By what law are you going to hang me?
- And we hang everyone according to one law - the law Gravity.

Law of gravity

The phenomenon of gravity is the law of universal gravitation. Two bodies act on each other with a force that is inversely proportional to the square of the distance between them and directly proportional to the product of their masses.

Mathematically, we can express this great law by the formula

Gravity acts over vast distances in the universe. But Newton argued that all objects are mutually attracted. Is it true that any two objects attract each other? Just imagine, it is known that the Earth attracts you sitting on a chair. But have you ever thought about the fact that a computer and a mouse attract each other? Or a pencil and pen on the table? In this case, we substitute the mass of the pen, the mass of the pencil into the formula, divide by the square of the distance between them, taking into account the gravitational constant, we obtain the force of their mutual attraction. But, it will come out so small (due to the small masses of the pen and pencil) that we do not feel its presence. Another thing is when we are talking about the Earth and the chair, or the Sun and the Earth. The masses are significant, which means that we can already evaluate the effect of force.

Let's think about free fall acceleration. This is the operation of the law of attraction. Under the action of a force, the body changes speed the slower, the greater the mass. As a result, all bodies fall to the Earth with the same acceleration.

What is the cause of this invisible unique power? To date, the existence of a gravitational field is known and proven. You can learn more about the nature of the gravitational field in additional material Topics.

Think about what gravity is. Where is it from? What does it represent? After all, it cannot be that the planet looks at the Sun, sees how far it is removed, calculates the inverse square of the distance in accordance with this law?

Direction of gravity

There are two bodies, let's say body A and B. Body A attracts body B. The force with which body A acts begins on body B and is directed towards body A. That is, it "takes" body B and pulls it towards itself. Body B "does" the same thing with body A.

Every body is attracted by the earth. The earth "takes" the body and pulls it towards its center. Therefore, this force will always be directed vertically downwards, and it is applied from the center of gravity of the body, it is called gravity.

The main thing to remember

Some methods of geological exploration, tide prediction and in recent times movement calculation artificial satellites and interplanetary stations. Early calculation of the position of the planets.

Can we set up such an experiment ourselves, and not guess whether planets, objects are attracted?

Such a direct experience made Cavendish (Henry Cavendish (1731-1810) - English physicist and chemist) using the device shown in the figure. The idea was to hang a rod with two balls on a very thin quartz thread and then bring two large lead balls to the side of them. The attraction of the balls will twist the thread slightly - slightly, because the forces of attraction between ordinary objects are very weak. With the help of such an instrument, Cavendish was able to directly measure the force, distance and magnitude of both masses and, thus, determine gravitational constant G.

The unique discovery of the gravitational constant G, which characterizes the gravitational field in space, made it possible to determine the mass of the Earth, the Sun and other celestial bodies. Therefore, Cavendish called his experience "weighing the Earth."

Interestingly, the various laws of physics have some common features. Let's turn to the laws of electricity (Coulomb force). Electric forces are also inversely proportional to the square of the distance, but already between the charges, and the thought involuntarily arises that this pattern hides deep meaning. Until now, no one has been able to represent gravity and electricity as two different manifestations the same entity.

The force here also varies inversely with the square of the distance, but the difference in the magnitude of electric forces and gravitational forces is striking. In trying to establish the common nature of gravity and electricity, we find such a superiority of electric forces over gravitational forces that it is difficult to believe that both have the same source. How can you say that one is stronger than the other? After all, it all depends on what is the mass and what is the charge. Arguing about how strong gravity acts, you have no right to say: "Let's take a mass of such and such a size," because you choose it yourself. But if we take what Nature herself offers us (her eigenvalues and measures that have nothing to do with our inches, years, our measures), then we can compare. We will take an elementary charged particle, such as, for example, an electron. Two elementary particles, two electrons, due to electric charge repel each other with a force inversely proportional to the square of the distance between them, and due to gravity attract each other again with a force inversely proportional to the square of the distance.

Question: What is the ratio of gravity to electrical force? Gravitation is related to electrical repulsion as one is to a number with 42 zeros. This is deeply puzzling. Where could such a huge number come from?

People are looking for this huge factor in other natural phenomena. They go through all sorts big numbers and if you need big number why not take, say, the ratio of the diameter of the Universe to the diameter of a proton - surprisingly, this is also a number with 42 zeros. And they say: maybe this coefficient is equal to the ratio of the diameter of the proton to the diameter of the universe? This is an interesting thought, but as the universe gradually expands, the constant of gravity must also change. Although this hypothesis has not yet been refuted, we do not have any evidence in its favor. On the contrary, some evidence suggests that the constant of gravity did not change in this way. This huge number remains a mystery to this day.

Einstein had to modify the laws of gravity in accordance with the principles of relativity. The first of these principles says that the distance x cannot be overcome instantly, while according to Newton's theory, forces act instantly. Einstein had to change Newton's laws. These changes, refinements are very small. One of them is this: since light has energy, energy is equivalent to mass, and all masses attract, light also attracts and, therefore, passing by the Sun, must be deflected. This is how it actually happens. The force of gravity is also slightly modified in Einstein's theory. But this very slight change in the law of gravity is just enough to explain some of the apparent irregularities in Mercury's motion.

Physical phenomena in the microcosm are subject to other laws than phenomena in the world of large scales. The question arises: how does gravity manifest itself in a world of small scales? The quantum theory of gravity will answer it. But quantum theory there is no gravity yet. People have not yet been very successful in creating a theory of gravity that is fully consistent with quantum mechanical principles and with the uncertainty principle.

DEFINITION

The law of universal gravitation was discovered by I. Newton:

Two bodies are attracted to each other with , which is directly proportional to their product and inversely proportional to the square of the distance between them:

Description of the law of gravity

The coefficient is the gravitational constant. In the SI system, the gravitational constant has the value:

This constant, as can be seen, is very small, so the gravitational forces between bodies with small masses are also small and practically not felt. However, the movement space bodies completely determined by gravity. The presence of universal gravitation or, in other words, gravitational interaction explains what the Earth and planets “hold” on, and why they move around the Sun along certain trajectories, and do not fly away from it. The law of universal gravitation allows us to determine many characteristics of celestial bodies - the masses of planets, stars, galaxies and even black holes. This law allows you to calculate the orbits of the planets with great accuracy and create mathematical model Universe.

With the help of the law of universal gravitation, it is also possible to calculate cosmic velocities. For example, the minimum speed at which a body moving horizontally above the Earth's surface will not fall on it, but will move in a circular orbit is 7.9 km/s (the first space velocity). In order to leave the Earth, i.e. to overcome its gravitational attraction, the body must have a speed of 11.2 km / s, (the second cosmic velocity).

Gravity is one of the most amazing natural phenomena. In the absence of gravitational forces, the existence of the Universe would be impossible, the Universe could not even arise. Gravity is responsible for many processes in the Universe - its birth, the existence of order instead of chaos. The nature of gravity is still not fully understood. To date, no one has been able to develop a worthy mechanism and model of gravitational interaction.

Gravity

A special case of the manifestation of gravitational forces is gravity.

Gravity is always directed vertically downward (toward the center of the Earth).

If the force of gravity acts on the body, then the body performs. The type of movement depends on the direction and module of the initial speed.

We deal with the force of gravity every day. , after a while it is on the ground. The book, released from the hands, falls down. Having jumped, a person does not fly into outer space and descends to the ground.

Considering the free fall of a body near the Earth's surface as a result of the gravitational interaction of this body with the Earth, we can write:

whence the acceleration free fall:

The free fall acceleration does not depend on the mass of the body, but depends on the height of the body above the Earth. Earth slightly flattened at the poles, so the bodies near the poles are located a little closer to the center of the Earth. In this regard, the acceleration of free fall depends on the latitude of the area: at the pole it is slightly greater than at the equator and other latitudes (at the equator m / s, at the North Pole equator m / s.

The same formula allows you to find the free fall acceleration on the surface of any planet with mass and radius .

Examples of problem solving

EXAMPLE 1 (the problem of "weighing" the Earth)

Exercise	The radius of the Earth is km, the acceleration of free fall on the surface of the planet is m/s. Using these data, estimate the approximate mass of the Earth.
Solution	Acceleration of free fall at the surface of the Earth: whence the mass of the Earth: In the C system, the radius of the Earth m. Substituting numerical values ​​into the formula physical quantities Let's estimate the mass of the Earth:
Answer	Mass of the Earth kg.

EXAMPLE 2

Exercise

An Earth satellite moves in a circular orbit at an altitude of 1000 km from the Earth's surface. How fast is the satellite moving? How long will it take the satellite to make one full turn around the Earth?

Solution

According to , the force acting on the satellite from the side of the Earth is equal to the product of the mass of the satellite and the acceleration with which it moves: From the side of the earth, the force of gravitational attraction acts on the satellite, which, according to the law of universal gravitation, is equal to: where and are the masses of the satellite and the Earth, respectively. Since the satellite is at a certain height above the surface of the Earth, the distance from it to the center of the Earth: where is the radius of the earth.

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 natural philosophy”: “A stone thrown horizontally will deflect
, \\
1
/ /
At
Rice. 3.2
under the influence of gravity rectilinear 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 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 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 tells the Moon centripetal acceleration. It is calculated by the formula
l 2
a \u003d - Tg
where B is the radius lunar orbit, equal to about 60 radii of the Earth, T \u003d 27 days 7 h 43 min \u003d 2.4 106 s - 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 reported by gravity, and consequently, the very force of attraction 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. Consequently, 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 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 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?

In 1667. Newton understood that in order for the Moon to revolve around the Earth, and the Earth and other planets around the Sun, there must be a force that keeps them in a circular orbit. He suggested that the force of gravity acting on all bodies on Earth and the force that keeps the planets in their circular orbits are one and the same force. This force is called gravitational force or gravitational force. This force is the force of attraction and acts between all bodies. Newton formulated law of gravity : two material points are attracted to each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

The proportionality factor G was unknown in Newton's time. It was first experimentally measured by the English scientist Cavendish. This ratio is called gravitational constant. Her contemporary meaning equals . The gravitational constant is one of the most fundamental physical constants. The law of universal gravitation can be written in vector form. If the force acting on the second point from the first is equal to F 21, and the radius vector of the second point relative to the first is equal to R21, then:

The presented form of the law of universal gravitation is valid only for the gravitational interaction of material points. It cannot be used for bodies of arbitrary shape and size. Calculation of the gravitational force in general case is a very difficult task. However, there are bodies that are not material points, for which gravitational force can be calculated according to the above formula. These are bodies that have spherical symmetry, for example, having the shape of a ball. For such bodies, the above law is valid if the distance R is understood as the distance between the centers of the bodies. In particular, the force of gravity acting on all bodies from the side of the Earth can be calculated using this formula, since the Earth has the shape of a ball, and all other bodies can be considered material points compared to the radius of the Earth.

Since gravity is gravitational force, then we can write that the force of gravity acting on a body of mass m is equal to

Where МЗ and RЗ are the mass and radius of the Earth. On the other hand, the force of gravity is equal to mg, where g is the acceleration due to gravity. So the free fall acceleration is

This is the formula for the free fall acceleration on the Earth's surface. If you move away from the surface of the Earth, then the distance to the center of the Earth will increase, and the acceleration of gravity will decrease accordingly. So at a height h above the Earth's surface, the free fall acceleration is:

The force of gravity

Newton discovered the laws of motion of bodies. According to these laws, movement with acceleration is possible only under the action of a force. Since falling bodies move with acceleration, they must be subjected to a force directed downward towards the Earth. Is it only the Earth that has the property of attracting bodies that are near its surface to itself? In 1667, Newton suggested that, in general, forces of mutual attraction act between all bodies. He called these forces the forces of universal gravitation.

Why do we not notice the mutual attraction between the bodies around us? Perhaps this is due to the fact that the forces of attraction between them are too small?

Newton managed to show that the force of attraction between bodies depends on the masses of both bodies and, as it turned out, reaches a noticeable value only when the interacting bodies (or at least one of them) have a sufficiently large mass.

"HOLES" IN SPACE AND TIME

Black holes are the product of gigantic gravitational forces. They arise when, in the course of a strong compression of a large mass of matter, its increasing gravitational field becomes so strong that it does not even let out light, nothing can come out of a black hole at all. You can only fall into it under the influence of huge gravitational forces, but there is no way out. modern science revealed the connection between time and physical processes, called to "probe" the first links of the chain of time in the past and follow its properties in the distant future.

The role of the masses of attracting bodies

The acceleration of free fall is distinguished by the curious feature that it is the same in a given place for all bodies, for bodies of any mass. How to explain this strange property?

The only explanation that can be found for the fact that the acceleration does not depend on the mass of the body is that the force F with which the Earth attracts the body is proportional to its mass m.

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 force F also by a factor of two, while the acceleration, which is equal to the ratio F/m, will remain unchanged. Newton made this only correct conclusion: the force of universal gravitation is proportional to the mass of the body on which it acts.

But after all, bodies are attracted mutually, and the forces of interaction are always of the same nature. Consequently, the force with which the body attracts the Earth is proportional to the mass of the Earth. According to Newton's third law, these forces are equal in absolute value. Hence, if one of them is proportional to the mass of the Earth, then the other force equal to it is also proportional to the mass of the Earth. From here it follows that the force of mutual attraction is proportional to the masses of both interacting bodies. And this means that it is proportional to the product of the masses of both bodies.

WHY IS GRAVITY IN SPACE NOT THE SAME AS ON EARTH?

Every object in the universe acts on another object, they attract each other. The force of attraction, or gravity, depends on two factors.

Firstly, it depends on how much substance the object, body, object contains. The greater the mass of a body, the stronger gravity. If a body has very little mass, its gravity is small. For example, the mass of the Earth is many times greater than the mass of the Moon, so the earth has a greater gravitational force than the Moon.

Secondly, the force of gravity depends on the distances between the bodies. The closer the bodies are to each other, the greater the force of attraction. The farther they are from each other, the less gravity.