GRAVITATIONAL INTERACTION elementary particles, the weakest of all known fundamental interactions, characterized by the participation of a gravitational field (gravitational field). By modern ideas, any interaction of particles is carried out through the exchange between them of virtual (or real) particles - carriers of interaction. In electromagnetic, weak and strong interactions the carriers are the photon, intermediate vector bosons and gluons, respectively. For gravitational interaction, the question of carriers is not simple, and the theory of gravitational interaction itself takes special place in the physical picture of the world.

According to the law universal gravity Newton, the force of interaction between two point masses (the dimensions of which are small compared to the distance r between them)

F g =Gm 1 m 2 /r 2 , (1)

where m 2 is the mass of particles, G = 6.67·10 -11 m 3 /kg?s 2 is the gravitational constant. The force of gravitational interaction between two protons is 10 36 times less Coulomb force electrostatic interaction between them. This relationship does not change when taking into account relativistic effects up to distances equal to the Compton wavelength of the proton. The quantity √Gm can be called “gravitational charge”. With this definition of “charge”, formula (1) coincides with Coulomb's law for the interaction of electric charges. The gravitational charge is proportional to the mass of the body, therefore, according to Newton’s second law (F = ma), the acceleration a caused by force (1) does not depend on the mass of the accelerated body. This fact, verified with great accuracy, is called the equivalence principle. In the relativistic theory of gravitational interaction, due to the relationship between mass and energy (E = mс 2), the gravitational charge is proportional to the energy, that is, the total mass m, and not the rest mass, as in formula (1). This determines the universality of gravitational interaction. There is no type of matter that has zero gravitational charge. It is this property of gravitational interaction that distinguishes it from other fundamental interactions of elementary particles. In addition, at high particle energies, gravitational interaction can no longer be considered weak. At energy >10 18 GeV, the gravitational charge of the particle √GE/c 2 becomes equal to its electric charge e, and at very high energies gravitational interaction may become the main one.

The most important property of the gravitational field is that it determines the geometry of space-time in which matter moves. The geometry of the world cannot be specified initially and changes with the movement of matter creating a gravitational field (see Gravity). A. Einstein made this conclusion from the property of the universality of gravitational interaction and built a relativistic theory of gravity - the general theory of relativity (GTR). Experiments confirm the validity of general relativity in the case of weak gravitational fields (when gravitational potential By absolute value much less with 2). For strong fields, general relativity has not yet been tested, so other theories of gravitational interaction are also possible.

GTR arose as a generalization special theory relativity. Other theories of gravity arise as a reflection of the successes of particle physics - theoretical and experimental. For example, the Einstein-Cartan-Troutman theory of gravity (the so-called gravity with torsion, Einstein, A. Cartan, A. Trautman, 1922-72) expands the equivalence principle in the sense that the gravitational field in it interacts not only with energy (the energy tensor -momentum) of particles, but also with their spin.

In the so-called f-g theories gravity by K. J. Isham, A. Salam and J. Strazdi (1973) assumes the existence of two gravitational fields: the carriers of one of them are massless particles with spin 2 (ordinary, “weak” gravity of general relativity), this field interacts with leptons; the other field is carried by massive particles (f-mesons) with spin 2 (“strong” gravity) and interacts with hadrons.

The Brans-Dicke-Jordan scalar-tensor theory of gravity (K. Brans, R. Dicke, P. Jordan, 1959-61) was a development of P. Dirac's idea about the change in fundamental physical constants and interaction constants over time.

A.D. Sakharov put forward (1967) the idea of ​​gravity as an induced interaction, by analogy with the van der Waals forces, which have electromagnetic nature. In this theory, gravitational interaction is not a fundamental interaction, but the result of quantum fluctuations of all other fields. Success quantum theory fields (QFT) made it possible to calculate the induced gravitational constant G, which in this case is expressed through the parameters of these quantum fields.

Theory of gravity - classical theory, the quantum theory of gravity has not yet been created. The need for quantization is caused by the fact that elementary particles are objects quantum nature, and therefore the connection of classical interaction and quantized sources of this interaction seems inconsistent.

The creation of a quantum theory of gravity encounters great mathematical difficulties arising from the nonlinearity of the zero equations. There are several methods for quantizing such complex mathematical objects; these methods are being developed and improved (see Quantum theory of gravity). As in quantum electrodynamics(QED), divergences appear during calculations, however, unlike QED, the quantum theory of gravity turns out to be non-renormalizable. There is an analogy here with the theory of weak interaction, which also, taken separately, without connection with other interactions, is non-renormalizable. But the unification of weak and electromagnetic interactions (based on the idea of ​​so-called spontaneous symmetry breaking) made it possible to construct a unified renormalizable theory of electroweak interaction. In this regard big hopes are assigned to supergravity - a theory that combines all interactions based on supersymmetry and in which, in addition to gravitons (massless particles with spin 2, bosons), there are other carriers of gravitational interaction - fermions, called gravitinos.

Interest in creating a quantum theory of gravity is not purely academic. The connection of gravitational interaction with all types of matter and with space-time diversity will inevitably lead in the future quantum theory to the quantization of space-time and to a change in our views not only on space and time at ultra-short distances and time intervals, but also on the concept of “particle”, on the measurement procedure in the microcosm, as well as on changes in the structure modern theory elementary particles.

Some outlines of these changes are already visible. This is, first of all, the problem of divergences in QFT. The divergence, for example, of the self-energy of an electrically charged particle appears already in classical electrodynamics. The total mass M of a classical charged thin sphere having a charge e and size r 0 is equal to

M = M 0 + e 2 /2r 0 s 2, (2)

where M 0 is the seed mass. As r 0 → 0, the mass M becomes infinite. This divergence is not eliminated in quantum theory either; it only becomes weaker - logarithmic. If we take into account gravitational interaction and the fact that it depends on total weight M, the divergence of self-energy disappears already in the classical theory.

The issue of divergences can be approached from a different angle. Interaction in QFT is an exchange virtual particles arbitrarily high energies. Therefore, when integrating over these energies, divergent expressions are obtained. In general relativity, particles cannot be pointlike. Their minimum size is determined by the gravitational radius r g . How more mass(energy), the larger the gravitational radius:

If a body of mass M is compressed to sizes smaller than rg, then it turns into black hole size r g . In quantum theory there is also a limit to the localization of a particle - its Compton wavelength l С = ћ/М с, which, obviously, cannot be less than the gravitational radius. Therefore, there is hope that in a theory that takes into account gravitational interaction, intermediate states with arbitrarily high energies will not arise and, therefore, the divergences will disappear. The maximum mass (energy) of particles corresponds to the equality l C = r g, and is equal to МР | =√ћc/G ≈ 10 -5 g. This value is called the Planck mass, and it corresponds to the Planck length l Р| = √ћG/c 3 ≈ 10 -33 cm.

M.A. Markov suggested (1965) that elementary particles of mass M P| and that these particles have the maximum possible mass for an elementary particle. He called these particles maximons. Markov called charged maximons with mass M = e/√G ≈ 10 -6 g friedmons. Freedmons and maximons have a number of unusual properties. Thus, the geometry inside these particles can differ significantly from the geometry outside, and one can imagine such friedmons and maximons, inside of which there are entire universes. It is quite possible that quantum formations similar to maximons and friedmons determined early stages evolution of the Universe and set the initial vacuum of a single interaction, which, during the expansion of the Universe, was divided, for example, through the mechanism of spontaneous symmetry breaking, into four interactions, currently known. The direction of development of elementary particle physics does not exclude, but rather assumes such a possibility.

Not only quantum gravity can have a significant impact on the theory of other interactions, and undoubtedly the opposite effect. Studies of QFT in curved space-time, studies of the evaporation of black holes and the birth of particles in cosmology show that QFT leads to a modification of Einstein's equations. In modern unified theories of interaction of elementary particles, the vacuum energy density can be non-zero and, therefore, have its own gravitational field. The dominance of this energy density leads to an acceleration of expansion modern universe. Finally, in models of multidimensional gravity, processes of non-gravitational interactions occur on a 4-dimensional brane (subspace) in multidimensional space-time. At energies that bring the particle to the brane boundary, a violation of Lorentz invariance can be observed, and the gravitational interaction ceases to be weak.

All this indicates that the creation of a quantum theory of gravitational interaction is impossible without taking into account other fundamental interactions and, conversely, the theory of other interactions will not be complete and free from internal contradictions without taking into account gravitational interaction. It may be possible to achieve such a unification of gravitational interaction with other interactions within the framework of intensive developing theory strings The study of such a unification is facilitated by the methods of cosmic microphysics, which studies the fundamental relationship between the micro and macro world in a combination of its physical, cosmological and astrophysical manifestations.

Lit.: Markov M. A. On the nature of matter. M., 1976; Mizner Ch., Thorne K., Wheeler J. Gravitation. M., 1977. T. 1-3; A. Einstein and the theory of gravity. M., 1979; Grib A. A., Mamaev S. G., Mostepanenko V. M. Quantum effects in intensive external fields. M., 1980; Rubakov V. A. Large and infinite additional dimensions // Advances in Physical Sciences. 2001. T. 171. Issue. 9; Landau L. D., Lifshits E. M. Field theory. 8th ed. M., 2003; Khlopov M. Yu. Fundamentals of cosmomicrophysics. M., 2004.

V. A. Berezin, M. Yu. Khlopov.

Gravity force

FORCE

The basis of mechanics is Newton's second law. When writing a law mathematically, the cause is written on the right, and the effect on the left. The cause is force, and the effect of forces is acceleration. Therefore, the second law is written like this:

The acceleration of a body is proportional to the resultant force acting on the body and inversely proportional to the mass of the body. Acceleration is directed in the direction of the resulting force. The resulting force is vector sum all forces acting on the body: .

Real powers characterize the measure of interaction between two bodies. In the future, we will consider several types of interactions - gravitational, electrical, molecular. Each type of interaction has its own strength. If there are no interactions, then there are no forces. Therefore, first of all, it is necessary to find out which bodies interact with each other.

Gravity force

The body is thrown and flies above the Earth (Fig. 1.1). There is only

Rice. 1.1. Forces acting on a thrown stone ( A), acceleration of the stone ( b) and its speed ( V)

the interaction of a body with the Earth, which is characterized by the gravitational force of attraction (gravity). According to the law of universal gravitation, the gravitational force is directed towards the center of the Earth and is equal to

Where M- mass of the Earth, T- body mass, r- distance from the center of the Earth to the body, γ - gravitational constant. There are no other interactions, therefore there are no other forces.

To find the acceleration of the stone, the gravitational force from formula 1.2 is substituted into formula 1.1 of Newton's second law. Obviously, the acceleration of the stone is always directed downward (Fig. 1.1, b). At the same time, the speed of the flying stone changes and at each point of the trajectory is directed tangentially to this trajectory (Fig. 1.1, V).

Newton's second law relates vector quantities - acceleration A and the resultant force. Any vector is given by magnitude (modulus) and direction. You can specify a vector with three projections onto coordinate axes, that is, three numbers. In this case, the choice of axes is determined by convenience. In Fig. 1.1 axis X can be directed downwards. Then the acceleration projections will be equal a x, 0, 0. If the axis X point upward, then the acceleration projections will become equal - a x,0,0. In the future we will choose the direction of the axis X so that it coincides in direction with the acceleration and for simplicity we will not write the quantity a x, but just A. So, the acceleration created by the gravitational force is

(1.3)

For bodies located near the Earth's surface, r» R(radius of the Earth R= 6400 km), therefore

m/s 2 (1.4)

Consequently, in the vertical direction the thrown body moves uniformly accelerated.

From formula 1.3 it follows that acceleration free fall does not depend on the mass of the flying (falling) body and is determined only by the mass of the planet M and the distance of the body from the center of the planet r. The farther the body is from the center of the planet, the lower the acceleration due to gravity.

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 types of 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 developed 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. Likewise, in the gravitational interaction of bodies with each other, each of them experiences such types gravitational forces, which 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 motion 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 carry out his famous experience by simultaneously dropping two cannonballs of different weights from the leaning 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

First quantity theory gravity, based on observations of the motion of planets, 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 values, reciprocals of the square distances. 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 was very successfully applied to solving problems of celestial mechanics during the 18th 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 other 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 from 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 gravitational field equations, the famous Einstein equations, and with their help calculate the correct value for the precession of the orbit of Mercury. They also give a measured value for the deflection of light rays that pass near the Sun, and there is no doubt that they imply correct results for macroscopic gravity. Einstein's theory of gravity, or general theory of relativity (GR), as he himself called it, is one of 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 coordinates that differ from point to point means that gravity is a geometric concept.

We say that spacetime is curved. Consider a ball on a 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, taking the shortest path. 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.

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
- a constant value 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 they act on given body other bodies 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 is numerically equal to the force of 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
acts between the surfaces of interacting bodies when the magnitude of external forces is insufficient to cause their relative movement.

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 static friction force

, (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. Numerical value rolling friction force is determined in accordance with Coulomb's law

, (2.55)

where k[m] – rolling friction coefficient.

 21.1. Newton's law of universal gravitation
Gravitational interactions are inherent in all material bodies (Fig. 111).

Rice. 111
The law describing these forces, discovered by I. Newton and published in 1687, was called the law of universal gravitation: two material points attract with forces proportional to the product of the masses of these points, inversely proportional to the square of the distance between the points and directed along the straight line connecting these points:

Since strength is vector quantity, then the formula that determines the force of attraction should be given a vector form.
To do this, we introduce the vector r 12, connecting the points 1 And 2 (Fig. 112).

rice. 112
Then the force of attraction acting on the second body can be written in the form

In formulas (1), (2), the proportionality coefficient b is called the gravitational constant. The value of this quantity cannot be found from others physical laws and determined experimentally. The numerical value of the gravitational constant depends on the choice of system of units, for example, in SI it is equal to:

The gravitational constant was first experimentally measured by the English physicist Henry Cavendish. In 1798, he constructed a torsion balance and used it to measure the force of attraction between two spheres, confirming the law of universal gravitation; determined the gravitational constant, mass and average density Earth.
The question of the nature of gravitational interaction is extremely complex. I. Newton himself gave a laconic answer to this question: “I do not invent hypotheses,” thereby refusing to even discuss this topic. It is enough that the law of universal gravitation high degree accurately quantitatively describes gravitational interaction. Tremendous successes Newtonian mechanics for almost two centuries predetermined a similar approach to all physical science, not only mechanics: it is enough to discover, find the laws that correctly describe physical phenomena, and learn to apply them to quantitative description these phenomena.
Thus, in the study of gravity, it was believed that in an incomprehensible way one body can influence another, and this influence is transmitted instantly, that is, a change in the position of one of the bodies instantly changes the forces acting on other bodies, regardless of the distance at which these bodies are located . This general approach to character physical interactions called the theory of long-range action. A similar view of the interactions of bodies was extended to electrical and magnetic phenomena, the study of which was actively carried out during the 18th – 19th centuries. Only in the 30s years XIX century English physicist M. Faraday for electromagnetic interactions the main provisions were formulated alternative theory short-range interaction: to transmit interaction, a “mediator” is required, a certain medium that transmits these interactions; the interactions themselves cannot be transmitted instantly, it requires certain time in order for a change in the position of one of the bodies to be “felt” by other interacting bodies. At the beginning of the 20th century German physicist A. Einstein built a new theory of gravity - the general theory of relativity. Within the framework of this theory, gravitational interactions are explained in the following way: each body with mass changes the properties of the space-time around itself (creates a gravitational field), while other bodies move in this changed space-time (in the gravitational field), which leads to the appearance of observable forces, acceleration, etc. From this point From a perspective, the expression “is in a gravitational field” is equivalent to the expression “gravitational forces act.”
We will turn to these questions later when studying the electromagnetic field.
The most striking thing about the phenomenon of gravity is that gravitational forces are proportional to the masses of bodies. Indeed, earlier we talked about mass as a measure of the inertia of a body. It turned out that mass also determines a fundamentally different property material bodies− is a measure of the ability to participate in gravitational interactions. Therefore, we can talk about two masses - inertial and gravitational. The law of universal gravitation states that these masses are proportional to each other. This statement has long been confirmed known fact: All bodies fall to the ground with the same acceleration. Experimented with high accuracy the proportionality of gravitational and inertial masses was confirmed in the works of the Hungarian physicist Lorand Eotvos. Subsequently, the proportionality of inertial and gravitational masses formed the basis new theory gravity − general theory relativity of A. Einstein.
In conclusion, we note that the law of universal gravitation can be used as the basis for determining the unit of mass (of course, gravitational). For example: two point bodies of unit gravitational mass, located at a distance of one meter, are attracted with a force of one N.

 Assignment for independent work : determine the masses of two point bodies located at a distance 1.0 m from each other and interacting with force 1.0 N.

For gravitational forces, the principle of superposition is valid: the force acting on a point body from several other bodies is equal to the sum of the forces acting from each body. This statement is also a generalization of experimental data and a fundamental property of gravitational interactions.
Let's look at the principle of superposition from a mathematical point of view: according to the law of universal gravitation, the force of gravitational interaction is proportional to the mass of these bodies. If the dependence on masses were nonlinear, then the principle of superposition would not apply. Indeed, let a body of mass m o interacts with two point bodies with masses m 1 And m 2. Let's mentally place the bodies m 1 And m 2 to one point (then they can be considered as one body). In this case, the force acting on the body m o, is equal to:

presented as the sum of forces acting on the part of two bodies − m 1 And m 2.
In the case of a nonlinear relationship between force and mass, the superposition principle would not be valid.
The law of universal gravitation for point bodies and the principle of superposition make it possible, in principle, to calculate the forces of interaction between bodies of finite sizes (Fig. 113).

rice. 113
To do this, it is necessary to mentally divide each of the bodies into small sections, each of which can be considered as a material point. Then calculate the double sum of the interaction forces between all pairs of points. IN general case calculating such a sum is a complex mathematical problem.
We emphasize that the force of interaction between bodies of finite sizes is calculated only by the method of breaking up bodies and subsequent summation. It is erroneous to say that the force of interaction between bodies can be calculated as the force of interaction, equal to strength interactions of point bodies located at centers of mass. To substantiate this statement, consider a simple example.
Let one of the interacting bodies be considered a material point of mass m o, and the second body can be represented as two material points equal masses m, located at a fixed distance a from each other (Fig. 114).

rice. 114
All material points are located on the same straight line, the distance from the first body to the center of the second is denoted by r. The force of attraction acting on a body m o, is equal to:

If we connect the material points that make up the second body into one mass 2m, located in the center of the body, then the interaction force will be equal to:

which is different from expression (3). Only when r >> a expression (3) goes into formula (2). Note that in this case the second body should be considered as a material point.