The force of gravity depends on the distance between bodies. The force of gravity

Why does a stone released from the hands fall to the ground? Because it is attracted by the Earth, each of you will say. In fact, the stone falls to the Earth with acceleration free fall. Consequently, a force directed towards the Earth acts on the stone from the side of the Earth. According to Newton's third law, the stone also acts on the Earth with the same modulus of force directed towards the stone. In other words, forces of mutual attraction act between the Earth and the stone.

Newton was the first who first guessed, and then strictly proved, that the reason causing the fall of a stone to the Earth, the movement of the Moon around the Earth and the planets around the Sun, is one and the same. This is the gravitational force acting between any bodies of the Universe. Here is the course of his reasoning, given in Newton's main work "Mathematical Principles natural philosophy»:

“A stone thrown horizontally will deviate under the influence of gravity from rectilinear path and, having described a curved trajectory, will finally fall to the Earth. If you throw it at a higher speed, then it will fall further” (Fig. 1).

Continuing this reasoning, Newton comes to the conclusion that if it were not for air resistance, then the trajectory of a stone thrown from high mountain with a certain speed, could become such that it would never reach the surface of the Earth at all, but would move around it "just as the planets describe their orbits in heavenly space."

Now we have become so accustomed to the movement of satellites around the Earth that there is no need to explain Newton's thought in more detail.

So, according to Newton, the movement of the Moon around the Earth or the planets around the Sun is also a free fall, but only a fall that lasts without stopping for billions of years. The reason for such a “fall” (whether we are really talking about the fall of an ordinary stone on Earth or the movement of the planets in their orbits) is the force gravity. What does this force depend on?

The dependence of the force of gravity on the mass of bodies

Galileo proved that during free fall, the Earth imparts the same acceleration to all bodies in a given place, regardless of their mass. But acceleration, according to Newton's second law, is inversely proportional to mass. How can one explain that the acceleration imparted to a body by the Earth's gravity is the same for all bodies? This is possible only if the force of attraction to the Earth is directly proportional to the mass of the body. In this case, an increase in the mass m, for example, by a factor of two will lead to an increase in the modulus of force F is also doubled, and the acceleration, which is equal to \(a = \frac (F)(m)\), will remain unchanged. Generalizing this conclusion for the forces of gravity between any bodies, we conclude that the force of universal gravitation is directly proportional to the mass of the body on which this force acts.

But at least two bodies participate in mutual attraction. Each of them, according to Newton's third law, is subject to the same modulus of gravitational forces. Therefore, each of these forces must be proportional both to the mass of one body and to the mass of the other body. Therefore, the force of universal gravitation between two bodies is directly proportional to the product of their masses:

\(F \sim m_1 \cdot m_2\)

The dependence of the force of gravity on the distance between bodies

It is well known from experience that the free fall acceleration is 9.8 m/s 2 and it is the same for bodies falling from a height of 1, 10 and 100 m, that is, it does not depend on the distance between the body and the Earth. This seems to mean that force does not depend on distance. But Newton believed that distances should be measured not from the surface, but from the center of the Earth. But the radius of the Earth is 6400 km. It is clear that several tens, hundreds or even thousands of meters above the Earth's surface cannot noticeably change the value of the free fall acceleration.

To find out how the distance between bodies affects the force of their mutual attraction, it would be necessary to find out what is the acceleration of bodies remote from the Earth at sufficiently large distances. However, it is difficult to observe and study the free fall of a body from a height of thousands of kilometers above the Earth. But nature itself came to the rescue here and made it possible to determine the acceleration of a body moving in a circle around the Earth and therefore possessing centripetal acceleration, caused, of course, by the same force of attraction to the Earth. Such a body is natural satellite Earth - Moon. If the force of attraction between the Earth and the Moon did not depend on the distance between them, then the centripetal acceleration of the Moon would be the same as the acceleration of a body freely falling near the surface of the Earth. In reality, the centripetal acceleration of the Moon is 0.0027 m/s 2 .

Let's prove it. The revolution of the Moon around the Earth occurs under the influence of the gravitational force between them. Approximately, the orbit of the Moon can be considered a circle. Therefore, the Earth imparts centripetal acceleration to the Moon. It is calculated by the formula \(a = \frac (4 \pi^2 \cdot R)(T^2)\), where R- radius lunar orbit, equal to about 60 radii of the Earth, T≈ 27 days 7 h 43 min ≈ 2.4∙10 6 s is the period of the Moon's revolution around the Earth. Given that the radius of the earth R h ≈ 6.4∙10 6 m, we get that the centripetal acceleration of the Moon is equal to:

\(a = \frac (4 \pi^2 \cdot 60 \cdot 6.4 \cdot 10^6)((2.4 \cdot 10^6)^2) \approx 0.0027\) m/s 2.

The found value of acceleration is less than the acceleration of free fall of bodies near the surface of the Earth (9.8 m/s 2) by approximately 3600 = 60 2 times.

Thus, an increase in the distance between the body and the Earth by 60 times led to a decrease in the acceleration reported by gravity, and consequently, the very force of attraction by 60 2 times.

This leads to an important conclusion: the acceleration imparted to bodies by the force of gravity towards the earth decreases in inverse proportion to the square of the distance to the center of the earth

\(F \sim \frac (1)(R^2)\).

Law of gravity

In 1667, Newton finally formulated the law of universal gravitation:

\(F = G \cdot \frac (m_1 \cdot m_2)(R^2).\quad (1)\)

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.

Proportionality factor G called gravitational constant.

Law of gravity is valid only for bodies whose dimensions are negligibly small compared to the distance between them. In other words, it is only fair for material points . In this case, the forces of gravitational interaction are directed along the line connecting these points (Fig. 2). Such forces are called central.

To find the gravitational force acting on a given body from the side of another, in the case when the size of the bodies cannot be neglected, proceed as follows. Both bodies are mentally divided into such small elements that each of them can be considered a point. Adding up the gravitational forces acting on each element of a given body from all the elements of another body, we obtain the force acting on this element (Fig. 3). Having done such an operation for each element of a given body and adding the resulting forces, they find the total gravitational force acting on this body. This task is difficult.

There is, however, one practically important case when formula (1) is applicable to extended bodies. It can be proved that spherical bodies, the density of which depends only on the distances to their centers, at distances between them that are greater than the sum of their radii, attract with forces whose modules are determined by formula (1). In this case R is the distance between the centers of the balls.

And finally, since the sizes of bodies falling to the Earth are many smaller sizes Earth, then these bodies can be considered as point bodies. Then under R in formula (1) one should understand the distance from a given body to the center of the Earth.

Between all bodies there are forces of mutual attraction, depending on the bodies themselves (their masses) and on the distance between them.

The physical meaning of the gravitational constant

From formula (1) we find

\(G = F \cdot \frac (R^2)(m_1 \cdot m_2)\).

It follows that if the distance between the bodies is numerically equal to one ( R= 1 m) and the masses of the interacting bodies are also equal to unity ( m 1 = m 2 = 1 kg), then the gravitational constant is numerically equal to the force modulus F. Thus ( physical meaning ),

the gravitational constant is numerically equal to the modulus of the gravitational force acting on a body of mass 1 kg from another body of the same mass with a distance between bodies equal to 1 m.

In SI, the gravitational constant is expressed as

Cavendish experience

The value of the gravitational constant G can only be found empirically. To do this, you need to measure the modulus of the gravitational force F, acting on the body mass m 1 side body weight m 2 at a known distance R between bodies.

The first measurements of the gravitational constant were made in mid-eighteenth in. Estimate, though very roughly, the value G at that time succeeded as a result of considering the attraction of the pendulum to the mountain, the mass of which was determined by geological methods.

Accurate measurements of the gravitational constant were first made in 1798 by the English physicist G. Cavendish using a device called a torsion balance. Schematically, the torsion balance is shown in Figure 4.

Cavendish fixed two small lead balls (5 cm in diameter and weighing m 1 = 775 g each) at opposite ends of a two meter rod. The rod was suspended on a thin wire. For this wire, the elastic forces arising in it when twisting through various angles were preliminarily determined. Two large lead balls (20 cm in diameter and weighing m 2 = 49.5 kg) could be brought close to small balls. Attractive forces from the large balls forced the small balls to move towards them, while the stretched wire twisted a little. The degree of twist was a measure of the force acting between the balls. The twisting angle of the wire (or the rotation of the rod with small balls) turned out to be so small that it had to be measured using an optical tube. The result obtained by Cavendish is only 1% different from the value of the gravitational constant accepted today:

G ≈ 6.67∙10 -11 (N∙m 2) / kg 2

Thus, the attraction forces of two bodies weighing 1 kg each, located at a distance of 1 m from each other, are only 6.67∙10 -11 N in modules. This is a very small force. Only in the case when bodies of enormous mass interact (or at least the mass of one of the bodies is large), the gravitational force becomes large. For example, the Earth pulls the Moon with force F≈ 2∙10 20 N.

Gravitational forces are the "weakest" of all the forces of nature. This is due to the fact that the gravitational constant is small. But for large masses space bodies The gravitational force becomes very strong. These forces keep all the planets near the Sun.

The meaning of the law of gravity

The law of universal gravitation underlies celestial mechanics - the science of planetary motion. With the help of this law, the positions of celestial bodies on the planet are determined with great accuracy. vault of heaven many decades ahead and their trajectories are calculated. The law of universal gravitation is also applied in motion calculations artificial satellites Earth and interplanetary automatic vehicles.

Disturbances in the motion of the planets. Planets do not move strictly according to Kepler's laws. Kepler's laws would be strictly observed for the motion of a given planet only if this planet alone revolved around the Sun. But there are many planets in the solar system, all of them are attracted by both the Sun and each other. Therefore, there are disturbances in the motion of the planets. In the solar system, perturbations are small, because the attraction of the planet by the Sun is much stronger than the attraction of other planets. When calculating the apparent position of the planets, perturbations must be taken into account. When launching artificial celestial bodies and when calculating their trajectories, they use an approximate theory of the motion of celestial bodies - perturbation theory.

Discovery of Neptune. One of clear examples The triumph of the law of universal gravitation is the discovery of the planet Neptune. In 1781, the English astronomer William Herschel discovered the planet Uranus. Its orbit was calculated and a table of the positions of this planet was compiled for many years to come. However, a check of this table, carried out in 1840, showed that its data differ from reality.

Scientists have suggested that the deviation in the motion of Uranus is caused by the attraction of an unknown planet, located even further from the Sun than Uranus. Knowing the deviations from the calculated trajectory (disturbances in the movement of Uranus), the Englishman Adams and the Frenchman Leverrier, using the law of universal gravitation, calculated the position of this planet in the sky. Adams completed the calculations earlier, but the observers to whom he reported his results were in no hurry to verify. Meanwhile, Leverrier, having completed his calculations, indicated to the German astronomer Halle the place where to look for an unknown planet. On the very first evening, September 28, 1846, Halle, pointing the telescope at the indicated place, discovered new planet. They named her Neptune.

In the same way, on March 14, 1930, the planet Pluto was discovered. Both discoveries are said to have been made "at the tip of a pen".

Using the law of universal gravitation, you can calculate the mass of the planets and their satellites; explain phenomena such as the ebb and flow of water in the oceans, and much more.

The forces of universal gravitation are the most universal of all the forces of nature. They act between any bodies that have mass, and all bodies have mass. There are no barriers to the forces of gravity. They act through any body.

Literature

Kikoin I.K., Kikoin A.K. Physics: Proc. for 9 cells. avg. school - M.: Enlightenment, 1992. - 191 p.
Physics: Mechanics. Grade 10: Proc. for in-depth study physics / M.M. Balashov, A.I. Gomonova, A.B. Dolitsky and others; Ed. G.Ya. Myakishev. – M.: Bustard, 2002. – 496 p.

In this section, we will talk about Newton's amazing conjecture, which led to the discovery of the law of universal gravitation.
Why does a stone released from the hands fall to the ground? Because it is attracted by the Earth, each of you will say. In fact, the stone falls to the Earth with free fall acceleration. Consequently, a force directed towards the Earth acts on the stone from the side of the Earth. According to Newton's third law, the stone also acts on the Earth with the same modulus of force directed towards the stone. In other words, forces of mutual attraction act between the Earth and the stone.
Newton's guess
Newton was the first who first guessed, and then strictly proved, that the reason causing the fall of a stone to the Earth, the movement of the Moon around the Earth and the planets around the Sun, is one and the same. This is the gravitational force acting between any bodies of the Universe. Here is the course of his reasoning, given in Newton's main work "Mathematical Principles of Natural Philosophy": "A stone thrown horizontally will deviate
, \\
1
/ /
At
Rice. 3.2
under the influence of gravity from a straight path and, having described a curved trajectory, will finally fall to the Earth. If you throw it with more speed, ! then it will fall further” (Fig. 3.2). Continuing these considerations, Newton \ comes to the conclusion that if it were not for air resistance, then the trajectory of a stone thrown from a high mountain at a certain speed could become such that it would never reach the surface of the Earth at all, but would move around it "just as the planets describe their orbits in celestial space."
Now we have become so accustomed to the movement of satellites around the Earth that there is no need to explain Newton's thought in more detail.
So, according to Newton, the movement of the Moon around the Earth or the planets around the Sun is also a free fall, but only a fall that lasts without stopping for billions of years. The reason for such a “fall” (whether we are really talking about the fall of an ordinary stone on the Earth or the movement of the planets in their orbits) is the force of universal gravitation. What does this force depend on?
The dependence of the force of gravity on the mass of bodies
In § 1.23 we talked about the free fall of bodies. Galileo's experiments were mentioned, which proved that the Earth communicates the same acceleration to all bodies in a given place, regardless of their mass. This is possible only if the force of attraction to the Earth is directly proportional to the mass of the body. It is in this case that the acceleration of free fall, equal to the ratio of the force of gravity to the mass of the body, is a constant value.
Indeed, in this case, an increase in the mass m, for example, by a factor of two will lead to an increase in the modulus of the force F also by a factor of two, and the acceleration
F
rhenium, which is equal to the ratio - , will remain unchanged.
Generalizing this conclusion for the forces of gravity between any bodies, we conclude that the force of universal gravitation is directly proportional to the mass of the body on which this force acts. But at least two bodies participate in mutual attraction. Each of them, according to Newton's third law, is subject to the same modulus of gravitational forces. Therefore, each of these forces must be proportional both to the mass of one body and to the mass of the other body.
Therefore, the force of universal gravitation between two bodies is directly proportional to the product of their masses:
F - here2. (3.2.1)
What else determines the gravitational force acting on a given body from another body?
The dependence of the force of gravity on the distance between bodies
It can be assumed that the force of gravity should depend on the distance between the bodies. To test the correctness of this assumption and to find the dependence of the force of gravity on the distance between bodies, Newton turned to the motion of the Earth's satellite - the Moon. Its motion was studied in those days much more accurately than the motion of the planets.
The revolution of the Moon around the Earth occurs under the influence of the gravitational force between them. Approximately, the orbit of the Moon can be considered a circle. Therefore, the Earth imparts centripetal acceleration to the Moon. It is calculated by the formula
l 2
a \u003d - Tg
where B is the radius of the lunar orbit, equal to approximately 60 radii of the Earth, T \u003d 27 days 7 h 43 min \u003d 2.4 106 s is the period of the Moon's revolution around the Earth. Taking into account that the radius of the Earth R3 = 6.4 106 m, we obtain that the centripetal acceleration of the Moon is equal to:
2 6 4k 60 ¦ 6.4 ¦ 10
M „ „„ „. , about
a = 2 ~ 0.0027 m/s*.
(2.4 ¦ 106 s)
The found value of acceleration is less than the acceleration of free fall of bodies near the Earth's surface (9.8 m/s2) by approximately 3600 = 602 times.
Thus, an increase in the distance between the body and the Earth by 60 times led to a decrease in the acceleration imparted by the Earth's gravity, and, consequently, the force of gravity itself, by 602 times.
This leads to an important conclusion: the acceleration imparted to bodies by the force of attraction to the Earth decreases in inverse proportion to the square of the distance to the center of the Earth:
ci
a = -k, (3.2.2)
R
where Сj - constant factor, the same for all bodies.
Kepler's laws
The study of the motion of the planets showed that this motion is caused by the force of gravity towards the Sun. Using careful long-term observations of the Danish astronomer Tycho Brahe, the German scientist Johannes Kepler in early XVII in. established the kinematic laws of planetary motion - the so-called Kepler's laws.
Kepler's first law
All planets move in ellipses with the Sun at one of the foci.
An ellipse (Fig. 3.3) is a flat closed curve, the sum of the distances from any point of which to two fixed points, called foci, is constant. This sum of distances is equal to the length of the major axis AB of the ellipse, i.e.
FgP + F2P = 2b,
where Fl and F2 are the foci of the ellipse, and b = ^^ is its semi-major axis; O is the center of the ellipse. The point of the orbit closest to the Sun is called perihelion, and the point farthest from it is called p.

AT
Rice. 3.4
"2
B A A aphelion. If the Sun is in focus Fr (see Fig. 3.3), then point A is perihelion, and point B is aphelion.
Kepler's second law
The radius vector of the planet for the same time intervals describes equal areas. So, if the shaded sectors (Fig. 3.4) have equal areas, then the paths si> s2> s3 will be covered by the planet in equal time intervals. It can be seen from the figure that Sj > s2. Hence, line speed the movements of the planet various points its orbit is not the same. At perihelion, the speed of the planet is greatest, at aphelion - the smallest.
Kepler's third law
The squares of the orbital periods of the planets around the Sun are related as the cubes of the semi-major axes of their orbits. Denoting the semi-major axis of the orbit and the period of revolution of one of the planets through bx and Tv and the other - through b2 and T2, Kepler's third law can be written as follows:

From this formula it can be seen that the farther the planet is from the Sun, the longer its period of revolution around the Sun.
Based on Kepler's laws, certain conclusions can be drawn about the accelerations imparted to the planets by the Sun. For simplicity, we will assume that the orbits are not elliptical, but circular. For planets solar system this substitution is not a very rough approximation.
Then the force of attraction from the side of the Sun in this approximation should be directed for all planets to the center of the Sun.
If through T we denote the periods of revolution of the planets, and through R the radii of their orbits, then, according to Kepler's third law, for two planets we can write
t\L? T2 R2
Normal acceleration when moving in a circle a = co2R. Therefore, the ratio of the accelerations of the planets
Q-i GlD.
7G=-2~- (3-2-5)
2t:r0
Using equation (3.2.4), we get
T2
Since Kepler's third law is valid for all planets, then the acceleration of each planet is inversely proportional to the square of its distance from the Sun:
Oh oh
a = -|. (3.2.6)
WT
The constant C2 is the same for all planets, but it does not coincide with the constant C2 in the formula for the acceleration given to bodies by the globe.
Expressions (3.2.2) and (3.2.6) show that the gravitational force in both cases (attraction to the Earth and attraction to the Sun) gives all bodies an acceleration that does not depend on their mass and decreases inversely with the square of the distance between them:
F~a~-2. (3.2.7)
R
Law of gravity
The existence of dependences (3.2.1) and (3.2.7) means that the force of universal gravitation 12
TP.L Sh
F~
R2? ТТТ-i ТПп
F=G
In 1667, Newton finally formulated the law of universal gravitation:
(3.2.8) R
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. The proportionality factor G is called the gravitational constant.
Interaction of point and extended bodies
The law of universal gravitation (3.2.8) is valid only for such bodies, the dimensions of which are negligible compared to the distance between them. In other words, it is valid only for material points. In this case, the forces of gravitational interaction are directed along the line connecting these points (Fig. 3.5). Such forces are called central.
To find the gravitational force acting on a given body from another, in the case when the size of the bodies cannot be neglected, proceed as follows. Both bodies are mentally divided into such small elements that each of them can be considered a point. Adding up the gravitational forces acting on each element of a given body from all the elements of another body, we obtain the force acting on this element (Fig. 3.6). Having done such an operation for each element of a given body and adding the resulting forces, they find the total gravitational force acting on this body. This task is difficult.
There is, however, one practically important case when formula (3.2.8) is applicable to extended bodies. It is possible to prove
m^
Fig. 3.5 Fig. 3.6
It can be stated that spherical bodies, the density of which depends only on the distances to their centers, at distances between them that are greater than the sum of their radii, are attracted with forces whose modules are determined by formula (3.2.8). In this case, R is the distance between the centers of the balls.
And finally, since the dimensions of the bodies falling to the Earth are much smaller than the dimensions of the Earth, these bodies can be considered as point ones. Then under R in the formula (3.2.8) one should understand the distance from the given body to the center of the Earth.
Between all bodies there are forces of mutual attraction, depending on the bodies themselves (their masses) and on the distance between them.
? 1. The distance from Mars to the Sun is 52% greater than the distance from the Earth to the Sun. What is the length of a year on Mars? 2. How will the force of attraction between the balls change if the aluminum balls (Fig. 3.7) are replaced by steel balls of the same mass? the same volume?

The law of universal gravitation was discovered by Newton in 1687 while studying the movement of the Moon's satellite around the Earth. The English physicist clearly formulated the postulate characterizing the forces of attraction. In addition, by analyzing Kepler's laws, Newton calculated that attractive forces must exist not only on our planet, but also in space.

Background

The law of universal gravitation was not born spontaneously. Since ancient times, people have studied the sky, mainly for compiling agricultural calendars, calculating important dates, religious holidays. Observations indicated that in the center of the "world" is the Luminary (Sun), around which orbits revolve celestial bodies. Subsequently, the dogmas of the church did not allow to think so, and people lost the knowledge accumulated over thousands of years.

In the 16th century, before the invention of telescopes, a galaxy of astronomers appeared who looked at the sky in a scientific way, rejecting the prohibitions of the church. T. Brahe, observing the cosmos for many years, systematized the movements of the planets with special care. These high-precision data helped I. Kepler subsequently discover three of his laws.

By the time of the discovery (1667) by Isaac Newton of the law of gravitation in astronomy, the heliocentric system of the world of N. Copernicus was finally established. According to it, each of the planets of the system revolves around the Sun in orbits, which, with an approximation sufficient for many calculations, can be considered circular. At the beginning of the XVII century. I. Kepler, analyzing the work of T. Brahe, established the kinematic laws that characterize the motions of the planets. The discovery became the foundation for clarifying the dynamics of the planets, that is, the forces that determine precisely this type of their movement.

Description of interaction

Unlike short-period weak and strong interactions, gravity and electromagnetic fields have properties long range: their influence is manifested at gigantic distances. On the mechanical phenomena 2 forces act in the macrocosm: electromagnetic and gravitational. The impact of planets on satellites, the flight of an abandoned or launched object, the floating of a body in a liquid - gravitational forces act in each of these phenomena. These objects are attracted by the planet, gravitate towards it, hence the name "law of universal gravitation".

It has been proven that between physical bodies There is certainly a force of mutual attraction. Such phenomena as the fall of objects on the Earth, the rotation of the Moon, planets around the Sun, occurring under the influence of the forces of universal attraction, are called gravitational.

Law of gravity: formula

Universal gravitation is formulated as follows: any two material object are attracted to each other with a certain force. The magnitude of this force is directly proportional to the product of the masses of these objects and inversely proportional to the square of the distance between them:

In the formula, m1 and m2 are the masses of the studied material objects; r is the distance determined between the centers of mass of the calculated objects; G is a constant gravitational quantity expressing the force with which the mutual attraction of two objects weighing 1 kg each, located at a distance of 1 m, is carried out.

What does the force of attraction depend on?

The law of universal gravitation works differently, depending on the region. Since the force of attraction depends on the values of latitude at a particular location, then similarly, the acceleration of gravity has different values in different places. Maximum value the force of gravity and, accordingly, the acceleration of free fall are at the poles of the Earth - the force of gravity at these points is equal to the force of attraction. The minimum values will be at the equator.

The globe is slightly flattened, its polar radius is less than the equatorial one by about 21.5 km. However, this dependence is less significant compared to the daily rotation of the Earth. Calculations show that due to the oblateness of the Earth at the equator, the value of the free fall acceleration is slightly less than its value at the pole by 0.18%, and after diurnal rotation- by 0.34%.

However, in the same place on the Earth, the angle between the direction vectors is small, so the discrepancy between the force of attraction and the force of gravity is insignificant, and it can be neglected in the calculations. That is, we can assume that the modules of these forces are the same - the acceleration of free fall near the surface of the Earth is the same everywhere and is approximately 9.8 m / s².

Conclusion

Isaac Newton was a scientist who made a scientific revolution, completely rebuilt the principles of dynamics and based on them created a scientific picture of the world. His discovery influenced the development of science, the creation of material and spiritual culture. It fell to Newton's fate to reconsider the results of his conception of the world. In the 17th century scientists completed the grandiose work of building the foundation new science- physics.

In this paragraph, we will remind you about gravity, centripetal acceleration and body weight.

Every body on the planet is affected by the Earth's gravity. The force with which the Earth attracts each body is determined by the formula

The point of application is at the center of gravity of the body. Gravity always pointing vertically down.

The force with which a body is attracted to the Earth under the influence of the Earth's gravitational field is called gravity. According to the law of universal gravitation, on the surface of the Earth (or near this surface), a body of mass m is affected by the force of gravity

F t \u003d GMm / R 2

where M is the mass of the Earth; R is the radius of the Earth.
If only gravity acts on the body, and all other forces are mutually balanced, the body is in free fall. According to Newton's second law and the formula F t \u003d GMm / R 2 free fall acceleration modulus g is found by the formula

g=F t /m=GM/R 2 .

From formula (2.29) it follows that the free fall acceleration does not depend on the mass m of the falling body, i.e. for all bodies in a given place on the Earth it is the same. From formula (2.29) it follows that Fт = mg. In vector form

F t \u003d mg

In § 5 it was noted that since the Earth is not a sphere, but an ellipsoid of revolution, its polar radius is less than the equatorial one. From the formula F t \u003d GMm / R 2 it can be seen that for this reason the force of gravity and the acceleration of free fall caused by it is greater at the pole than at the equator.

The force of gravity acts on all bodies in the Earth's gravitational field, but not all bodies fall to the Earth. This is due to the fact that the movement of many bodies is hindered by other bodies, such as supports, suspension threads, etc. Bodies that restrict the movement of other bodies are called connections. Under the action of gravity, the bonds are deformed and the reaction force of the deformed bond, according to Newton's third law, balances the force of gravity.

The acceleration of free fall is affected by the rotation of the Earth. This influence is explained as follows. The frames of reference associated with the surface of the Earth (except for the two associated with the poles of the Earth) are not, strictly speaking, inertial frames of reference - the Earth rotates around its axis, and such frames of reference move along circles with centripetal acceleration along with it. This non-inertiality of reference systems is manifested, in particular, in the fact that the value of the acceleration of free fall turns out to be different in different places on the Earth and depends on the geographical latitude of the place where the reference frame associated with the Earth is located, relative to which the acceleration of gravity is determined.

Measurements taken on different latitudes, showed that numerical values free fall accelerations differ little from each other. Therefore, with not very accurate calculations, one can neglect the non-inertial reference systems associated with the Earth's surface, as well as the difference in the shape of the Earth from a spherical one, and assume that the acceleration of free fall in any place on the Earth is the same and equal to 9.8 m / s 2.

From the law of universal gravitation it follows that the force of gravity and the acceleration of free fall caused by it decrease with increasing distance from the Earth. At a height h from the Earth's surface, the gravitational acceleration module is determined by the formula

g=GM/(R+h) 2.

It has been established that at a height of 300 km above the Earth's surface, the free fall acceleration is less than at the Earth's surface by 1 m/s2.
Consequently, near the Earth (up to heights of several kilometers), the force of gravity practically does not change, and therefore the free fall of bodies near the Earth is a uniformly accelerated motion.

Body weight. Weightlessness and overload

The force in which, due to attraction to the Earth, the body acts on its support or suspension, is called body weight. Unlike gravity, which is gravitational force applied to the body, the weight is elastic force applied to a support or suspension (i.e., to a connection).

Observations show that the weight of the body P, determined on a spring balance, is equal to the force of gravity F t acting on the body only if the balance with the body relative to the Earth is at rest or moving uniformly and rectilinearly; In this case

P \u003d F t \u003d mg.

If the body is moving with acceleration, then its weight depends on the value of this acceleration and on its direction relative to the direction of free fall acceleration.

When a body is suspended on a spring balance, two forces act on it: the force of gravity F t =mg and the elastic force F yp of the spring. If at the same time the body moves vertically up or down relative to the direction of free fall acceleration, then the vector sum of the forces F t and F yn gives the resultant, which causes the acceleration of the body, i.e.

F t + F pack \u003d ma.

According to the above definition of the concept of "weight", we can write that P=-F yp. From the formula: F t + F pack \u003d ma. taking into account the fact that F t =mg, it follows that mg-ma=-F yp . Therefore, P \u003d m (g-a).

The forces F t and F yn are directed along one vertical straight line. Therefore, if the acceleration of the body a is directed downward (i.e., it coincides in direction with the acceleration of free fall g), then modulo

P=m(g-a)

If the acceleration of the body is directed upwards (i.e., opposite to the direction of free fall acceleration), then

P \u003d m \u003d m (g + a).

Consequently, the weight of a body whose acceleration coincides in direction with the acceleration of free fall is less than the weight of a body at rest, and the weight of a body whose acceleration is opposite to the direction of acceleration of free fall is greater than the weight of a body at rest. The increase in body weight caused by accelerated movement, called overload.

In free fall a=g. From the formula: P=m(g-a)

it follows that in this case P=0, i.e., there is no weight. Therefore, if bodies move only under the influence of gravity (i.e., fall freely), they are in a state weightlessness. characteristic feature this state is the absence of deformations in freely falling bodies and internal stresses, which are caused in resting bodies by gravity. The reason for the weightlessness of bodies is that the force of gravity imparts the same accelerations to a freely falling body and its support (or suspension).

In nature there are various powers, which characterize the interaction of bodies. Consider those forces that occur in mechanics.

gravitational forces. Probably, the very first force, the existence of which was realized by a person, was the force of attraction acting on bodies from the side of the Earth.

And it took many centuries for people to understand that the force of gravity acts between any bodies. And it took many centuries for people to understand that the force of gravity acts between any bodies. The first to understand this fact English physicist Newton. Analyzing the laws that govern the motion of the planets (Kepler's laws), he came to the conclusion that the observed laws of planetary motion can only be fulfilled if there is an attractive force between them, which is directly proportional to their masses and inversely proportional to the square of the distance between them.

Newton formulated law of gravity. Any two bodies are attracted to each other. The force of attraction between point bodies is directed along the straight line connecting them, is directly proportional to the masses of both and inversely proportional to the square of the distance between them:

Under the point bodies in this case understand bodies whose dimensions are many times smaller than the distance between them.

The force of gravity is called gravitational forces. The coefficient of proportionality G is called the gravitational constant. Its value was determined experimentally: G = 6.7 10¯¹¹ N m² / kg².

gravity acting near the surface of the Earth, is directed towards its center and is calculated by the formula:

where g is the free fall acceleration (g = 9.8 m/s²).

The role of gravity in living nature is very significant, since the size, shape and proportions of living beings largely depend on its magnitude.

Body weight. Consider what happens when a load is placed on horizontal plane(support). At the first moment after the load is lowered, it begins to move downward under the action of gravity (Fig. 8).

The plane bends and there is an elastic force (reaction of the support), directed upwards. After the elastic force (Fy) balances the force of gravity, the lowering of the body and the deflection of the support will stop.

The deflection of the support arose under the action of the body, therefore, a certain force (P) acts on the support from the side of the body, which is called the weight of the body (Fig. 8, b). According to Newton's third law, the weight of a body is equal in magnitude to the support reaction force and is directed in the opposite direction.

P \u003d - Fu \u003d F heavy.

body weight called the force P, with which the body acts on a horizontal support that is stationary relative to it.

Since gravity (weight) is applied to the support, it deforms and, due to elasticity, counteracts the force of gravity. The forces developed in this case from the side of the support are called the forces of the reaction of the support, and the very phenomenon of the development of counteraction is called the reaction of the support. According to Newton's third law, the reaction force of the support is equal in magnitude to the force of gravity of the body and opposite to it in direction.

If a person on a support moves with the acceleration of the links of his body directed away from the support, then the reaction force of the support increases by the value ma, where m is the mass of the person, and are the accelerations with which the links of his body move. These dynamic effects can be recorded using strain gauge devices (dynamograms).

Weight should not be confused with body mass. The mass of a body characterizes its inertial properties and does not depend on either the gravitational force or the acceleration with which it moves.

The weight of the body characterizes the force with which it acts on the support and depends both on the force of gravity and on the acceleration of movement.

For example, on the Moon, the weight of a body is about 6 times less than the weight of a body on Earth. The mass is the same in both cases and is determined by the amount of matter in the body.

In everyday life, technology, sports, weight is often indicated not in newtons (N), but in kilograms of force (kgf). The transition from one unit to another is carried out according to the formula: 1 kgf = 9.8 N.

When the support and the body are motionless, then the mass of the body is equal to the force of gravity of this body. When the support and the body move with some acceleration, then, depending on its direction, the body may experience either weightlessness or overload. When the acceleration coincides in direction and is equal to the acceleration of gravity, the weight of the body will be zero, so a state of weightlessness occurs (ISS, high-speed elevator when lowering down). When the acceleration of the movement of the support is opposite to the acceleration of free fall, the person experiences an overload (start from the Earth's surface of a manned spaceship, high-speed elevator going up).