When he came to a great result: the same cause causes phenomena of an amazingly wide range - from the fall of a thrown stone to 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 mass of the Moon, then the Moon, according to Newton’s third law, must act on the Earth with the same force. Moreover, this force must be proportional to the mass of the Earth. If the force of gravity is truly universal, then from the side of a given body a force must act on any other body proportional to the mass of this other body. Consequently, the force of universal gravity must be proportional to the product of the masses of interacting bodies. This leads to the formulation law of universal gravitation.
Definition of the law of universal gravitation
The force of mutual attraction between 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:
Proportionality factor 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. After all, when m 1 =m 2=1 kg and R=1 m we get G=F(numerically).
It must 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 ( Fig.4.2). This kind of force is called central.
It can be shown that homogeneous bodies shaped like a ball (even if they cannot be considered material points) also interact with the force determined by formula (4.5). In this case R- 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.) The bodies that we usually consider falling on the Earth have dimensions much smaller than the Earth’s radius ( R≈6400 km). Such bodies can, regardless of their shape, be considered as material points and determine the force of their attraction to the Earth using the law (4.5), keeping 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 to find the gravitational constant. First of all, we 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 provides 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 SI unit of gravitational constant:
N m 2 / kg 2 = m 3 / (kg s 2).
For quantification 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 the masses of the planets, the Sun, and the Earth can only be determined on the basis of the law of universal gravitation itself, if the value of the gravitational constant is known. The experiment must be carried out on Earth with bodies whose masses can be measured on a scale.
The difficulty is 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, fairly subtle experiments are needed.
The gravitational constant was first 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 light rocker with two identical weights at the ends is suspended from a thin elastic thread. Two heavy balls are fixed motionless nearby. Gravitational forces act between the weights and the stationary balls. Under the influence of these forces, the rocker turns and twists the thread. By the angle of twist you can determine the force of attraction. 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 enormous mass interact (or at least the mass of one of the bodies is very large) does the gravitational force reach a large value. For example, the Earth and the Moon are attracted to each other with a force F≈2 10 20 H.
Dependence of the acceleration of free falling bodies on geographic latitude
One of the reasons for the increase in the acceleration of gravity when the point where the body is located moves 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 inertial 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 weight? Everyone will say that this is impossible. But the Earth is just such an “extraordinary football player” with the only difference that its effect on bodies is not of the nature 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 inertial properties of the body, that is, its ability to acquire a certain acceleration under the influence of a given force. It is natural to call this mass inert mass and denote by m and.
It would seem, what relation can it have to 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, i.e. that
Equality (4.6) is a direct consequence of experiment. It means that we can simply talk about the mass of a body as a quantitative measure of both its inertial and gravitational properties.
The law of universal gravitation is one of the most universal laws of nature. It is valid for any bodies 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 does not have the possibility of its application everywhere. 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 having 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 movement 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 sea tides arise. Over time, thanks to the work 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 forecasts of solar and lunar eclipses and accurately calculate the movements of spacecraft.
The forces of universal gravity are the most universal of all the forces of nature. After all, their action extends to the interaction between any bodies that have mass. And as you know, any body has mass. The forces of gravity act through any body, since there are no barriers to 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 force of gravity acting on the rocket at a height h has decreased compared to the force of gravity mg acting on it at the surface of the Earth? Radius of the Earth R = 6400 km. Let us denote by m the mass of the rocket, and by M the mass of the Earth.
At height h the force of gravity is:
From here we calculate:
Substituting the value will give the result:
The legend about how Newton discovered the law of universal gravitation after hitting the top of his head with an apple was invented by Voltaire. Moreover, Voltaire himself assured that this true story was told to him by Newton’s beloved niece Katherine Barton. It’s just strange that neither the niece herself nor her very close friend Jonathan Swift ever mentioned the fateful apple in their memoirs about Newton. By the way, Isaac Newton himself, writing 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, not to mention an apple. However, this did not stop Newton’s descendants from taking tourists around the garden on the Woolstock estate and showing them that same apple tree before the storm destroyed it.
Yes, there was an apple tree, and apples probably fell from it, but how great was the merit of the apple in the discovery of the law of universal gravitation?
The debate about the apple has not subsided for 300 years, just like the debate about the law of universal gravitation itself or about who has the priority of discovery.uk
G.Ya.Myakishev, B.B.Bukhovtsev, N.N.Sotsky, Physics 10th grade
Law of Gravity
Gravity (universal gravitation, gravitation)(from Latin gravitas - “gravity”) - a 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, it imparts the same acceleration to all bodies without exception, regardless of their mass. Mainly gravity plays a decisive role on a cosmic scale. Term gravity also used as the name of the branch of physics that studies gravitational interaction. The most successful modern physical theory in classical physics that describes gravity is the general theory of relativity; the quantum theory of gravitational interaction has not yet been constructed.
Gravitational interaction
Gravitational interaction is one of the four fundamental interactions in our world. Within the framework of classical mechanics, gravitational interaction is described law of universal gravitation Newton, who states that the force of gravitational attraction between two material points of mass m 1 and m 2 separated by distance R, is 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, 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 the sphere with increasing radius, which leads to a quadratic decrease in the contribution of any unit area to area of the entire sphere.
The simplest problem of celestial mechanics is the gravitational interaction of two bodies in empty space. This problem is solved analytically to the end; the result of its solution is often formulated in the form of Kepler's three laws.
As the number of interacting bodies increases, the task becomes dramatically more complicated. Thus, 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, instability of the solutions relative to the initial conditions occurs quite quickly. When applied to the Solar System, this instability makes it impossible to predict the motion of planets on scales larger than a hundred million years.
In some special cases, it is possible to find an approximate solution. The most important case is when the mass of one body is significantly greater than the mass of other bodies (examples: the solar system and the dynamics of the rings of Saturn). 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 may arise, such as resonances, attractors, chaos, etc. A clear example of such phenomena is the non-trivial structure of the rings of Saturn.
Despite attempts to describe the behavior of a system of a large number of attracting bodies of approximately the same mass, this cannot be done due to the phenomenon of dynamic chaos.
Strong gravitational fields
In strong gravitational fields, when moving at relativistic speeds, the effects of general relativity begin to appear:
- deviation of the law of gravity from Newton's;
- delay of potentials 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 no longer holds true;
- 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: energy losses in the binary system with the pulsar PSR B1913+16 - the Hulse-Taylor pulsar - are in good agreement with a 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. Gravity power l-field source is proportional (v / c) 2l + 2 , if the multipole is of electric type, and (v / c) 2l + 4 - if the multipole is of magnetic type, where v is the characteristic speed of movement of sources in the radiating system, and c- 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- quadrupole moment tensor of the mass distribution of the radiating system. Constant 
(1/W) allows us to estimate the order of magnitude of the radiation power.
From 1969 (Weber's experiments) to the present (February 2007), attempts have been made to directly detect gravitational radiation. In the USA, Europe and Japan, there are currently several operating ground-based detectors (GEO 600), as well as a project for a space gravitational detector of the Republic of Tatarstan.
Subtle effects of gravity
In addition to the classical effects of gravitational attraction and time dilation, the general theory of relativity predicts the existence of other manifestations of gravity, which under 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 entrainment of inertial frames of reference (or the Lense-Thirring effect) and the gravitomagnetic field. In 2005, NASA's unmanned Gravity Probe B conducted an unprecedented precision experiment to measure these effects near Earth, but its full results have not yet been 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 constructed. However, at low energies, in the spirit of quantum field theory, gravitational interaction can be represented as an exchange of gravitons - gauge bosons with spin 2.
Standard theories of gravity
Due to the fact that 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 vast majority of cases one can limit oneself to the classical description of gravitational interaction.
There is a modern canonical classical theory of gravity - general theory of relativity, and many hypotheses and theories of varying degrees of development that clarify it, competing with each other (see the article Alternative theories of gravity). All of these theories make very similar predictions within the approximation in which experimental tests are currently carried out. The following are several basic, most well-developed or known theories of gravity.
- 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 flat Minkowski space, in which the laws of conservation of energy-momentum and angular momentum are unambiguously satisfied. 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, the graviton mass should be taken into account, and gauge conditions associated with the Minkowski space metric should be used. This does not allow the gravitational field to be destroyed even locally by choosing some suitable reference frame.
As in general relativity, in RTG matter refers to 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 by opponents of RTG, which boil down to the following points:
A similar thing occurs in RTG, where the second tensor equation is introduced to take into account the connection between non-Euclidean space and Minkowski space. Due to the presence of a dimensionless fitting 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.
| Theories of gravity | ||||||||
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Sources and notes
Literature
- Vizgin V. P. Relativistic theory of gravity (origins and formation, 1900-1915). M.: Nauka, 1981. - 352c.
- Vizgin V. P. 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. - 200 p.
see also
- Gravimeter
Links
- The law of universal gravitation or “Why doesn’t the Moon fall to Earth?” - Just about the complex
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I. Newton was able to derive from Kepler's laws one of the fundamental laws of nature - the law of universal gravitation. Newton knew that for all planets in the solar system, acceleration is inversely proportional to the square of the distance from the planet to the Sun and the coefficient of proportionality is the same for all planets.
From here it follows, first of all, that the force of attraction acting from the Sun on a planet must be proportional to the mass of this planet. In fact, if the acceleration of the planet is given by formula (123.5), then the force causing the acceleration
where is the mass of this planet. On the other hand, Newton knew the acceleration that the Earth imparts to the Moon; it was determined from observations of the movement of the Moon as it revolves around the Earth. This acceleration is approximately one times less than the acceleration imparted by the Earth to bodies located near the Earth's surface. The distance from the Earth to the Moon is approximately equal to the Earth's radii. In other words, the Moon is several times farther from the center of the Earth than bodies located on the surface of the Earth, and its acceleration is several times less.
If we accept that the Moon moves under the influence of the Earth's gravity, then it follows that the force of the Earth's gravity, like the force of the Sun's gravity, decreases in inverse proportion to the square of the distance from the center of the Earth. Finally, the force of gravity of the Earth is directly proportional to the mass of the attracted body. Newton established this fact in experiments with pendulums. He discovered that the period of swing of a pendulum does not depend on its mass. This means that the Earth imparts the same acceleration to pendulums of different masses, and, consequently, the force of gravity of the Earth is proportional to the mass of the body on which it acts. The same, of course, follows from the same acceleration of gravity for bodies of different masses, but experiments with pendulums make it possible to verify this fact with greater accuracy.
These similar features of the gravitational forces of the Sun and the Earth led Newton to the conclusion that the nature of these forces is the same and that there are forces of universal gravity acting between all bodies and decreasing in inverse proportion to the square of the distance between the bodies. In this case, the gravitational force acting on a given body of mass must be proportional to the mass.
Based on these facts and considerations, Newton formulated the law of universal gravitation in this way: any two bodies are attracted to each other with a force that is directed along the line connecting them, directly proportional to the masses of both bodies and inversely proportional to the square of the distance between them, i.e. mutual gravitational force
where and are the masses of bodies, is the distance between them, and is the coefficient of proportionality, called the gravitational constant (the method of measuring it will be described below). Combining this formula with formula (123.4), we see that , where is the mass of the Sun. The forces of universal gravity satisfy Newton's third law. This was confirmed by all astronomical observations of the movement of celestial bodies.
In this formulation, the law of universal gravitation is applicable to bodies that can be considered material points, i.e., to bodies the distance between which is very large compared to their sizes, otherwise it would be necessary to take into account that different points of bodies are separated from each other at different distances . For homogeneous spherical bodies, the formula is valid for any distance between the bodies, if we take the distance between their centers as the value. In particular, in the case of attraction of a body by the Earth, the distance must be counted from the center of the Earth. This explains the fact that the force of gravity almost does not decrease as the height above the Earth increases (§ 54): since the radius of the Earth is approximately 6400, then when the position of the body above the Earth’s surface changes within even tens of kilometers, the force of gravity of the Earth remains practically unchanged.
The gravitational constant can be determined by measuring all other quantities included in the law of universal gravitation for any specific case.
It was possible for the first time to determine the value of the gravitational constant using torsion balances, the structure of which is schematically shown in Fig. 202. A light rocker, at the ends of which two identical balls of mass are attached, is hung on a long and thin thread. The rocker arm is equipped with a mirror, which allows optical measurement of small rotations of the rocker arm around the vertical axis. Two balls of significantly greater mass can be approached from different sides to the balls.
Rice. 202. Scheme of torsion balances for measuring the gravitational constant
The forces of attraction of small balls to large ones create a pair of forces that rotate the rocker clockwise (when viewed from above). By measuring the angle at which the rocker arm rotates when approaching the balls of the balls, and knowing the elastic properties of the thread on which the rocker arm is suspended, it is possible to determine the moment of the pair of forces with which the masses are attracted to the masses. Since the masses of the balls and the distance between their centers (at a given position of the rocker) are known, the value can be found from formula (124.1). It turned out to be equal
After the value was determined, it turned out to be possible to determine the mass of the Earth from the law of universal gravitation. Indeed, in accordance with this law, a body of mass located at the surface of the Earth is attracted to the Earth with a force
where is the mass of the Earth, and is its radius. On the other hand, we know that . Equating these quantities, we find
.
Thus, although the forces of universal gravity acting between bodies of different masses are equal, a body of small mass receives significant acceleration, and a body of large mass experiences low acceleration.
Since the total mass of all the planets of the Solar System is slightly more than the mass of the Sun, the acceleration that the Sun experiences as a result of the action of gravitational forces on it from the planets is negligible compared to the accelerations that the gravitational force of the Sun imparts to the planets. The gravitational forces acting between the planets are also relatively small. Therefore, when considering the laws of planetary motion (Kepler's laws), we did not take into account the motion of the Sun itself and approximately assumed that the trajectories of the planets were elliptical orbits, in one of the foci of which the Sun was located. However, in accurate calculations it is necessary to take into account those “perturbations” that gravitational forces from other planets introduce into the movement of the Sun itself or any planet.
124.1. How much will the force of gravity acting on a rocket projectile decrease when it rises 600 km above the Earth's surface? The radius of the Earth is taken to be 6400 km.
124.2. The mass of the Moon is 81 times less than the mass of the Earth, and the radius of the Moon is approximately 3.7 times less than the radius of the Earth. Find the weight of a man on the Moon if his weight on Earth is 600N.
124.3. The mass of the Moon is 81 times less than the mass of the Earth. Find on the line connecting the centers of the Earth and the Moon the point at which the gravitational forces of the Earth and the Moon acting on a body placed at this point are equal to each other.
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) :.
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Subtitles
Now let's learn a little about gravity, or gravitation. So there's no need to worry about these units: just know that we'll have to work with meters, seconds, and kilograms. Next, we calculate the square of 6.37 using a calculator and get... Square 6.37. And it's 40.58. 40.58.
Properties of Newtonian gravity
In Newtonian theory, each massive body generates a force field of attraction towards 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, when the density of a substanceρ (\displaystyle \rho )
distributed randomly, satisfies the Poisson equation:Δ φ = − 4 π G ρ (r) .
(\displaystyle \Delta \varphi =-4\pi G\rho (r).)Where 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,) 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. The trajectory of a material point in a gravitational field created by a much larger material point obeys Kepler's laws. In particular, planets and comets in the Solar System move in 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 to measure the quadrupole interaction of a rotating body and a stationary antenna showed that the incrementδ (\displaystyle \delta ) in the expression for the dependence of the Newtonian potential. Other experiments also confirmed the absence of modifications in the law of universal gravitation.
Newton's law of universal gravitation in 2007 was tested at distances smaller 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 conclusively connect 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 of 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 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 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. At the end of the 19th century, another problem emerged: 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, the general theory of 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 of quantum gravity, gravitational interaction occurs 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 wavelengths, and at large distances only 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
DEFINITION
The law of universal gravitation was discovered by I. Newton:
Two bodies attract each other with , directly proportional to their product and inversely proportional to the square of the distance between them:
Description of the law of universal gravitation
The coefficient is the gravitational constant. In the SI system, the gravitational constant has the meaning:
This constant, as can be seen, is very small, therefore the gravitational forces between bodies with small masses are also small and practically not felt. However, the movement of cosmic bodies is completely determined by gravity. The presence of universal gravitation or, in other words, gravitational interaction explains what the Earth and planets are “supported” by, and why they move around the Sun along certain trajectories, and do not fly away from it. The law of universal gravitation allows us to determine many characteristics of celestial bodies - the masses of planets, stars, galaxies and even black holes. This law makes it possible to calculate the orbits of planets with great accuracy and create a mathematical model of the Universe.
Using the law of universal gravitation, cosmic velocities can also be calculated. For example, the minimum speed at which a body moving horizontally above the Earth’s surface will not fall on it, but will move in a circular orbit is 7.9 km/s (first escape velocity). In order to leave the Earth, i.e. to overcome its gravitational attraction, the body must have a speed of 11.2 km/s (second escape velocity).
Gravity is one of the most amazing natural phenomena. In the absence of gravitational forces, the existence of the Universe would be impossible; the Universe could not even arise. Gravity is responsible for many processes in the Universe - its birth, the existence of order instead of chaos. The nature of gravity is still not fully understood. Until now, no one has been able to develop a decent mechanism and model of gravitational interaction.
Gravity
A special case of the manifestation of gravitational forces is the force of gravity.
Gravity is always directed vertically downward (toward the center of the Earth).
If the force of gravity acts on a body, then the body does . The type of movement depends on the direction and magnitude of the initial speed.
We encounter the effects of gravity every day. , after a while he finds himself on the ground. The book, released from the hands, falls down. Having jumped, a person does not fly into outer space, but falls down to the ground.
Considering the free fall of a body near the Earth's surface as a result of the gravitational interaction of this body with the Earth, we can write:
where does the acceleration of free fall come from:
The acceleration of gravity does not depend on the mass of the body, but depends on the height of the body above the Earth. The globe is slightly flattened at the poles, so bodies located near the poles are located a little closer to the center of the Earth. In this regard, the acceleration of gravity depends on the latitude of the area: at the pole it is slightly greater than at the equator and other latitudes (at the equator m/s, at the North Pole equator m/s.
The same formula allows you to find the acceleration of gravity on the surface of any planet with mass and radius.
Examples of problem solving
EXAMPLE 1 (problem about “weighing” the Earth)
| Exercise | The radius of the Earth is km, the acceleration of gravity on the surface of the planet is m/s. Using these data, estimate approximately the mass of the Earth. |
| Solution | Acceleration of gravity at the Earth's surface: where does the Earth's mass come from: In the C system, the radius of the Earth Substituting numerical values of physical quantities into the formula, we estimate the mass of the Earth:
|
| Answer | Earth mass kg. |
m.
EXAMPLE 2
| Exercise | An Earth satellite moves in a circular orbit at an altitude of 1000 km from the Earth's surface. At what speed is the satellite moving? How long will it take the satellite to complete one revolution around the Earth? |
| Solution | According to , the force acting on the satellite from the Earth is equal to the product of the mass of the satellite and the acceleration with which it moves:
The force of gravitational attraction acts on the satellite from the side of the earth, which, according to the law of universal gravitation, is equal to: where and are the masses of the satellite and the Earth, respectively. Since the satellite is at a certain height above the Earth's surface, the distance from it to the center of the Earth is: where is the radius of the Earth. |
