heavy rigid body with a fixed point.
11.1 General formulation of the problem of the motion of a heavy rigid body around.
fixed point. Euler's dynamic equations and their first integrals.
11.2 Qualitative analysis of the motion of a rigid body in the Lagrange case.
11.3 Forced regular precession of a dynamically symmetric rigid body.
11.4 Basic formula of gyroscopy.
11.5 The concept of the elementary theory of gyroscopes.

12. Dynamics of a point in the central field.
12.1 Binet's equation.
12.2 Orbital equation. Kepler's laws.
12.3 Scattering problem.
12.4 Two-body problem. Equations of motion. Area integral, energy integral, Laplace integral.

13. Dynamics of systems of variable composition.
13.1 Basic concepts and theorems on changes in basic dynamic quantities in systems of variable composition.
13.2 Movement of a material point of variable mass.
13.3 Equations of motion of a body of variable composition.

14. Theory of impulsive movements.
14.1 Basic concepts and axioms of the theory of impulsive movements.
14.2 Theorems on changes in basic dynamic quantities during impulsive movement.
14.3 Impulsive motion of a rigid body.
14.4 Collision of two rigid bodies.
14.5 Carnot's theorems.

15. Test

Learning outcomes

As a result of mastering the discipline, the student must:

• Know:
  • basic concepts and theorems of mechanics and the resulting methods for studying the motion of mechanical systems;
• Be able to:
  • correctly formulate problems in terms of theoretical mechanics;
  • develop mechanical and mathematical models that adequately reflect the basic properties of the phenomena under consideration;
  • apply the acquired knowledge to solve relevant specific tasks;
• Own:
  • skills in solving classical problems of theoretical mechanics and mathematics;
  • skills in studying mechanics problems and constructing mechanical and mathematical models that adequately describe various mechanical phenomena;
  • skills in the practical use of methods and principles of theoretical mechanics when solving problems: force calculations, determining the kinematic characteristics of bodies when in various ways tasks of motion, determination of the law of motion of material bodies and mechanical systems under the influence of forces;
  • skills to independently master new information in the process of production and scientific activity using modern educational and information technologies;

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 material point with a radius vector from other points - this 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 such a transformation, the equations of motion retain their form, but the system of forces acting on the body can be transformed into a more simple 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 vector sum The forces acting on it are 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 - scalar product 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 property 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 value moment:
.
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).

Center of gravity position for different geometric shapes can be found in the relevant 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 theoretical mechanics, "Higher School", 2010.

List of exam questions

1. Technical mechanics, its definition. Mechanical movement and mechanical interaction. Material point, mechanical system, absolutely rigid body.

Technical mechanics – the science of mechanical movement and interaction of material bodies.

Mechanics is one of the most ancient sciences. The term “Mechanics” was introduced by the outstanding ancient philosopher Aristotle.

The achievements of scientists in the field of mechanics make it possible to solve complex practical problems in the field of technology and, in essence, not a single natural phenomenon can be understood without understanding it from the mechanical side. And not a single creation of technology can be created without taking into account certain mechanical laws.

Mechanical movement is change over time mutual position in the space of material bodies or the relative position of parts of a given body.

Mechanical interaction - these are the actions of material bodies on each other, as a result of which there is a change in the movement of these bodies or a change in their shape (deformation).

Basic concepts:

Material point is a body whose dimensions can be neglected under given conditions. It has mass and the ability to interact with other bodies.

Mechanical system is a set of material points, the position and movement of each of which depend on the position and movement of other points of the system.

Absolutely solid body (ATB) is a body whose distance between any two points always remains unchanged.

1. Theoretical mechanics and its sections. Problems of theoretical mechanics.

Theoretical mechanics is a branch of mechanics in which the laws of motion of bodies and the general properties of these movements are studied.

Theoretical mechanics consists of three sections: statics, kinematics and dynamics.

Statics examines the equilibrium of bodies and their systems under the influence of forces.

Kinematics examines the general geometric properties of the motion of bodies.

Dynamics studies the movement of bodies under the influence of forces.

Statics tasks:

1. Transformation of systems of forces acting on the ATT into systems equivalent to them, i.e. bringing this system of forces to its simplest form.

2. Determination of the equilibrium conditions for the system of forces acting on the ATT.

To solve these problems, two methods are used: graphical and analytical.

1. Equilibrium. Force, system of forces. Resultant force, concentrated force and distributed forces.

Equilibrium - This is the state of rest of a body in relation to other bodies.

Force – this is the main measure of the mechanical interaction of material bodies. It is a vector quantity, i.e. Strength is characterized by three elements:

Application point;

Line of action (direction);

Modulus (numeric value).

Force system – this is the totality of all forces acting on the considered absolutely rigid body (ATB)

The system of forces is called convergent , if the lines of action of all forces intersect at one point.

The system is called flat , if the lines of action of all forces lie in the same plane, otherwise spatial.

The system of forces is called parallel , if the lines of action of all forces are parallel to each other.

The two systems of forces are called equivalent , if one system of forces acting on an absolutely rigid body can be replaced by another system of forces without changing the state of rest or motion of the body.

Balanced or equivalent to zero is called a system of forces under the influence of which free ATT can be at rest.

Resultant force is a force whose action on a body or material point is equivalent to the action of a system of forces on the same body.

By external forces

The force exerted on a body at any one point is called concentrated .

Forces acting on all points of a certain volume or surface are called distributed .

A body that is not prevented from moving in any direction by any other body is called free.

1. External and internal forces. Free and unfree body. The principle of liberation from ties.

By external forces are the forces with which the parts of a given body act on each other.

When solving most problems of statics, it is necessary to represent a non-free body as free, which is done using the principle of liberation, which is formulated as follows:

any unfree body can be considered as free if we discard connections and replace them with reactions.

As a result of the application of this principle, a body is obtained that is free from connections and is under the influence of a certain system of active and reactive forces.

1. Axioms of statics.

Conditions under which a body can be in equal vesii, are derived from several basic provisions, accepted without evidence, but confirmed by experiments , and called axioms of statics. The basic axioms of statics were formulated by the English scientist Newton (1642-1727), and therefore they are named after him.

Axiom I (axiom of inertia or Newton's first law).

Every body maintains its state of rest or rectilinear uniform motion, so far some Powers will not bring him out of this state.

The ability of a body to maintain its state of rest or linear uniform motion is called inertia. Based on this axiom, we consider a state of equilibrium to be a state when the body is at rest or moves rectilinearly and uniformly (i.e., by inertia).

Axiom II (axiom of interaction or Newton's third law).

If one body acts on the second with a certain force, then the second body simultaneously acts on the first with a force equal in magnitude to opposite in direction.

The set of forces applied to a given body (or system of bodies) is called system of forces. The force of action of a body on a given body and the reaction force of a given body do not represent a system of forces, since they are applied to different bodies.

If any system of forces has such a property that, after application to a free body, it does not change its state of equilibrium, then such a system of forces is called balanced.

Axiom III (condition of equilibrium of two forces).

For the equilibrium of a free rigid body under the action of two forces, it is necessary and sufficient that these forces be equal in magnitude and act in one straight line in opposite directions.

necessary to balance the two forces. This means that if a system of two forces is in equilibrium, then these forces must be equal in magnitude and act in one straight line in opposite directions.

The condition formulated in this axiom is sufficient to balance the two forces. This means that the reverse formulation of the axiom is valid, namely: if two forces are equal in magnitude and act along one straight line in opposite directions, then such a system of forces is necessarily in equilibrium.

In the following, we will get acquainted with the equilibrium condition, which will be necessary, but not sufficient for equilibrium.

Axiom IV.

The equilibrium of a solid body will not be disturbed if a system of balanced forces is applied to it or removed.

Corollary of the axioms III And IV.

The equilibrium of a rigid body will not be disturbed by the transfer of force along the line of its action.

Parallelogram axiom. This axiom is formulated as follows:

Resultant of two forces applied To body at one point, is equal in magnitude and coincides in direction with the diagonal of a parallelogram constructed on these forces, and is applied at the same point.

1. Connections, reactions of connections. Examples of connections.

Connections are called bodies that limit the movement of a given body in space. The force with which a body acts on a connection is called pressure; the force with which a bond acts on a body is called reaction. According to the axiom of interaction, reaction and pressure modulo equal and act in one straight line in opposite directions. Reaction and pressure are applied to various bodies. External forces acting on a body are divided into active And reactive. Active forces tend to move the body to which they are applied, and reactive forces, through connections, prevent this movement. The fundamental difference between active forces and reactive forces is that the magnitude of reactive forces, generally speaking, depends on the magnitude of active forces, but not vice versa. Active forces are often called

The direction of reactions is determined by the direction in which this connection prevents the movement of the body. The rule for determining the direction of reactions can be formulated as follows:

the direction of the reaction of the connection is opposite to the direction of movement destroyed by this connection.

1. Perfectly smooth plane

In this case the reaction R directed perpendicular to the reference plane towards the body.

2. Ideally smooth surface (Fig. 16).

In this case, the reaction R is directed perpendicular to the tangent plane t - t, i.e., normal to the supporting surface towards the body.

3. Fixed point or corner edge (Fig. 17, edge B).

In this case the reaction R in directed normal to the surface of an ideally smooth body towards the body.

4. Flexible connection (Fig. 17).

The reaction T of the flexible connection is directed along s v i z i. From Fig. 17 it can be seen that a flexible connection thrown over the block changes the direction of the transmitted force.

5. Ideally smooth cylindrical hinge (Fig. 17, hinge A; rice. 18, bearing D).

In this case, it is only known in advance that the reaction R passes through the hinge axis and is perpendicular to this axis.

6. Ideally smooth thrust bearing (Fig. 18, thrust bearing A).

The thrust bearing can be considered as a combination of a cylindrical hinge and a supporting plane. Therefore we will

7. Perfectly smooth ball joint (Fig. 19).

In this case, it is only known in advance that the reaction R passes through the center of the hinge.

8. A rod fixed at two ends in perfectly smooth hinges and loaded only at the ends (Fig. 18, rod BC).

In this case, the reaction of the rod is directed along the rod, since, according to Axiom III, the reactions of the hinges B and C when in equilibrium, the rod can only be directed along the line sun, i.e. along the rod.

1. System of converging forces. Addition of forces applied at one point.

Converging are called forces whose lines of action intersect at one point.

This chapter examines systems of converging forces whose lines of action lie in the same plane (plane systems).

Let's imagine that a flat system of five forces acts on the body, the lines of action of which intersect at point O (Fig. 10, a). In § 2 it was established that the force is sliding vector. Therefore, all forces can be transferred from the points of their application to the point O of the intersection of the lines of their action (Fig. 10, b).

Thus, any system of converging forces applied to various points bodies can be replaced by an equivalent system of forces applied to one point. This system of forces is often called a bundle of strength.