You already know that there are attractive forces between all bodies, called forces universal gravity .
Their action is manifested, for example, in the fact that bodies fall to the Earth, the Moon revolves around the Earth, and the planets revolve around the Sun. If gravitational forces disappeared, the Earth would fly away from the Sun (Fig. 14.1).
The law of universal gravitation was formulated in the second half of the 17th century by Isaac Newton.
Two material points of mass m 1 and m 2 located at a distance R are attracted with forces directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Modulus of each force
The proportionality factor G is called gravitational constant. (From the Latin “gravitas” - heaviness.) Measurements showed that
G = 6.67 * 10 -11 N * m 2 / kg 2. (2)
The law of universal gravitation reveals another important property body mass: it is a measure of not only the inertia of the body, but also its gravitational properties.
1. What are the forces of attraction between two material points weighing 1 kg each, located at a distance of 1 m from each other? How many times is this force greater or less than the weight of a mosquito whose mass is 2.5 mg?
Such a small value of the gravitational constant explains why we do not notice gravitational attraction between the objects around us.
Gravitational forces manifest themselves noticeably only when at least one of the interacting bodies has a huge mass - for example, it is a star or a planet.
3. How will the force of attraction between two material points change if the distance between them is increased by 3 times?
4. Two material points of mass m each are attracted with a force F. With what force are material points of mass 2m and 3m, located at the same distance, attracted?
2. The movement of planets around the Sun
The distance from the Sun to any planet is many times greater than the size of the Sun and the planet. Therefore, when considering the movement of planets, they can be considered material points. Therefore, the force of attraction of the planet to the Sun
where m is the mass of the planet, M С is the mass of the Sun, R is the distance from the Sun to the planet.
We will assume that the planet moves around the Sun uniformly in a circle. Then the speed of the planet’s movement can be found if we take into account that the acceleration of the planet a = v 2 /R is due to the action of the gravitational force F of the Sun and the fact that, according to Newton’s second law, F = ma.
5. Prove that the speed of the planet
the larger the orbital radius, the slower the planet's speed.
6. The radius of Saturn’s orbit is approximately 9 times larger than the radius of the Earth’s orbit. Find orally what is approximately the speed of Saturn if the Earth moves in its orbit at a speed of 30 km/s?
In a time equal to one orbital period T, the planet, moving at speed v, travels a distance equal to length circle of radius R.
7. Prove that the planet’s orbital period
From this formula it follows that the larger the orbital radius, the longer period planetary revolutions.
9. Prove that for all planets of the Solar system
Clue. Use formula (5).
From formula (6) it follows that For all planets in the Solar System, the ratio of the cube of the orbital radius to the square of the orbital period is the same. This pattern (it is called Kepler's third law) was discovered by the German scientist Johann Kepler based on the results of many years of observations by the Danish astronomer Tycho Brahe.
3. Conditions for the applicability of the formula for the law of universal gravitation
Newton proved that the formula
F = G(m 1 m 2 /R 2)
For the force of attraction between two material points, you can also use:
- For homogeneous balls and spheres (R is the distance between the centers of balls or spheres, Fig. 14.2, a);
– for a homogeneous ball (sphere) and a material point (R is the distance from the center of the ball (sphere) to the material point, Fig. 14.2, b). 
4. Gravity and the law of universal gravitation
The second of the above conditions means that using formula (1) you can find the force of attraction of a body of any shape to a homogeneous ball, which is much larger than this body. Therefore, using formula (1), it is possible to calculate the force of attraction to the Earth of a body located on its surface (Fig. 14.3, a). We get an expression for gravity:
(The Earth is not a homogeneous sphere, but it can be considered spherically symmetrical. This is sufficient for the possibility of applying formula (1).) 
10. Prove that near the surface of the Earth
Where M Earth is the mass of the Earth, R Earth is its radius.
Clue. Use formula (7) and the fact that F t = mg.
Using formula (1), we can find the acceleration free fall at height h above surface of the Earth(Fig. 14.3, b).
11. Prove that
12. What is the acceleration of gravity at a height above the Earth’s surface equal to its radius?
13. How many times is the acceleration of gravity on the surface of the Moon less than on the surface of the Earth?
Clue. Use formula (8), in which replace the mass and radius of the Earth with the mass and radius of the Moon.
14. Star radius white dwarf may be equal to the radius of the Earth, and its mass - equal mass Sun. Why equal to weight kilogram weight on the surface of such a “dwarf”?
5. First escape velocity
Let's imagine that very high mountain They installed a huge cannon and fired it in a horizontal direction (Fig. 14.4). 
The greater the initial speed of the projectile, the further it will fall. It will not fall at all if its initial speed is selected so that it moves around the Earth in a circle. Flying in a circular orbit, the projectile will then become an artificial satellite of the Earth.
Let our satellite projectile move in low Earth orbit (this is the name for an orbit whose radius can be taken equal to the radius Earth R Earth).
With uniform motion in a circle, the satellite moves with centripetal acceleration a = v2/REarth, where v is the speed of the satellite. This acceleration is due to the action of gravity. Consequently, the satellite moves with gravitational acceleration directed towards the center of the Earth (Fig. 14.4). Therefore a = g.
15. Prove that when moving in low Earth orbit, the speed of the satellite
Clue. Use the formula a = v 2 /r for centripetal acceleration and the fact that when moving in an orbit of radius R Earth, the acceleration of the satellite is equal to the acceleration of gravity.
The speed v 1 that must be imparted to a body so that it moves under the influence of gravity in a circular orbit near the surface of the Earth is called the first escape velocity. It is approximately equal to 8 km/s.
16. Express the first escape velocity in terms of the gravitational constant, mass and radius of the Earth.
Clue. In the formula obtained in the previous task, replace the mass and radius of the Earth with the mass and radius of the Moon.
In order for a body to leave the vicinity of the Earth forever, it must be given a speed of approximately 11.2 km/s. It is called the second escape velocity.
6. How the gravitational constant was measured
If we assume that the gravitational acceleration g near the Earth's surface, the mass and radius of the Earth are known, then the value of the gravitational constant G can be easily determined using formula (7). The problem, however, is that until the end of the 18th century the mass of the Earth could not be measured.
Therefore, in order to find the value of the gravitational constant G, it was necessary to measure the force of attraction of two bodies of known mass located at a certain distance from each other. At the end of the 18th century, the English scientist Henry Cavendish was able to carry out such an experiment.
He suspended a light horizontal rod with small metal balls a and b on a thin elastic thread and, using the angle of rotation of the thread, measured the attractive forces acting on these balls from large metal balls A and B (Fig. 14.5). The scientist measured small angles of rotation of the thread by the displacement of the “bunny” from the mirror attached to the thread. 
Cavendish's experiment was figuratively called the "weighing of the Earth" because this experiment made it possible for the first time to measure the mass of the Earth.
18. Express the mass of the Earth in terms of G, g and R Earth.
Additional questions and tasks
19. Two ships weighing 6000 tons each are attracted by forces of 2 mN. What is the distance between the ships?
20. With what force does the Sun attract the Earth?
21. With what force does a person weighing 60 kg attract the Sun?
22. What is the acceleration of gravity at a distance from the Earth’s surface equal to its diameter?
23. How many times is the acceleration of the Moon, due to the Earth’s gravity, less than the acceleration of gravity on the Earth’s surface?
24. The acceleration of free fall on the surface of Mars is 2.65 times less than the acceleration of free fall on the surface of the Earth. The radius of Mars is approximately 3400 km. How many times is the mass of Mars less than the mass of Earth?
25. Why equal to the period revolutions of an artificial Earth satellite in low Earth orbit?
26. What is the first escape velocity for Mars? The mass of Mars is 6.4 * 10 23 kg, and the radius is 3400 km.
Newton's law of gravity the law of universal gravitation, one of the universal laws of nature; according to N. z. i.e. all material bodies attract each other, and the magnitude of the gravitational force does not depend on physical and chemical properties bodies, on the state of their motion, on the properties of the environment where the bodies are located. On Earth, gravity manifests itself primarily in the existence of gravity, which is the result of the attraction of any material body by the Earth. Associated with this is the term “gravity” (from the Latin gravitas - heaviness), equivalent to the term “gravity”. Gravitational interaction in accordance with the New Law. t. plays main role in motion star systems such as double and multiple stars, inside star clusters and galaxies. However, gravitational fields inside star clusters and galaxies are very complex character, have not yet been sufficiently studied, as a result of which the movements inside them are studied using methods other than celestial mechanics(see Stellar astronomy). Gravitational interaction also plays a significant role in all cosmic processes in which accumulations of large masses of matter participate. N. z. t. is the basis for studying the movement of artificial celestial bodies, in particular artificial satellites of the Earth and the Moon, space probes. On N. z. t. relies on Gravimetry. The forces of attraction between ordinary macroscopic material bodies on Earth can be detected and measured, but do not play any noticeable practical role. In the microcosm, the forces of attraction are negligible compared to intramolecular and intranuclear forces. Newton left open the question of the nature of gravity. The assumption about the instantaneous propagation of gravity in space (i.e., the assumption that with a change in the positions of bodies the gravitational force between them instantly changes), which is closely related to the nature of gravity, was also not explained. The difficulties associated with this were eliminated only in Einstein's theory of gravitation, which represented a new stage in the knowledge of the objective laws of nature. Lit.: Isaac Newton. 1643-1727. Sat. Art. to the tercentenary of his birth, ed. acad. S. I. Vavilova, M. - L., 1943; Berry A., Brief History astronomy, trans. from English, M. - L., 1946; Subbotin M.F., Introduction to theoretical astronomy, M., 1968. Yu. A. Ryabov.
Great Soviet Encyclopedia. - M.: Soviet encyclopedia . 1969-1978 .
See what "Newton's law of gravity" is in other dictionaries:
- (law of universal gravitation), see Art. (see GRAVITY). Physical encyclopedic dictionary. M.: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1983 ... Physical encyclopedia
NEWTON'S LAW OF GRAVITY, the same as the law of universal gravitation... Modern encyclopedia
The same as the law of universal gravitation... Big Encyclopedic Dictionary
Newton's law of gravity- NEWTON'S LAW OF GRAVITY, the same as the law of universal gravitation. ... Illustrated Encyclopedic Dictionary
NEWTON'S LAW OF GRAVITY- the same as (see) ...
Same as the law of universal gravitation. * * * NEWTON'S LAW OF GRAVITY NEWTON'S LAW OF GRAVITY, the same as the law of universal gravitation (see UNIVERSAL GRAVITATION LAW) ... Encyclopedic Dictionary
Newton's law of gravity- Niutono gravitacijos dėsnis statusas T sritis fizika atitikmenys: engl. Newton's law of gravitation vok. Newtonsches Gravitationsgesetz, n; Newtonsches Massenanziehungsgesetz, n rus. Newton's law of gravity, m; Newton's law of gravity, m pranc.… … Fizikos terminų žodynas
Gravity (universal gravitation, gravitation) (from Latin gravitas “heaviness”) long-range fundamental interaction in nature, to which all material bodies are subject. According to modern data, it is a universal interaction in that... ... Wikipedia
LAW OF GRAVITY- (Newton’s law of gravity) all material bodies attract each other with forces directly proportional to their masses and inversely proportional to the square of the distance between them: where F is the modulus of the gravitational force, m1 and m2, the masses of interacting bodies, R... ... Big Polytechnic Encyclopedia
Law of Gravity- I. Newton’s law of gravitation (1643–1727) in classical mechanics, according to which the force of gravitational attraction of two bodies with masses m1 and m2 is inversely proportional to the square of the distance r between them; proportionality coefficient G gravitational... Concepts modern natural science. Glossary of basic terms
By what law are you going to hang me?
- And we hang everyone according to one law - the law of Universal Gravity.
Law of Gravity
The phenomenon of gravity is the law of universal gravitation. Two bodies act on each other with a force that is inversely proportional to the square of the distance between them and directly proportional to the product of their masses.
Mathematically we can express this great law by the formula
Gravity acts on vast distances in the Universe. But Newton argued that all objects are mutually attracted. Is it true that any two objects attract each other? Just imagine, it is known that the Earth attracts you sitting on a chair. But have you ever thought that a computer and a mouse attract each other? Or a pencil and pen lying on the table? In this case, we substitute the mass of the pen, the mass of the pencil into the formula, divide by the square of the distance between them, taking into account the gravitational constant, we obtain their force mutual attraction. But it will turn out to be so small (due to the small masses of the pen and pencil) that we do not feel its presence. It's another matter when we're talking about about the Earth and the chair, or the Sun and the Earth. The masses are significant, which means we can already evaluate the effect of the force.
Let's remember the acceleration of free fall. This is the effect of the law of attraction. Under the influence of force, a body changes speed the more slowly, the greater its mass. As a result, all bodies fall to Earth with the same acceleration.
What causes this invisible unique force? Today the existence of gravitational field. You can learn more about the nature of the gravitational field in additional material topics.
Think about it, what is gravity? Where is it from? What is it? Surely it cannot be that the planet looks at the Sun, sees how far away it is, and calculates the inverse square of the distance in accordance with this law?
Direction of gravity
There are two bodies, let’s say body A and B. Body A attracts body B. The force with which body A acts begins on body B and is directed towards body A. That is, it “takes” body B and pulls it towards itself. Body B “does” the same thing to body A.
Every body is attracted by the Earth. The earth “takes” the body and pulls it towards its center. Therefore, this force will always be directed vertically downward, and it is applied from the center of gravity of the body, it is called the force of gravity.
The main thing to remember
Some methods of geological exploration, tide prediction and, more recently, calculation of the movement of artificial satellites and interplanetary stations. Advance calculation of planetary positions.
Can we carry out such an experiment ourselves, and not guess whether planets and objects are attracted?
Such direct experience made Cavendish (Henry Cavendish (1731-1810) - English physicist and chemist) using the device shown in the figure. The idea was to hang a rod with two balls on a very thin quartz thread and then bring two large lead balls towards them from the side. The attraction of the balls will twist the thread slightly - slightly, because the forces of attraction between ordinary objects are very weak. With the help of such a device, Cavendish was able to directly measure the force, distance and magnitude of both masses and, thus, determine gravitational constant G.
The unique discovery of the gravitational constant G, which characterizes the gravitational field in space, made it possible to determine the mass of the Earth, the Sun and other celestial bodies. Therefore, Cavendish called his experience "weighing the Earth."
It's interesting that various laws physicists have some common features. Let's turn to the laws of electricity (Coulomb force). Electric forces are also inversely proportional to the square of the distance, but between charges, and the thought involuntarily arises that this pattern conceals deep meaning. Until now, no one has been able to imagine gravity and electricity as two different manifestations the same entity.
The force here also varies inversely with the square of the distance, but the difference in magnitude electrical forces and the forces of gravity are amazing. Trying to install general nature gravity and electricity, we discover such a superiority of electrical forces over the forces of gravity that it is difficult to believe that both have the same source. How can you say that one is more powerful than the other? After all, everything depends on what the mass is and what the charge is. When discussing how strongly gravity acts, you have no right to say: “Let's take a mass of such and such a size,” because you choose it yourself. But if we take what Nature itself offers us (her eigenvalues and measures that have nothing to do with our inches, years, with our measures), then we can compare. We take an elementary charged particle, such as an electron. Two elementary particles, two electrons, due to an electric charge, repel each other with a force inversely proportional to the square of the distance between them, and due to gravity they are attracted to each other again with a force inversely proportional to the square of the distance.
Question: What is the ratio of gravitational force to electrical force? Gravity is to electrical repulsion as one is to a number with 42 zeros. This causes deepest bewilderment. Where could such a huge number come from?
People look for this huge coefficient in other natural phenomena. They try all kinds of big numbers, and if you need a big number, why not take, say, the ratio of the diameter of the Universe to the diameter of a proton - surprisingly, this is also a number with 42 zeros. And so they say: maybe this coefficient is equal to the ratio of the diameter of the proton to the diameter of the Universe? This is an interesting idea, but as the Universe gradually expands, the gravitational constant must also change. Although this hypothesis has not yet been refuted, we do not have any evidence in its favor. On the contrary, some evidence suggests that the gravitational constant did not change in this way. This huge number remains a mystery to this day.
Einstein had to modify the laws of gravity in accordance with the principles of relativity. The first of these principles states that a distance x cannot be overcome instantly, whereas according to Newton's theory, forces act instantly. Einstein had to change Newton's laws. These changes and clarifications are very small. One of them is this: since light has energy, energy is equivalent to mass, and all masses are attracted, light is also attracted and, therefore, passing by the Sun, must be deflected. This is how it actually happens. The force of gravity is also slightly modified in Einstein's theory. But this very slight change in the law of gravitation is just sufficient to explain some of the apparent irregularities in the motion of Mercury.
Physical phenomena in the microworld are subject to different laws than phenomena in the world on a large scale. The question arises: how does gravity manifest itself in the world of small scales? The quantum theory of gravity will answer it. But quantum theory there is no gravity yet. People have not yet been very successful in creating a theory of gravity that is fully consistent with quantum mechanical principles and with the uncertainty principle.
So, the movement of planets, for example the Moon around the Earth or the Earth around the Sun, is the same fall, but only a fall that lasts indefinitely (in any case, if we ignore the transition of energy into “non-mechanical” forms).
The conjecture about the unity of causes governing the movement of planets and the fall of earthly bodies was expressed by scientists long before Newton. Apparently, the first to clearly express this idea was the Greek philosopher Anaxagoras, a native of Asia Minor, who lived in Athens almost two thousand years ago. He said that the Moon, if it did not move, would fall to the Earth.
However, Anaxagoras’s brilliant guess, apparently, did not have any practical impact on the development of science. She was destined to be misunderstood by her contemporaries and forgotten by her descendants. Ancient and medieval thinkers, whose attention was attracted by the movement of the planets, were very far from the correct (and more often than not any) interpretation of the causes of this movement. After all, even the great Kepler, who managed, at the cost of enormous labor, to formulate precise mathematical laws movement of the planets, believed that the cause of this movement was the rotation of the Sun.
According to Kepler's ideas, the Sun, rotating, constantly pushes the planets into rotation. True, it remained unclear why the time of revolution of the planets around the Sun differs from the period of revolution of the Sun around its own axis. Kepler wrote about this: “if the planets did not have natural resistance, then it would be impossible to give reasons why they should not follow exactly the rotation of the Sun. But although in reality all the planets move in the same direction in which the rotation of the Sun occurs, the speed of their movement is not the same. The fact is that they mix, in certain proportions, the inertia of their own mass with the speed of their movement.”
Kepler failed to understand that the coincidence of the directions of motion of the planets around the Sun with the direction of rotation of the Sun around its axis is not associated with the laws of planetary motion, but with the origin of our solar system. Artificial planet can be launched both in the direction of rotation of the Sun and against this rotation.
Robert Hooke came much closer than Kepler to the discovery of the law of attraction of bodies. Here are his actual words from a work entitled An Attempt to Study the Motion of the Earth, published in 1674: “I will develop a theory which is in every respect consistent with the generally accepted rules of mechanics. This theory is based on three assumptions: firstly, that all celestial bodies, without exception, have a gravity directed towards their center, due to which they attract not only their own parts, but also all celestial bodies within their sphere of action. According to the second assumption, all bodies moving in a rectilinear and uniform manner will move in a straight line until they are deflected by some force and begin to describe trajectories in a circle, an ellipse, or some other less simple curve. According to the third assumption, the forces of attraction act the more strongly, the closer to them the bodies on which they act are located. I have not yet been able to establish by experience what the different degrees of attraction are. But if we develop this idea further, astronomers will be able to determine the law according to which all celestial bodies move.”
Truly, one can only be amazed that Hooke himself did not want to engage in the development of these ideas, citing being busy with other work. But a scientist appeared who made a breakthrough in this area
The history of Newton's discovery of the law of universal gravitation is quite well known. For the first time, the idea that the nature of the forces that make a stone fall and determine the movement of celestial bodies is the same arose with Newton the student, that the first calculations did not give the correct results, since the data available at that time on the distance from the Earth to the Moon were inaccurate, that 16 years later new, corrected information about this distance appeared. To explain the laws of planetary motion, Newton applied the laws of dynamics he created and the law of universal gravitation that he himself established.
He named the Galilean principle of inertia as the first law of dynamics, including it in the system of basic laws-postulates of his theory.
At the same time, Newton had to eliminate the error of Galileo, who believed that uniform motion in a circle was motion by inertia. Newton pointed out (and this is the second law of dynamics) that the only way to change the motion of a body - the value or direction of the velocity - is to act on it with some force. In this case, the acceleration with which a body moves under the influence of a force is inversely proportional to the mass of the body.
According to Newton's third law of dynamics, “to every action there is always an equal and opposite reaction.”
Consistently applying the principles - the laws of dynamics, he first calculated centripetal acceleration of the Moon as it moves in orbit around the Earth, and then was able to show that the ratio of this acceleration to the acceleration of free fall of bodies near the Earth’s surface is equal to the ratio of the squares of the Earth’s radii and lunar orbit. From this Newton concluded that the nature of gravity and the force that holds the Moon in orbit are the same. In other words, according to his conclusions, the Earth and the Moon are attracted to each other with a force inversely proportional to the square of the distance between their centers Fg ≈ 1∕r2.
Newton was able to show that the only explanation for the independence of the acceleration of free fall of bodies from their mass is the proportionality of the force of gravity to the mass.
Summarizing the findings, Newton wrote: “there can be no doubt that the nature of gravity on other planets is the same as on Earth. In fact, let us imagine that the earth's bodies are raised to the orbit of the Moon and sent together with the Moon, also devoid of any movement, to fall to the Earth. Based on what has already been proven (meaning the experiments of Galileo), there is no doubt that at the same times they will pass through the same spaces as the Moon, for their masses are related to the mass of the Moon in the same way as their weights are to its weight.” So Newton discovered and then formulated the law of universal gravitation, which is rightfully the property of science.
2. Properties of gravitational forces.
One of the most remarkable properties of the forces of universal gravity, or, as they are often called, gravitational forces, is reflected in the very name given by Newton: universal. These forces, so to speak, are the “most universal” among all the forces of nature. Everything that has mass - and mass is inherent in any form, any kind of matter - must experience gravitational influences. Even light is no exception. If you visualize gravitational forces with the help of threads that stretch from one body to another, then an innumerable number of such threads would have to permeate space anywhere. At the same time, it is worth noting that it is impossible to break such a thread and protect yourself from gravitational forces. There are no barriers to universal gravity; their radius of action is unlimited (r = ∞). Gravitational forces are long-range forces. This is the “official name” of these forces in physics. Due to long-range action, gravity connects all bodies of the Universe.
The relative slowness of the decrease in forces with distance at each step is manifested in our terrestrial conditions: after all, all bodies do not change their weight, being transferred, from one height to another (or, to be more precise, they change, but extremely insignificantly), precisely because with a relatively small change in distance - in in this case from the center of the Earth - gravitational forces practically do not change.
By the way, it is for this reason that the law of measuring gravitational forces with distance was discovered “in the sky.” All the necessary data was drawn from astronomy. One should not, however, think that a decrease in gravity with height cannot be detected under terrestrial conditions. So, for example, a pendulum clock with an oscillation period of one second will fall behind a day by almost three seconds if it is raised from the basement to the top floor of Moscow University (200 meters) - and this is only due to a decrease in gravity.
Heights at which they move artificial satellites, are already comparable to the radius of the Earth, so to calculate their trajectory, taking into account the change in force gravity with distance is absolutely necessary.
Gravitational forces have another very interesting and unusual property, which will be discussed now.
For many centuries, medieval science accepted as an unshakable dogma Aristotle's statement that a body falls the faster the greater its weight. Even everyday experience confirms this: it is known that a feather falls slower than a stone. However, as Galileo was able to show for the first time, the whole point here is that air resistance, coming into play, radically distorts the picture that would be if only earthly gravity acted on all bodies. There is a remarkable experiment with the so-called Newton tube, which makes it possible to very easily evaluate the role of air resistance. Here is a short description of this experience. Imagine an ordinary glass tube (so that you can see what is happening inside) in which various items: pellets, pieces of cork, feathers or fluffs, etc. If you turn the tube over so that all this can fall, then the pellet will flash quickly, followed by pieces of cork and, finally, the fluff will smoothly fall. But let’s try to follow the fall of the same objects when the air is pumped out of the tube. The fluff, having lost its former slowness, rushes along, keeping pace with the pellet and the cork. This means that its movement was delayed by air resistance, which had a lesser effect on the movement of the plug and even less on the movement of the pellet. Consequently, if it were not for air resistance, if only the forces of universal gravity acted on bodies - in a particular case, gravity - then all bodies would fall exactly the same, accelerating at the same pace.
But “there is nothing new under the sun.” Two thousand years ago, Lucretius Carus wrote in his famous poem “On the Nature of Things”:
everything that falls in rare air,
Should fall faster according to its own weight
Only because water or air is a subtle essence
I am not able to put obstacles in the way of things that are the same,
But it is more likely to yield to those with greater severity.
On the contrary, I am never capable of anything anywhere
The thing holds the emptiness and appears as some kind of support,
By nature, constantly giving in to everything.
Therefore, everything, rushing through the void without obstacles,
Have the same speed despite the difference in weight.
Of course, these wonderful words were a great guess. To turn this guess into a reliably established law, it took many experiments, starting with the famous experiments of Galileo, who studied the fall of balls of the same size, but made of various materials(marble, wood, lead, etc.), and ending with the most complex modern measurements of the influence of gravity on light. And all this variety of experimental data persistently strengthens us in the belief that gravitational forces impart equal acceleration to all bodies; in particular, the acceleration of free fall caused by gravity is the same for all bodies and does not depend on the composition, structure, or mass of the bodies themselves.
This seemingly simple law expresses perhaps the most remarkable feature of gravitational forces. There are literally no other forces that accelerate all bodies equally, regardless of their mass.
So, this property of the forces of universal gravity can be compressed into one short statement: the gravitational force is proportional to the mass of bodies. Let us emphasize that here we are talking about the very mass that acts as a measure of inertia in Newton’s laws. It is even called inert mass.
The four words “gravitational force is proportional to mass” contain a surprisingly deep meaning. Large and small bodies, hot and cold, of all kinds chemical composition, any structure - they all experience the same gravitational interaction if their masses are equal.
Or maybe this law is really simple? After all, Galileo, for example, considered it almost self-evident. Here is his reasoning. Let two bodies of different weights fall. According to Aristotle, a heavy body should fall faster even in vacuum. Now let's connect the bodies. Then, on the one hand, the bodies should fall faster, since the total weight has increased. But, on the other hand, adding to a heavy body a part that falls more slowly should slow down this body. There is a contradiction that can be eliminated only if we assume that all bodies under the influence of gravity alone fall with the same acceleration. It's like everything is consistent! However, let us think again about the above reasoning. It is based on the common method of proof “by contradiction”: by assuming that a heavier body falls faster than a lighter one, we have arrived at a contradiction. And from the very beginning there was an assumption that the acceleration of free fall is determined by weight and only weight. (Strictly speaking, not by weight, but by mass.)
But this is not at all obvious in advance (i.e., before the experiment). What if this acceleration was determined by the volume of the bodies? Or temperature? Let's imagine that there is a gravitational charge, similar to an electric charge and, like the latter, completely unrelated directly to mass. The comparison with electric charge is very useful. Here are two specks of dust between the charged plates of a capacitor. Let these specks of dust equal charges, and the masses are related as 1 to 2. Then the accelerations must differ by a factor of two: the forces determined by the charges are equal, and with equal forces the body is twice greater mass accelerates at half the rate. If you connect dust particles, then, obviously, the acceleration will have a new, intermediate value. No speculative approach without an experimental study of electrical forces can give anything here. The picture would be exactly the same if the gravitational charge were not associated with mass. But only experience can answer the question of whether such a connection exists. And we now understand that it was the experiments that proved the identical acceleration due to gravity for all bodies that essentially showed that the gravitational charge (gravitational or heavy mass) is equal to the inertial mass.
Experience and only experience can serve as a basis for physical laws, and the criterion of their fairness. Let us at least recall the record-breaking precision experiments conducted under the leadership of V.B. Braginsky at Moscow State University. These experiments, in which an accuracy of about 10-12 was obtained, once again confirmed the equality of heavy and inert mass.
It is on experience, on the wide testing of nature - from the modest scale of a small laboratory of a scientist to the grandiose cosmic scale - that the law of universal gravitation is based, which (to summarize everything said above) says:
The force of mutual attraction of any two bodies whose dimensions are much smaller than the distance between them is proportional to the product of the masses of these bodies and inversely proportional to the square of the distance between these bodies.
The proportionality coefficient is called the gravitational constant. If we measure length in meters, time in seconds, and mass in kilograms, the gravitational force will always be equal to 6.673*10-11, and its dimension will be m3/kg*s2 or N*m2/kg2, respectively.
G=6.673*10-11 N*m2/kg2
3. Gravitational waves.
Newton's law of universal gravitation does not say anything about the time of transmission of gravitational interaction. It is implicitly assumed that it occurs instantly, no matter how large the distances between the interacting bodies are. This view is generally typical of supporters of action at a distance. But from " special theory Einstein's relativity implies that gravity is transmitted from one body to another at the same speed as the light signal. If some body moves from its place, then the curvature of space and time caused by it does not change instantly. First, this will affect the immediate vicinity of the body, then the change will affect more and more distant areas, and, finally, a new distribution of curvature will be established throughout space, corresponding to the changed position of the body.
And here we come to a problem that has caused and continues to cause greatest number disputes and disagreements - the problem of gravitational radiation.
Can gravity exist if there is no mass creating it? According to Newton's law, definitely not. It makes no sense to even raise such a question there. However, as soon as we agreed that gravitational signals are transmitted, although very large, but still not infinite speed, everything is changing radically. Indeed, imagine that at first the mass causing gravity, for example a ball, was at rest. All bodies around the ball will be affected by ordinary Newtonian forces. Now let’s remove the ball from its original place with great speed. At first, the surrounding bodies will not feel this. After all, gravitational forces do not change instantly. It takes time for changes in the curvature of space to spread in all directions. This means that the surrounding bodies will experience the same influence of the ball for some time, when the ball itself is no longer there (at least, in the same place).
It turns out that the curvatures of space acquire a certain independence, that it is possible to tear a body out of the region of space where it caused the curvatures, and in such a way that these curvatures themselves, at least over large distances, will remain and develop in their own way internal laws. Here is gravity without gravitating mass! We can go further. If you make the ball oscillate, then, as it turns out from Einstein’s theory, on Newton's picture gravity, a kind of ripple is superimposed - waves of gravity. To better imagine these waves, you need to use a model - a rubber film. If you not only press your finger on this film, but at the same time make it oscillatory movements, then these vibrations will begin to be transmitted along the stretched film in all directions. This is an analogue of gravitational waves. The farther from the source, the weaker such waves are.
And now at some point we will stop putting pressure on the film. The waves won't go away. They will exist independently, scattering further and further across the film, causing geometry to bend along the way.
In exactly the same way, waves of space curvature - gravitational waves - can exist independently. Many researchers draw this conclusion from Einstein’s theory.
Of course, all these effects are very weak. For example, the energy released when one match burns is many times greater than the energy of gravitational waves emitted by our entire solar system during the same time. But what is important here is not the quantitative, but the principled side of the matter.
Proponents of gravitational waves - and they seem to be in the majority now - predict one more thing. amazing phenomenon; the transformation of gravity into particles such as electrons and positrons (they must be born in pairs), protons, antitrons, etc. (Ivanenko, Wheeler, etc.).
It should look something like this. A wave of gravity reached a certain area of space. At a certain moment, this gravity sharply, abruptly, decreases and at the same time, say, an electron-positron pair appears there. The same can be described as an abrupt decrease in the curvature of space with the simultaneous birth of a pair.
There are many attempts to translate this into quantum mechanical language. Particles are introduced into consideration - gravitons, which are compared to the non-quantum image of a gravitational wave. In the physical literature, the term “transmutation of gravitons into other particles” is in circulation, and these transmutations are mutual transformations– are possible between gravitons and, in principle, any other particles. After all, there are no particles that are insensitive to gravity.
Even if such transformations are unlikely, that is, they happen extremely rarely, cosmic scale they may turn out to be fundamental.
4. Curvature of space-time by gravity,
"Eddington's Parable"
A parable by the English physicist Eddington from the book “Space, Time and Gravity” (retelling):
“In an ocean that has only two dimensions, there once lived a breed of flat fish. It was observed that the fish generally swam in straight lines as long as they did not encounter obvious obstacles in their path. This behavior seemed quite natural. But there was a mysterious area in the ocean; when the fish fell into it, they seemed enchanted; some sailed through this area but changed the direction of their movement, others endlessly circled around this area. One fish (almost Descartes) proposed a theory of vortices; she said that in this area there are whirlpools that make everything that gets into them spin. Over time, a much more advanced theory was proposed (Newton's theory); they said that all fish are attracted to a very large fish - the sun fish, dormant in the middle of the region - and this explained the deviation of their paths. At first this theory seemed perhaps a little strange; but it was confirmed with amazing accuracy by a wide variety of observations. All fish have been found to have this attractive property, proportionate to their size; the law of attraction (analogous to the law of universal gravitation) was extremely simple, but despite this, it explained all movements with such precision that precision had never reached before scientific research. True, some fish, grumbling, declared that they did not understand how such an action at a distance was possible; but everyone agreed that this action was carried out by the ocean, and that it would be easier to understand when the nature of water was better studied. Therefore, almost every fish that wanted to explain gravity began by suggesting some mechanism by which it spread through water.
But there was a fish who looked at things differently. She noticed the fact that the big fish and the small ones always moved along the same paths, although it might seem that it would take a lot of force to deflect the big fish from its path. (The sunfish imparted equal accelerations to all bodies.) Therefore, instead of trying, she began to study in detail the paths of movement of fish and thus came to an astonishing solution to the problem. There was a high place in the world where the sunfish lay. The fish could not directly notice this because they were two-dimensional; but when the fish in its movement fell on the slope of this elevation, then although it tried to swim in a straight line, it involuntarily turned a little to the side. This was the secret of the mysterious attraction or curvature of paths that occurred in the mysterious area. »
This parable shows how the curvature of the world in which we live can give the illusion of gravity, and we see that an effect like gravity is the only way such curvature can manifest itself.
Briefly, this can be formulated as follows. Since gravity bends the paths of all bodies in the same way, we can think of gravity as the curvature of space-time.
5. Gravity on Earth.
If you think about the role that gravitational forces play in the life of our planet, entire oceans open up. And not only oceans of phenomena, but also oceans in the literal sense of the word. Oceans of water. Air ocean. Without gravity they would not exist.
A wave in the sea, the movement of every drop of water in the rivers that feed this sea, all currents, all winds, clouds, the entire climate of the planet are determined by the play of two main factors: solar activity and gravity.
Gravity not only holds people, animals, water and air on Earth, but also compresses them. This compression at the Earth's surface is not so great, but its role is important.
The ship is sailing on the sea. What prevents him from drowning is known to everyone. This is the famous buoyant force of Archimedes. But it appears only because the water is compressed by gravity with a force that increases with increasing depth. Inside spaceship in flight there is no buoyant force, just as there is no weight. The globe itself is compressed by gravitational forces to colossal pressures. At the center of the Earth, the pressure appears to exceed 3 million atmospheres.
Under the influence of long-acting pressure forces under these conditions, all substances that we are accustomed to consider solid behave like pitch or resin. Heavy materials sink to the bottom (if you can call the center of the Earth that way), and light materials float to the surface. This process has been going on for billions of years. It has not ended, as follows from Schmidt’s theory, even now. Concentration heavy elements in the region of the center of the Earth slowly increases.
Well, how does the attraction of the Sun and the closest celestial body of the Moon manifest itself on Earth? Only residents of the ocean coasts can observe this attraction without special instruments.
The sun acts in almost the same way on everything on and inside the Earth. The force with which the Sun attracts a person at noon, when he is closest to the Sun, is almost the same as the force acting on him at midnight. After all, the distance from the Earth to the Sun is ten thousand times greater than the Earth’s diameter, and an increase in the distance by one ten-thousandth when the Earth rotates half a turn around its axis practically does not change the force of gravity. Therefore, the Sun imparts almost identical accelerations to all parts globe and all bodies on its surface. Almost, but still not quite the same. Because of this difference, the ebb and flow of the ocean occurs.
In an area facing the sun earth's surface the force of attraction is somewhat greater than that necessary for the movement of this section along an elliptical orbit, and on the opposite side of the Earth it is somewhat less. As a result, according to Newton's laws of mechanics, the water in the ocean bulges slightly in the direction facing the Sun, and on the opposite side it recedes from the Earth's surface. Tidal forces, as they say, arise, stretching the globe and giving, roughly speaking, the surface of the oceans the shape of an ellipsoid.
The smaller the distances between interacting bodies, the greater the tidal forces. This is why the Moon has a greater influence on the shape of the world's oceans than the Sun. More precisely, tidal influence is determined by the ratio of the mass of a body to the cube of its distance from the Earth; this ratio for the Moon is approximately twice that for the Sun.
If there were no cohesion between the parts of the globe, then tidal forces would tear it apart.
Perhaps this happened to one of Saturn's moons when it came close to this large planet. That fragmented ring that makes Saturn such a remarkable planet may be debris from the satellite.
So, the surface of the world's oceans is similar to an ellipsoid, major axis which is facing the Moon. The earth rotates around its axis. Therefore, a tidal wave moves along the surface of the ocean towards the direction of rotation of the Earth. When it approaches the shore, the tide begins. In some places the water level rises to 18 meters. Then the tidal wave goes away and the tide begins to ebb. The water level in the ocean fluctuates, on average, with a period of 12 hours. 25min. (half a lunar day).
This simple picture is greatly distorted by the simultaneous tidal action of the Sun, water friction, continental resistance, and the complexity of the configuration of the ocean shores and bottom in coastal areas and some other private effects.
It is important that the tidal wave slows down the Earth's rotation.
True, the effect is very small. Over 100 years, the day increases by a thousandth of a second. But, acting for billions of years, the braking forces will lead to the fact that the Earth will always be turned to the Moon with one side, and the Earth’s day will become equal to the lunar month. This has already happened to Luna. The Moon is slowed down so much that it always faces the Earth with one side. To "look" at reverse side Moon, we had to send a spaceship around it.
Newton's classical theory of gravity (Newton's Law of Universal Gravitation)- a law describing gravitational interaction within the framework of classical mechanics. This law was discovered by Newton around 1666. It says that strength F (\displaystyle F) gravitational attraction between two material points of mass m 1 (\displaystyle m_(1)) And m 2 (\displaystyle m_(2)), separated by distance R (\displaystyle R), is proportional to both masses and inversely proportional to the square of the distance between them - that is:
F = G ⋅ m 1 ⋅ m 2 R 2 (\displaystyle F=G\cdot (m_(1)\cdot m_(2) \over R^(2)))
Here G (\displaystyle G)- gravitational constant equal to 6.67408(31)·10 −11 m³/(kg·s²) :.
Encyclopedic YouTube
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✪ Introduction to Newton's law of universal gravitation
✪ Law of Gravity
✪ physics LAW OF UNIVERSAL GRAVITY 9th grade
✪ About Isaac Newton (Brief History)
✪ Lesson 60. The law of universal gravitation. Gravitational constant
Subtitles
Now let's learn a little about gravity, or gravitation. As you know, gravitation, especially in a beginner or even in a fairly advanced course of physics, is a concept that can be calculated and the basic parameters by which it is determined are known, but in fact, gravitation is not entirely understandable. Even if you are familiar with the general theory of relativity, if you are asked what gravity is, you can answer: it is the curvature of space-time and the like. However, it is still difficult to get an intuition as to why two objects, simply because they have so-called mass, are attracted to each other. At least for me it's mystical. Having noted this, let us begin to consider the concept of gravity. We will do this by studying Newton's law of universal gravitation, which is valid for most situations. This law states: the force of mutual gravitational attraction F between two material points with masses m₁ and m₂ is equal to the product of the gravitational constant G by the mass of the first object m₁ and the second object m₂, divided by the square of the distance d between them. This is a fairly simple formula. Let's try to transform it and see if we can get some results that are familiar to us. We use this formula to calculate the acceleration of gravity near the Earth's surface. Let's draw the Earth first. Just to understand what we are talking about. This is our Earth. Let's say we need to calculate the gravitational acceleration acting on Sal, that is, on me. Here I am. Let's try to apply this equation to calculate the magnitude of the acceleration of my fall towards the center of the Earth, or to the center Earth masses. The quantity indicated by the capital letter G is the universal gravitational constant. Once again: G is the universal gravitational constant. Although, as far as I know, although I am not an expert on this matter, it seems to me that its value can change, that is, it is not a real constant, and I assume that its value differs in different measurements. But for our needs, as well as in most physics courses, this is a constant, a constant equal to 6.67 * 10^(−11) cubic meters, divided by kilogram per second squared. Yes, its dimension looks strange, but it is enough for you to understand that these are conventional units necessary to, as a result of multiplying by the masses of objects and dividing by the square of the distance, obtain the dimension of force - newton, or kilogram per meter divided by second squared. So there's no need to worry about these units: just know that we'll have to work with meters, seconds, and kilograms. Let's substitute this number into the formula for force: 6.67 * 10^(−11). Since we need to know the acceleration acting on Sal, m₁ is equal to the mass of Sal, that is, me. I wouldn’t like to expose how much I weigh in this story, so let’s leave this mass as a variable, denoting ms. The second mass in the equation is the mass of the Earth. Let's write down its meaning by looking at Wikipedia. So, the mass of the Earth is 5.97 * 10^24 kilograms. Yes, the Earth is more massive than Sal. By the way, weight and mass are different concepts. So, the force F is equal to the product of the gravitational constant G by the mass ms, then by the mass of the Earth, and divide all this by the square of the distance. You may object: what is the distance between the Earth and what stands on it? After all, if objects touch, the distance is zero. It is important to understand here: the distance between two objects in this formula is the distance between their centers of mass. In most cases, a person's center of mass is located about three feet above the surface of the Earth, unless the person is very tall. Anyway, my center of mass may be three feet above the ground. Where is the center of mass of the Earth? Obviously in the center of the Earth. What is the radius of the Earth? 6371 kilometers, or approximately 6 million meters. Since the height of my center of mass is about one millionth the distance to the center of mass of the Earth, it can be neglected in this case. Then the distance will be equal to 6 and so on, like all other quantities, you need to write it in standard form - 6.371 * 10^6, since 6000 km is 6 million meters, and a million is 10^6. We write, rounding all fractions to the second decimal place, the distance is 6.37 * 10^6 meters. The formula contains the square of the distance, so let's square everything. Let's try to simplify now. First, let's multiply the values in the numerator and move forward the variable ms. Then the force F is equal to the entire mass of Sal top part, let's calculate it separately. So 6.67 times 5.97 equals 39.82. 39.82. This is a product of significant parts, which should now be multiplied by 10 to the required power. 10^(−11) and 10^24 have the same base, so to multiply them it is enough to add the exponents. Adding 24 and −11, we get 13, resulting in 10^13. Let's find the denominator. It is equal to 6.37 squared times 10^6 also squared. As you remember, if the number written in as a degree, is raised to another power, then the exponents are multiplied, which means that 10^6 squared is equal to 10 to the power of 6 multiplied by 2, or 10^12. Next, we calculate the square of 6.37 using a calculator and get... Square 6.37. And it's 40.58. 40.58. All that remains is to divide 39.82 by 40.58. Divide 39.82 by 40.58, which equals 0.981. Then we divide 10^13 by 10^12, which is equal to 10^1, or just 10. And 0.981 times 10 is 9.81. After simplification and simple calculations, we found that the gravitational force near the Earth’s surface acting on Sal is equal to Sel’s mass multiplied by 9.81. What does this give us? Is it now possible to calculate gravitational acceleration? It is known that force is equal to the product of mass and acceleration, therefore the gravitational force is simply equal to the product of Sal’s mass and gravitational acceleration, which is usually denoted lowercase letter g. So, on the one hand, the force of gravity is equal to 9.81 times Sal's mass. On the other hand, it is equal to Sal’s mass per gravitational acceleration. Dividing both sides of the equation by Sal’s mass, we find that the coefficient 9.81 is the gravitational acceleration. And if we included in the calculations full entry units of dimension, then, having reduced the kilograms, we would see that gravitational acceleration is measured in meters divided by a second squared, like any acceleration. You can also notice that the resulting value is very close to the one we used when solving problems about the motion of a thrown body: 9.8 meters per second squared. This is impressive. Let's do another quick gravity problem because we have a couple of minutes left. Let's say we have another planet called Baby Earth. Let the Baby's radius rS be doubled less than radius The Earth is rE, and its mass mS is also equal to half the Earth's mass mE. What will be the force of gravity acting here on any object, and how much less is it than the force of gravity? Although, let's leave the problem for next time, then I'll solve it. See you. Subtitles by the Amara.org community
Properties of Newtonian gravity
In Newtonian theory, every massive body generates a force field of attraction to this body, which is called a gravitational field. This field is potential, and the function of gravitational potential for a material point with mass M (\displaystyle M) is determined by the formula:
φ (r) = − G M r . (\displaystyle \varphi (r)=-G(\frac (M)(r)).)IN general case, when the density of the substance ρ (\displaystyle \rho ) distributed randomly, satisfies the Poisson equation:
Δ φ = − 4 π G ρ (r) . (\displaystyle \Delta \varphi =-4\pi G\rho (r).)The solution to this equation is written as:
φ = − G ∫ ρ (r) d V r + C , (\displaystyle \varphi =-G\int (\frac (\rho (r)dV)(r))+C,)Where r (\displaystyle r) - distance between volume element d V (\displaystyle dV) and the point at which the potential is determined φ (\displaystyle \varphi ), C (\displaystyle C) - arbitrary constant.
The force of attraction acting in a gravitational field on a material point with mass m (\displaystyle m), is related to the potential by the formula:
F (r) = − m ∇ φ (r) . (\displaystyle F(r)=-m\nabla \varphi (r).)A spherically symmetrical body creates the same field outside its boundaries as a material point of the same mass located in the center of the body.
Trajectory of a material point in a gravitational field created by a much larger mass material point, obeys Kepler's laws. In particular, planets and comets in solar system move along ellipses or hyperbolas. The influence of other planets, which distorts this picture, can be taken into account using perturbation theory.
Accuracy of Newton's law of universal gravitation
An experimental assessment of the degree of accuracy of Newton's law of gravitation is one of the confirmations of the general theory of relativity. Experiments on measuring the quadrupole interaction of a rotating body and a stationary antenna showed that the increment δ (\displaystyle \delta ) in the expression for the dependence of the Newtonian potential r − (1 + δ) (\displaystyle r^(-(1+\delta))) at distances of several meters is within (2 , 1 ± 6 , 2) ∗ 10 − 3 (\displaystyle (2.1\pm 6.2)*10^(-3)). Other experiments also confirmed the absence of modifications in the law of universal gravitation.
Newton's law of universal gravitation in 2007 was also tested at distances less than one centimeter (from 55 microns to 9.53 mm). Taking into account the experimental errors, no deviations from Newton's law were found in the studied range of distances.
Precision laser ranging observations of the Moon's orbit confirm the law of universal gravitation at the distance from the Earth to the Moon with precision 3 ⋅ 10 − 11 (\displaystyle 3\cdot 10^(-11)).
Connection with the geometry of Euclidean space
Fact of equality with very high accuracy 10 − 9 (\displaystyle 10^(-9)) exponent of the distance in the denominator of the expression for the force of gravity to the number 2 (\displaystyle 2) reflects the Euclidean nature of the three-dimensional physical space of Newtonian mechanics. In three-dimensional Euclidean space, the surface area of a sphere is exactly proportional to the square of its radius
Historical sketch
The very idea of the universal force of gravity was repeatedly expressed before Newton. Previously, Epicurus, Gassendi, Kepler, Borelli, Descartes, Roberval, Huygens and others thought about it. Kepler believed that gravity is inversely proportional to the distance to the Sun and extends only in the ecliptic plane; Descartes considered it to be the result of vortices in the ether. There were, however, guesses with a correct dependence on distance; Newton, in a letter to Halley, mentions Bulliald, Wren and Hooke as his predecessors. But before Newton, no one was able to clearly and mathematically prove the connection between the law of gravity (a force inversely proportional to the square of the distance) and the laws of planetary motion (Kepler’s laws).
- law of gravitation;
- law of motion (Newton's second law);
- system of methods for mathematical research(mathematical analysis).
Taken together, this triad is sufficient for a complete study of the most complex movements celestial bodies, thereby creating the foundations of celestial mechanics. Before Einstein, no fundamental amendments to this model were needed, although the mathematical apparatus turned out to be necessary to significantly develop.
Note that Newton's theory of gravity was no longer, strictly speaking, heliocentric. Already in the two-body problem, the planet rotates not around the Sun, but around general center gravity, since not only the Sun attracts the planet, but the planet also attracts the Sun. Finally, it became clear that it was necessary to take into account the influence of the planets on each other.
During the 18th century, the law of universal gravitation was the subject of active debate (it was opposed by supporters of the Descartes school) and careful testing. By the end of the century, it became generally accepted that the law of universal gravitation makes it possible to explain and predict the movements of celestial bodies with great accuracy. Henry Cavendish in 1798 carried out a direct test of the validity of the law of gravity in terrestrial conditions, using extremely sensitive torsion balances. An important step was the introduction by Poisson in 1813 of the concept of gravitational potential and the Poisson equation for this potential; this model made it possible to study the gravitational field with an arbitrary distribution of matter. After this, Newton's law began to be regarded as a fundamental law of nature.
At the same time, Newton's theory contained a number of difficulties. The main one is the inexplicable long-range action: the force of attraction was transmitted incomprehensibly through completely empty space, and infinitely quickly. Essentially, Newton's model was purely mathematical, without any physical content. In addition, if the Universe, as was then assumed, is Euclidean and infinite, and at the same time the average density of matter in it is non-zero, then a gravitational paradox arises. IN late XIX century, another problem was discovered: the discrepancy between the theoretical and observed displacement of the perihelion of Mercury.
Further development
General theory of relativity
For more than two hundred years after Newton, physicists proposed various ways to improve Newton's theory of gravity. These efforts were crowned with success in 1915, with the creation of Einstein's general theory of relativity, in which all these difficulties were overcome. Newton's theory, in full agreement with the correspondence principle, turned out to be an approximation of a more general theory, applicable when two conditions are met:
In weak stationary gravitational fields, the equations of motion become Newtonian (gravitational potential). To prove this, we show that the scalar gravitational potential in weak stationary gravitational fields satisfies the Poisson equation
Δ Φ = − 4 π G ρ (\displaystyle \Delta \Phi =-4\pi G\rho ).It is known (Gravitational potential) that in this case the gravitational potential has the form:
Φ = − 1 2 c 2 (g 44 + 1) (\displaystyle \Phi =-(\frac (1)(2))c^(2)(g_(44)+1)).Let us find the component of the energy-momentum tensor from the gravitational field equations of the general theory of relativity:
R i k = − ϰ (T i k − 1 2 g i k T) (\displaystyle R_(ik)=-\varkappa (T_(ik)-(\frac (1)(2))g_(ik)T)),Where R i k (\displaystyle R_(ik))- curvature tensor. For we can introduce the kinetic energy-momentum tensor ρ u i u k (\displaystyle \rho u_(i)u_(k)). Neglecting quantities of the order u/c (\displaystyle u/c), you can put all the components T i k (\displaystyle T_(ik)), except T 44 (\displaystyle T_(44)), equal to zero. Component T 44 (\displaystyle T_(44)) equal to T 44 = ρ c 2 (\displaystyle T_(44)=\rho c^(2)) and therefore T = g i k T i k = g 44 T 44 = − ρ c 2 (\displaystyle T=g^(ik)T_(ik)=g^(44)T_(44)=-\rho c^(2)). Thus, the gravitational field equations take the form R 44 = − 1 2 ϰ ρ c 2 (\displaystyle R_(44)=-(\frac (1)(2))\varkappa \rho c^(2)). Due to the formula
R i k = ∂ Γ i α α ∂ x k − ∂ Γ i k α ∂ x α + Γ i α β Γ k β α − Γ i k α Γ α β β (\displaystyle R_(ik)=(\frac (\partial \ Gamma _(i\alpha )^(\alpha ))(\partial x^(k)))-(\frac (\partial \Gamma _(ik)^(\alpha ))(\partial x^(\alpha )))+\Gamma _(i\alpha )^(\beta )\Gamma _(k\beta )^(\alpha )-\Gamma _(ik)^(\alpha )\Gamma _(\alpha \beta )^(\beta ))value of the curvature tensor component R 44 (\displaystyle R_(44)) can be taken equal R 44 = − ∂ Γ 44 α ∂ x α (\displaystyle R_(44)=-(\frac (\partial \Gamma _(44)^(\alpha ))(\partial x^(\alpha )))) and since Γ 44 α ≈ − 1 2 ∂ g 44 ∂ x α (\displaystyle \Gamma _(44)^(\alpha )\approx -(\frac (1)(2))(\frac (\partial g_(44) )(\partial x^(\alpha )))), R 44 = 1 2 ∑ α ∂ 2 g 44 ∂ x α 2 = 1 2 Δ g 44 = − Δ Φ c 2 (\displaystyle R_(44)=(\frac (1)(2))\sum _(\ alpha )(\frac (\partial ^(2)g_(44))(\partial x_(\alpha )^(2)))=(\frac (1)(2))\Delta g_(44)=- (\frac (\Delta \Phi )(c^(2)))). Thus, we arrive at the Poisson equation:
Δ Φ = 1 2 ϰ c 4 ρ (\displaystyle \Delta \Phi =(\frac (1)(2))\varkappa c^(4)\rho ), Where ϰ = − 8 π G c 4 (\displaystyle \varkappa =-(\frac (8\pi G)(c^(4))))Quantum gravity
However general theory relativity is not the final theory of gravity, since it unsatisfactorily describes gravitational processes on quantum scales (at distances on the order of the Planck distance, about 1.6⋅10 −35). The construction of a consistent quantum theory of gravity is one of the most important unsolved problems of modern physics.
From the point of view quantum gravity, gravitational interaction is carried out through the exchange of virtual gravitons between interacting bodies. According to the uncertainty principle, the energy of a virtual graviton is inversely proportional to the time of its existence from the moment of emission by one body to the moment of absorption by another body. The lifetime is proportional to the distance between the bodies. Thus, at short distances, interacting bodies can exchange virtual gravitons with short and long lengths waves, and at large distances only by long-wave gravitons. From these considerations we can obtain the law of inverse proportionality of the Newtonian potential to distance. The analogy between Newton's law and Coulomb's law is explained by the fact that the graviton mass, like the mass
