What is the force of gravitational interaction. Gravity force

To the question “What is force?” physics answers this way: “Force is a measure of the interaction of material bodies with each other or between bodies and other material objects - physical fields" All forces in nature can be classified into four fundamental species interactions: strong, weak, electromagnetic and gravitational. Our article talks about what gravitational forces are - a measure of the last and, perhaps, most widespread type of these interactions in nature.

Let's start with the gravity of the earth

Everyone alive knows that there is a force that attracts objects to the earth. It is commonly referred to as gravity, gravity, or gravity. Thanks to its presence, a person arose the concepts of “up” and “down”, which determine the direction of movement or the location of something relative to earth's surface. So in a particular case, on the surface of the earth or near it, gravitational forces manifest themselves, which attract objects with mass to each other, manifesting their effect at any distance, both small and very large, even by cosmic standards.

Gravity and Newton's third law

As is known, any force, if it is considered as a measure of the interaction of physical bodies, is always applied to one of them. So in the gravitational interaction of bodies with each other, each of them experiences such types of gravitational forces that are caused by the influence of each of them. If there are only two bodies (it is assumed that the action of all others can be neglected), then each of them, according to Newton’s third law, will attract the other body with the same force. So the Moon and the Earth attract each other, resulting in the ebb and flow of the Earth's seas.

Every planet in solar system experiences several forces of attraction from the Sun and other planets at once. Of course, it determines the shape and dimensions of its orbit precisely force of gravity The sun, but also the influence of others celestial bodies astronomers take into account their movement trajectories in their calculations.

Which will fall to the ground faster from a height?

The main feature of this force is that all objects fall to the ground at the same speed, regardless of their mass. Once upon a time, right up to the 16th century, it was believed that everything was the other way around - heavier bodies should fall faster than lighter ones. To dispel this misconception, Galileo Galilei had to perform his famous experiment of simultaneously dropping two cannonballs of different weights from the leaning Tower of Pisa. Contrary to the expectations of witnesses to the experiment, both nuclei reached the surface at the same time. Today every schoolchild knows that this happened due to the fact that gravity imparts to any body the same acceleration of free fall g = 9.81 m/s 2 regardless of the mass m of this body, and its value according to Newton’s second law is equal to F = mg.

Gravitational forces on the Moon and other planets have different meanings this acceleration. However, the nature of the action of gravity on them is the same.

Gravity and body weight

If the first force is applied directly to the body itself, then the second to its support or suspension. In this situation, elastic forces always act on the bodies from the supports and suspensions. Gravitational forces applied to the same bodies act towards them.

Imagine a weight suspended above the ground by a spring. Two forces are applied to it: the elastic force of the stretched spring and the force of gravity. According to Newton's third law, the load acts on the spring with a force equal and opposite to the elastic force. This force will be its weight. A load weighing 1 kg has a weight equal to P = 1 kg ∙ 9.81 m/s 2 = 9.81 N (newton).

Gravitational forces: definition

The first quantitative theory of gravity, based on observations of planetary motion, was formulated by Isaac Newton in 1687 in his famous “Principles of Natural Philosophy.” He wrote that the gravitational forces that act on the Sun and planets depend on the amount of matter they contain. They extend to long distances and always decrease as the reciprocal of the square of the distance. How can we calculate these gravitational forces? The formula for the force F between two objects with masses m 1 and m 2 located at a distance r is:

• F=Gm 1 m 2 /r 2 ,
  where G is a constant of proportionality, a gravitational constant.

Physical mechanism of gravity

Newton was not completely satisfied with his theory, since it assumed interaction between attracting bodies at a distance. The great Englishman himself was sure that there must be some physical agent responsible for transferring the action of one body to another, which he quite clearly stated in one of his letters. But the time when the concept of a gravitational field that permeates all space was introduced came only four centuries later. Today, speaking about gravity, we can talk about the interaction of any (cosmic) body with the gravitational field of other bodies, the measure of which is the gravitational forces arising between each pair of bodies. The law of universal gravitation, formulated by Newton in the above form, remains true and is confirmed by many facts.

Gravity theory and astronomy

It has been very successfully applied to problem solving celestial mechanics during the XVIII and early XIX century. For example, mathematicians D. Adams and W. Le Verrier, analyzing disturbances in the orbit of Uranus, suggested that it is subject to gravitational forces of interaction with an as yet unknown planet. They indicated its expected position, and soon Neptune was discovered there by astronomer I. Galle.

There was still one problem though. Le Verrier in 1845 calculated that the orbit of Mercury precesses by 35" per century, in contrast to the zero value of this precession obtained from Newton's theory. Subsequent measurements gave more exact value 43"". (The observed precession is actually 570"/century, but a careful calculation to subtract the influence from all other planets gives a value of 43".)

It was not until 1915 that Albert Einstein was able to explain this discrepancy within the framework of his theory of gravity. It turned out that the massive Sun, like any other massive body, bends space-time in its vicinity. These effects cause deviations in the orbits of planets, but on Mercury, as the smallest planet and closest to our star, they are most pronounced.

Inertial and gravitational masses

As noted above, Galileo was the first to observe that objects fall to the ground at the same speed, regardless of their mass. In Newton's formulas the concept of mass comes from two different equations. His second law says that a force F applied to a body with mass m gives acceleration according to the equation F = ma.

However, the force of gravity F applied to a body satisfies the formula F = mg, where g depends on the other body interacting with the one in question (the earth usually when we talk about gravity). In both equations m there is a coefficient of proportionality, but in the first case it is inertial mass, and in the second it is gravitational, and there is no obvious reason that they must be the same for any physical object.

However, all experiments show that this is indeed the case.

Einstein's theory of gravity

He took the fact of equality of inertial and gravitational masses as a starting point for his theory. He managed to construct the equations of the gravitational field, famous equations Einstein, and with their help calculate correct value for the precession of Mercury's orbit. They also give a measured value for the deflection of light rays that pass near the Sun, and there is no doubt that they give the correct results for macroscopic gravity. Einstein's theory of gravity, or general theory of relativity (GR) as he called it, is one of his greatest triumphs modern science.

Are gravitational forces acceleration?

If you cannot distinguish inertial mass from gravitational mass, then you cannot distinguish gravity from acceleration. The gravitational field experiment can instead be performed in an accelerating elevator in the absence of gravity. When an astronaut in a rocket accelerates away from the earth, he experiences a force of gravity that is several times greater than Earth's, with the vast majority of it coming from acceleration.

If no one can distinguish gravity from acceleration, then the former can always be reproduced by acceleration. A system in which acceleration replaces gravity is called inertial. Therefore, the Moon in near-Earth orbit can also be considered as an inertial system. However, this system will differ from point to point as the gravitational field changes. (In the example of the Moon, the gravitational field changes direction from one point to another.) The principle that one can always find an inertial system at any point in space and time at which physics obeys the laws in the absence of gravity is called the equivalence principle.

Gravity as a manifestation of the geometric properties of space-time

The fact that gravitational forces can be considered as accelerations in inertial systems ah coordinates that differ from point to point means that gravity is a geometric concept.

We say that spacetime is curved. Consider a ball on flat surface. It will rest or, if there is no friction, move uniformly in the absence of any forces acting on it. If the surface is curved, the ball will accelerate and move to the lowest point, selecting shortest way. Similarly, Einstein's theory states that four-dimensional space-time is curved, and a body moves in this curved space along geodetic line, which corresponds to the shortest path. Therefore, the gravitational field and the forces acting on it physical bodies gravitational forces are geometric quantities that depend on the properties of space-time, which change most strongly near massive bodies.

6.7 Potential energy of gravitational attraction.

All bodies with mass attract each other with a force that obeys the law universal gravity I. Newton. Consequently, attracting bodies have interaction energy.

Let us show that the work of gravitational forces does not depend on the shape of the trajectory, that is, gravitational forces are also potential. To do this, consider the motion of a small body with a mass m, interacting with another massive body masses M, which we will assume to be motionless (Fig. 90). As follows from Newton's law, the force \(~\vec F\) acting between bodies is directed along the line connecting these bodies. Therefore, when the body moves m along an arc of a circle with the center at the point where the body is located M, the work done by the gravitational force is zero, since the vectors of forces and displacement remain mutually perpendicular all the time. When moving along a segment directed towards the center of the body M, the displacement and force vectors are parallel, therefore, in this case, when the bodies approach each other, the work of the gravitational force is positive, and when the bodies move away, it is negative. Next, we note that during radial motion, the work of the attractive force depends only on the initial and final distance between the bodies. So when moving along segments (see Fig. 91) DE And D 1 E 1, the work done is equal, since the laws of change of forces from distance on both segments are the same. Finally, an arbitrary body trajectory m can be divided into a set of arc and radial sections (for example, a broken line ABCDE). When moving along arcs, the work is zero; when moving along radial segments, the work does not depend on the position of this segment - therefore, the work of the gravitational force depends only on the initial and final distance between the bodies, which was what needed to be proven.

Note that when proving potentiality, we only used the fact that gravitational forces are central, that is, directed along the straight line connecting the bodies, and did not mention specific form dependence of force on distance. Hence, All central forces are potential.

We have proven the potentiality of the gravitational interaction between two point bodies. But for gravitational interactions the principle of superposition is valid - the force acting on a body from a system of point bodies is equal to the sum of the forces of pair interactions, each of which is potential, therefore, their sum is also potential. Indeed, if the work of each pair interaction force does not depend on the trajectory, then their sum also does not depend on the shape of the trajectory. Thus, all gravitational forces are potential.

We just need to get a specific expression for potential energy gravitational interaction.

To calculate the work of attraction between two point bodies, it is enough to calculate this work when moving along a radial segment when changing the distance from r 1 to r 2 (Fig. 92).

We'll use it next time graphical method, for which we plot the dependence of the force of attraction \(~F = G \frac(mM)(r^2)\) on the distance r between bodies, then the area under the graph of this relationship in within specified limits and will be equal to the required work (Fig. 93). Calculating this area is not very difficult task, which, however, requires certain mathematical knowledge and skills. Without going into details of this calculation, we present final result, for a given dependence of force on distance, the area under the graph, or the work done by the force of attraction, is determined by the formula

\(~A_(12) = GmM \left(\frac(1)(r_2) - \frac(1)(r_1) \right)\) .

Since we have proven that gravitational forces are potential, this work is equal to the decrease in the potential interaction energy, that is

\(~A_(12) = GmM \left(\frac(1)(r_2) - \frac(1)(r_1) \right) = -\Delta U = -(U_2 - U_1)\) .

From this expression we can determine the expression for the potential energy of gravitational interaction

\(~U(r) = - G \frac(mM)(r)\) . (1)

With this definition, potential energy is negative and tends to zero at an infinite distance between bodies \(~U(\infty) = 0\) . Formula (1) determines the work done by the force gravitational attraction with increasing distance from r to infinity, since with such movement the vectors of force and displacement are directed in opposite sides, then this work is negative. With the opposite movement, when bodies approach from infinite distance to distance, the work of the attractive force will be positive. This work can be calculated by the definition of potential energy \(~A_(\infty \to r)U(r) = - (U(\infty)- U(r)) = G \frac(mM)(r)\) .

We emphasize that potential energy is a characteristic of the interaction of at least two bodies. It is impossible to say that the energy of interaction “belongs” to one of the bodies, or how to “divide this energy between the bodies.” Therefore, when we talk about a change in potential energy, we mean a change in the energy of a system of interacting bodies. However, in some cases it is still permissible to talk about a change in the potential energy of one body. So, when describing the movement of a small body, compared to the Earth, in the gravitational field of the Earth, we talk about the force acting on the body from the Earth, as a rule, without mentioning or taking into account the equal force acting from the body on the Earth. The fact is that with the enormous mass of the Earth, the change in its velocity is small. Therefore, a change in the potential interaction energy leads to a noticeable change kinetic energy body and an infinitesimal change in the kinetic energy of the Earth. In such a situation, it is permissible to talk about the potential energy of a body near the Earth’s surface, that is, to “attribute” all the energy of gravitational interaction to a small body. IN general case we can talk about the potential energy of an individual body if the other interacting bodies are motionless.

We have repeatedly emphasized that the point at which potential energy is accepted equal to zero, is chosen arbitrarily. IN in this case such a point turned out to be infinite remote point. In a sense, this unusual conclusion can be considered reasonable: indeed, at an infinite distance, interaction disappears and potential energy also disappears. From this point of view, the sign of potential energy also looks logical. Indeed, in order to separate two attracting bodies, external forces must do positive work, therefore, in such a process, the potential energy of the system must increase: so it increases, increases and... becomes equal to zero! If the attracting bodies are in contact, then the force of attraction cannot do positive work, but if the bodies are separated, then such work can be done when the bodies come together. Therefore it is often said that attracting bodies have negative energy, and the energy of repelling bodies is positive. This statement is true only if the zero level of potential energy is chosen at infinity.

So, if two bodies are connected by a spring, then as the distance between the bodies increases, an attractive force will act between them, however, the energy of their interaction is positive. Do not forget that zero level potential energy corresponds to the state of an undeformed spring (and not to infinity).

1. Introduction

All weighty bodies mutually experience gravity; this force determines the movement of planets around the sun and satellites around planets. The theory of gravity, a theory created by Newton, stood at the cradle of modern science. Another theory of gravity, developed by Einstein, is the greatest achievement of theoretical physics of the 20th century. Over the centuries of human development, people have observed the phenomenon mutual attraction bodies and measured its size; they tried to put this phenomenon at their service, to surpass its influence, and finally, already at the very Lately calculate it with extreme accuracy during the first steps deep into the Universe.

The immense complexity of the bodies around us is due primarily to such a multi-stage structure, the final elements of which - elementary particles - have relatively little a large number types of interaction. But these types of interactions differ sharply in their strength. The particles that form atomic nuclei are bound together by the most powerful forces known to us; In order to separate these particles from each other, it is necessary to expend a colossal amount of energy. The electrons in an atom are bound to the nucleus by electromagnetic forces; it is enough to give them a very modest energy (usually enough energy chemical reaction) as electrons are already separated from the nucleus. If we talk about elementary particles atoms, then for them the weakest interaction is the gravitational interaction.

When compared with the interaction of elementary particles, gravitational forces are so weak that it is difficult to imagine. Nevertheless, they and only they completely regulate the movement of celestial bodies. This happens because gravity combines two features, due to which its effect intensifies when we move to large bodies. Unlike atomic interaction, the forces of gravitational attraction are noticeable even at great distances from the bodies that create them. In addition, gravitational forces are always attractive forces, that is, bodies are always attracted to each other.

The development of the theory of gravity occurred at the very beginning of the development of modern science using the example of the interaction of celestial bodies. The task was made easier by the fact that celestial bodies move in the vacuum of world space without the side influence of other forces. Brilliant astronomers - Galileo and Kepler - with their works prepared the ground for further discoveries in this area. Further the great Newton managed to come up with a complete theory and give it a mathematical form.

2. Newton and his predecessors

Among all the forces that exist in nature, the force of gravity is distinguished primarily by the fact that it manifests itself everywhere. All bodies have mass, which is defined as the ratio of the force applied to the body to the acceleration that the body acquires under the influence of this force. The force of attraction acting between any two bodies depends on the masses of both bodies; it is proportional to the product of the masses of the bodies under consideration. In addition, the force of gravity is characterized by the fact that it obeys the law of inverse proportionality to the square of the distance. Other forces may depend on distance quite differently; Many such forces are known.

One aspect of universal gravity - the surprising dual role played by mass - has served cornerstone to construct the general theory of relativity. According to Newton's second law, mass is a characteristic of any body, which shows how the body will behave when a force is applied to it, regardless of whether it is gravity or some other force. Since all bodies, according to Newton, accelerate (change their speed) in response to an external force, the mass of a body determines what acceleration the body experiences when a given force is applied to it. If the same force is applied to a bicycle and a car, each will reach a certain speed at a different time.

But in relation to gravity, mass also plays another role, completely different from the one it played as the ratio of force to acceleration: mass is the source of mutual attraction of bodies; If we take two bodies and look at the force with which they act on a third body located at the same distance, first from one and then from the other body, we will find that the ratio of these forces is equal to the ratio of the first two masses. In fact, it turns out that this force is proportional to the mass of the source. Similarly, according to Newton's third law, the forces of attraction experienced by two different bodies and under the influence of the same source of attraction (at the same distance from it), are proportional to the ratio of the masses of these bodies. In engineering and everyday life, the force with which a body is attracted to the ground is referred to as the weight of the body.

So, mass enters into the connection that exists between force and acceleration; on the other hand, mass determines the magnitude of the force of attraction. This dual role of mass leads to the fact that the acceleration of different bodies in the same gravitational field turns out to be the same. Indeed, let us take two different bodies with masses m and M, respectively. Let them both fall freely to Earth. The ratio of the forces of attraction experienced by these bodies is equal to the ratio of the masses of these bodies m/M. However, the acceleration acquired by them turns out to be the same. Thus, the acceleration acquired by bodies in a gravitational field turns out to be the same for all bodies in the same gravitational field and does not depend at all on the specific properties of the falling bodies. This acceleration depends only on the masses of the bodies creating the gravitational field, and on the location of these bodies in space. The dual role of mass and the resulting equality of acceleration of all bodies in the same gravitational field is known as the principle of equivalence. This name has historical origin, emphasizing the fact that the effects of gravity and inertia are to a certain extent equivalent.

On the Earth's surface, the acceleration due to gravity is, roughly speaking, 10 m/sec2. The speed of a freely falling body, if you do not take into account air resistance during the fall, increases by 10 m/sec. Every second. For example, if a body begins to freely fall from a state of rest, then by the end of the third second its speed will be 30 m/sec. Usually acceleration free fall denoted by the letter g. Due to the fact that the shape of the Earth does not strictly coincide with a sphere, the value of g on Earth is not the same everywhere; it is greater at the poles than at the equator, and less on the tops of large mountains than in the valleys. If the value of g is determined with sufficient accuracy, then it is affected even geological structure. This explains the fact that geological methods for searching for oil and other minerals also include an accurate determination of the value of g.

What's in this place all bodies experience the same acceleration - a characteristic feature of gravity; No other forces have such properties. And although Newton had no choice but to describe this fact, he understood the universality and unity of the acceleration of gravity. The German physicist and theorist Albert Einstein (1870 - 1955) had the honor of discovering the principle on the basis of which this property of gravity, the principle of equivalence, could be explained. Einstein also belongs to the foundations of the modern understanding of the nature of space and time.

3. Special theory of relativity

Since the time of Newton, it was believed that all reference systems are a set of rigid rods or some other objects that make it possible to establish the position of bodies in space. Of course, in each reference system such bodies were chosen differently. At the same time, it was assumed that all observers had the same time. This assumption seemed so intuitively obvious that it was not specifically stated. In everyday practice on Earth, this assumption is confirmed by all our experience.

But Einstein was able to show that comparisons of clock readings, if we take them into account relative motion, does not require special attention only in the case when relative speeds hours is significantly less than the speed of light in a vacuum. So, the first result of Einstein's analysis was the establishment of the relativity of simultaneity: two events occurring at a sufficient distance from each other can appear simultaneous for one observer, but for an observer moving relative to him, occurring at different times. Therefore, the assumption of a uniform time cannot be justified: it is impossible to specify a specific procedure that allows any observer to establish such universal time regardless of the movement in which he participates. The reference system must also contain a clock, moving with the observer and synchronized with the observer's clock.

The next step taken by Einstein was to establish new relationships between the results of measurements of distance and time in two different inertial frames of reference. The special theory of relativity, instead of “absolute lengths” and “absolute time”, brought to light a different “absolute value”, which is usually called the invariant space-time interval. For two given events occurring at some distance from each other, the spatial distance between them is not an absolute (i.e., independent of the reference system) value, even in the Newtonian scheme, if there is a certain time interval between the occurrence of these events. Indeed, if two events do not occur simultaneously, an observer moving with a certain frame of reference in one direction and finding himself at the point where the first event occurred can, during the period of time separating these two events, end up at the place where the second event occurs; for this observer both events will occur in the same place in space, although for an observer moving in opposite direction, they may seem to occur at a considerable distance from each other.

4. Relativity and gravity

The deeper they go Scientific research into the final constituent substances and the smaller the number of particles and forces acting between them remains, the more insistent the demands for a comprehensive understanding of the action and structure of each component of matter become. It is for this reason that when Einstein and other physicists became convinced that the special theory of relativity had replaced Newtonian physics, they began again fundamental properties particles and force fields. Most important object, requiring revision was gravity.

But why shouldn't the discrepancy between the relativity of time and Newton's law of gravitation be resolved as simply as in electrodynamics? It would be necessary to introduce the concept of a gravitational field, which would propagate in approximately the same way as electric and magnetic field, and which would turn out to be a mediator in the gravitational interaction of bodies, in accordance with the concepts of the theory of relativity. This gravitational interaction would be reduced to Newton's law of gravitation, when the relative speeds of the bodies in question would be small compared to the speed of light. Einstein tried to build a relativistic theory of gravity on this basis, but one circumstance did not allow him to carry out this intention: no one knew anything about the propagation of gravitational interaction with high speed, there was only some information regarding the effects associated with high speeds of movement of the sources of the gravitational field - the masses.

The effect of high speeds on masses is different from the effect of high speeds on charges. If electric charge bodies remains the same for all observers, the mass of bodies depends on their speed relative to the observer. The higher the speed, the greater the observed mass. For a given body, the smallest mass will be determined by the observer relative to whom the body is at rest. This mass value is called the rest mass of the body. For all other observers, the mass will be greater than the rest mass by an amount equal to the kinetic energy of the body divided by c. The value of the mass would become infinite in the frame of reference in which the speed of the body would become equal speed Sveta. One can speak only conditionally about such a reference system. Since the magnitude of the gravitational source depends so significantly on the frame of reference in which its value is determined, the field generated by the mass must be more complex than the electromagnetic field. Einstein therefore concluded that the gravitational field appears to be a so-called tensor field, described by a larger number of components than the electromagnetic field.

As the next starting principle, Einstein postulated that the laws of the gravitational field should be obtained on the basis of a mathematical procedure similar to the procedure leading to the laws electromagnetic theory; the laws of the gravitational field obtained in this way must obviously be similar in form to the laws of electromagnetism. But even taking all these considerations into account, Einstein found that he could construct several different theories that equally satisfy all requirements. A different point of view was needed to unambiguously arrive at the relativistic theory of gravity. Einstein found one new point view of the principle of equivalence, according to which the acceleration acquired by a body in the field of gravitational forces does not depend on the characteristics of this body.

5. Relativity of free fall

IN special theory relativity, as in Newtonian physics, postulates the existence of inertial reference systems, i.e. systems relative to which bodies move without acceleration when they are not acted upon by external forces. The experimental discovery of such a system depends on whether we can place test bodies in conditions where no external forces act on them, and there must be experimental confirmation of the absence of such forces. But if the presence, for example, of an electric (or any other force) field can be detected by the difference in the effect that these fields have on various test particles, then all test particles placed in the same gravitational field acquire the same acceleration.

However, even in the presence of a gravitational field, there is a certain class of reference systems that can be identified by purely local experiments. Since all gravitational accelerations at a given point ( small area) all bodies are identical both in magnitude and direction, they will all be equal to zero with respect to the reference frame, which is accelerated along with other physical objects that are under the influence only of gravity. Such a reference frame is called a freely falling reference frame. Such a system cannot be extended indefinitely to all space and all moments of time. It can be uniquely determined only in the vicinity of the world point, in a limited region of space and for a limited period of time. In this sense, freely falling reference frames can be called local reference frames. In relation to freely falling reference frames, material bodies that are not acted upon by any forces other than gravitational forces do not experience acceleration.

Freely falling frames of reference in the absence of gravitational fields are identical to inertial frames of reference; in this case they are indefinitely extendable. But such unlimited distribution of systems becomes impossible when gravitational fields. The fact that freely falling systems generally exist, even if only as local frames of reference, is a direct consequence of the principle of equivalence, to which all gravitational effects are subject. But the same principle is responsible for the fact that it is impossible to construct inertial frames of reference in the presence of gravitational fields by any local procedures.

Einstein considered the equivalence principle to be the most fundamental property of gravity. He realized that the idea of ​​infinitely extendable inertial frames of reference should be abandoned in favor of local freely falling frames of reference; and only by doing so can the principle of equivalence be accepted as a fundamental part of the foundation of physics. This approach has enabled physicists to look deeper into the nature of gravity. The presence of gravitational fields turns out to be equivalent to the impossibility of propagation in space and time of a local freely falling reference frame; Thus, when studying gravitational fields, attention should be focused not so much on the local field magnitude as on the inhomogeneity of gravitational fields. The value of this approach, which ultimately denies the universality of the existence of inertial reference frames, is that it makes clear that there is no reason to accept without reflection the possibility of constructing inertial reference frames, despite the fact that such frames have been used for several centuries.

6. Gravity in time and space

In Newton's theory of gravity, the gravitational acceleration caused by a given large mass is proportional to that mass and inversely proportional to the square of the distance from that mass. The same law can be formulated a little differently, but at the same time we can reach relativistic law gravity. This different formulation is based on the idea of ​​the gravitational field as something that is imprinted in the vicinity of a large gravitating mass. The field can be completely described by specifying at each point in space a vector whose magnitude and direction correspond to that gravitational acceleration. Which is acquired by any test body placed at this point. It is possible to describe the gravitational field graphically by drawing curves in it, the tangent to which at each point in space coincides with the direction of the local gravitational field (acceleration); these curves are drawn with density ( certain number curves per unit area cross section, rice. 2) equal to the local field value. If one large mass is considered, such curves - they are called lines of force - turn out to be straight lines; these straight lines point directly to the body creating the gravitational field.

Back proportional dependence from the square of the distance is expressed graphically as follows: all power lines start at infinity and end at large masses. If the density of field lines is equal to the magnitude of the acceleration, the number of lines passing through spherical surface, the center of which is located on a large mass, is exactly equal to the density of field lines multiplied by the area of ​​a spherical surface of radius r; The area of ​​a spherical surface is proportional to the square of its radius. In general, Newton's law of inverse dependence on the square of the distance can be given in a form that is equally suitable for the source of gravity in the form of one large mass and for random distribution masses: all lines of force of the gravitational field begin at infinity and end at the masses themselves. The total number of field lines ending in some region containing masses is proportional to gross weight contained in this area. In addition, the gravitational field is a conservative field: lines of force cannot take the form of closed curves, and moving a test body along a closed curve cannot lead to either gain or loss of energy.

In the relativistic theory of gravity, the role of sources is assigned to combinations of mass and momentum (momentum acts as a link between the state of the same object in different four-dimensional or Lorentzian reference systems). The inhomogeneities of the relativistic gravitational field are described by the curvature tensor. A tensor is a mathematical object obtained by generalizing the idea of ​​vectors. In a manifold described by coordinates, tensors can be associated with components that completely define the tensor. The relativistic theory connects the curvature tensor with the tensor that describes the behavior of gravitational sources. These tensors are proportional to each other. The proportionality coefficient is determined from the requirement: the law of gravitation in tensor form must be reduced to the Newtonian law of gravitation for weak gravitational fields and at low velocities of bodies; this coefficient of proportionality, up to world constants, is equal to Newton's constant of gravity. With this step, Einstein completed the construction of the theory of gravity, otherwise called general theory relativity.

7. Conclusion

The general theory of relativity made it possible to take a slightly different look at issues related to gravitational interactions. It included all Newtonian mechanics only as special case at low speeds of bodies. This opened up a very wide area for exploring the Universe, where gravitational forces play a decisive role.

LITERATURE:

P. BERGMAN “THE MYSTERY OF GRAVITY” LOGUNOV “RELATIVISTIC THEORY OF GRAVITY”

VLADIMIROV “SPACE, TIME, GRAVITY”

Gravitational interaction manifests itself in the attraction of bodies to each other. This interaction is explained by the presence of a gravitational field around each body.

Modulus of the force of gravitational interaction between two material points of mass m 1 and m 2 located at a distance from each other

(2.49)

where F 1,2,F 2,1 – interaction forces directed along the connecting straight line material points,G= 6,67
– gravitational constant.

Relationship (2.3) is called law of universal gravitation discovered by Newton.

Gravitational interaction is valid for material points and bodies with a spherically symmetric distribution of masses, the distance between which is measured from their centers.

If we take one of the interacting bodies to be the Earth, and the second is a body with mass m, located near or on its surface, then an attractive force acts between them

, (2.50)

where M 3 ,R 3 – mass and radius of the Earth.

Ratio
- constant equal to 9.8 m/s 2, denoted g, has the dimension of acceleration and is called acceleration of free fall.

Product of body mass m and free fall acceleration , called gravity

. (2.51)

Unlike the force of gravitational interaction gravity module
depends on geographical latitude location of the body on Earth. At the poles
, and at the equator it decreases by 0.36%. This difference is due to the fact that the Earth rotates on its axis.

With the body removed relative to the Earth's surface to a height gravity decreases

, (2.52)

Where
– acceleration of free fall at a height h from the Earth.

Mass in formulas (2.3-2.6) is a measure of gravitational interaction.

If you hang a body or place it on a fixed support, it will be at rest relative to the Earth, because the force of gravity is balanced by the reaction force acting on the body from the support or suspension.

Reaction force- the force with which other bodies act on a given body, limiting its movement.

Force normal reaction supportsattached to the body and directed perpendicular to the plane of support.

Thread reaction force(suspension) directed along the thread (suspension)

Body weight –the force with which the body presses on the support or stretches the thread of the suspension and is applied to the support or suspension.

Weight numerically equal to force gravity if the body is on a horizontal surface of a support in a state of rest or uniform linear motion. In other cases, the weight of the body and the force of gravity are not equal in magnitude.

2.6.3.Friction forces

Friction forces arise as a result of the interaction of moving and resting bodies in contact with each other.

There are external (dry) and internal (viscous) friction.

External dry friction divided by:

The listed types of external friction correspond to the forces of friction, rest, sliding, and rolling.

WITH

static friction

If an increasing external force is applied to a body in contact with another body , parallel to the plane of contact (Fig. 2.2.a), then when changing from zero to some value
body movement does not occur. The body begins to move at F F tr. max.

Maximum strength static friction

, (2.53)

Where – coefficient of static friction, N – modulus of the normal reaction force of the support.

Static friction coefficient can be determined experimentally by finding the tangent of the angle of inclination to the horizon of the surface from which the body begins to roll under the influence of its gravity.

When F>
bodies slide relative to each other at a certain speed (Fig. 2.11 b).

The sliding friction force is directed against the speed . The modulus of the sliding friction force at low speeds is calculated in accordance with Amonton's law

, (2.54)

Where – dimensionless coefficient of sliding friction, depending on the material and state of the surface of the contacting bodies, and is always less .

The rolling friction force occurs when a body in the shape of a cylinder or ball of radius R rolls along the surface of a support. The numerical value of the rolling friction force is determined in accordance with Coulomb's law

, (2.55)

where k[m] – rolling friction coefficient.