The force of gravity

The force of gravity mg is the sum of the gravitational attraction of the planet GMm / r ² and the centrifugal inertia force m ω ² a .

Gravity - a force acting on any physical body located near the surface of the Earth or other astronomical body .

By definition, gravity on the surface of a planet is composed of the gravitational attraction of the planet and centrifugal inertia caused by the diurnal rotation of the planet ^[1] ^[2] .

The remaining forces (for example, the attraction of the Moon and the Sun), due to their smallness, are not taken into account or studied separately as temporary changes in the Earth’s gravitational field ^[3] ^[4] ^[5] .

Gravity gives all bodies, regardless of their mass, the same acceleration ^[6] and is a conservative force ^[7] .

The force of gravity ${\vec {P}}$ ${\ displaystyle {\ vec {P}}}$ ${\ vec P}$ acting on a mass point mass $m$ ${\ displaystyle m}$ $m$ calculated by the formula ^[6] : ${\vec {P}}=m{\vec {g}}$ ${\ displaystyle {\ vec {P}} = m {\ vec {g}}}$ ${\ displaystyle {\ vec {P}} = m {\ vec {g}}}$ where ${\vec {g}}$ ${\ displaystyle {\ vec {g}}}$ ${\ vec g}$ - the acceleration imparted to the body by gravity, which is called the acceleration of gravity ^[8] .

If the field of gravity is uniform within the extended body, then the resultant of gravity acting on the elements of this body is applied to the center of mass of the body ^[9] .

Besides bodies of gravity, Coriolis force ^[10] ^[11] ^[12] also acts on bodies moving relative to the Earth’s surface.

Content

History

Aristotle explained gravity by the movement of heavy physical elements (earth, water) to his natural place (the center of the Universe inside the Earth), and the greater the speed, the closer the heavy body is to it ^[13] .

Archimedes considered the issue of the center of gravity of a parallelogram, triangle, trapezoid and parabolic segment. In the work "On Floating Bodies" Archimedes proved the law of hydrostatics , bearing his name ^[13] .

Jordan Nemorariy in the essay “On Weights”, when examining loads on an inclined plane, decomposed their gravity into normal and parallel components on an inclined plane, was close to determining the static moment ^[14] .

Stevin experimentally determined that bodies of different masses fall with the same acceleration , established theorems on the fluid pressure in vessels (pressure depends only on the depth and does not depend on the size, shape and volume of the vessel) and on the balance of goods on an inclined plane (on inclined planes of equal height forces acting on the side of balancing weights along inclined planes are inversely proportional to the lengths of these planes). He proved the theorem according to which, in the case of equilibrium, the center of gravity of a homogeneous floating body should be higher than the center of gravity of the displaced fluid ^[15] .

Galileo experimentally investigated the laws of falling bodies ( acceleration does not depend on body weight), oscillations of pendulums (the oscillation period does not depend on the weight of the pendulum) and motion along an inclined plane ^[16] .

Huygens created the classical theory of pendulum motion, which had a significant impact on the theory of gravity ^[16] .

Descartes developed the kinetic theory of gravity, explaining the force of gravity by the interaction of bodies with a heavenly fluid, put forward a hypothesis about the dependence of gravity on the distance between a heavy body and the center of the earth ^[16] .

Newton from the equality of the accelerations of falling bodies and Newton’s second law concluded that gravity is proportional to the masses of bodies and established that gravity is one of the manifestations of gravity ^[17] ^[18] . To test this idea, he compared the acceleration of gravity of bodies at the surface of the Earth with the acceleration of the moon in orbit along which it moves relative to the Earth. ^[nineteen]

Einstein explained the fact that the accelerations of falling bodies are equal regardless of their mass (equivalence of inert and heavy mass) as a consequence of the principle of equivalence of a uniformly accelerated reference frame and a reference frame located in a gravitational field ^[20] .

Spherically Symmetric Body

In accordance with the law of universal gravitation , the force of gravitational attraction acting on a mass point by mass $m$ ${\ displaystyle m}$ on the surface of a spherically symmetric astronomical body having mass $M$ ${\ displaystyle M}$ is determined by the ratio:

F=G\cdot {M\cdot m \over R^{2}},

{\ displaystyle F = G \ cdot {M \ cdot m \ over R ^ {2}},}

Where $G$ ${\ displaystyle G}$ - gravitational constant equal to 6.67384 (80) · 10 ⁻¹¹ m ³ · s ⁻² · kg ⁻¹ , and $R$ ${\ displaystyle R}$ - body radius. This relation is valid under the assumption that the mass distribution over the body volume is spherically symmetric. In this case, the force of gravitational attraction is directed to the center of the body.

The module of centrifugal inertia $Q$ ${\ displaystyle Q}$ acting on a material particle is expressed by the formula:

Q=ma\omega ^{2},

{\ displaystyle Q = ma \ omega ^ {2},}

Where $a$ ${\ displaystyle a}$ - the distance between the particle and the axis of rotation of the considered astronomical body, and $\omega$ ${\ displaystyle \ omega}$ - the angular velocity of its rotation. The centrifugal inertia is perpendicular to the axis of rotation and directed away from it.

The corrections introduced by the general theory of relativity into Newton’s law of universal gravitation in the conditions of the Earth and other planets are extremely small (the module of the gravitational potential on the Earth’s surface is equal to half the square of the second cosmic velocity $v_{II}$ ${\ displaystyle v_ {II}}$ extremely small compared to the square of the speed of light $c$ ${\ displaystyle c}$ : ${\frac {v_{II}^{2}}{2c^{2}}}\sim 10^{-10}$ ${\ displaystyle {\ frac {v_ {II} ^ {2}} {2c ^ {2}}} \ sim 10 ^ {- 10}}$ ) ^[21] .

Earth

The shape of the Earth ( geoid ) differs from spherical and is close to a flattened ellipsoid . In this case, the force of gravitational attraction acting on a material point by mass $m$ ${\ displaystyle m}$ is defined by a more complex expression than before:

${\vec {F}}=Gm\int \limits _{M}{{dM} \over {R^{2}}}{{\vec {R}} \over R}.$ ${\ displaystyle {\ vec {F}} = Gm \ int \ limits _ {M} {{dM} \ over {R ^ {2}}} {{\ vec {R}} \ over R}.}$

Here $d M$ ${\ displaystyle dM}$ - an element of the mass of the Earth, ${\vec {R}}={\vec {r}}-{\vec {r}}',$ ${\ displaystyle {\ vec {R}} = {\ vec {r}} - {\ vec {r}} ',}$ but ${\vec {r}}$ ${\ displaystyle {\ vec {r}}}$ and ${{\vec {r}}'}$ ${\ displaystyle {{\ vec {r}} '}}$ Are the radius vectors of the measuring point and the Earth element, respectively. Integration in this case is carried out over the entire mass of the Earth.

In vector form, the expression for centrifugal inertia can be written as

{\vec {Q}}=m\omega ^{2}{{\vec {R}}_{0}},

{\ displaystyle {\ vec {Q}} = m \ omega ^ {2} {{\ vec {R}} _ {0}},}

Where ${{\vec {R}}_{0}}$ ${\ displaystyle {{\ vec {R}} _ {0}}}$ Is a vector perpendicular to the axis of rotation and drawn from it to a given material point located near the surface of the Earth.

In this case, gravity ${\vec {P}}$ ${\ displaystyle {\ vec {P}}}$ , as before, is equal to the sum ${\vec {F}}$ ${\ displaystyle {\ vec {F}}}$ and ${\vec {Q}}$ ${\ displaystyle {\ vec {Q}}}$ :

{\vec {P}}={\vec {F}}+{\vec {Q}}.

{\ displaystyle {\ vec {P}} = {\ vec {F}} + {\ vec {Q}}.}

The force of gravity acting near the surface of the Earth depends on the latitude of the place $\varphi$ ${\ displaystyle \ varphi}$ and his heights $H$ ${\ displaystyle H}$ above sea level. The approximate expression for the absolute value of gravity in the SI system is ^[8] :

P=9{,}780318(1+0{,}005302\sin \varphi -0{,}000006\sin ^{2}2\varphi )m-0{,}000003086Hm.

{\ displaystyle P = 9 {,} 780318 (1 + 0 {,} 005302 \ sin \ varphi -0 {,} 000006 \ sin ^ {2} 2 \ varphi) m-0 {,} 000003086Hm.}

Angle $\alpha$ ${\ displaystyle \ alpha}$ between gravity ${\vec {P}}$ ${\ displaystyle {\ vec {P}}}$ and gravitational pull to Earth ${\vec {F}}$ ${\ displaystyle {\ vec {F}}}$ equal to ^[22] :

\alpha \approx 0{,}0018\sin {2\varphi }

{\ displaystyle \ alpha \ approx 0 {,} 0018 \ sin {2 \ varphi}}

.

It varies from zero (at the equator , where $\varphi =0^{\circ }$ ${\ displaystyle \ varphi = 0 ^ {\ circ}}$ and at the poles where $\varphi =90^{\circ }$ ${\ displaystyle \ varphi = 90 ^ {\ circ}}$ ) before $0{,}0018$ ${\ displaystyle 0 {,} 0018}$ glad or $6'$ ${\ displaystyle 6 '}$ (at latitude $45^{\circ }$ ${\ displaystyle 45 ^ {\ circ}}$ )

The movement of bodies under the action of gravity

In the case when the module of the body’s movement is much less than the distance to the center of the Earth, then the force of gravity can be considered constant, and the body’s motion uniformly accelerated. If the initial velocity of the body is nonzero and its vector is not directed vertically, then under the influence of gravity the body moves along a parabolic trajectory.

When a body is thrown from a certain height parallel to the surface of the Earth, the flight range increases with increasing initial velocity. For large values of the initial velocity, to calculate the trajectory of the body, it is necessary to take into account the spherical shape of the Earth and the change in the direction of gravity at different points of the trajectory.

At a certain value of velocity, called the first cosmic velocity , a body thrown along a tangent to the Earth’s surface, under the influence of gravity in the absence of resistance from the atmosphere, can move around the Earth in a circle without falling to the Earth. At a speed exceeding the second cosmic velocity , the body moves away from the Earth’s surface to infinity along a hyperbolic trajectory. At speeds intermediate between the first and second cosmic, the body moves around the Earth along an elliptical trajectory ^[23] .

Potential energy of a body raised above the Earth

The potential energy of a body raised above the Earth is the work of gravity taken with the opposite sign, performed when the body moves from the Earth's surface to this position. She is equal $E_{p}=\gamma Mm({\frac {1}{R_{z}}}-{\frac {1}{R}})$ ${\ displaystyle E_ {p} = \ gamma Mm ({\ frac {1} {R_ {z}}} - {\ frac {1} {R}}}$ where $\gamma$ ${\ displaystyle \ gamma}$ - gravitational constant $M$ ${\ displaystyle M}$ - mass of land $m$ ${\ displaystyle m}$ - body mass, $R_{z}$ ${\ displaystyle R_ {z}}$ is the radius of the Earth, $R$ ${\ displaystyle R}$ - the distance to the center of the earth of the body.

When the body is removed, the distances from the Earth’s surface that are not small compared to the Earth’s radius can be considered uniform, that is, the acceleration of gravity is constant. In this case, when lifting a body mass $m$ ${\ displaystyle m}$ to the height $h$ ${\ displaystyle h}$ from the surface of the earth, gravity does the work $A=-mgh$ ${\ displaystyle A = -mgh}$ . Therefore, the potential energy of the body: $E_{p}=mgh$ ${\ displaystyle E_ {p} = mgh}$ . The potential energy of the body can have both positive and negative values. Deep body $h$ ${\ displaystyle h}$ from the Earth’s surface has a negative value of potential energy $E_{p}=-mgh$ ${\ displaystyle E_ {p} = - mgh}$ ^[24] .

When water evaporates from the Earth's surface, solar radiation is transformed into the potential energy of water vapor in the atmosphere. Then, when precipitation falls to land, it passes into kinetic energy during runoff and performs erosive work in the process of transferring denudation material of all land and makes life possible for the organic world on Earth ^[25] .

The potential energy of rock masses moved by tectonic processes is mainly spent on moving rock destruction products from elevated surface areas to lower-lying ones ^[26] .

Significance in nature

Gravity plays an important role in the evolution of stars. For stars at the stage of the main sequence of their evolution, gravity is one of the important factors providing the conditions necessary for thermonuclear fusion . At the final stages of the evolution of stars, in the process of their collapse, due to gravity not compensated by the forces of internal pressure, the stars turn into neutron stars or black holes .

Gravity is very important for the formation of the structure of the internal structure of the Earth and other planets and the tectonic evolution of its surface ^[27] . The greater the gravity, the greater the mass of meteorite material falls on a unit of its surface ^[28] . During the Earth’s existence, its mass has significantly increased due to gravity: annually 30-40 million tons of meteorite material are deposited on the Earth, mainly in the form of dust, which significantly exceeds the scattering of light components of the Earth’s upper atmosphere in space ^[29] .

Without the potential energy of gravity, continuously turning into kinetic energy, the circulation of matter and energy on Earth would be impossible ^[30] .

Gravity plays a very important role for life on Earth ^[31] . Thanks to her, the Earth has an atmosphere. Due to gravity acting on air, atmospheric pressure exists ^[32] .

All living organisms with a nervous system have receptors that determine the magnitude and direction of gravity and serve to orient in space. In vertebral organisms, including humans, the size and direction of gravity determines the vestibular apparatus ^[33] .

The presence of gravity led to the emergence in all multicellular terrestrial organisms of the strong skeletons necessary to overcome it. In aquatic living organisms, the force of gravity is balanced by hydrostatic force ^[34] .

The role of gravity in the life processes of organisms is studied by gravitational biology ^[35] .

Application in technology

Gravity and the principle of equivalence of inert and gravitational masses are used to determine the masses of objects by weighing them on a scale. Gravity is used in the settling separation of gas and liquid mixtures, in some types of watches , in plumb and counterweights , Atwood machine, Oberbek machine and liquid barometers .

Accurate measurements of gravity and its gradient ( gravimetry ) are used in the study of the internal structure of the Earth and in the gravity exploration of various minerals ^[36] .

Stability of the body in the field of gravity

For a body in a gravitational field based on one point (for example, when hanging the body at one point or placing the ball on a plane) for stable equilibrium it is necessary that the center of gravity of the body occupies the lowest position compared to all possible neighboring positions ^[37] .

For a body in the field of gravity, resting on several points (for example, a table) or on an entire platform (for example, a box on a horizontal plane) for stable equilibrium it is necessary that the vertical drawn through the center of gravity passes inside the area of the body support. The area of the body support is the contour connecting the support points or inside the platform on which the body rests ^[37] .

Gravity Measurement Methods

Gravity is measured by dynamic and static methods. Dynamic methods use the observation of body motion under the action of gravity and measure the time it takes for the body to transition from one predetermined position to another. They use: pendulum vibrations, free fall of the body, string vibrations with a load. Static methods use the observation of changes in the equilibrium position of the body under the action of gravity and some balancing force and measure the linear or angular displacement of the body.

Gravity measurements are absolute and relative. Absolute measurements determine the full value of gravity at a given point. Relative measurements determine the difference in gravity at a given point and some other, previously known value. Instruments designed for relative measurements of gravity are called gravimeters .

Dynamic methods for determining gravity can be both relative and absolute, static methods can only be relative.

Gravity on other planets

**Gravity on the surface ^{[39] of} some celestial bodies, gravity on Earth is accepted as 1** ^[40]
Land	1.00	The sun	27.85
Moon	0.165	Mercury	0.375-0.381
Venus	0,906	Mars	0.394
Jupiter	2,442	Saturn	1,065
Uranus	0,903	Neptune	1,131

Notes

↑ Sivukhin D.V. General course of physics. - M .: Fizmatlit , 2005. - T. I. Mechanics. - S. 372. - 560 p. - ISBN 5-9221-0225-7 .
↑ Targ S. M. Gravity // Physical Encyclopedia / Ch. ed. A.M. Prokhorov . - M .: Big Russian Encyclopedia , 1994. - T. 4. - S. 496. - 704 p. - 40,000 copies. - ISBN 5-85270-087-8 .
↑ Mironov, 1980 , p. 49.
↑ The maximum change in gravity due to the attraction of the moon is approximately $0{,}25\cdot 10^{-5}$ ${\ displaystyle 0 {,} 25 \ cdot 10 ^ {- 5}}$ m / s ² , of the Sun $0{,}1\cdot 10^{-5}$ ${\ displaystyle 0 {,} 1 \ cdot 10 ^ {- 5}}$ m / s ²
↑ Mironov, 1980 , p. 71.
↑ ¹ ² Saveliev, 1987 , p. 70.
↑ Saveliev, 1987 , p. 82-83.
↑ ¹ ² Acceleration of free fall // Physical Encyclopedia / Ch. ed. A.M. Prokhorov . - M .: Big Russian Encyclopedia , 1998. - V. 5. - S. 245-246. - 760 s. - ISBN 5-85270-101-7 .
↑ Saveliev, 1987 , p. 156.
↑ Tarasov, 2012 , p. 200, 270.
↑ Saveliev, 1987 , p. 128.
↑ Butenin, 1971 , p. 253-259.
↑ ¹ ² Zubov V.P. Physical Ideas of Antiquity // Otv. ed. Grigoryan A. T. , Polak L. S. Essays on the development of basic physical ideas. - M., Academy of Sciences of the USSR, 1959. - S. 38, 54-55;
↑ Zubov V.P. Physical Ideas of the Middle Ages // Otv. ed. Grigoryan A. T. , Polak L. S. Essays on the development of basic physical ideas. - M., Academy of Sciences of the USSR, 1959. - S. 114;
↑ Zubov V.P. Physical Ideas of the Renaissance // Otv. ed. Grigoryan A. T. , Polak L. S. Essays on the development of basic physical ideas. - M., Academy of Sciences of the USSR, 1959. - S. 151;
↑ ¹ ² ³ Kuznetsov B.G. Genesis of a mechanical explanation of physical phenomena and the idea of Cartesian physics // Otv. ed. Grigoryan A. T. , Polak L. S. Essays on the development of basic physical ideas. - M., Academy of Sciences of the USSR, 1959. - S. 160-161, 169-170, 177;
↑ Newton, 1989 , p. 7.
↑ B. G. Kuznetsov. Fundamentals of Newtonian physics // Res. ed. Grigoryan A. T. , Polak L. S. Essays on the development of basic physical ideas. - M., Academy of Sciences of the USSR, 1959. - S. 189-191;
↑ Sivukhin D.V. General course of physics. Mechanics. - M., Nauka, 1979. - Circulation of 50,000 copies. - with. 323
↑ Ivanenko D. D. Basic ideas of the general theory of relativity // Otv. ed. Grigoryan A. T. , Polak L. S. Essays on the development of basic physical ideas. - M., Academy of Sciences of the USSR, 1959. - S. 300;
↑ Grischuk L.P. , Zeldovich Ya. B. Gravitation // Space Physics. Little Encyclopedia. - M., Soviet Encyclopedia, 1986. - S. 676
↑ Saveliev, 1987 , p. 122.
↑ Zhirnov N.I. Classical mechanics. - M., Enlightenment , 1980. - Circulation 28000 copies. - with. 121
↑ Kabardin O.F., Orlov V.A., Ponomareva A.V. Optional physics course. 8th grade. - M .: Education , 1985. - Circulation 143,500 copies. - S. 151 - 152
↑ Krivolutsky, 1985 , p. 307.
↑ Krivolutsky, 1985 , p. 70, 234.
↑ Krivolutsky, 1985 , p. 208.
↑ Krivolutsky, 1985 , p. 77.
↑ Krivolutsky, 1985 , p. 48, 237-238.
↑ Krivolutsky, 1985 , p. 289.
↑ Zelmanov A. L. The diversity of the material world and the problem of the infinity of the Universe // Infinity and the Universe. - M., Thought, 1969. - Circulation 12000 copies. - S. 283
↑ Khromov S.P. , Petrosyants M.A. Meteorology and climatology. - M., Moscow State University, 2006. - ISBN 5-211-05207-2 . - C. 67
↑ Yuri Frolov. https://www.nkj.ru/archive/articles/21172/ Our gravitational compass] // Science and life . - 2012. - No. 10 .
↑ P. Kemp, K. Arms Introduction to Biology. - M .: Mir, 1988 .-- ISBN 5-03-001286-9 . - The circulation of 125,000 copies. - S. 75
↑ Lozovskaya E. Life with and without gravity // Science and Life . - 2004. - No. 9 .
↑ Mironov, 1980 , p. 1-543.
↑ ¹ ² Landsberg G.S. Elementary textbook of physics. Volume 1. Mechanics, heat, molecular physics. - M., Science , 1975. - Circulation 350 000 copies. - S. 189-190
↑ Mironov, 1980 , p. 94-262.
↑ For gas giants, “surface” is understood as a region of heights in the atmosphere, where the pressure is equal to the atmospheric pressure on Earth at sea level ( 1.013 × 10 ⁵ Pa ).
↑ Data taken from Wikipedia article Acceleration of free fall

Literature

Newton I. Mathematical principles of natural philosophy. - M .: Nauka, 1989 .-- 688 p. - ISBN 5-02-000747-1 .
Saveliev I.V. Course in General Physics. T. 1. Mechanics. Molecular physics. - M .: Nauka, 1987 .-- 688 p.
Krivolutsky A.E. Blue planet. Earth among the planets. The geographical aspect .. - M .: Thought , 1985. - 335 p.
Mironov V.S. Gravity survey course. - L .: Nedra, 1980 .-- 543 p.
Tarasov V.N., Boyarkina I.V., Kovalenko M.V., Fedorchenko N.P., Fisenko N.I. Theoretical mechanics. - M .: TransLit, 2012 .-- 560 p.
Butenin N.V. Introduction to analytical mechanics. - M .: Nauka , 1971. - 264 p. - 25,000 copies.

[1] Sivukhin D.V. General course of physics. - M .: Fizmatlit , 2005. - T. I. Mechanics. - S. 372. - 560 p. - ISBN 5-9221-0225-7 .

[2] Targ S. M. Gravity // Physical Encyclopedia / Ch. ed. A.M. Prokhorov . - M .: Big Russian Encyclopedia , 1994. - T. 4. - S. 496. - 704 p. - 40,000 copies. - ISBN 5-85270-087-8 .

[_6fa75f00a10108b6-3] Mironov, 1980 , p. 49.

[4] The maximum change in gravity due to the attraction of the moon is approximately $0{,}25\cdot 10^{-5}$ ${\ displaystyle 0 {,} 25 \ cdot 10 ^ {- 5}}$ m / s ² , of the Sun $0{,}1\cdot 10^{-5}$ ${\ displaystyle 0 {,} 1 \ cdot 10 ^ {- 5}}$ m / s ²

[_6fa76000a1010a73-5] Mironov, 1980 , p. 71.

[_6bcf5dabfaf3d9ae-6] ¹ ² Saveliev, 1987 , p. 70.

[_14691824fdae7a91-7] Saveliev, 1987 , p. 82-83.

[ФЭ5-8] ¹ ² Acceleration of free fall // Physical Encyclopedia / Ch. ed. A.M. Prokhorov . - M .: Big Russian Encyclopedia , 1998. - V. 5. - S. 245-246. - 760 s. - ISBN 5-85270-101-7 .

[_253cc23b6c60da75-9] Saveliev, 1987 , p. 156.

[_ede8d8a9dde76bbe-10] Tarasov, 2012 , p. 200, 270.

[_253cc73b6c60e2fa-11] Saveliev, 1987 , p. 128.

[_e0ee91ec57b71245-12] Butenin, 1971 , p. 253-259.

[Zub1-13] ¹ ² Zubov V.P. Physical Ideas of Antiquity // Otv. ed. Grigoryan A. T. , Polak L. S. Essays on the development of basic physical ideas. - M., Academy of Sciences of the USSR, 1959. - S. 38, 54-55;

[Zub2-14] Zubov V.P. Physical Ideas of the Middle Ages // Otv. ed. Grigoryan A. T. , Polak L. S. Essays on the development of basic physical ideas. - M., Academy of Sciences of the USSR, 1959. - S. 114;

[Zub3-15] Zubov V.P. Physical Ideas of the Renaissance // Otv. ed. Grigoryan A. T. , Polak L. S. Essays on the development of basic physical ideas. - M., Academy of Sciences of the USSR, 1959. - S. 151;

[Kuz1-16] ¹ ² ³ Kuznetsov B.G. Genesis of a mechanical explanation of physical phenomena and the idea of Cartesian physics // Otv. ed. Grigoryan A. T. , Polak L. S. Essays on the development of basic physical ideas. - M., Academy of Sciences of the USSR, 1959. - S. 160-161, 169-170, 177;

[_fa39e0b048535f68-17] Newton, 1989 , p. 7.

[Kuz2-18] B. G. Kuznetsov. Fundamentals of Newtonian physics // Res. ed. Grigoryan A. T. , Polak L. S. Essays on the development of basic physical ideas. - M., Academy of Sciences of the USSR, 1959. - S. 189-191;

[19] Sivukhin D.V. General course of physics. Mechanics. - M., Nauka, 1979. - Circulation of 50,000 copies. - with. 323

[Ivan-20] Ivanenko D. D. Basic ideas of the general theory of relativity // Otv. ed. Grigoryan A. T. , Polak L. S. Essays on the development of basic physical ideas. - M., Academy of Sciences of the USSR, 1959. - S. 300;

[21] Grischuk L.P. , Zeldovich Ya. B. Gravitation // Space Physics. Little Encyclopedia. - M., Soviet Encyclopedia, 1986. - S. 676

[_253cc73b6c60e2f0-22] Saveliev, 1987 , p. 122.

[23] Zhirnov N.I. Classical mechanics. - M., Enlightenment , 1980. - Circulation 28000 copies. - with. 121

[24] Kabardin O.F., Orlov V.A., Ponomareva A.V. Optional physics course. 8th grade. - M .: Education , 1985. - Circulation 143,500 copies. - S. 151 - 152

[_9167d8980edd72be-25] Krivolutsky, 1985 , p. 307.

[_9ac32e0212030742-26] Krivolutsky, 1985 , p. 70, 234.

[_916452980eda5938-27] Krivolutsky, 1985 , p. 208.

[_fd8ffc5cbeb5cb98-28] Krivolutsky, 1985 , p. 77.

[_f70b6d3c6d227bfa-29] Krivolutsky, 1985 , p. 48, 237-238.

[_91645a980eda66a1-30] Krivolutsky, 1985 , p. 289.

[31] Zelmanov A. L. The diversity of the material world and the problem of the infinity of the Universe // Infinity and the Universe. - M., Thought, 1969. - Circulation 12000 copies. - S. 283

[32] Khromov S.P. , Petrosyants M.A. Meteorology and climatology. - M., Moscow State University, 2006. - ISBN 5-211-05207-2 . - C. 67

[NKJ201210-33] Yuri Frolov. https://www.nkj.ru/archive/articles/21172/ Our gravitational compass] // Science and life . - 2012. - No. 10 .

[34] P. Kemp, K. Arms Introduction to Biology. - M .: Mir, 1988 .-- ISBN 5-03-001286-9 . - The circulation of 125,000 copies. - S. 75

[NKJ200409-35] Lozovskaya E. Life with and without gravity // Science and Life . - 2004. - No. 9 .

[_64460712b4b620f5-36] Mironov, 1980 , p. 1-543.

[Land-37] ¹ ² Landsberg G.S. Elementary textbook of physics. Volume 1. Mechanics, heat, molecular physics. - M., Science , 1975. - Circulation 350 000 copies. - S. 189-190

[_d6323242e69368b5-38] Mironov, 1980 , p. 94-262.

[39] For gas giants, “surface” is understood as a region of heights in the atmosphere, where the pressure is equal to the atmospheric pressure on Earth at sea level ( 1.013 × 10 ⁵ Pa ).

[40] Data taken from Wikipedia article Acceleration of free fall