The law of universal gravitation

Gravity (universal gravity, gravity)(from Lat. 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 the sense that, unlike any other forces, all bodies without exception, regardless of their mass, are given the same acceleration. Mainly gravity plays a decisive role on a cosmic scale. Term gravity is also used as the name of the branch of physics that studies gravitational interaction. The most successful modern physical theory in classical physics describing gravity is general relativity; the quantum theory of gravitational interaction has not yet been built.

Gravitational interaction

Gravitational interaction is one of four fundamental interactions in our world. Within the framework of classical mechanics, the gravitational interaction is described the law of gravity Newton, who states that the force of gravitational attraction between two material points of mass m 1 and m 2 separated by distance R, proportional to both masses and inversely proportional to the square of the distance - that is

Here G- gravitational constant equal to approximately m³ / (kg s²). The minus sign means that the force acting on the body is always equal in direction to the radius vector directed to the body, that is, the gravitational interaction always leads to the attraction of any bodies.

The law of universal gravitation is one of the applications of the inverse square law, which also occurs in the study of radiation (see, for example, Light pressure), and is a direct consequence of the quadratic increase in the area of ​​a sphere with increasing radius, which leads to a quadratic decrease in the contribution of any unit area to the area of ​​the entire sphere.

The simplest problem of celestial mechanics is the gravitational interaction of two bodies in empty space. This task is solved analytically to the end; the result of its solution is often formulated in the form of Kepler's three laws.

With an increase in the number of interacting bodies, the task becomes much more complicated. So, the already famous three-body problem (that is, the motion of three bodies with non-zero masses) cannot be solved analytically in a general form. With a numerical solution, the instability of the solutions with respect to the initial conditions sets in rather quickly. Applied to the solar system, this instability makes it impossible to predict the motion of the planets on scales exceeding a hundred million years.

In some special cases, it is possible to find an approximate solution. The most important is the case when the mass of one body is significantly greater than the mass of other bodies (examples: the solar system and the dynamics of Saturn's rings). In this case, as a first approximation, we can assume that light bodies do not interact with each other and move along Keplerian trajectories around the massive body. The interactions between them can be taken into account within the framework of perturbation theory and averaged over time. In this case, non-trivial phenomena such as resonances, attractors, chaos, etc. can arise. An illustrative example of such phenomena is the nontrivial structure of Saturn's rings.

Despite attempts to describe the behavior of a system of a large number of attracting bodies of approximately the same mass, this has not been possible due to the phenomenon of dynamic chaos.

Strong gravitational fields

In strong gravitational fields, when moving with relativistic speeds, the effects of the general theory of relativity begin to manifest themselves:

• deviation of the law of gravitation from Newtonian;
• potential lag associated with the finite speed of propagation of gravitational disturbances; the appearance of gravitational waves;
• nonlinearity effects: gravitational waves tend to interact with each other, so the principle of superposition of waves in strong fields is no longer fulfilled;
• changing the geometry of space-time;
• the emergence of black holes;

Gravitational radiation

One of the important predictions of general relativity is gravitational radiation, the presence of which has not yet been confirmed by direct observations. However, there is indirect observational evidence in favor of its existence, namely: the energy losses in the binary system with the PSR B1913 + 16 pulsar - the Huls-Taylor pulsar - are in good agreement with the model in which this energy is carried away by gravitational radiation.

Gravitational radiation can only be generated by systems with variable quadrupole or higher multipole moments, this fact suggests that the gravitational radiation of most natural sources is directional, which significantly complicates its detection. Gravitational power l-the field source is proportional to (v / c) 2l + 2 if the multipole is of electrical type, and (v / c) 2l + 4 - if the multipole is magnetic type, where v is the characteristic speed of movement of sources in the emitting system, and c is the speed of light. Thus, the dominant moment will be the quadrupole moment of the electric type, and the power of the corresponding radiation is equal to:

where Q ij is the tensor of the quadrupole moment of the mass distribution of the emitting system. Constant (1 / W) allows you to estimate the order of magnitude of the radiation power.

From 1969 (Weber's experiments) to the present day (February 2007), attempts have been made to directly detect gravitational radiation. In the USA, Europe and Japan at the moment there are several operating ground-based detectors (GEO 600), as well as the project of the space gravitational detector of the Republic of Tatarstan.

Subtle effects of gravity

In addition to the classical effects of gravitational attraction and time dilation, general relativity predicts the existence of other manifestations of gravity, which in terrestrial conditions are very weak and their detection and experimental verification are therefore very difficult. Until recently, overcoming these difficulties seemed beyond the capabilities of experimenters.

Among them, in particular, we can name the dragging of inertial frames of reference (or the Lense-Thirring effect) and the gravitomagnetic field. In 2005, NASA's robotic Gravity Probe B conducted an unprecedentedly accurate experiment to measure these effects near Earth, but the full results have yet to be published.

Quantum theory of gravity

Despite more than half a century of attempts, gravity is the only fundamental interaction for which a consistent renormalizable quantum theory has not yet been built. However, at low energies, in the spirit of quantum field theory, the gravitational interaction can be represented as an exchange of gravitons - gauge bosons with spin 2.

Standard theories of gravity

Due to the fact that the quantum effects of gravity are extremely small even under the most extreme experimental and observational conditions, there are still no reliable observations of them. Theoretical estimates show that in the overwhelming majority of cases one can restrict oneself to the classical description of the gravitational interaction.

There is a modern canonical classical theory of gravity - the general theory of relativity, and many hypotheses that refine it and theories of varying degrees of elaboration, competing with each other (see the article Alternative theories of gravity). All these theories give very similar predictions within the framework of the approximation in which experimental tests are currently being carried out. Several of the main, most well-developed or known theories of gravity are described below.

• Gravity is not a geometric field, but a real physical force field described by a tensor.

• Gravitational phenomena should be considered within the framework of the flat Minkowski space, in which the laws of conservation of energy-momentum and angular momentum are unambiguously fulfilled. Then the motion of bodies in Minkowski space is equivalent to the motion of these bodies in effective Riemannian space.

• In tensor equations to determine the metric, one should take into account the graviton mass, and also use the gauge conditions associated with the metric of the Minkowski space. This does not allow annihilating the gravitational field even locally by choosing some suitable frame of reference.

As in general relativity, in RTG, matter is understood as all forms of matter (including the electromagnetic field), with the exception of the gravitational field itself. The consequences of the RTG theory are as follows: black holes as physical objects predicted in general relativity do not exist; The universe is flat, homogeneous, isotropic, stationary and Euclidean.

On the other hand, there are no less convincing arguments from the opponents of RTG, which boil down to the following provisions:

A similar situation takes place in the RTG, where the second tensor equation is introduced to take into account the connection between the non-Euclidean space and the Minkowski space. Due to the presence of a dimensionless adjustable parameter in the Jordan - Brans - Dicke theory, it becomes possible to choose it so that the results of the theory coincide with the results of gravitational experiments.

Sources and Notes

Literature

• V.P. Vizgin Relativistic theory of gravitation (origins and formation, 1900-1915). M .: Nauka, 1981 .-- 352c.
• V.P. Vizgin Unified theories in the 1st third of the twentieth century. M .: Nauka, 1985 .-- 304c.
• Ivanenko D. D., Sardanashvili G. A. Gravity, 3rd ed. M.: URSS, 2008 .-- 200p.

see also

• Gravimeter

Links

• The law of universal gravitation or "Why does the moon not fall to the earth?" - Just about the difficult

This article will focus on the history of the discovery of the law of universal gravitation. Here we will get acquainted with biographical information from the life of the scientist who discovered this physical dogma, consider its main provisions, the relationship with quantum gravity, the course of development and much more.

Genius

Sir Isaac Newton is a scientist from England. At one time, he devoted a lot of attention and energy to such sciences as physics and mathematics, and also brought a lot of new things to mechanics and astronomy. He is rightfully considered one of the first founders of physics in its classical model. He is the author of the fundamental work "Mathematical Principles of Natural Philosophy", where he presented information about the three laws of mechanics and the law of universal gravitation. Isaac Newton laid the foundations of classical mechanics with these works. He also developed an integral type, a light theory. He also made major contributions to physical optics and developed many other theories in physics and mathematics.

Law

The law of universal gravitation and the history of its discovery go back to a distant beginning. Its classical form is the law by which the interaction of a gravitational type is described, which does not go beyond the framework of mechanics.

Its essence was that the indicator of the force F of the gravitational thrust arising between 2 bodies or points of matter m1 and m2, separated from each other by a certain distance r, observes proportionality with respect to both indicators of mass and is inversely proportional to the square of the distance between the bodies:

F = G, where by G we denote the constant of gravity, equal to 6.67408 (31) .10 -11 m 3 / kgf 2.

Newton's gravity

Before considering the history of the discovery of the law of universal gravitation, let's get acquainted in more detail with its general characteristics.

In the theory created by Newton, all bodies with a large mass should generate a special field around them, which attracts other objects to itself. It is called the gravitational field, and it has potential.

A body with spherical symmetry forms a field outside of itself, similar to that created by a material point of the same mass located in the center of the body.

The direction of the trajectory of such a point in the gravitational field, created by a body with a much larger mass, obeys the Objects of the universe, such as, for example, a planet or a comet, also obeys it, moving along an ellipse or hyperbole. Allowance for the distortion created by other massive bodies is taken into account using the provisions of the perturbation theory.

Analyzing accuracy

After Newton discovered the law of universal gravitation, it had to be tested and proven many times. For this, a series of calculations and observations were made. Having come to an agreement with its provisions and proceeding from the accuracy of its indicator, the experimental form of estimation serves as a clear confirmation of general relativity. Measurement of the quadrupole interactions of a body that rotates, but its antennas remain stationary, show us that the process of building up δ depends on the potential r - (1 + δ), at a distance of several meters and is in the limit (2.1 ± 6.2) .10 -3. A number of other practical confirmations allowed this law to be established and take a single form, without the presence of modifications. In 2007, this dogma was rechecked at a distance less than a centimeter (55 μm-9.59 mm). Taking into account the experimental errors, the scientists examined the range of the distance and found no obvious deviations in this law.

Observation of the Moon's orbit in relation to the Earth also confirmed its validity.

Euclidean space

Newton's classical theory of gravitation is associated with Euclidean space. The actual equality with a sufficiently high accuracy (10 -9) of the exponents of the measure of distance in the denominator of the equality considered above shows us the Euclidean basis of the space of Newtonian mechanics, with a three-dimensional physical form. At such a point of matter, the area of ​​a spherical surface has exact proportionality with respect to the magnitude of the square of its radius.

Historical data

Consider a brief summary of the history of the discovery of the law of universal gravitation.

Ideas were also put forward by other scientists who lived before Newton. Reflections about her were visited by Epicurus, Kepler, Descartes, Roberval, Gassendi, Huygens and others. Kepler put forward the assumption that the gravitational force has an inverse proportion to the distance from the Sun's star and has propagation only in the ecliptic planes; according to Descartes, it was a consequence of the activity of vortices in the thickness of the ether. There were a number of guesses that reflected the correct guesses about distance dependence.

A letter from Newton to Halley contained information that the predecessors of Sir Isaac himself were Hooke, Ren and Buyo Ismael. However, before him no one managed to clearly, using mathematical methods, connect the law of gravitation and planetary motion.

The history of the discovery of the law of universal gravitation is closely connected with the work "Mathematical Principles of Natural Philosophy" (1687). In this work, Newton was able to derive the law under consideration thanks to Kepler's empirical law, which was already known by that time. He shows us that:

• the form of movement of any visible planet indicates the presence of a central force;
• the gravitational force of the central type forms elliptical or hyperbolic orbits.

About Newton's theory

Inspection of the brief history of the discovery of the law of universal gravitation can also point us to a number of differences that set it apart from previous hypotheses. Newton was engaged not only in the publication of the proposed formula for the phenomenon under consideration, but also proposed a model of a mathematical type in its entirety:

• provision on the law of gravitation;
• regulation on the law of traffic;
• systematics of methods of mathematical research.

This triad could quite accurately investigate even the most complex movements of celestial objects, thus creating the basis for celestial mechanics. Until the beginning of Einstein's activity, this model did not require a fundamental set of corrections. Only the mathematical apparatus had to be significantly improved.

Object for discussion

Discovered and proven law throughout the eighteenth century became a well-known subject of active controversy and scrupulous checks. However, the century ended with a general agreement with his postulates and statements. Using the calculations of the law, it was possible to accurately determine the paths of motion of bodies in heaven. A direct check was made in 1798. He did this using a torsion balance with great sensitivity. In the history of the discovery of the universal law of gravitation, it is necessary to highlight a special place for the interpretations introduced by Poisson. He developed the concept of the potential of gravity and the Poisson equation, with which it was possible to calculate this potential. This type of model made it possible to study the gravitational field in the presence of an arbitrary distribution of matter.

There were many difficulties in Newton's theory. The main one could be considered the inexplicability of long-range action. It was impossible to accurately answer the question of how the forces of attraction are sent through vacuum space at infinite speed.

"Evolution" of the law

Over the next two hundred years, and even more, many physicists have attempted to suggest various ways to improve Newton's theory. These efforts ended in a triumph in 1915, namely the creation of the General Theory of Relativity, which was created by Einstein. He was able to overcome the whole set of difficulties. In accordance with the correspondence principle, Newton's theory turned out to be an approximation to the beginning of work on a theory in a more general form, which can be applied under certain conditions:

1. The potential of a gravitational nature cannot be too large in the systems under study. The solar system is an example of the observance of all the rules for the movement of the celestial type of bodies. The relativistic phenomenon finds itself in a noticeable manifestation of the displacement of the perihelion.
2. The indicator of the speed of movement in this group of systems is insignificant in comparison with the light speed.

The proof that in a weak stationary gravitational field the calculations of general relativity take the form of Newtonian ones is the presence of a scalar potential of gravity in a stationary field with weakly pronounced characteristics of forces, which is able to satisfy the conditions of the Poisson equation.

Quantum scale

However, in history, neither the scientific discovery of the law of universal gravitation, nor the General Theory of Relativity could serve as the final gravitational theory, since both do not adequately describe processes of the gravitational type on the scale of quanta. An attempt to create a quantum-gravitational theory is one of the most important tasks of modern physics.

From the point of view of quantum gravity, the interaction between objects is created through the interchange of virtual gravitons. In accordance with the uncertainty principle, the energy potential of virtual gravitons is inversely proportional to the period of time in which it existed, from the point of radiation by one object to the moment in which it was absorbed by another point.

In view of this, it turns out that on a small scale of distances, the interaction of bodies entails the exchange of virtual gravitons. Thanks to these considerations, it is possible to conclude a provision on Newton's law of potential and its dependence in accordance with the inverse exponent of proportionality with respect to distance. The existence of an analogy between the laws of Coulomb and Newton is explained by the fact that the weight of gravitons is equal to zero. The weight of the photons is of the same importance.

Delusion

In the school curriculum, the answer to the question from history, how Newton discovered the law of universal gravitation, is the story of a falling apple. According to this legend, it fell on the scientist's head. However, this is a widespread misconception, and in reality everything could do without such a case of possible head injury. Newton himself sometimes confirmed this myth, but in reality the law was not a spontaneous discovery and did not come in a burst of momentary insight. As it was written above, it was developed for a long time and was presented for the first time in the works on the "Mathematical Principles", which were released to the public in 1687.

When he came to a great result: one and the same cause causes phenomena of an amazingly wide range - from the fall of a thrown stone on the Earth to the movement of huge cosmic bodies. Newton found this reason and was able to accurately express it in the form of one formula - the law of universal gravitation.

Since the force of universal gravitation imparts the same acceleration to all bodies, regardless of their mass, it must be proportional to the mass of the body on which it acts:

But since, for example, the Earth acts on the Moon with a force proportional to the Moon's mass, then the Moon, according to Newton's third law, must act on the Earth with the same force. Moreover, this force should be proportional to the mass of the Earth. If the gravitational force is truly universal, then from the side of a given body, a force proportional to the mass of this other body must act on any other body. Consequently, the force of universal gravity should be proportional to the product of the masses of the interacting bodies. Hence follows the formulation the law of universal gravitation.

Definition of the law of universal gravitation

The force of mutual attraction of two bodies is directly proportional to the product of the masses of these bodies and inversely proportional to the square of the distance between them:

Aspect ratio G called gravitational constant.

The gravitational constant is numerically equal to the force of attraction between two material points weighing 1 kg each, if the distance between them is 1 m. m 1 = m 2= 1 kg and R= 1 m we get G = F(numerically).

It should be borne in mind that the law of universal gravitation (4.5) as a universal law is valid for material points. In this case, the forces of gravitational interaction are directed along the line connecting these points ( Figure 4.2). These kinds of forces are called central.

It can be shown that homogeneous bodies having the shape of a ball (even if they cannot be considered material points) also interact with the force determined by formula (4.5). In this case R is the distance between the centers of the balls. The forces of mutual attraction lie on a straight line passing through the centers of the balls. (Such forces are called central forces.) The bodies, the fall of which to the Earth we usually consider, have dimensions much smaller than the Earth's radius ( R≈6400 km). Such bodies, regardless of their shape, can be considered as material points and the force of their attraction to the Earth can be determined using the law (4.5), bearing in mind that R is the distance from a given body to the center of the Earth.

Determination of the gravitational constant

Now let's find out how you can find the gravitational constant. First of all, note that G has a specific name. This is due to the fact that the units (and, accordingly, the names) of all quantities included in the law of universal gravitation have already been established earlier. The law of gravitation gives a new connection between known quantities with certain names of units. That is why the coefficient turns out to be a named quantity. Using the formula of the law of universal gravitation, it is easy to find the name of the unit of the gravitational constant in SI:

N m 2 / kg 2 = m 3 / (kg s 2).

For quantitative determination G it is necessary to independently determine all the quantities included in the law of universal gravitation: both masses, force and distance between bodies. It is impossible to use astronomical observations for this, since it is possible to determine the masses of the planets, the Sun, and the Earth only on the basis of the law of universal gravitation itself, if the value of the gravitational constant is known. The experiment should be carried out on Earth with bodies whose masses can be measured on a balance.

The difficulty lies in the fact that the gravitational forces between bodies of small masses are extremely small. It is for this reason that we do not notice the attraction of our body to surrounding objects and the mutual attraction of objects to each other, although gravitational forces are the most universal of all forces in nature. Two people with masses of 60 kg at a distance of 1 m from each other are attracted with a force of only about 10 -9 N. Therefore, to measure the gravitational constant, rather subtle experiments are needed.

For the first time the gravitational constant was measured by the English physicist G. Cavendish in 1798 using an instrument called a torsion balance. The diagram of the torsion balance is shown in Figure 4.3. A lightweight rocker with two identical weights at the ends is suspended on a thin elastic thread. Two heavy balls are fixed nearby. The forces of gravity act between the weights and the stationary balls. Under the influence of these forces, the yoke turns and twists the thread. The angle of twist can be used to determine the force of gravity. To do this, you only need to know the elastic properties of the thread. The masses of the bodies are known, and the distance between the centers of interacting bodies can be directly measured.

From these experiments, the following value for the gravitational constant was obtained:

Only in the case when bodies of huge masses interact (or at least the mass of one of the bodies is very large), the force of gravity reaches a large value. For example, the Earth and the Moon are attracted to each other with force F≈2 10 20 H.

Dependence of the acceleration of free fall of bodies on the geographical latitude

One of the reasons for the increase in the acceleration of gravity when the point where the body is located, from the equator to the poles, is that the globe is somewhat flattened at the poles and the distance from the center of the Earth to its surface at the poles is less than at the equator. Another, more significant reason is the rotation of the Earth.

Equality of inert and gravitational masses

The most striking property of gravitational forces is that they impart the same acceleration to all bodies, regardless of their masses. What would you say about a football player whose kick would be equally accelerated by an ordinary leather ball and a two-pound kettlebell? Everyone will say that this is impossible. But the Earth is just such an "extraordinary footballer" with the only difference that its effect on the body does not have the character of a short-term blow, but continues continuously for billions of years.

The extraordinary property of gravitational forces, as we have already said, is explained by the fact that these forces are proportional to the masses of both interacting bodies. This fact cannot but cause surprise if you think about it carefully. After all, the mass of a body, which is included in Newton's second law, determines the inert properties of the body, that is, its ability to acquire a certain acceleration under the action of a given force. It is natural to call this mass inert mass and denote by m and.

It would seem, what does it have to do with the ability of bodies to attract each other? The mass that determines the ability of bodies to attract each other should be called gravitational mass m g.

It does not at all follow from Newtonian mechanics that the inertial and gravitational masses are the same, that is, that

Equality (4.6) is a direct consequence of experience. It means that we can simply talk about the mass of a body as a quantitative measure of both its inert and gravitational properties.

The law of gravity is one of the most universal laws of nature. It is valid for any body with mass.

The meaning of the law of universal gravitation

But if we approach this topic more radically, it turns out that the law of universal gravitation is not everywhere possible to apply it. This law has found its application for bodies that have the shape of a ball, it can be used for material points, and it is also acceptable for a ball with a large radius, where this ball can interact with bodies much smaller than its size.

As you may have guessed from the information provided in this lesson, the law of universal gravitation is the basis in the study of celestial mechanics. And as you know, celestial mechanics studies the motion of planets.

Thanks to this law of universal gravitation, it became possible to more accurately determine the location of celestial bodies and the ability to calculate their trajectory.

But for a body and an infinite plane, as well as for the interaction of an infinite rod and a ball, this formula cannot be applied.

With the help of this law, Newton was able to explain not only how the planets move, but also why the ebb and flow of the sea occurs. Over the course of time, thanks to the labors of Newton, astronomers managed to discover such planets of the solar system as Neptune and Pluto.

The importance of the discovery of the law of universal gravitation lies in the fact that with its help it became possible to make predictions of solar and lunar eclipses and accurately calculate the movements of spaceships.

The forces of gravity are the most universal of all the forces of nature. After all, their action extends to the interaction between any bodies with mass. And as you know, any body has a mass. The forces of gravity act through any body, since there are no barriers for the forces of gravity.

Task

And now, in order to consolidate knowledge about the law of universal gravitation, let's try to consider and solve an interesting problem. The rocket rose to a height h equal to 990 km. Determine how much the gravity force acting on the rocket at height h has decreased in comparison with the gravity force mg acting on it at the Earth's surface? The radius of the Earth is R = 6400 km. Let us denote by m the mass of the rocket, and by M the mass of the Earth.

At height h, gravity is equal to:

From here we calculate:

Substituting a value will give the result:

The legend about how Newton discovered the law of universal gravitation, having received an apple on the top of his head, was invented by Voltaire. Moreover, Voltaire himself assured that this true story was told to him by Newton's beloved niece Catherine Barton. It's just strange that neither the niece herself, nor her very close friend Jonathan Swift, in their memoirs about Newton, never mentioned the fateful apple. By the way, Isaac Newton himself, writing down in detail in his notebooks the results of experiments on the behavior of different bodies, noted only vessels filled with gold, silver, lead, sand, glass, water or wheat, or about an apple. However, this did not stop Newton's descendants from taking tourists around the garden at the Woolstock estate and showing them that same apple tree until a storm broke it.

Yes, there was an apple tree, and apples probably fell from it, but how great is the apple's merit in the discovery of the law of universal gravitation?

The controversy over the apple has not subsided for 300 years, as have the controversy over the law of gravity itself and the belief about who has priority for the discovery.

G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky, Physics Grade 10

The most important phenomenon constantly studied by physicists is movement. Electromagnetic phenomena, laws of mechanics, thermodynamic and quantum processes - all this is a wide range of fragments of the universe studied by physics. And all these processes come down, one way or another, to one thing - to.

In contact with

Everything in the universe is moving. Gravity is a familiar phenomenon for all people since childhood, we were born in the gravitational field of our planet, this physical phenomenon is perceived by us at the deepest intuitive level and, it would seem, does not even require study.

But, alas, the question is why and how all bodies are attracted to each other, remains to this day not fully disclosed, although it has been studied up and down.

In this article we will look at what Newton's universal attraction is - the classical theory of gravity. However, before moving on to formulas and examples, let us talk about the essence of the attraction problem and give it a definition.

Perhaps the study of gravity was the beginning of natural philosophy (the science of understanding the essence of things), perhaps natural philosophy gave rise to the question of the essence of gravity, but, one way or another, the question of gravitation of bodies interested in ancient Greece.

Movement was understood as the essence of the sensory characteristics of the body, or rather, the body moved while the observer sees it. If we cannot measure, weigh, feel a phenomenon, does this mean that this phenomenon does not exist? Naturally, it doesn't. And since Aristotle realized this, began to think about the essence of gravity.

As it turned out today, after many tens of centuries, gravity is the basis not only of the earth's attraction and the attraction of our planet to, but also the basis of the origin of the Universe and almost all available elementary particles.

Movement task

Let's do a thought experiment. Take a small ball in our left hand. Let's take the same on the right. Let go of the right ball and it will start falling down. At the same time, the left one remains in the hand, it is still motionless.

Let us stop mentally the passage of time. The falling right ball "hangs" in the air, the left one still remains in the hand. The right ball is endowed with "energy" of movement, the left one is not. But what is the deep, meaningful difference between them?

Where, in what part of the falling ball is it written that it should move? It has the same mass, the same volume. He has the same atoms, and they are no different from the atoms of the resting ball. Ball possesses? Yes, this is the correct answer, but how does the ball know that it has potential energy, where is it fixed in it?

This is precisely the task set before themselves by Aristotle, Newton and Albert Einstein. And all three brilliant thinkers have partly solved this problem for themselves, but today there are a number of issues that need to be resolved.

Newton's gravity

In 1666, the greatest English physicist and mechanic I. Newton discovered a law capable of quantitatively calculating the force due to which all matter in the Universe tends to each other. This phenomenon is called universal gravitation. When asked: "Formulate the law of universal gravitation", your answer should sound like this:

The force of gravitational interaction, contributing to the attraction of two bodies, is in direct proportional relationship with the masses of these bodies and inversely proportional to the distance between them.

Important! Newton's law of attraction uses the term "distance". This term should not be understood as the distance between the surfaces of bodies, but the distance between their centers of gravity. For example, if two balls of radii r1 and r2 lie on top of each other, then the distance between their surfaces is zero, but there is an attractive force. The thing is that the distance between their centers r1 + r2 is nonzero. On a cosmic scale, this clarification is not important, but for a satellite in orbit, this distance is equal to the height above the surface plus the radius of our planet. The distance between the Earth and the Moon is also measured as the distance between their centers, not surfaces.

For the law of gravitation, the formula is as follows:

,

• F is the force of attraction,
• - masses,
• r - distance,
• G - gravitational constant equal to 6.67 · 10−11 m³ / (kg · s²).

What is weight if we have just considered the force of gravity?

Force is a vector quantity, but in the law of universal gravitation it is traditionally written as a scalar. In a vector picture, the law will look like this:

.

But this does not mean that the force is inversely proportional to the cube of the distance between the centers. The ratio should be understood as a unit vector directed from one center to another:

.

The law of gravitational interaction

Weight and gravity

Having considered the law of gravity, one can understand that there is nothing surprising in the fact that we personally we feel the attraction of the sun much weaker than the earth... The massive Sun, although it has a large mass, is very far from us. is also far from the Sun, but it is attracted to it, since it has a large mass. How to find the force of attraction of two bodies, namely how to calculate the force of gravity of the Sun, Earth and you and me - we will deal with this issue a little later.

As far as we know, the force of gravity is:

where m is our mass and g is the acceleration of the Earth's gravity (9.81 m / s 2).

Important! There are no two, three, ten types of attraction forces. Gravity is the only force that quantifies attraction. Weight (P = mg) and gravity are the same thing.

If m is our mass, M is the mass of the earth, R is its radius, then the gravitational force acting on us is equal to:

Thus, since F = mg:

.

The masses m contract, and the expression for the acceleration of gravity remains:

As you can see, the acceleration of gravity is really a constant value, since its formula includes constant values ​​- the radius, the mass of the Earth and the gravitational constant. Substituting the values ​​of these constants, we will make sure that the acceleration due to gravity is 9.81 m / s 2.

At different latitudes, the radius of the planet is somewhat different, since the Earth is still not a perfect ball. Because of this, the acceleration of gravity is different at different points in the world.

Let's go back to the attraction of the Earth and the Sun. Let us try to prove by example that the globe attracts you and me more than the Sun.

For convenience, let's take the mass of a person: m = 100 kg. Then:

• The distance between man and the earth is equal to the radius of the planet: R = 6.4 ∙ 10 6 m.
• The mass of the Earth is: M ≈ 6 ∙ 10 24 kg.
• The mass of the Sun is equal to: Mc ≈ 2 ∙ 10 30 kg.
• Distance between our planet and the Sun (between the Sun and man): r = 15 ∙ 10 10 m.

Gravitational attraction between man and Earth:

This result is fairly obvious from a simpler weight expression (P = mg).

The force of gravitational attraction between man and the Sun:

As you can see, our planet attracts us almost 2000 times stronger.

How to find the force of attraction between the Earth and the Sun? In the following way:

Now we see that the Sun attracts our planet more than a billion billion times stronger than the planet attracts you and me.

First space speed

After Isaac Newton discovered the law of universal gravitation, he became interested in how fast a body needs to be thrown so that it, having overcome the gravitational field, leaves the globe forever.

True, he imagined it somewhat differently, in his understanding there was not a vertically standing rocket aimed at the sky, but a body that horizontally makes a jump from the top of the mountain. This was a logical illustration, since at the top of the mountain, the force of gravity is slightly less.

So, at the top of Everest, the acceleration of gravity will be equal not to the usual 9.8 m / s 2, but almost m / s 2. It is for this reason that there is so rarefied, air particles are no longer so attached to gravity as those that "fell" to the surface.

Let's try to find out what cosmic speed is.

The first cosmic speed v1 is the speed at which the body leaves the surface of the Earth (or another planet) and enters a circular orbit.

Let's try to find out the numerical value of this value for our planet.

Let's write Newton's second law for a body that revolves around the planet in a circular orbit:

,

where h is the height of the body above the surface, R is the radius of the Earth.

In orbit, centrifugal acceleration acts on the body, thus:

.

The masses are reduced, we get:

,

This speed is called the first cosmic speed:

As you can see, the cosmic speed is absolutely independent of the body mass. Thus, any object accelerated to a speed of 7.9 km / s will leave our planet and enter its orbit.

First space speed

Second space speed

However, even having accelerated the body to the first cosmic speed, we will not be able to completely break its gravitational connection with the Earth. For this, the second cosmic speed is needed. Upon reaching this speed, the body leaves the gravitational field of the planet and all possible closed orbits.

Important! By mistake, it is often believed that in order to get to the moon, astronauts had to reach the second cosmic speed, because they had to first "disconnect" from the gravitational field of the planet. This is not so: the pair "Earth - Moon" are in the gravitational field of the Earth. Their common center of gravity is within the globe.

In order to find this speed, let us set the problem a little differently. Let's say a body flies from infinity to the planet. The question is: what speed will be achieved on the surface upon landing (excluding the atmosphere, of course)? It is this speed and it will take the body to leave the planet.

The law of universal gravitation. Physics grade 9

The law of universal gravitation.

Output

We learned that although gravity is the main force in the Universe, many of the reasons for this phenomenon are still a mystery. We learned what Newton's gravitational force is, learned to count it for various bodies, and also studied some useful consequences that follow from such a phenomenon as the universal law of gravitation.

Newton's classical theory of gravity (Newton's law of universal gravitation)- the law describing the gravitational interaction in the framework of classical mechanics. This law was discovered by Newton around 1666. It says that the 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), 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).

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✪ Introduction to Newton's Law of Gravitation

✪ The Law of Universal Gravity

✪ physics LAW OF GRAVITY 9th grade

✪ About Isaac Newton (A Brief History)

✪ Lesson 60. The law of universal gravitation. Gravitational constant

Subtitles

Now let's learn a little about gravity, or gravity. As you know, gravitation, especially in the initial or even in a fairly advanced physics course, is such a concept that can be calculated and found out the main parameters by which it is caused, but in fact, gravitation is not completely understandable. Even if you are familiar with the general theory of relativity - if you are asked what gravitation is, you can answer: it is a curvature of space-time and the like. However, it is still difficult to get an intuitive idea why two objects, just because they have a so-called mass, are attracted to each other. At least for me this is mysticism. Having noted this, we proceed to consider the concept of gravitation. We will do this by studying Newton's law of universal gravitation, which is true for most situations. This law says: 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 straightforward formula. Let's try to transform it and see if we can get some well-known results. 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 to the center of the Earth, or to the center of mass of the Earth. 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 in 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 with different measurements. But for our needs, and in most physics courses, it's a constant, a constant equal to 6.67 * 10 ^ (- 11) cubic meters divided by kilogram per second squared. Yes, its dimension looks strange, but you just need to understand that these are conditional units necessary to multiply by the masses of objects and divide by the square of the distance to obtain the dimension of force - newton, or kilogram per meter divided by a second squared. So there is no need to worry about these units: just know that we will have to work with meters, seconds and kilograms. Let's substitute this number in the formula for force: 6.67 * 10 ^ (- 11). Since we need to know the acceleration acting on Sal, then m₁ is equal to the mass of Sal, that is, me. I would not like to expose in this plot how much I weigh, so we will leave this mass to 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 all this is divided by the square of the distance. You might argue: what is the distance between the Earth and what stands on it? After all, if objects are in contact, 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 Earth's surface, unless the person is too tall. However, my center of mass can be three feet above the ground. And where is the center of mass of the Earth? Obviously in the center of the earth. And the radius of the Earth is equal to what? 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 6, and so on, like all other values, you need to write it down in the 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 is the square of the distance, so let's square everything. Let's try to simplify now. First, we multiply the values ​​in the numerator and move the ms variable forward. Then the force F is equal to the mass of Sal for the entire upper part, we calculate it separately. So 6.67 times 5.97 equals 39.82. 39.82. This is the product of the significant parts, which should now be multiplied by 10 to the desired 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, in the end we have 10 ^ 13. Find the denominator. It is equal to 6.37 squared, multiplied by 10 ^ 6 also squared. As you remember, if a number written as a power is raised to another power, then the exponents are multiplied, which means that 10 ^ 6 squared is 10 to the power of 6 times 2, or 10 ^ 12. Next, let's calculate the square of the number 6.37 using a calculator and get ... Let us square 6.37. And that's 40.58. 40.58. It remains 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 10 ^ 1, or just 10. And 0.981 multiplied by 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 Sal's mass multiplied by 9.81. What does it give us? Can gravitational acceleration be calculated now? It is known that the force is equal to the product of mass and acceleration, therefore, the gravitational force is simply equal to the product of Sal's mass by the gravitational acceleration, which is usually denoted by the lowercase letter g. So, on the one hand, the force of gravity is equal to 9.81 times the mass of Sal. On the other hand, it is equal to the mass of Sal for gravitational acceleration. Dividing both sides of the equality by the Sal mass, we get that the coefficient 9.81 is the gravitational acceleration. And if we included in the calculations the full record of units of dimension, then, having reduced the kilograms, we would see that the gravitational acceleration is measured in meters divided by a second squared, like any acceleration. You can also notice that the obtained value is very close to that which we used when solving problems about the motion of an abandoned body: 9.8 meters per second squared. This is impressive. Let's solve one more short gravity problem, because we have a couple of minutes left. Let's say we have another planet called Earth Baby. Let Baby's radius rS be half the Earth's radius rE, and her mass mS is also equal to half the Earth's mass mE. What will be equal to the force of gravity acting here on any object, and how much less than the force of gravity? Although, let's leave the problem for the next time, then I will solve it. See you. Subtitles by the Amara.org community

Properties of Newtonian gravitation

In Newtonian theory, each massive body generates a force field of attraction to this body, which is called the gravitational field. This field is potential, and the function of the gravitational potential for a material point with mass M (\ displaystyle M) is defined by the formula:

In the general case, when the density of the substance ρ (\ displaystyle \ rho) distributed arbitrarily, satisfies the Poisson equation:

The solution to this equation is written in the form:

where r (\ displaystyle r) - the distance between the volume element d V (\ displaystyle dV) and the point at which the potential is determined φ (\ displaystyle \ varphi), C (\ displaystyle C) is an 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:

A spherically symmetrical body creates outside its limits the same field as a material point of the same mass, located in the center of the body.

The trajectory of a material point in a gravitational field created by a material point much larger in mass obeys Kepler's laws. In particular, planets and comets in the 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 estimate 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 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 in the investigated range of distances, no deviations from Newton's law were found.

Precision laser ranging observations of the Moon's orbit confirm the law of universal gravitation at a distance from the Earth to the Moon with accuracy 3 ⋅ 10 - 11 (\ displaystyle 3 \ cdot 10 ^ (- 11)).

Connection with the geometry of Euclidean space

The fact of equality with very high precision 10 - 9 (\ displaystyle 10 ^ (- 9)) the exponent of the distance in the denominator of the expression for the gravitational force 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 gravitation 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 propagates only in the plane of the ecliptic; Descartes considered it to be the result of vortices in the ether. There were, however, guesses with the 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 conclusively link the law of gravitation (force inversely proportional to the square of the distance) and the laws of planetary motion (Kepler's laws).

• law of gravitation;
• the 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 motions of celestial bodies, thereby creating the foundations of celestial mechanics. Before Einstein, no fundamental amendments to this model were needed, although it turned out to be necessary to significantly develop the mathematical apparatus.

Note that Newton's theory of gravitation was no longer, strictly speaking, heliocentric. Already in the problem of two bodies, the planet revolves not around the Sun, but around a common center of 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 discussion (opposed by the supporters of the Descartes school) and thorough checks. 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 carried out a direct test of the law of gravitation in earthly conditions in 1798, using an extremely sensitive torsion balance. An important stage was the introduction by Poisson in 1813 of the concept of the 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 that, Newton's law began to be viewed as a fundamental law of nature.

At the same time, Newtonian theory contained a number of difficulties. The main one is inexplicable long-range action: the force of gravity was transmitted incomprehensibly through a completely empty space, and infinitely fast. Essentially, the Newtonian model was purely mathematical, without any physical content. In addition, if the Universe, as it was then assumed, is Euclidean and infinite, and the average density of matter in it is nonzero, then a gravitational paradox arises. At the end of the 19th 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 have proposed various ways to improve Newton's theory of gravitation. 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 under two conditions:

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

It is known (Gravitational potential) that in this case the gravitational potential has the form:

Let us find the component of the energy-momentum tensor from the equations of the gravitational field of the general theory of relativity:

where R i k (\ displaystyle R_ (ik)) is the 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 of 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)) is 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 equations of the gravitational field 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

curvature tensor component value 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 come to the Poisson equation:

Quantum gravity

However, the general theory of relativity is not the final theory of gravity, since it describes gravitational processes in a quantum scale unsatisfactorily (at distances of the order of the Planckian, about 1.6⋅10 −35). The construction of a consistent quantum theory of gravity is one of the most important unsolved problems in modern physics.

From the point of view of 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 it is emitted by one body to the moment it is absorbed by another body. The lifetime is proportional to the distance between the bodies. Thus, at small distances, interacting bodies can exchange virtual gravitons with short and long wavelengths, and at large distances, only long-wave gravitons. From these considerations, it is possible to obtain the law of inverse proportionality of the Newtonian potential on the distance. The analogy between Newton's law and Coulomb's law is explained by the fact that the mass of the graviton, like the mass