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 - this is the section theoretical mechanics, which studies methods for converting force systems into equivalent systems and establishes equilibrium conditions for forces applied to a solid body.

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.

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, saw an apple falling, and it was this phenomenon that prompted him to discover the law universal gravity. 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. 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 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.

Theoretical mechanics is a section of mechanics that sets out the basic laws of mechanical motion and mechanical interaction of material bodies.

Theoretical mechanics is a science that studies the movement of bodies over time (mechanical movements). It serves as the basis for other branches of mechanics (theory of elasticity, strength of materials, theory of plasticity, theory of mechanisms and machines, hydroaerodynamics) and many technical disciplines.

Mechanical movement- this is a change over time in the relative position in space of material bodies.

Mechanical interaction- this is an interaction as a result of which the mechanical movement changes or the relative position of body parts changes.

Rigid body statics

Statics is a section of theoretical mechanics that deals with problems of equilibrium of solid bodies and the transformation of one system of forces into another, equivalent to it.

Basic concepts and laws of statics
• Absolutely rigid body(solid body, body) is a material body, the distance between any points in which does not change.
• Material point is a body whose dimensions, according to the conditions of the problem, can be neglected.
• Free body- this is a body on the movement of which no restrictions are imposed.
• Unfree (bound) body is a body whose movement is subject to restrictions.
• Connections– these are bodies that prevent the movement of the object in question (a body or a system of bodies).
• Communication reaction is a force that characterizes the action of a bond on a solid body. If we consider the force with which a solid body acts on a bond to be an action, then the reaction of the bond is a reaction. In this case, the force - action is applied to the connection, and the reaction of the connection is applied to the solid body.
• Mechanical system is a collection of interconnected bodies or material points.
• Solid can be considered as a mechanical system, the positions and distances between points of which do not change.
• Force is a vector quantity that characterizes the mechanical action of one material body on another.
  Force as a vector is characterized by the point of application, direction of action and absolute value. The unit of force modulus is Newton.
• Line of action of force is a straight line along which the force vector is directed.
• Focused Power– force applied at one point.
• Distributed forces (distributed load)- these are forces acting on all points of the volume, surface or length of a body.
  The distributed load is specified by the force acting per unit volume (surface, length).
  The dimension of the distributed load is N/m 3 (N/m 2, N/m).
• External force is a force acting from a body that does not belong to the mechanical system under consideration.
• Inner strength is a force acting on a material point of a mechanical system from another material point belonging to the system under consideration.
• Force system is a set of forces acting on a mechanical system.
• Flat force system is a system of forces whose lines of action lie in the same plane.
• Spatial system of forces is a system of forces whose lines of action do not lie in the same plane.
• System of converging forces is a system of forces whose lines of action intersect at one point.
• Arbitrary system of forces is a system of forces whose lines of action do not intersect at one point.
• Equivalent force systems- these are systems of forces, the replacement of which one with another does not change the mechanical state of the body.
  Accepted designation: .
• Equilibrium- this is a state in which a body, under the action of forces, remains motionless or moves uniformly in a straight line.
• Balanced system of forces- this is a system of forces that, when applied to a free solid body, does not change its mechanical state (does not throw it out of balance).
  .
• Resultant force is a force whose action on a body is equivalent to the action of a system of forces.
  .
• Moment of power is a quantity characterizing the rotating ability of a force.
• Couple of forces is a system of two parallel forces of equal magnitude and oppositely directed.
  Accepted designation: .
  Under the influence of a pair of forces, the body will perform a rotational movement.
• Projection of force on the axis- this is a segment enclosed between perpendiculars drawn from the beginning and end of the force vector to this axis.
  The projection is positive if the direction of the segment coincides with the positive direction of the axis.
• Projection of force onto a plane is a vector on a plane, enclosed between perpendiculars drawn from the beginning and end of the force vector to this plane.
• Law 1 (law of inertia). An isolated material point is at rest or moves uniformly and rectilinearly.
  The uniform and rectilinear motion of a material point is motion by inertia. The state of equilibrium of a material point and a rigid body is understood not only as a state of rest, but also as motion by inertia. For a solid body there are different kinds motion by inertia, for example, uniform rotation of a rigid body around a fixed axis.
• Law 2. A rigid body is in equilibrium under the action of two forces only if these forces are equal in magnitude and directed in opposite directions along a common line of action.
  These two forces are called balancing.
  In general, forces are called balanced if the solid body to which these forces are applied is at rest.
• Law 3. Without disturbing the state (the word “state” here means the state of motion or rest) of a rigid body, one can add and reject balancing forces.
  Consequence. Without disturbing the state of the solid body, the force can be transferred along its line of action to any point of the body.
  Two systems of forces are called equivalent if one of them can be replaced by the other without disturbing the state of the solid body.
• Law 4. The resultant of two forces applied at one point, applied at the same point, is equal in magnitude to the diagonal of a parallelogram constructed on these forces, and is directed along this
  diagonals.
  The absolute value of the resultant is:
  
• Law 5 (law of equality of action and reaction). The forces with which two bodies act on each other are equal in magnitude and directed in opposite directions along the same straight line.
  It should be kept in mind that action- force applied to the body B, And opposition- force applied to the body A, are not balanced, since they are applied to different bodies.
• Law 6 (law of solidification). The equilibrium of a non-solid body is not disturbed when it solidifies.
  It should not be forgotten that the equilibrium conditions, which are necessary and sufficient for a solid body, are necessary but insufficient for the corresponding non-solid body.
• Law 7 (law of emancipation from ties). A non-free solid body can be considered as free if it is mentally freed from bonds, replacing the action of the bonds with the corresponding reactions of the bonds.

Connections and their reactions
• Smooth surface limits movement normal to the support surface. The reaction is directed perpendicular to the surface.
• Articulated movable support limits the movement of the body normal to the reference plane. The reaction is directed normal to the support surface.
• Articulated fixed support counteracts any movement in a plane perpendicular to the axis of rotation.
• Articulated weightless rod counteracts the movement of the body along the line of the rod. The reaction will be directed along the line of the rod.
• Blind seal counteracts any movement and rotation in the plane. Its action can be replaced by a force represented in the form of two components and a pair of forces with a moment.

Kinematics

Kinematics- a section of theoretical mechanics that examines the general geometric properties of mechanical motion as a process occurring in space and time. Moving objects are considered as geometric points or geometric bodies.

Basic concepts of kinematics
• Law of motion of a point (body)– this is the dependence of the position of a point (body) in space on time.
• Point trajectory– this is the geometric location of a point in space during its movement.
• Speed ​​of a point (body)– this is a characteristic of the change in time of the position of a point (body) in space.
• Acceleration of a point (body)– this is a characteristic of the change in time of the speed of a point (body).

Determination of kinematic characteristics of a point
• Point trajectory
  In a vector reference system, the trajectory is described by the expression: .
  In the coordinate reference system, the trajectory is determined by the law of motion of the point and is described by the expressions z = f(x,y)- in space, or y = f(x)- in a plane.
  IN natural system The reference trajectory is set in advance.
• Determining the speed of a point in a vector coordinate system
  When specifying the movement of a point in a vector coordinate system, the ratio of movement to a time interval is called the average value of speed over this time interval: .
  Taking the time interval to be an infinitesimal value, we obtain the speed value at a given time (instantaneous speed value): .
  Vector average speed is directed along the vector in the direction of the point’s movement, the instantaneous velocity vector is directed tangentially to the trajectory in the direction of the point’s movement.
  Conclusion: the speed of a point is a vector quantity equal to the time derivative of the law of motion.
  Derivative property: the derivative of any quantity with respect to time determines the rate of change of this quantity.
• Determining the speed of a point in a coordinate reference system
  Rate of change of point coordinates:
  .
  The modulus of the total velocity of a point with a rectangular coordinate system will be equal to:
  .
  The direction of the velocity vector is determined by the cosines of the direction angles:
  ,
  where are the angles between the velocity vector and the coordinate axes.
• Determining the speed of a point in a natural reference system
  The speed of a point in the natural reference system is defined as the derivative of the law of motion of the point: .
  According to previous conclusions, the velocity vector is directed tangentially to the trajectory in the direction of the point’s movement and in the axes is determined by only one projection.

Rigid body kinematics
• In the kinematics of rigid bodies, two main problems are solved:
  1) setting the movement and determining the kinematic characteristics of the body as a whole;
  2) determination of kinematic characteristics of body points.
• Translational motion of a rigid body
  Translational motion is a motion in which a straight line drawn through two points of a body remains parallel to its original position.
  Theorem: during translational motion, all points of the body move along identical trajectories and at each moment of time have the same magnitude and direction of speed and acceleration.
  Conclusion: the translational motion of a rigid body is determined by the movement of any of its points, and therefore, the task and study of its motion is reduced to the kinematics of the point.
• Rotational motion of a rigid body around a fixed axis
  Rotational motion of a rigid body around a fixed axis is the motion of a rigid body in which two points belonging to the body remain motionless during the entire time of movement.
  The position of the body is determined by the angle of rotation. The unit of measurement for angle is radian. (A radian is the central angle of a circle, the arc length of which is equal to the radius; the total angle of the circle contains 2π radian.)
  The law of rotational motion of a body around a fixed axis.
  We determine the angular velocity and angular acceleration of the body using the differentiation method:
  — angular velocity, rad/s;
  — angular acceleration, rad/s².
  If you dissect the body with a plane perpendicular to the axis, select a point on the axis of rotation WITH and an arbitrary point M, then point M will describe around a point WITH circle radius R. During dt there is an elementary rotation through an angle , and the point M will move along the trajectory a distance .
  Linear speed module:
  .
  Point acceleration M with a known trajectory, it is determined by its components:
  ,
  Where .
  As a result, we get the formulas
  tangential acceleration: ;
  normal acceleration: .

Dynamics

Dynamics is a section of theoretical mechanics in which the mechanical movements of material bodies are studied depending on the causes that cause them.

Basic concepts of dynamics
• Inertia- this is the property of material bodies to maintain a state of rest or uniform rectilinear movement until external forces change this state.
• Weight is a quantitative measure of the inertia of a body. The unit of mass is kilogram (kg).
• Material point- this is a body with mass, the dimensions of which are neglected when solving this problem.
• Center of mass of a mechanical system- a geometric point whose coordinates are determined by the formulas:
  
  Where m k , x k , y k , z k— mass and coordinates k-that point of the mechanical system, m— mass of the system.
  In a uniform field of gravity, the position of the center of mass coincides with the position of the center of gravity.
• Moment of inertia of a material body relative to an axis is a quantitative measure of inertia during rotational motion.
  The moment of inertia of a material point relative to the axis is equal to the product of the mass of the point by the square of the distance of the point from the axis:
  .
  The moment of inertia of the system (body) relative to the axis is equal to arithmetic sum moments of inertia of all points:
  
• Inertia force of a material point is a vector quantity equal in modulus to the product of the mass of a point and the acceleration modulus and directed opposite to the acceleration vector:
• The force of inertia of a material body is a vector quantity equal in modulus to the product of the body mass and the modulus of acceleration of the center of mass of the body and directed opposite to the acceleration vector of the center of mass: ,
  where is the acceleration of the center of mass of the body.
• Elementary impulse of force is a vector quantity equal to the product of the force vector and an infinitesimal period of time dt:
  .
  The total force impulse for Δt is equal to the integral of the elementary impulses:
  .
• Elementary work of force is a scalar quantity dA, equal to the scalar proi

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• Levi-Civita T., Amaldi U. Course in theoretical mechanics. Volume 2, part 2: Dynamics of systems with a finite number of degrees of freedom. M.: From foreign. literature, 1951 (djvu)
• Leach J.W. Classical mechanics. M.: Foreign. literature, 1961 (djvu)
• Lunts Ya.L. Introduction to the theory of gyroscopes. M.: Nauka, 1972 (djvu)
• Lurie A.I. Analytical mechanics. M.: GIFML, 1961 (djvu)
• Lyapunov A.M. General problem of motion stability. M.-L.: GITTL, 1950 (djvu)
• Markeev A.P. Dynamics of a body in contact with a solid surface. M.: Nauka, 1992 (djvu)
• Markeev A.P. Theoretical Mechanics, 2nd edition. Izhevsk: RHD, 1999 (djvu)
• Martynyuk A.A. Stability of motion of complex systems. Kyiv: Nauk. Dumka, 1975 (djvu)
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• Mechanics in the USSR for 50 years. Volume 1. General and applied mechanics. M.: Nauka, 1968 (djvu)
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• Collection of scientific and methodological articles on theoretical mechanics. Issue 2. M.: Higher. school, 1971 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 3. M.: Higher. school, 1972 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 4. M.: Higher. school, 1974 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 5. M.: Higher. school, 1975 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 6. M.: Higher. school, 1976 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 7. M.: Higher. school, 1976 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 8. M.: Higher. school, 1977 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 9. M.: Higher. school, 1979 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 10. M.: Higher. school, 1980 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 11. M.: Higher. school, 1981 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 12. M.: Higher. school, 1982 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 13. M.: Higher. school, 1983 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 14. M.: Higher. school, 1983 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 15. M.: Higher. school, 1984 (djvu)
• Collection of scientific and methodological articles on theoretical mechanics. Issue 16. M.: Vyssh. school, 1986

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 outstanding philosopher antiquity by 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 - this is a change over time in the relative position in 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 different points of the body can be replaced by an equivalent system of forces applied to one point. This system of forces is often called a bundle of strength.