Movement of bodies under the action of gravitational forces. Movement of bodies under the action of gravity

The movement of the body under the action of gravity is one of the central themes in dynamic physics. The dynamics section is based on three knows even the usual schoolboy. Let's try to disassemble this topic thoroughly, and the article, describing in detail each example, will help us to study the movement of the body under the action of gravity as useful as possible.

A bit of history

People with curiosity watched various phenomenaoriginating in our lives. Humanity for a long time could not understand the principles and the device of many systems, but the long path of studying the surrounding world brought our ancestors to the scientific coup. Nowadays, when technologies are developing with an incredible speed, people almost do not think about how those or other mechanisms work.

Meanwhile, our ancestors were always interested in riddles of natural processes and the device of the world, they were looking for answers to the most difficult issues and did not stop studying until they found answers. So, for example, the famous scientist Galileo Galiley back in the 16th century asked: "Why does the bodies always fall down, what power attracts them to the ground?" In 1589, he set a number of experiments whose results were very valuable. He studied in detail the patterns of free fall of various bodies, dropping items with the famous tower in the city of Pisa. The laws that he brought were improved and described in more detail by the formulas by another famous English scientist - Sir Isaac Newton. It is he who owns three laws, on which almost all modern physics is based.

The fact that the patterns of motion of the tel, described more than 500 years ago, are relevant to this day, means that our planet is subject to constant laws. Modern man It is necessary to at least superficially examine the basic principles of the arrangement of the world.

The main and auxiliary concepts of speakers

In order to fully understand the principles of such a movement, you should first get acquainted with some concepts. So, the most necessary theoretical terms:

• The interaction is the effect of bodies on each other, at which the change or the beginning of their movement relative to each other occurs. There are four types of interaction: electromagnetic, weak, strong and gravitational.
• Speed \u200b\u200bis physical quantitydenoting the speed with which the body moves. Speed \u200b\u200bis a vector, that is, it has not only a value, but also the direction.
• Acceleration is the value that shows us the speed of changing body velocity in a period of time. It is also
• The trajectory of the path is a curve, and sometimes a straight line that outlines the body when moving. With uniform straight movement The trajectory can coincide with the value of moving.
• The path is the length of the trajectory, that is, exactly as much as the body passed for a certain amount of time.
• The inertial reference system is a medium in which the first Newton law is performed, that is, the body maintains its inertia, provided that all external forces are completely absent.

The above concepts are enough to correctly draw or present in the head modeling the body movement under the action of gravity.

What does strength mean?

Let's go to the basic concept of our topic. So, force is the magnitude, the meaning of which is the effects or influence of one body to another quantitatively. And the power of gravity is the power that acts absolutely to each body located on the surface or near our planet. The question arises: where does this very power come from? The answer lies in the law of global gravity.

What is the power of gravity?

On any body from the ground, the influence of the gravitational force, which tells him some acceleration. The strength of gravity always has a vertical direction down to the center of the planet. In other words, the strength of gravity attracts objects to the ground, which is why the items always fall down. It turns out that the strength of gravity is a special case of the strength of the world. Newton brought one of the main formulas to find the force attraction between two bodies. It looks in this way: f \u003d g * (M 1 x m 2) / R 2.

What is the acceleration of free fall?

The body that was released from some height always flies down under the action of attraction force. The body movement under the action of gravity vertically up and down can be described by the equations where the main constant will be the value of the acceleration "G". This value is due exclusively by the action of the force of attraction, and its value is approximately 9.8 m / s 2. It turns out that the body thrown from the height without the initial speed will move down with acceleration equal value "G".

Body movement under the action of gravity: Formulas for solving problems

The main formula for the force of gravity looks like this: F severity \u003d M x G, where M is a body weight, which is valid, and "G" - acceleration of free fall (it is considered to be 10 m / s to simplify the tasks 2) .

There are several more formulas used to find a particular unknown body at the free movement. So, for example, in order to calculate the path passed by the body, it is necessary to substitute the known values \u200b\u200bin this formula: S \u003d V 0 x T + A x T 2/2 (path equal to sum Works of the initial speed multiplied by time and acceleration per square of time divided by 2).

Equations for describing the vertical body movement

The body movement under the action of gravity vertically can be described by the equation that looks like this: x \u003d x 0 + v 0 x T + A x T 2 / 2. Using this expression, the coordinates of the body in famous moment time. It is necessary to simply substitute the values \u200b\u200bknown in the problem: the initial location, the initial speed (if the body is not just released, and pushed with some force) and acceleration, in our case it will be accelerated g.

In the same way, the body speed can also be found, which moves under the action of the force of attraction. The expression for finding an unknown value at any time: v \u003d v 0 + g x t (the value of the initial speed can be zero, then the speed will be equal to the product of accelerating the free fall to the value of the time for which the body makes movement).

Movement of bodies under the action of gravity: tasks and ways to solve them

When solving many tasks related to gravity, we recommend using the following plan:

1. Determine for yourself a convenient inertial reference system, it is usually taken to choose land, because it meets many requirements for ISO.
2. Draw a small drawing or picture, which depicts the main forces acting on the body. The body movement under the action of gravity implies an outline or a diagram on which the body is moving in which direction is moving if acceleration is acting equal to G.
3. Then you should select the direction for projecting forces and the accelerations obtained.
4. Record unknown values \u200b\u200band determine their direction.
5. Finally, using the above formulas for solving problems, calculate all unknown values, substituting the data into the equations for finding the acceleration or the traveled path.

Ready solution easy task

When it comes to such a phenomenon, as a body movement under the action of how practical to solve the task, it can be difficult. However, there are several tricks using which, you can easily solve even the most difficult task. So, we will analyze on the living examples, how to solve this or that task. Let's start with easy to understand the task.

Some body was released from a height of 20 m without initial speed. Determine for how much time it reaches the surface of the Earth.

Solution: we know the path traveled by the body, it is known that the initial speed was equal to 0. We can also determine that only the power of gravity acts on the body, it turns out that this is the movement of the body under the action of gravity, and therefore this formula should be used: S \u003d V 0 x T + A x T 2/2. Since in our case a \u003d g, then after some transformations we obtain the following equation: S \u003d G x T 2 / 2. Now it remains only to express time through this formula, we obtain that T 2 \u003d 2S / g. We substitute the known values \u200b\u200b(we believe that G \u003d 10 m / s 2) T 2 \u003d 2 x 20/10 \u003d 4. Therefore, T \u003d 2 s.

So, our answer: the body will fall to Earth in 2 seconds.

The trick that allows you to quickly solve the problem, is as follows: it can be noted that the described movement of the body in the given task occurs in one direction (vertically down). It is very similar to an equilibrium movement, since no force acts on the body, except for gravity (the power of the air resistance is neglecting). Thanks to this, it is possible to use a light formula to find the path with an equilibrium movement, bypassing images of drawings with the arrangement of the forces acting on the body.

An example of solving a more complex task

And now let's see how it is better to solve the problem of body movement under the action of gravity, if the body is moving not vertically, but has a more complex nature of movement.

For example, the following task. Some object weighing M moves with an unknown acceleration down on the inclined plane, the friction coefficient of which is k. Determine the value of acceleration, which is available when the body is moving if the angle of inclination α is known.

Solution: You should use the plan, which is described above. First of all, draw the pattern of the inclined plane with the image of the body and all the forces acting on it. It turns out that there are three components: the strength of gravity, friction and the strength of the support reaction. Looks like general equation equally existing forces So: F friction + n + Mg \u003d Ma.

The main highlight of the task is the condition of inclination at an angle α. When OX and OY axis, it is necessary to take into account this condition, then we will have the following expression: Mg x Sin α - F friction \u003d Ma (for the axis OH) and N - Mg x Cos α \u003d F friction (for Oy axis).

F friction is easy to calculate by the formula for finding the friction force, it is equal to k x Mg (the friction coefficient multiplied by the body mass and accelerate free fall). After all calculations, it remains only to substitute the values \u200b\u200bfound in the formula, it will be a simplified equation to calculate the acceleration, with which the body moves along the inclined plane.

Based on the interpretation of the second law of Newton, it can be concluded that the change in movement occurs by force. Mechanics considers the strength of various physical nature. Many of them are determined by the action of the forces of gravity.

In 1862, the law of the World Health I. Newton was opened. He suggested that the forces holding the moon, the same nature as the strength forcing the apple to fall on the ground. The sense of the hypothesis consists in the presence of the action of attraction forces directed along the line and connecting the centers of the masses as shown in Figure 1. 10 . one . The spherical body has a mass center coinciding with a ball center.

Picture 1 . 10 . 1 . Gravitational forces of attraction between bodies. F 1 → \u003d - F 2 →.

Definition 1.

With the well-known directions of movements, Newton planets tried to find out what forces they act on them. This process was called the inverse problem of mechanics.

The main task of mechanics is to determine the coordinates of the body of the known mass with its speed at any time with the help of the known forces acting on the body, and a given condition (direct task). The reverse is performed with the definition of the current forces on the body with its known direction. Such tasks led a scientist to the discovery of the definition of the law of the World Committee.

All bodies are attracted to each other with strength, directly proportional to their masses and inversely proportional to the square of the distance between them.

F \u003d G M 1 m 2 R 2.

The value G determines the proportionality coefficient of all bodies in nature, called the gravitational constant and denoted by the formula G \u003d 6, 67 · 10 - 11 N · m 2 / K 2 (C and).

Most of the phenomena in nature are explained by the presence of the world's strength. The movement of planets, artificial satellites of the Earth, the trajectory of the flight of ballistic missiles, the movement of the tel near the surface of the Earth - everything is explained by the law of gravity and dynamics.

Definition 3.

The manifestation of the force is characterized by the presence forces of gravity. So called the force of attraction of bodies to the ground and near her surface.

When M is denoted as the mass of the Earth, R h is the radius, M - body weight, the formula of gravity takes the form:

F \u003d g m r z 2 m \u003d m g.

Where G is the acceleration of the free fall, equal to G \u003d G M R h 2.

The power of gravity is directed towards the center of the Earth, as shown in the example of the Moon-Earth. In the absence of action of other forces, the body moves with the acceleration of free fall. Its average equals 9, 81 m / s 2. With the known G and radius R 3 \u003d 6, 38 · 10 6 M, calculations of the mass of the Earth M by the formula are made:

M \u003d g R 3 2 g \u003d 5, 98 · 10 24)

If the body is removed from the surface of the Earth, then the action of the force and acceleration of the free fall is changed inversely in proportion to the square R distance R to the center. Picture 1 . 10 . 2 shows how the strength of the ship is changing, acting on the cosmonaut of the ship, when removing from the ground. Obviously, F attracting it to Earth is 700 N.

Picture 1 . 10 . 2 . Changes in force acting on the astronaut when removing from the ground.

Example 1.

Earth-moon is suitable as an example of the interaction of a system of two bodies.

The distance to the moon - R l \u003d 3, 84 · 10 6 m. It is 60 times more than the radius of the Earth R z. So, in the presence of terrestrial attraction, the acceleration of the free fall α l of the moon orbits will be α l \u003d g r z r l 2 \u003d 9, 81 m / s 2 60 2 \u003d 0, 0027 m / s 2.

It is directed towards the center of the Earth and received the name of the centripetal. The calculation is made according to the formula A L \u003d υ 2 R l \u003d 4 π 2 R L T 2 \u003d 0, 0027 m / s 2, where T \u003d 27, 3 days - the period of circulation of the moon around the Earth. Results and calculations made in different ways, they suggest that Newton was right in his assumption of a single nature of the force holding the moon in orbit, and gravity.

The moon has its own gravitational field, which determines the acceleration of the free fall G l on the surface. The mass of the moon is 81 times less than the mass of the Earth, and the radius is 3, 7 times. It can be seen that the acceleration G L should be determined from the expression:

g l \u003d g m l r l 2 \u003d g m z 3, 7 2 t 3 2 \u003d 0, 17 g \u003d 1, 66 m / s 2.

Such weak gravity is characteristic of astronauts located on the moon. Therefore, you can make huge jumps and steps. A jump up on the meter on the ground corresponds to the seventener on the moon.

The movement of artificial satellites is recorded outside the earth's atmosphere, so they have the effect of the Earth's gravity. The trajectory of the cosmic body may vary depending on the initial speed. Movement of artificial satellite near-earth orbit Approximately accepted as a distance to the center of the Earth, equal to the radius R b. They fly at altitudes 200 - 300 to m.

Definition 4.

Hence it follows that centripetal acceleration The satellite, which is communicated with the forces of gravity, is equal to the acceleration of the free fall g. The satellite speed will take the designation υ 1. It is called the first space speed.

Applying the kinematic formula for centripetal acceleration, we get

a n \u003d υ 1 2 R z \u003d g, υ 1 \u003d g r s \u003d 7, 91 · 10 3 m / s.

At such a speed, the satellite was able to fly to the land in a time equal to T 1 \u003d 2 πr s υ 1 \u003d 84 m and H 12 s.

But the period of circulation of the satellite in a circular orbit near the ground is much larger than indicated above, since there is a distinction between the radius of the real orbit and the radius of the Earth.

The satellite moves according to the principle of free fall, remotely similar to the trajectory of the projectile or ballistic missile. The difference is B. high speed The satellite, and the radius of curvature of its trajectory reaches the length of the radius of the Earth.

Satellites that move around the circular trajectories at large distances have a weakened earthly attraction, inversely proportional to the square of the R radius R pathor. Then the satellite speed is followed by condition:

υ 2 K \u003d G R 3 2 R 2, υ \u003d G R 3 R z r \u003d υ 1 r 3 r.

Therefore, the presence of satellites in high orbits speaks of a lower speed of their movement than with an near-earth orbit. The formula of the appeal period is:

T \u003d 2 πR υ \u003d 2 πR υ 1 R R z \u003d 2 πr z υ 1 r R 3 3/2 \u003d T 1 2 π R z

T 1 takes the value of the satellite handling period of an near-earth orbit. T increases with the sizes of the radius of the orbit. If R is 6, 6 R 3, then the satellite is 24 hours. When it starts in the equator plane, it will be observed as hanging over a certain point ground surface. The use of such satellites is known in the Space Radiocommunication system. Orbit having radius R \u003d 6, 6 R z is called geostationary.

Picture 1 . 10 . 3 . Model movement of satellites.

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Gravity, it is an attraction or a lot - this is a universal property of matter that all items and bodies in the universe are possessed. The essence of gravity is incorporated in that all material bodies attract all other bodies around.

Force of gravity

If gravity is general concept And the quality that all items in the universe are possessed, then the earthly attraction is a special case of this comprehensive phenomenon. The land attracts all the material objects on it. Thanks to this, people and animals can safely move on the ground, rivers, the sea and oceans - to stay within their shores, and the air is not to fly over the endless spaces of space, but to form an atmosphere of our planet.

There is a fair question: if all items have gravity, why land attracts people and animals to themselves, and not vice versa? First, we also attract the earth to themselves, simply, compared with her strength of attraction, our gravity is negligible. Secondly, the force of gravity is directly proportional to the mass of the body: the less the mass of the body, the lower its gravitational forces.

The second indicator on which the attraction force depends is the distance between the items: the more distance, the less the action of gravity. Including this, planets move on their orbits, and do not fall on each other.

It is noteworthy that with its spherical form of land, the moon, the sun and other planets are obliged precisely by force. It acts in the direction of the center, tightening the substance to it, which makes the "body" of the planet.

Gravitational field of land

The gravitational field of the Earth is a powerful energy field that is formed around our planet due to the action of two forces:

• gravity;
• centrifugal strength, which is obliged to rotate the Earth around its axis (daily rotation).

Since gravity and centrifugal force Permanent, then the gravitational field is permanent.

Significant impact on the field have the strength of the sun, the moon and some others heavenly Tel, as well as the atmospheric masses of the Earth.

World Act and Sir Isaac Newton

English physicist, Sir Isaac Newton, according to a famous legend, once walking around the garden in the afternoon, saw the moon in the sky. At the same time, an apple fell from the branch. Newton then studied the law of movement and knew that the apple falls under the influence of the gravitational field, and the moon rotates in orbit around the Earth.

And here, in the head, a brilliant scientist, insult, came to the idea that, perhaps, the apple falls to the ground, obeying the same force, thanks to which the moon is in his orbit, and not worn randomly throughout the Galaxy. So the law of global gravity was opened, he is the third law of Newton.

In language mathematical formulas This law looks like this:

F.= GMM / D 2 ,

where F. - the power of mutual gravity between the two bodies;

M. - the mass of the first body;

m. - the mass of the second body;

D 2. - distance between two bodies;

G. - gravitational constant, equal to 6.67x10 -11.

Why the stone released from the hands falls to the ground? Because the earth attracts him, each of you will say. In fact, the stone falls on the ground with an acceleration of free fall. Consequently, the stone on the ground is the power aimed towards the Earth.

According to the third law of Newton and the stone acts on Earth with the same module by the force pointing to the stone. In other words, the strength of mutual attraction act between the Earth and Stone.

Caught Newton

Newton was the first one who first guessed, and then strictly proved that the reason that causes the fall of the stone to the ground, the movement of the moon around the earth and the planets around the Sun, the same. This is the force of gravity acting between any bodies of the universe. Here is the course of his reasoning given in Newton's main work " Mathematical principles Natural philosophy ":" Abandoned horizontal stone will deviate under the action of gravity from the straight way and, describing the curve of the trajectory, will finally fall to the ground. If you throw it with greater speed, it will fall further "(Fig. 3.2). Continuing these arguments, Newton comes to the conclusion that if it were not for the resistance of the air, the trajectory of the stone abandoned with a high mountain at a certain speed could be such that he would never achieve the surface of the earth, and moved around her "like that How planets describe their orbits in the heavenly space. "

Fig. 3.2

Now we have become so familiar to the movement of satellites around the Earth, which is not necessary to explain the idea of \u200b\u200bNewton.

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, billions of years. The reason for such a "fall" (whether it is really about the fall of ordinary stone on earth or the movement of the planets on their orbits) is the power of global. What does this power depends on?

The dependence of the force of body mass

In § 1.23 it was referred to a free drop of tel. Galilee's experiments were mentioned, which proved that the Earth reports to all the bodies in this place the same acceleration regardless of their mass. This is possible only if the force of attraction to the ground 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 power of the earth's attraction to the mass of the body is a constant value.

Indeed, in this case, the increase in mass M, for example, halfway will lead to an increase in the module of force, too twice, and the acceleration that is equal to the ratio remains unchanged.

Summarizing this conclusion for the forces of gravity between any bodies, we conclude that the world's strength is directly proportional to the mass of the body, which this force is valid. But in mutual attraction, at least two bodies participate. For each of them, according to the third law of Newton, the same in the module of the force is valid. Therefore, each of these forces should be proportional to both the mass of one body and the mass of another body.

therefore the world's strength between two bodies is directly proportional to the work of their masses.:

What does the force acting on this body on the part of another body depends?

The dependence of the force against the distance between the bodies

It can be assumed that the force of gravity should depend on the distance between the bodies. To verify the correctness of this assumption and find the dependence of the force from the distance between the bodies, Newton turned to the movement of the Earth's satellite - the Moon. Its movement was much more accurate in those days than the movement of the planets.

The appeal of the moon around the Earth occurs under the influence of the force between them. An approximate orbit of the moon can be considered a circle. Consequently, the Earth reports the moon centripetal acceleration. It is calculated by the formula

where R is the radius of the lunar orbit, equal to about 60 radius of the Earth, T \u003d 27 days 7 h 43 min \u003d 2.4 10 6 C - the period of circulation of the moon around the Earth. Considering that the radius of the Earth R 3 \u003d 6.4 10 6 m, we obtain that the centripetal acceleration of the moon is:

The value of the acceleration is less than accelerating the free fall of the bodies at the surface of the Earth (9.8 m / s 2) approximately 3600 \u003d 60 2 times.

Thus, an increase in the distance between the body and the ground 60 times led to a decrease in the acceleration reported by the earthly attraction, and therefore the force of attraction in 60 2 times (1).

This implies an important conclusion: acceleration, which informs the power of attraction to earth, decreases inversely proportional to the square of the distance to the center of the Earth:

where C 1 is a permanent coefficient, the same for all bodies.

Kepler laws

The study of the motion of the planets showed that this movement caused by the force of attraction to the Sun. Using thorough perennial observations of Danish astronomer Quiet Brage, German scientist Johann Kepler in beginning of XVII in. I installed the kinematic laws of the motion of the planets - the so-called laws of Kepler.

First law of Kepler

All planets move along the ellipses, in one of the focus of which the sun is located.

The ellipse (Fig. 3.3) is called a flat closed curve, the amount of distances from any point of which up to two fixed points, called focus, is constant. This amount of distances is equal to the length of the large axis of AB ellipses, i.e.

where F 1 and F 2 - the focuses of the ellipse, and B \u003d - its large semi-axis; About the center of the ellipse. The nearest orbit dot to the sun is called the periecelium, and the most distant point from it is Aplia. If the sun is in focus F 1 (see Fig. 3.3), then the point A is perigelius, and the point in the aphelius.

Fig. 3.3.

Second law of Kepler

The radius-vector of the planet for the same intervals describes equal Square . So, if the shaded sectors (Fig. 3.4) have the same area, the paths S 1, S 2, S 3 will be covered with planet in equal intervals. From the figure it is seen that S 1\u003e S 2. Therefore, the linear velocity of the planet at various points of its orbits is not the same. In the perihelion, the speed of the planet is the greatest, in Aflia - the smallest.

Fig. 3.4.

The third law of Kepler

Squares of circulation periods of planets around the Sun belong as cubes of large semi-axes of their orbits. By designating the larger part of the orbit and the period of treatment of one of the planets through B 1 and T 1 and the other - through B 2 and T 2, the third law of the Kepler can be written as follows:

Based on the laws of the Kepler, certain conclusions on accelerations reported by the planets of the Sun can be made. For simplicity, we will consider orbits not elliptic, but circular. For planets Solar system This replacement is not too rough approximation.

Then the strength of attraction from the sun in this approximation should be directed for all planets to the center of the Sun.

If it means to designate the periods of circulation of planets, and through R radii their orbits, then, according to the third law of the Kepler, for two planets can be recorded

Normal acceleration when driving around the circle A \u003d Ω 2 R. Therefore, the ratio of accelerations of the planets

Using equation (3.2.4), we get

Since the third Capler's law is fair for all planets, then the acceleration of each planet is inversely proportional to the square of her distance to the Sun:

Permanent C 2 is the same for all planets, but does not coincide with a constant C 1 in the formula to accelerate, reported by the bodies of the globe.

Expressions (3.2.2) and (3.2.6) show that the force of gravity in both cases (attraction to the ground and attraction to the Sun) informs all bodies acceleration, independent of their mass and decreasing inversely in proportion to the square of the distance between them:

The Law of the World Health

The existence of dependencies (3.2.1) and (3.2.7) means that the power of world

In 1667, Newton finally formulated the world of world gravity:

The power of mutual attraction of two bodies is directly proportional to the mass of the masses of these bodies and is inversely proportional to the square of the distance between them. The ratio of the proportionality G is called gravitational (2) constant.

Interaction of point and extended bodies

The law of world community (3.2.8) is valid only for such bodies, the sizes of which are negligible compared to the distance between them. In other words, it is just for material dots. In this case, the forces of gravitational interaction are directed along the line connecting these points (Fig. 3.5). This kind of force is called central.

Fig. 3.5

To find the strength of the force acting on this body from the other, in the case when the sizes of bodies cannot neglect, they are applied as follows. Both bodies mentally separated on so small elements so that each of them can be considered point. Folding the forces acting on each element of this body by all elements of another body receive the force acting on this element (Fig. 3.6). Having done such an operation for each element of this body and folding the forces received, find full strength The body acting on this body. The task is complicated.

Fig. 3.6.

There are, however, one is almost an important case, when Formula (3.2.8) is applicable to extended bodies. It can be proved that spherical bodies of which depends only on distances to their centers, with distances between them, large sums of their radii, are attracted with the forces whose modules are determined by formula (3.2.8). In this case, R is the distance between the centers of the balls.

Finally, since the dimensions of the body falling on earth a lot less than the dimensions of the Earth, then these bodies can be considered as point. Then under R in formula (3.2.8), it is necessary to understand the distance from this body to the center of the Earth.

Questions for self-test

1. Distance from Mars to Sun by 52% more distances From the ground to the Sun. What is the duration of the year on Mars?
2. How will the attraction force change between the balls, if aluminum balls (Fig. 3.7) replace with steel balls of the same mass? of the same volume?

Fig. 3.7.

(1) I wonder what, being a student, Newton realized that the moon was moving under the influence of attraction to Earth. But at that time, the radius of the Earth was known inaccurately, and the calculations did not lead to the correct result. Only after 16 years later, new, corrected data appeared, and the law of global gravity was published.

(2) From the Latin word Gravitas - the severity.

The action of the world's strength in nature is explained by many phenomena: the movement of the planets in the solar system, artificial satellites of the Earth, the trajectory of the flight of ballistic missiles, the movement of the telephone near the surface of the Earth - they all find an explanation based on the law of world-wide and the laws of dynamics.

The law of global gravity explains the mechanical device of the solar system, and the laws of the Kepler, describing the trajectories of the motion of the planets, can be derived from it. For Kepler, his laws were purely descriptive in nature - the scientist simply summarized his observations in mathematical form, not submitting no theoretical grounds under the formula. In the Great System of the World Australia on Newton, the laws of Kepler become a direct consequence of the universal laws of mechanics and the law of world community. That is, we again observe how the empirical conclusions obtained at one level turn into strictly substantiated logic conclusions when moving to the next level of deepening of our knowledge about the world.

Newton first expressed the idea that gravitational forces define not only the movement of the planets of the solar system; They act between any bodies of the Universe. One of the manifestations of the world's strength is gravity - so it is customary to call the power of attraction to the ground near her surface.

If M is the mass of the Earth, RZ is its radius, M - the mass of this body, then the strength of gravity is equal

where G is the acceleration of free fall;

at the surface of the Earth

Gravity is directed towards the center of the Earth. In the absence of other forces, the body freely falls on the ground with an acceleration of free fall.

The average to accelerate the free fall for various points of the earth's surface is 9.81 m / s2. Knowing the acceleration of free fall and the radius of the Earth (RZ \u003d 6.38 · 106 m), one can calculate the mass of the earth

The picture of the device of the solar system arising from these equations and uniting earth and celestial gravity, can be understood on a simple example. Suppose we stand at the edge of the sheer cliff, next to the gun and a slide of cannon nuclei. If you just reset the kernel from the edge of the cliff vertically, it will start falling down and equally. Its movement will be described by Newton's laws for an equilibrium body movement with acceleration g. If you now release the kernel from the gun in the direction of the horizon, it will fly - and will fall on the arc. And in this case, its movement will be described by Newton's laws, only now they are applied to the body moving under the influence of gravity and having some kind of initial speed in the horizontal plane. Now, once over time, charging in the gun is increasingly heavy core and shooting, you will find that, since each next kernel flies out of the trunk with a greater initial speed, the kernel fall further and further from the foot of the rock.

Now imagine that we scored so much powder into the gun that the velocity of the nucleus is enough to fly around around the globe. If you neglect the air resistance, the kernel, wailing around the earth, will return to the starting point exactly at the same speed, with which it originally flew out of the gun. What will happen next, it is clear: the kernel will not stop on it and will continue to wink the circle around the planet.

In other words, we get artificial satelliteAround the Earth in orbit, like a natural satellite - the moon.

So in stages, we switched from the description of the movement of the body, falling solely under the influence of the "earthly" gravity (Newtonian apple), to the description of the movement of the satellite (moon) in orbit, without changing the nature of the gravitational impact with the "earth" on the "heavenly". This is the insight and allowed Newton to tie together the two forces of gravitational attraction in their nature.

When the earth is removed from the surface of the Earth and the acceleration of the free fall is changed inversely proportional to the square R distance R to the center of the Earth. An example of a system of two interacting bodies can serve as the Earth-Moon system. The moon is from the ground at a distance of RL \u003d 3.84 · 106 m. This distance is approximately 60 times higher than the radius of the Earth RZ. Consequently, the acceleration of the free fall of AL, due to the earthly attraction, in the orbit of the moon is

With such an acceleration directed towards the center of the Earth, the moon is moving in orbit. Consequently, this acceleration is a centripetal acceleration. It can be calculated by the kinematic formula for the centripetal acceleration

where t \u003d 27.3 days is the period of circulation of the moon around the Earth.

The coincidence of the results of calculations made in different ways, confirms the assumption of Newton on the unified nature of the force holding the moon in orbit, and gravity.

The self-gravitational field of the moon determines the acceleration of the free fall of the GL on its surface. Moon weight is 81 times less than the mass of the Earth, and its radius is approximately 3.7 times less than the radius of the Earth.

Therefore, the acceleration of the GL is determined by the expression

In the conditions of such a weak gravity, astronauts were found on the moon. A person in such conditions can perform gigantic jumps. For example, if a person jershits 1 m in earthly conditions, then on the moon he could jump at a height of more than 6 m.

Consider the question of artificial satellites of the Earth. Artificial satellites of the Earth are moving outside the earth's atmosphere, and they only have the forces on the part of the Earth.

Depending on the initial speed, the trajectory of the cosmic body may be different. Consider the event of the movement of an artificial satellite on a circular near-earth orbit. Such satellites fly at altitudes of about 200-300 km, and it is possible to approximately take the distance to the center of the Earth equal to its RZ radius. Then the centripetal acceleration of the satellite reported to him by the forces of gravity is approximately equal to the acceleration of the free fall g. Denote the satellite rate at an near-earth orbit through υ1 - this speed is called the first space rate. Using the kinematic formula for centripetal acceleration, we get

Moving at such speed, the satellite would take lump in the time

In fact, the period of satellite circular orbit near the surface of the earth slightly exceeds the specified value due to the differences between the radius of the real orbit and the radius of the Earth. The movement of the satellite can be viewed as a free drop, similar to the movement of shells or ballistic missiles. The difference lies only in the fact that the satellite speed is so high that the radius of curvature of its trajectory is equal to the radius of the Earth.

For satellites moving around the circular trajectories at a significant distance from the ground, the earth's attraction is weakening inversely proportional to the square of the R radius R pathor. Thus, in high orbits, the speed of movement of satellites is less than in an near-earth orbit.

The period of satellite circulation grows with an increase in the radius of the orbit. It is easy to calculate that with a radius R orbit, equal to about 6.6 Rz, the period of circulation of the satellite will be 24 hours. Satellite with such a period of treatment, launched in the equator plane, will be motionless to hang over some point of the earth's surface. Such satellites are used in Space Radio Systems. Orbit with radius R \u003d 6.6 RЗ is called geostationary.

The second space rate is called the minimum speed that the spacecraft should be informed at the surface of the Earth so that it, overcoming the earthly attraction, turned into an artificial satellite of the Sun (artificial planet). At the same time, the ship will be removed from the ground in a parabolic path.

Figure 5 illustrates cosmic speeds. If speed spacecraft equal to υ1 \u003d 7.9 · 103 m / s and directed parallel to the surface of the Earth, then the ship will move along a circular orbit on a small height above the ground. At initial speeds exceeding υ1, but smaller υ2 \u003d 11.2 · 103 m / s, the orbit of the ship will be elliptical. At the initial speed 2, the ship will move along Parabola, and with even greater initial speed - by hyperbola.

Space speeds

Speed \u200b\u200bnear the ground surface are indicated: 1) υ \u003d υ1 - the circular trajectory;

2) υ1< υ < υ2 – эллиптическая траектория; 3) υ = 11,1·103 м/с – сильно вытянутый эллипс;

4) υ \u003d υ2 - a parabolic trajectory; 5) υ\u003e υ2 - hyperbolic trajectory;

6) the trajectory of the moon

Thus, we found out that all movements in the solar system are subject to the law of Newton's world.

Based on the small mass of the planets and all the more other bodies of the solar system, it is possible to approximately believe that the movements in the near-free space are obeyed by the laws of the Kepler.

All bodies move around the Sun on elliptical orbits, in one of the focus of which the sun is located. The closer to the Sun, the heavenly body, the faster its speed of movement in orbit (Planet Pluto, the most distant from the famous, moves 6 times slower than the Earth).

Bodies can move on open orbits: parabole or hyperbola. This happens if the body's velocity is equal to or exceeds the value of the second cosmic velocity for the Sun on this distance from central Svetila. If we are talking about the satellite of the planet, then space speed It is necessary to calculate relative to the mass of the planet and the distance to its center.