Newton took Galileo's principle of inertia (1632) as his first law and supplemented it with the concept of an inertial frame of reference. According to Galileo's principle of inertia, a free body maintains a state of rest or uniform, rectilinear motion until the influence of other bodies brings him out of this state.

From this principle it follows that a state of rest or uniform rectilinear motion does not require any external influences to maintain it. This reveals a special dynamic property of bodies called inertia. Therefore, Newton’s first law is called the law of inertia, and the movement of a body in the absence of influences from other bodies is movement by inertia.

Newton's first law does not hold true in all frames of reference. Those systems in which it runs are called inertial reference systems.

It has been experimentally established that the practically inertial reference system is the heliocentric reference system, the origin of which is located at the center of the Sun, and the axes are drawn in the direction of three distant stars, chosen, for example, so that they are mutually perpendicular.

For many practical purposes, the motion of macroscopic bodies uses the Earth-related frame as a reference frame. Such a reference system is considered approximately inertial due to the influence of daily and annual rotation Earth.

Thus, we can give the following formulation of Newton's first law: there are such reference systems in which the body maintains a state of rest or uniform rectilinear motion until the influence of other bodies takes it out of this state.

Let us show that any reference system that moves uniformly and rectilinearly relative to an inertial frame is also inertial. Let body A be at rest in the inertial frame of reference K (Fig. 3.1). The reference system K" moves relative to the system K uniformly and rectilinearly with speed. Body A relative to the system K" moves uniformly and rectilinearly with speed - , which also satisfies Newton's first law. Consequently, the reference system K" is inertial. Thus, using a known one inertial reference system, you can construct as many of them as you like using the method described above.

3.1.2. Newton's second law

This law is the basic law of the dynamics of a material point and a rigid body moving translationally.

The law establishes the relationship between force, mass and acceleration.

Experience shows that any change in the magnitude or direction of the speed of movement of a body is caused by its interaction with other bodies.

In mechanics, force is defined as a quantitative measure of the interaction of bodies, which leads to a change in their speed or deformation.

Force is characterized by magnitude, direction and point of application. Hence, force is a vector quantity.

According to modern ideas, based on experience, all interactions observed in nature can be reduced to four fundamental ones: gravitational, weak, electromagnetic and strong.

Gravitational interaction inherent in all material objects. It is determined by the presence of mass in material bodies and obeys Newton’s law of universal gravitation. The range of gravitational interaction is unlimited. In the microworld region, the role of gravitational interaction is negligible.

Weak interaction- short-range, exists in the microcosm and manifests itself in what leads to a certain type of instability of elementary particles.

Electromagnetic interaction manifests itself during the interaction of currents and charges. The range of electromagnetic interaction is unlimited. It is decisive in the formation of atoms, molecules and macroscopic bodies.

Nuclear or strong interaction is the most intense. The radius of the strong interaction is very small ~ 10 -15 m. Thanks to this interaction, protons and neutrons are retained in the nuclei, despite the strong repulsion of protons.

Non-fundamental forces include the forces of elasticity, friction, resistance and others. All these forces can be reduced to electromagnetic or gravitational, however, this leads to a significant complication of solving mechanics problems. For this reason, in mechanics, elasticity and friction forces are considered along with fundamental ones.

Another important property of forces, which manifests itself during mechanical interaction, has been experimentally established. Forces in mechanics obey superposition principle which is as follows: simultaneous interaction of particle M with several othersnparticles with forces

equivalent to the action of one force , equal to their vector sum.

. (3.1)

Strength called the resultant.

As experience shows, all bodies have the property of preventing changes in the magnitude and direction of velocity. This property is called inertia.

Mass can be determined in two ways. The first of them is as follows. A reference body is selected whose mass m fl is taken as a unit of mass. The mass m of the body under study is determined from the following relation established experimentally:

,

Where A And A et - accelerations caused by the action of the same force on the reference and test bodies. In this case, the so-called inert mass.

The second method is based on the use of the law of universal gravitation. In this case, the so-called gravitational mass.

A. Einstein formulated the principle of equivalence of gravitational and inertial mass: the inertial and gravitational masses of the same body are the same.

Equivalence of inert and gravitational mass allows you to select one unit of measurement for them. The SI unit of mass is the kilogram (kg) - the mass of a standard platinum-iridium body stored in France at the International Bureau of Weights and Measures.

The dynamic effect of a moving body on other bodies depends on speed and mass. Therefore, as a dynamic characteristic of traffic intensity, we introduce vector quantity , called the momentum (or momentum) of the body and equal to the product of its mass and speed:

. (3.2)

The unit of impulse is kilogram meter divided by second (kg m/s).

According to Newton's second law, the time derivative of the momentum of a body is equal to the resultant of all forces applied to it:

. (3.3)

From (3.3) it follows that the change in momentum occurs in the direction of the resultant force . Note that Newton's second law in the form (3.3) allows for a description of the motion of a body with variable mass. If the mass of the body is constant, then from (3.2) and (3.3) we obtain the equation of Newton’s second law in the form

, (3.4)

whence, taking into account formula (2.21), we obtain:

. (3.5)

The SI unit of force is a derived unit, the definition of which is based on formula (3.5). Unit of force - 1 Newton (N), this is the force that imparts an acceleration of 1 m to a body with a mass of 1 kg/ With 2 .

Newton's second law is often called the fundamental law of translational dynamics. With the help of this law, mechanics solves two main tasks:

1. Direct main task -establishment of differential equations of motion of a body (point) and their solution.

2. Inverse main problem- finding the dependence of the forces of interaction between bodies on their coordinates, velocities and time, that is, establishing the laws of interaction.

Formulation first law Newton , the experimental basis of which was created by experiments, Galileo back in 1636 was changed several times, but its essence remained the same. Currently, two formulations of this law are used. The most commonly used is the following:

There are such reference systems relative to which a translationally moving body retains its speed constant if other bodies do not act on it or the action of other bodies is compensated.

Newton's first law is formulated differently.

The body maintains a state of rest or uniform rectilinear motion until it is subjected to uncompensated influence from other bodies or physical fields.

Physical meaning: 1) The law states what will happen to the body if other bodies do not act on it or the action of other bodies is compensated. 2) Of all reference systems, Newton’s first law distinguishes those where it is satisfied; such reference systems are called inertial

The phenomenon of a body that is not subject to uncompensated external influences maintaining its speed constant (including equal to zero if the body is at rest) is called inertia, and the reference systems relative to which such bodies move with constant speed or rest, are called inertial. In this regard, Newton's first law is often called the law of inertia. Rectilinear uniform motion of a body in an inertial frame of reference is called motion by inertia. The concept of an inertial reference frame is fundamental in physics in general and mechanics in particular.

The laws of mechanics do not depend on which inertial frame of reference they relate to. In other words, everything inertial systems reference points for any mechanical phenomena are equal, i.e. there is no special, “main” inertial frame of reference, the movement relative to which could be considered as “absolute movement”.

8. Strength. Newton's second law.

Newton's first law indicates that to change the speed of a body relative to an inertial reference frame, i.e. For the accelerated movement of a body, it is necessary that some other body acts on this body. This effect is called by force . The nature of forces can be different, but any force is characterized by two basic properties.

1. Strength is a physical quantity, i.e. it can be characterized not only from the qualitative side, distinguishing it from other physical quantities, but can also be expressed in a certain quantitative way. This is confirmed by the experimental fact that different forces cause different accelerations.

2. Force is a vector quantity. As a result of the action of a force on a body, it acquires acceleration, which is a vector quantity. Consequently, force is also a vector quantity: by changing the direction of the force, we change the direction of acceleration. The magnitude of the force vector determines the extent of the action of other bodies on a given body.

Thus, force - vector physical quantity, characterizing the action of one body on another, which, being uncompensated, leads to a change in the acceleration of this body and is a measure of such influence. In the SI system, force is 1 N. Force is characterized by: point of application, magnitude, direction.

A direct quantitative relationship between the force acting on a body and the acceleration of this body is established Newton's second law :

The acceleration that a body acquires under the influence of a force is directly proportional to this force, and its direction coincides with the direction of this force. Or: The resultant of all forces acting on a body is equal to the product of the body’s mass and acceleration.

Physical meaning: 1) The law connects the kinematic and dynamic characteristics of one body; 2) The law states what will happen to a body if other bodies or fields act on it 3) The unit of force is set to 1 Newton

Body (material point), not subject to external influences, is either at rest or moves rectilinearly and uniformly. Such a body is called free. The movement of such a body is called free movement or coasting.

There is a frame of reference in which everything free bodies move straight and evenly

There are such reference systems, called inertial ones, relative to which a material point, in the absence of external influences, retains the magnitude and direction of its speed indefinitely.

Such systems are called inertial reference systems - Newton's first law.

Newton's second law

Every body exhibits resistance when trying to set it in motion, i.e. give it some acceleration. This property of bodies is called inertia. Measure of inertia - weight.

A system of bodies that is not influenced by other bodies is called closed system or isolated system. In such systems, bodies can only interact with each other. Let a closed system consist of two bodies (two material points). The speed of bodies and , and the increment of these speeds over the same period of time. Vectors and have opposite directions and are related by the relation . The coefficients are both constant and have identical signs and are called masses or inertial masses of bodies 1 and 2.

Pulse or momentum of a material point- vector, equal to the product the mass of a point to its speed.

System impulse- vector sum of the impulses of individual material points that make up the system: for a system consisting of material points.

The momentum of an isolated system remains constant over time - Law of conservation of momentum.

Force (in mechanics)- any reason that changes the momentum of the body (this is a qualitative characteristic). Quantitative characteristics expressed by the equation:

This equation is only valid if m does not depend on speed.

In an inertial reference frame, the derivative of the momentum of a material point with respect to time is equal to the force acting on it.

In an inertial reference frame, the acceleration that a material point receives is directly proportional to the resultant of all forces applied to it and inversely proportional to its mass

The above statements are nothing more than two formulations Newton's second law. The equation corresponding to the definition of the law is equation of motion of a material point.

Newton's third law

The forces of interaction between two material points are equal in magnitude, oppositely directed and act along the straight line connecting these material points.

Every action has an equal and opposite reaction.

Material points act on each other in pairs with forces having same nature, directed along the straight line connecting these points, equal in magnitude and opposite in direction: . Or, if the system consists of many material points, then, i.e. material points interact in pairs. Both forces are directed along the straight line connecting these points.

These three expressions are different formulations Newton's third law.

Any system moving with acceleration relative to an inertial reference frame is non-inertial.

Dynamics is the branch of mechanics that studies different kinds mechanical movements taking into account the interaction of bodies with each other. The foundations of dynamics are Newton's three laws, which are the result of a generalization of observations and experiments in the field of mechanical phenomena that were known before Newton and carried out by Newton himself. Newton's laws of dynamics (otherwise known as classical dynamics) have limited area applicability. They are valid for macroscopic bodies moving at speeds much lower than the speed of light in vacuum. The phenomenon of inertia Let us observe the behavior of various bodies relative to the Earth, choosing a fixed reference system associated with the Earth's surface. We will discover that the speed of any body changes only under the influence of other bodies. For example, let the body stand on a stationary cart. We push the cart and the body will tip against the movement. If, on the contrary, you suddenly stop a moving cart with a body, it will tip over in the direction of movement. Obviously, if there were no friction between the cart and the body, the body would not tip over. In the first case, the following would happen: since the speed standing body is zero, and the speed of the cart began to increase, the cart would slide forward from under the stationary body. In the second case, when the cart was braking, the body standing on it would maintain its speed of movement and slide forward from the stopped cart.

Another example. A metal ball rolls down an inclined chute onto a horizontal plane from the same height h (Fig. 16), therefore, its speed at the point at which it starts horizontal movement, is always the same. Let the horizontal surface be sprinkled with sand first. The ball will travel a short distance s1 and stop. Let's replace the sandy surface with a smooth board. The ball will travel a significantly greater distance s2 before stopping. Let's replace the board with ice. The ball will roll for a very long time and will travel a distance s3 >> s2 before stopping. This sequence of experiments shows that if we reduce the influence environment on a moving body, its horizontal motion relative to the Earth indefinitely approaches uniform and rectilinear. (When the body moves along horizontal surface the attraction of this body by the Earth is compensated by the elasticity of the support - boards, ice, etc.) The fact that the body tends to maintain not any movement, namely rectilinear, is evidenced, for example, by the following experiment (Fig. 17). A ball moving rectilinearly along a flat horizontal surface, colliding with an obstacle having a curved shape, is forced to move in an arc under the influence of this obstacle. However, when the ball reaches the edge of the obstacle, it stops moving curvilinearly and starts moving in a straight line again. Summarizing the results of the above-mentioned (and similar) observations, we can conclude that if a given body is not acted upon by other bodies or their actions are mutually compensated, this body is at rest or the speed of its movement remains unchanged relative to the frame of reference, fixedly connected with the surface of the Earth. The phenomenon of a body maintaining a state of rest or rectilinear uniform motion in the absence or compensation of external influences on this body is called inertia.

Galileo and then Newton first came to the conclusion about the existence of the phenomenon of inertia. This conclusion is formulated in the form of Newton’s first law (law of inertia): there are such reference systems relative to which a body (material point), in the absence of external influences on it (or with their mutual compensation), maintains a state of rest or uniform rectilinear motion. Frames of reference in which Newton's first law is satisfied are called inertial. Consequently, inertial systems are those reference systems relative to which a material point, in the absence of external influences on it or their mutual compensation, is at rest or moves uniformly and rectilinearly.

Observations show that with very high degree accuracy can be considered an inertial reference frame heliocentric system, in which the origin of coordinates is connected to the Sun, and the axes are directed to certain “fixed” stars. Reference systems rigidly connected to the Earth's surface, strictly speaking, are not inertial, since the Earth moves in an orbit around the Sun and at the same time rotates around its axis. However, when describing movements that do not have a global (i.e., worldwide) scale, reference systems associated with the Earth can be considered inertial with sufficient accuracy. Reference systems that move uniformly and rectilinearly relative to some inertial reference system are also inertial (see below). Galileo established that no mechanical experiments carried out inside an inertial reference system can establish whether this system is at rest or moves uniformly and rectilinearly. This statement is called Galileo's principle of relativity or the mechanical principle of relativity. This principle was subsequently developed by A. Einstein and is one of the postulates special theory relativity. Inertial frames of reference play an exclusive role in physics important role, since, according to Einstein’s principle of relativity, mathematical expression any law of physics has the same form in each inertial frame of reference. In what follows, we will use only inertial systems (without mentioning this every time). Frames of reference in which Newton's first law is not satisfied are called non-inertial. Such systems include any reference system moving with acceleration relative to an inertial reference system.

I.2.1 NEWTON'S FIRST LAW. INERTIAL REFERENCE SYSTEMS.

Newton's first law: every body maintains its state of rest or uniform rectilinear motion until an external influence forces it to change this state.

Newton's first law states that a state of rest or uniform linear motion does not require any external influences to maintain it. This reveals a special dynamic property of bodies called inertia . Accordingly, Newton's first law is also called law of inertia , and the movement of a body free from external influences is coasting .

In the above formulation of Newton’s first law, it is implied that the body is not deformed, i.e. absolutely solid, and that in the absence of external influences, it moves forward. In addition, a rigid body can also rotate uniformly by inertia. If in Newton’s first law we speak not about a “body”, but about a material point, which, by its very definition, can neither deform nor rotate, then the need for all these reservations disappears. Considering all that has been said, we can give the following formulation of this law: There are such reference systems, called inertial, relative to which a material point, in the absence of external influences, retains the magnitude and direction of its speed indefinitely. The law is also true in situations where external influences are present, but are mutually compensated (this follows from Newton’s 2nd law, since compensated forces impart zero total acceleration to the body).

The fact that the body remains at rest (i.e. maintains speed, equal to zero) until another body acts on it - is quite understandable and confirmed by everyday observations. The stone itself will not move from its place until it is moved by someone or something. But it is difficult for us to believe that a body can forever maintain uniform and rectilinear motion. A thrown stone experiences air resistance and gravity towards the ground. If these influences did not exist, the body would maintain a state of uniform and rectilinear motion (i.e., it would maintain the magnitude and direction of its speed). Or another example: after running, a person cannot instantly stop or instantly turn to the side. To go around a pole while running, a person instinctively grabs it with his hand, i.e. resorts to the influence of another body (column) to change the direction of its speed.

I.2.2 POWER

By force is called a vector physical quantity, which is a measure of influence on material point or a body from the side of other bodies or fields.

Special shape matter that binds particles of matter into unified systems and transmitting with terminal speed the action of some particles on others is called physical field.

A field acting on a material point with a force is called stationary field, if it does not change over time, i.e. if at any point in the field

Interaction between distant bodies is carried out through gravitational and electromagnetic fields.

Gravitational interaction – arises between bodies in accordance with the law of universal gravitation.

Electromagnetic interaction – occurs between bodies or particles with electrical charges.

In addition, there are also strong interaction, existing, for example, between the particles that make up the nuclei of atoms and weak interaction, characterizing, for example, the processes of transformation of some elementary particles.

Mechanics problems take into account gravitational forces (gravity) and two varieties electromagnetic forces – elastic forces And friction forces.

The forces of interaction between parts of some system of bodies under consideration are called internal forces.

The forces of influence on bodies of a given system from other bodies not included in this system are called external forces.

Totality physical bodies, which have interactions with external bodies absent or compensated, called closed(isolated) system.

A force is completely defined if its magnitude, direction, and point of application are given. The straight line along which the force is directed is called line of action of force.

The simultaneous action of several forces ( , ..., ) on a material point is equivalent to the action of one force, called resultant or resulting strength and equal to them geometric sum:

Formula (I.48) is principle of superposition of forces.

TYPES OF FORCES IN NATURE

Most simple types forces are those that are caused by direct mechanical action of one body on another when they come into contact, these include: forces of traction, friction, pressure, elasticity, tension.

Let's look at just a few of them.

Elastic forces. The forces arising during elastic deformation of bodies are called elastic forces . These forces act between the contacting layers of the deformable body, as well as at the point of contact of the deformable body with the body causing the deformation.

For example, from the side of an elastically deformed board, an elastic force acts on a block lying on it (Fig. 25). Elastic forces are forces of electromagnetic nature.

The elastic force acting on the body considered in this problem from the side of the support or suspension is called ground reaction force(suspension) or suspension tension force. In Fig. Figure 26 shows examples of the application of support reaction forces (force ) and suspension tension force (force ) to bodies.

The elastic force depends only on changes in the distances between the interacting parts of a given elastic body. The work of the elastic force does not depend on the shape of the trajectory and when moving along a closed trajectory is equal to zero. Therefore, the elastic forces are potential forces (the concept of work and potential forces will be discussed in chapter I.3 (§ I.3.1, p. 41), (§ I.3.2, p. 45)).

Hooke's Law: The elastic force is proportional to the elongation (compression) vector and is opposite to it in the direction:

, (I.49)

Where - body rigidity– value determined by the elastic force arising at a unit deformation given body;

The elongation vector is a quantity characterizing one-dimensional (linear) stretching (compression).

Friction forces. Whenever one body moves across the surface of another, resistance arises to this movement, which we imagine as friction force directed against this movement.

A distinction is made between external and internal friction. External friction is the mechanical resistance that occurs during the relative movement of two contacting bodies in the plane of their contact. For example, external friction exists between a block and inclined plane, on which the block lies or from which it slides. Under certain conditions, external friction turns into internal friction, in which there is no jump in speed in the contact zone when moving from one body to another.

The friction between the surfaces of two contacting solid bodies in the absence of a liquid or gaseous layer between them is called dry friction. Friction between surface solid and the surrounding liquid or gaseous medium in which the body moves is called liquid or viscous friction.

Dry friction is divided into:

§ static friction– friction in the absence of relative movement of contacting bodies;

§ sliding friction– friction during relative motion of contacting bodies.

The frictional force that prevents the movement of one body on the surface of another is called static friction force.

Usually, when speaking about the force of static friction, they mean ultimate static friction force. Let us denote by external force applied to a body in contact with another body. This force is parallel to the plane of contact. Relative motion of a body occurs under the condition . The force of static friction is caused by the engagement of uneven surfaces of bodies, elastic deformations of these unevennesses and adhesion (adhesion) of bodies in those places where the distances between their particles are small and sufficient for the occurrence of intermolecular attraction. In this regard, the force of static friction can be considered as a type of manifestation of elastic forces.

It has been experimentally established that the maximum static friction force () does not depend on the area of ​​contact of the bodies and is approximately proportional to the modulus of the normal pressure force (), pressing the rubbing surfaces to each other: .

The dimensionless factor is called the coefficient of static friction. It depends on the nature and condition of the rubbing surfaces.

Sliding friction is explained by the roughness of the rubbing surfaces. Big role Intermolecular interaction forces also play a role.

Laws of sliding friction.

I. Ratio of friction force to pressure force (those. to the force that presses rubbing surfaces against each other) there is a constant value for these surfaces. The first law of friction can be formulated as follows: friction force is directly proportional to pressure force. It has been experimentally shown that the sliding friction force is proportional to the normal pressure force: .

II. The friction coefficient depends on the materials of the rubbing surfaces.

III. The friction coefficient does not depend on the size of the rubbing surfaces. If the surface area is very small, so that a moving body can leave a scratch on a stationary one (for example, the tip of a nail), then this law loses its force.

IV. The coefficient of friction decreases with increasing speed. This is explained by the fact that at high speeds, not all protrusions of rough surfaces have time to engage each other deeply enough.

Figure 27 shows a graph of the friction coefficient depending on the speed of movement.

From the graph it is clear that highest coefficient friction (and therefore greatest strength friction) exists at rest. This is briefly expressed as follows: the maximum value of the static friction force is greater than the sliding friction force. Laws I, II and III were found by Coulomb from experiments with a tribometer.

Note: in the simplest cases, the friction force and the normal pressure force are related by inequality, which turns into equality only in the presence of relative motion. This ratio is called