The history of the discovery of the law of universal gravitation. The meaning of the law of universal gravitation Application of the law in the discovery of new planets

23.12.2020

Lesson 1(write the topic and purpose of the lesson in notebooks)

The law of universal gravitation. Acceleration of free fall on Earth and other planets

The purpose of the lesson:

To study the law of universal gravitation, to show its practical significance.

During the classes

I. new material(Make notes in notebooks)

The Danish astronomer Tycho Brahe, observing the motion of the planets for many years, accumulated numerous data, but was unable to process them. This was done by his student Johannes Kepler. Using the idea of Copernicus about the heliocentric system and the results of Tycho Brahe's observations, Kepler established the laws of planetary motion around the Sun. But Kepler failed to explain the dynamics of motion. Why do the planets revolve around the Sun according to such laws? Isaac Newton was able to answer this question using the laws of motion established by Kepler and the general laws of dynamics.

Newton suggested that a number of phenomena that seemed to have nothing in common (the fall of bodies to the Earth, the revolution of the planets around the Sun, the movement of the Moon around the Earth, the tides, etc.) are caused by one reason. After numerous calculations, Newton came to the conclusion that celestial bodies are attracted to each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Let us show how Newton came to this conclusion.

From the second law of "dynamics, it follows that the acceleration that a body receives under the action of a force is inversely proportional to the mass of the body. But the acceleration of free fall does not depend on the mass of the body. This is only possible if the force with which the Earth attracts the body changes proportionally body weight.

According to the third law, the forces with which bodies interact are equal. If the force acting on one body is proportional to the mass of this body, then the force equal to it acting on the second body is obviously proportional to the mass of the second body. But the forces acting on both bodies are equal, therefore, they are proportional to the mass of both the first and second bodies.

Newton calculated the ratio of the radius of the Moon's orbit to that of the Earth. The ratio was 60. And the ratio of the acceleration of free fall on the Earth to the centripetal acceleration with which the Moon revolves around the Earth was 3600. Therefore, the acceleration is inversely proportional to the square of the distance between the bodies.

But according to Newton's second law, the force and acceleration are directly related, therefore, the force is inversely proportional to the square of the distance between the bodies.

Isaac Newton discovered this law at the age of 23, but did not publish it for 9 years, since incorrect data on the distance between the Earth and the Moon did not confirm his idea. And only when this distance was specified, Newton in 1667 published the law of universal gravitation.

Force of gravitational interaction of two bodies (material points) with masses T 1 and T 2 is equal to:

where G is the gravitational constant, r- distance between bodies.

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

The gravitational constant was first measured by the English physicist G. Cavendish in 1788 using a device called a torsion balance. G. Cavendish fixed two small lead balls (5 cm in diameter and 775 g each) at opposite ends of a two-meter rod. The rod was suspended on a thin wire. Two large lead balls (20 cm in diameter and weighing 45.5 kg) were brought close to the small ones. The forces of attraction from the large balls forced the small ones to move, while the wire twisted. The degree of twist was a measure of the force acting between the balls. The experiment showed that the gravitational constant G = 6.66 1011 Nm2/kg2.

Limits of applicability of the law

The law of universal gravitation is applicable only for material points, i.e., for bodies whose dimensions are much smaller than the distances between them; spherical bodies; for a ball of large radius interacting with bodies whose dimensions are much smaller than the dimensions of the ball.

But the law is not applicable, for example, to the interaction of an infinite rod and a ball. In this case, the force of gravity is only inversely proportional to the distance, not the square of the distance. And the force of attraction between a body and an infinite plane does not depend on the distance at all.

Gravity

A special case of gravitational forces is the force of attraction of bodies to the Earth. This force is called gravity. In this case, the law of universal gravitation has the form:

where T- body weight [kg],

M- mass of the Earth [kg],

R- radius of the Earth [m],

h- height above the surface [m].

But gravity F T= mg, hence , and the acceleration of free fall .

on the surface of the earth ( h = 0) .

Free fall acceleration depends

♦ from the height above the Earth's surface;

♦ on the latitude of the area (the Earth is a non-inertial frame of reference);

♦ on rock density earth's crust;

♦ from the shape of the Earth (flattened at the poles).

In the above formula for g, the last three dependences are not taken into account. At the same time, we emphasize once again that the acceleration of free fall does not depend on the mass of the body.

Application of the law in the discovery of new planets

When the planet Uranus was discovered, its orbit was calculated on the basis of the law of universal gravitation. But the true orbit of the planet did not coincide with the calculated one. It was assumed that the perturbation of the orbit was caused by the presence of another planet located behind Uranus, which, with its gravitational force, changes its orbit. To find a new planet, it was necessary to solve a system of 12 differential equations with 10 unknowns. This task was carried out by an Ang - Yai student Adams; he sent the solution to the English Academy of Sciences. But there, no attention was paid to his work. And the French mathematician Le Verrier, having solved the problem, sent the result to the Italian astronomer Galle. And he, on the very first evening, pointing his pipe at the indicated point, discovered a new planet. She was given the name Neptune. In a similar way, in the 30s of the twentieth century, the 9th planet was discovered solar system- Pluto.

When asked about the nature of the forces of gravity, Newton replied: “I don’t know, but I don’t want to invent hypotheses.”

III. Exercises and questions for review (oral)

How is the law of universal gravitation formulated?

What is the formula for the law of universal gravitation for material points?

What is called the gravitational constant? What is her physical meaning? What is the meaning in SI?

What is a gravitational field?

Does the force of gravity depend on the properties of the environment in which the bodies are located?

Does the free fall acceleration depend on its mass?

Is the force of gravity the same various points the globe?

Explain the effect of the rotation of the Earth around its axis on the acceleration of free fall.

How does the acceleration of free fall change with distance from the Earth's surface?

Why doesn't the moon fall to earth? ( The moon revolves around the earth, held by the force of gravity. The moon does not fall to the Earth, because, having an initial speed, it moves by inertia. If the force of attraction of the Moon to the Earth ceases, the Moon will rush in a straight line into the abyss of outer space. Stop moving by inertia - and the moon would fall to the Earth. The fall would have lasted four days, nineteen hours, fifty-four minutes, seven seconds. This is how Newton calculated.)

IV. Problem solving (Written in notebooks with design!!!)

Task 1

At what distance is the force of attraction of two balls with masses of 1 g equal to 6.7 10-17 N?

Task 2

At what height from the surface of the earth did spaceship, if the instruments noted a decrease in the acceleration of free fall to 4.9 m/s2?

Task 3

The gravitational force between two balls is 0.0001 N. What is the mass of one of the balls if the distance between their centers is 1 m, and the mass of the other ball is 100 kg?

Homework

1. Learn §11;

2. Perform exercise 5.1-5.10 (orally), 5.11-5.5.20 (in writing in notebooks with registration);

3. Answer the micro test question:

The space rocket is moving away from the Earth. How will the gravitational force acting from the Earth on the rocket change with an increase in the distance to the center of the Earth by 3 times?

a) will increase by 3 times; b) will decrease by 3 times;

c) decrease by 9 times; d) will not change.

One of the clearest examples of 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 to the indicated place, discovered a new planet. They named her Neptune.

In the same way, on March 14, 1930, the planet Pluto was discovered. The discovery of Neptune, made, in the words of Engels, at the "tip of a pen", is the most convincing proof of the validity of Newton's law of universal gravitation.

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.

Determination of the mass of celestial bodies

Newton's law of universal gravitation allows one to measure one of the most important physical characteristics celestial body-- its mass.

The mass of a celestial body can be determined:

a) from measurements of gravity on the surface of a given body (gravimetric method);

b) according to the third (refined) Kepler's law;

c) from an analysis of the observed perturbations produced by a celestial body in the movements of other celestial bodies.

The first method is applicable so far only to the Earth, and is as follows.

Based on the law of gravity, the acceleration of gravity on the surface of the Earth is easily found from formula (1.3.2).

The acceleration of gravity g (more precisely, the acceleration of the gravity component due only to the force of attraction), as well as the radius of the Earth R, is determined from direct measurements on the surface of the Earth. The gravitational constant G is determined quite accurately from the experiments of Cavendish and Yolli, well known in physics.

With the currently accepted values of g, R and G, formula (1.3.2) yields the mass of the Earth. Knowing the mass of the Earth and its volume, it is easy to find the average density of the Earth. It is equal to 5.52 g / cm 3

The third, refined Kepler's law allows you to determine the relationship between the mass of the Sun and the mass of the planet, if the latter has at least one satellite and its distance from the planet and the period of revolution around it are known.

Indeed, the motion of the satellite around the planet obeys the same laws as the motion of the planet around the Sun and, therefore, the third Kepler equation can be written in this case as follows:

where M is the mass of the Sun, kg;

m is the mass of the planet, kg;

m c - satellite mass, kg;

T is the period of revolution of the planet around the Sun, s;

t c - period of revolution of the satellite around the planet, s;

a is the distance of the planet from the Sun, m;

and c is the distance of the satellite from the planet, m;

Dividing the numerator and denominator of the left side of the fraction of this equation pa m and solving it for the masses, we get

The ratio for all the planets is very great; the ratio, on the contrary, is small (except for the Earth and its satellite, the Moon) and can be neglected. Then in equation (2.2.2) there will be only one unknown relation, which is easily determined from it. For example, for Jupiter, the inverse ratio determined in this way is 1: 1050.

Since the mass of the Moon, the only satellite of the Earth, is quite large compared to the mass of the Earth, the ratio in equation (2.2.2) cannot be neglected. Therefore, to compare the mass of the Sun with the mass of the Earth, it is necessary to first determine the mass of the Moon. The exact determination of the mass of the Moon is a rather difficult task, and it is solved by analyzing those perturbations in the motion of the Earth, which are caused by the Moon.

Under the influence of lunar attraction, the Earth should describe an ellipse around the common center of mass of the Earth-Moon system within a month.

By precise determinations of the apparent positions of the Sun in its longitude, changes with a monthly period, called "lunar inequality", were discovered. The presence of “lunar inequality” in the apparent motion of the Sun indicates that the center of the Earth really describes a small ellipse during the month around the common center of mass “Earth - Moon”, located inside the Earth, at a distance of 4650 km from the center of the Earth. This made it possible to determine the ratio of the mass of the Moon to the mass of the Earth, which turned out to be equal. The position of the center of mass of the system "Earth - Moon" was also found from observations minor planet Eros in 1930-1931 These observations gave a value for the ratio of the masses of the Moon and the Earth. Finally, according to the perturbations in the movements artificial satellites Earth, the ratio of the masses of the Moon and the Earth turned out to be equal. The last value is the most accurate, and in 1964 the International Astronomical Union accepted it as the final one among other astronomical constants. This value was confirmed in 1966 by calculating the mass of the Moon from the orbital parameters of its artificial satellites.

With the known ratio of the masses of the Moon and the Earth, from equation (2.26) it turns out that the mass of the Sun M ? 333,000 times the mass of the Earth, i.e.

Mz \u003d 2 10 33 g.

Knowing the mass of the Sun and the ratio of this mass to the mass of any other planet that has a satellite, it is easy to determine the mass of this planet.

The masses of planets that do not have satellites (Mercury, Venus, Pluto) are determined from the analysis of the perturbations they produce in the motion of other planets or comets. So, for example, the masses of Venus and Mercury are determined by the perturbations that they cause in the motion of the Earth, Mars, some minor planets (asteroids) and the Encke-Backlund comet, as well as by the perturbations they produce on each other.

earth planet universe gravity

This article will focus on the history of the discovery of the law of universal gravitation. Here we will get acquainted with biographical information from the life of a scientist who discovered this physical dogma, we will consider its main provisions, the relationship with quantum gravity, the course of development, and much more.

Genius

Sir Isaac Newton is an English scientist. At one time, he devoted much attention and effort to such sciences as physics and mathematics, and also brought a lot of new things to mechanics and astronomy. He is rightfully considered one of the first founders of physics in its classical model. He is the author of the fundamental work "Mathematical Principles of Natural Philosophy", where he presented information about the three laws of mechanics and the law of universal gravitation. Isaac Newton laid the foundations with these works classical mechanics. He also developed an integral type, the light theory. He also made many contributions to physical optics and developed many other theories in physics and mathematics.

Law

The law of universal gravitation and the history of its discovery go far back in time. Its classical form is a law that describes the interaction of a gravitational type that does not go beyond the framework of mechanics.

Its essence was that the indicator of the force F of the gravitational pull arising between 2 bodies or points of matter m1 and m2, separated from each other by a certain distance r, is proportional to both mass indicators and is inversely proportional to the square of the distance between the bodies:

F = G, where by the symbol G we denote the gravitational constant equal to 6.67408(31).10 -11 m 3 /kgf 2.

Newton's gravity

Before considering the history of the discovery of the law of universal gravitation, let's take a closer look at its general characteristics.

In the theory created by Newton, all bodies with a large mass must generate a special field around them, which attracts other objects to itself. It's called the gravitational field, and it has potential.

A body with spherical symmetry forms a field outside of itself, similar to that which creates material point the same mass, located in the center of the body.

The direction of the trajectory of such a point in the gravitational field, created by a body with a much larger mass, obeys. Objects of the universe, such as, for example, a planet or a comet, also obey it, moving along an ellipse or hyperbola. Accounting for the distortion that other massive bodies create is taken into account using the provisions of the perturbation theory.

Analyzing Accuracy

After Newton discovered the law of universal gravitation, it had to be tested and proved many times over. For this, a number of calculations and observations were made. Having come to agreement with its provisions and proceeding from the accuracy of its indicator, the experimental form of estimation serves as a clear confirmation of GR. Measurement of the quadrupole interactions of a body that rotates, but its antennas remain stationary, show us that the process of increasing δ depends on the potential r - (1 + δ) , at a distance of several meters and is in the limit (2.1±6.2) .10 -3 . A number of other practical confirmations allowed this law to be established and take a single form, without any modifications. In 2007, this dogma was rechecked at a distance less than a centimeter (55 microns-9.59 mm). Taking into account the experimental errors, the scientists examined the distance range and found no obvious deviations in this law.

Observation of the Moon's orbit with respect to the Earth also confirmed its validity.

Euclidean space

Newton's classical theory of gravity is related to Euclidean space. The actual equality with a sufficiently high accuracy (10 -9) of the distance measures in the denominator of the equality discussed above shows us the Euclidean basis of the space of Newtonian mechanics, with a three-dimensional physical form. At such a point in matter, the area of a spherical surface is exactly proportional to the square of its radius.

Data from history

Consider summary history of the discovery of the law of universal gravitation.

Ideas were put forward by other scientists who lived before Newton. Epicurus, Kepler, Descartes, Roberval, Gassendi, Huygens and others visited reflections on it. Kepler put forward the assumption that the gravitational force is inversely proportional to the distance from the star of the Sun and has distribution only in the ecliptic planes; according to Descartes, it was a consequence of the activity of vortices in the thickness of the ether. There was a series of guesses that contained a reflection of the correct guesses about the dependence on distance.

A letter from Newton to Halley contained information that Hooke, Wren and Buyo Ismael were the predecessors of Sir Isaac himself. However, before him, no one succeeded clearly, with the help of mathematical methods, link the law of gravity and planetary motion.

The history of the discovery of the law of universal gravitation is closely connected with the work "Mathematical Principles of Natural Philosophy" (1687). In this work, Newton was able to derive the law in question thanks to Kepler's empirical law, which was already known by that time. He shows us that:

the form of movement of any visible planet testifies to the presence of a central force;
the attractive force of the central type forms elliptical or hyperbolic orbits.

About Newton's theory

Inspection brief history the discovery of the law of universal gravitation can also point us to a number of differences that distinguished it from the background of previous hypotheses. Newton was engaged not only in the publication of the proposed formula of the phenomenon under consideration, but also proposed a model of a mathematical type in a holistic form:

position on the law of gravity;
position on the law of motion;
systematics of methods of mathematical research.

This triad was able to investigate even the most complex movements of celestial objects to a fairly accurate extent, thus creating the basis for celestial mechanics. Up to the beginning of Einstein's activity in this model, the presence of a fundamental set of corrections was not required. Only mathematical apparatus had to be significantly improved.

Object for discussion

The discovered and proven law became, throughout the eighteenth century, a well-known subject of active controversy and scrupulous scrutiny. However, the century ended with a general agreement with his postulates and statements. Using the calculations of the law, it was possible to accurately determine the paths of the movement of bodies in heaven. A direct check was made in 1798. He did this using a torsion-type balance with great sensitivity. In the history of discovery world law gravity must be distinguished special place interpretations introduced by Poisson. He developed the concept of the potential of gravity and the Poisson equation, with which it was possible to calculate this potential. This type of model made it possible to study the gravitational field in the presence of an arbitrary distribution of matter.

There were many difficulties in Newton's theory. The main one could be considered the inexplicability of long-range action. There was no exact answer to the question of how attractive forces are sent through vacuum space at infinite speed.

"Evolution" of the law

Over the next two hundred years, and even more, attempts were made by many physicists to propose various ways to improve Newton's theory. These efforts ended in a triumph in 1915, namely the creation of the General Theory of Relativity, which was created by Einstein. He was able to overcome the whole set of difficulties. In accordance with the correspondence principle, Newton's theory turned out to be an approximation to the beginning of work on the theory in more general view, which can be used under certain conditions:

The potential of the gravitational nature cannot be too large in the systems under study. The solar system is an example of compliance with all the rules for the movement of celestial bodies. The relativistic phenomenon finds itself in a noticeable manifestation of the shift of the perihelion.
The indicator of the speed of movement in this group of systems is insignificant in comparison with the speed of light.

The proof that in a weak stationary field of gravitation GR calculations take the form of Newtonian ones is the presence of a scalar gravitational potential in a stationary field with weakly expressed force characteristics, which is able to satisfy the conditions of the Poisson equation.

Quantum Scale

However, in history, neither the scientific discovery of the law of universal gravitation, nor the General Theory of Relativity could serve as the final gravitational theory, since both do not adequately describe the processes of the gravitational type on the quantum scale. An attempt to create a quantum gravitational theory is one of the most important tasks of contemporary physics.

From the point of view of quantum gravity, the interaction between objects is created by the interchange of virtual gravitons. In accordance with the uncertainty principle, the energy potential of virtual gravitons is inversely proportional to the time interval in which it existed, from the point of emission by one object to the point in time at which it was absorbed by another point.

In view of this, it turns out that on a small scale of distances, the interaction of bodies entails the exchange of virtual type gravitons. Thanks to these considerations, it is possible to conclude the provision on the law of Newton's potential and its dependence in accordance with the reciprocal of proportionality with respect to distance. The analogy between the laws of Coulomb and Newton is explained by the fact that the weight of gravitons is equal to zero. The weight of photons has the same meaning.

Delusion

V school curriculum The answer to the question from the story of how Newton discovered the law of universal gravitation is the story of the falling apple fruit. According to this legend, it fell on the head of a scientist. However, this is a widespread misconception, and in fact, everything was able to do without a similar case of a possible head injury. Newton himself sometimes confirmed this myth, but in reality the law was not a spontaneous discovery and did not come in a burst of momentary insight. As it was written above, it was developed for a long time and was presented for the first time in the works on the "Principles of Mathematics", which appeared on public display in 1687.

2.1 Discovery of Neptune

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.

Astronomy

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Gravity measurements

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solar system

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The purpose of the lesson:

create conditions for the formation of cognitive interest, activity of students;
derive the law of universal gravitation;
promote the development of convergent thinking;
promote aesthetic education of students;
formation of communication communication;

Equipment: interactive complex SMART Board Notebook.

Lesson teaching method: in the form of a conversation.

Lesson Plan

Class organization
Frontal survey
Learning new material
Anchoring
Fixing homework

The purpose of the lesson- learn to model the conditions of the problem and master various ways to solve them.

1 slide title

2-6 slide - how the law of universal gravitation was discovered

The Danish astronomer Tycho Brahe (1546-1601), who observed the motion of the planets for many years, accumulated a huge amount of interesting data, but failed to process them.

Johannes Kepler (1571-1630), using the idea of Copernicus about the heliocentric system and the results of Tycho Brahe's observations, established the laws of planetary motion around the Sun, however, he could not explain the dynamics of this motion .

Isaac Newton discovered this law at the age of 23, but did not publish it for 9 years, since the then incorrect data on the distance between the Earth and the Moon did not confirm his idea. Only in 1667, after clarifying this distance, law of gravity was finally published.

Casting a single mind's eye on the "earthly" and "heavenly", Newton suggested that there is a single law of universal gravitation, which is subject to all bodies in the universe - from apples to planets!

In 1667, Newton suggested that forces of mutual attraction act between all bodies, which he called the forces of universal gravitation.

Isaac Newton is an English physicist and mathematician, the creator of the theoretical foundations of mechanics and astronomy. He discovered the law of universal gravitation, developed differential and integral calculus, invented the mirror telescope, and was the author of the most important experimental work in optics. Newton is rightfully considered the creator of "classical physics".

7-8 slide - the law of gravity

In 1687, Newton established one of the fundamental laws of mechanics, called the law of universal gravitation: “Any two bodies are attracted to each other with a force whose modulus is directly proportional to the product of their masses and inversely proportional to the square of the distance between them”

where m 1 and m 2 are the masses of interacting bodies, r is the distance between the bodies, G is the coefficient of proportionality, which is the same for all bodies in nature and is called the universal gravitational constant or the gravitational constant.

9 slide - Remember

Gravitational interaction is an interaction inherent in all bodies of the Universe and manifested in their mutual attraction to each other.
A gravitational field is a special kind of matter that performs gravitational interaction.

10 slide - the mechanism of gravitational interaction

At present, the mechanism of gravitational interaction is represented as follows: Each body with a mass M creates a field around itself, which is called gravitational. If a test body with mass is placed at some point of this field T, then the gravitational field acts on this body with a force F, depending on the properties of the field at this point and on the mass of the test body.

11 slide - Henry Cavendish's experiment to determine the gravitational constant.

The English physicist Henry Cavendish determined how strong the force of attraction between two objects is. As a result, the gravitational constant was determined quite accurately, which allowed Cavendish to determine the mass of the Earth for the first time.

12 slide - gravitational constant

G is the gravitational constant, it is numerically equal to the force of gravitational attraction of two bodies, each weighing 1 kg. Each located at a distance of 1 m from one another.

G is the universal gravitational constant

G \u003d 6.67 * 10 -11 N m 2 / kg 2

The force of mutual attraction is always directed along the straight line connecting the bodies.

13 slide - the limits of applicability of the law

The law of universal gravitation has certain limits of applicability; it is applicable for:

1) material points;

2) bodies having the shape of a ball;

3) a ball of large radius interacting with bodies whose dimensions are much smaller than the dimensions of the ball.

The law is not applicable, for example, to the interaction of an infinite rod and a ball.

The gravitational force is very small and becomes noticeable only when at least one of the interacting bodies has a very large mass (planet, star).

14 slide - why do we not notice the gravitational attraction between the bodies around us?

Let's use the law of universal gravitation and do some calculations:

Two ships weighing 50,000 tons each are in the roadstead at a distance of 1 km from each other. What is the force of attraction between them?

15 slide - task

It is known that the period of revolution of the Moon around the Earth is 27.3 days, the average distance between the centers of the Moon and the Earth is 384,000 kilometers. Calculate the acceleration of the Moon and find how many times it differs from the acceleration of free fall of a stone near the surface of the Earth, that is, at a distance equal to the radius Earth (6400 kilometers).

16 slide - derivation of the law

On the other hand, the ratio of the distances from the Moon and the stone to the center of the Earth is:

It is easy to see that

17 slide - directly proportional dependence

It follows from Newton's second law that there is a directly proportional relationship between a force and the acceleration it causes:

Therefore, the gravitational force, as well as acceleration, is inversely proportional to the square of the distance between the body and the center of the Earth:

18-19 slide - directly proportional dependence

Galileo Galilei experimentally proved that all bodies fall to the Earth with the same acceleration, called free fall acceleration(experiment with the fall of various bodies in a tube with evacuated air)

Why is this acceleration the same for all bodies?

This is possible only if the force of gravity is proportional to the mass of the body: F

m . Indeed, then, for example, an increase or decrease in mass by a factor of two will cause a corresponding change in the gravitational force by a factor of two, but the acceleration according to Newton's second law will remain the same

On the other hand, two bodies always participate in the interaction, each of which, according to Newton's third law, is affected by forces of the same modulus:

Therefore, the gravitational force must be proportional to the mass of both bodies.

So Newton came to the conclusion that the gravitational force between the body and the Earth is directly proportional to the product of their masses:

20 slide - the results of the lesson

Summarizing all the above regarding the gravitational force of the planet Earth and any body, we come to the following statement: the gravitational force between the body and the Earth is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers, which can be written as

Does this law hold only for the Earth or is it universal?

To answer this question, Newton used the kinematic laws of motion of the planets in the solar system, formulated by the German scientist Johannes Kepler on the basis of many years of astronomical observations by the Danish scientist Tycho Brahe.

21-22 slide - Think and answer

Why doesn't the moon fall to earth?
Why do we notice the force of attraction of all bodies to the Earth, but do not notice the mutual attraction between these bodies themselves?
How would the planets move if the sun's gravity suddenly disappeared?
How would the moon move if it stopped in orbit?
Does a person standing on its surface attract the Earth? Flying plane? An astronaut on an orbital station?

Some bodies (balloons, smoke, airplanes, birds) rise up despite gravity. Why do you think? Is there a violation of the law of universal gravitation here?

What should be done to increase the force of gravity between two bodies?

What force causes the ebb and flow in the seas and oceans of the Earth?

Why do we not notice the gravitational attraction between the bodies around us?

23 slide - Question-answer

Make up questions and then give an answer to figures 1-4.

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Presentation "Discovery and application of the law of universal gravitation"

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Razumov Viktor Nikolaevich,

teacher MOU "Bolsheyelkhovskaya secondary school"

Lyambirsky municipal district Republic of Mordovia

Law of gravity

All bodies in the universe 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.

where m1 and m2 are the masses of the bodies;

r is the distance between the bodies;

The discovery of the law of universal gravitation was largely facilitated by

Kepler's laws of planetary motion

and other achievements of astronomy of the XVII century.

Knowing the distance to the Moon allowed Isaac Newton to prove the identity of the force that holds the Moon as it moves around the Earth, and the force that causes bodies to fall to the Earth.

Since the force of gravity varies inversely with the square of the distance, as follows from the law of universal gravitation, the Moon,

located at a distance of about 60 of its radii from the Earth,

should experience an acceleration 3600 times smaller,

than the acceleration of gravity on the surface of the Earth, equal to 9.8 m/s.

Therefore, the acceleration of the Moon must be 0.0027 m/s2.

At the same time, the Moon, like any body moving uniformly in a circle, has an acceleration

where ? - her angular velocity, r is the radius of its orbit.

then the radius of the lunar orbit will be

r= 60 6 400 000 m = 3.84 10 m.

Sidereal period of the moon T= 27.32 days,

in seconds is 2.36 10 s.

Then the acceleration of the moon's orbital motion

The equality of these two values of acceleration proves that the force holding the Moon in orbit is the force of the earth's attraction, weakened by 3600 times compared to that acting on the surface of the Earth.

Isaac Newton (1643–1727)

When the planets move, in accordance with Kepler's third law, their acceleration and the force of attraction of the Sun acting on them are inversely proportional to the square of the distance, as follows from the law of universal gravitation.

Indeed, according to Kepler's third law, the ratio of the cubes of the semi-major axes of the orbits d and squares of circulation periods T is a constant value:

So, the force of interaction between the planets and the Sun satisfies the law of universal gravitation.

The acceleration of the planet is

From Kepler's third law it follows

so the acceleration of the planet is

Perturbations in the motions of the bodies of the solar system

The motion of the planets of the solar system does not exactly obey Kepler's laws due to their interaction not only with the Sun, but also with each other.

Deviations of bodies from moving along ellipses are called perturbations.

The perturbations are small, since the mass of the Sun is much greater than the mass of not only an individual planet, but all the planets as a whole.

Particularly noticeable are the deviations of asteroids and comets during their passage near Jupiter, whose mass is 300 times the mass of the Earth.

In the 19th century the calculation of perturbations made it possible to discover the planet Neptune.

William Herschel in 1781 discovered the planet Uranus.

Even when perturbations from all known planets were taken into account, the observed motion of Uranus did not agree with the calculated one.

Based on the assumption of the presence of another "transuranium" planet John Adams in England and Urbain Le Verrier in France independently made calculations of its orbit and position in the sky.

Based on Le Verrier's calculations, a German astronomer Johann Galle On September 23, 1846, he discovered a previously unknown planet in the constellation Aquarius - Neptune.

According to the perturbations of Uranus and Neptune, it was predicted, and in 1930 it was discovered dwarf planet Pluto.

The discovery of Neptune was a triumph for the heliocentric system,

the most important confirmation of the validity of the law of universal gravitation.

Mass and density of the Earth

In accordance with the law of universal gravitation, the acceleration of free fall:

Knowing the mass and volume of the globe, we can calculate its average density:

With depth, due to an increase in pressure and the content of heavy elements, the density increases

The law of universal gravitation made it possible to determine the mass of the Earth.

Determination of the mass of celestial bodies

A more accurate formula of Kepler's third law, which was obtained by Newton, makes it possible to determine the mass of a celestial body.

Angular velocity of revolution around the center of mass:

Centripetal accelerations of bodies:

Let two mutually attracting bodies circulate in a circular orbit with a period T around a common center of mass. Distance between their centers R = r1 + r2.

The right side of the expression contains only constant values, therefore it is valid for any system of two bodies interacting according to the law of gravity and revolving around a common center of mass - the Sun and the planet, the planet and the satellite.

Equating the expressions obtained for the accelerations, expressing from them r1 and r1 and adding them term by term, we get:

Based on the law of universal gravitation, the acceleration of each of these bodies is:

Neglecting the mass of the Earth, which is negligible compared to the mass of the Sun, and the mass of the Moon, which is 81 times less than the mass of the Earth, we get:

Substituting the appropriate values into the formula and taking the mass of the Earth as a unit, we get that the Sun is 333 thousand times larger in mass than the Earth.

Let's determine the mass of the Sun from the expression:

where M is the mass of the Sun; and are the masses of the Earth and the Moon;

and is the period of revolution of the Earth around the Sun (year) and

the major semiaxis of its orbit; and - circulation period

Moon around the Earth and the semi-major axis of the lunar orbit.

The masses of planets that do not have satellites are determined by the perturbations they have on the motion of asteroids, comets, or spacecraft flying in their vicinity.

Under the influence of mutual attraction of particles, the body tends to take the form of a ball. If these bodies rotate, they are deformed, compressed along the axis of rotation.

In addition, a change in their shape also occurs under the action of mutual attraction, which is caused by phenomena called tides.

The gravity of the Sun also causes tides, but due to its greater remoteness, they are smaller than those caused by the Moon.

Between the huge masses of water involved in tidal phenomena, and the ocean floor arises tidal friction.

Tidal friction slows down the rotation of the Earth and causes an increase in the length of the day, which in the past were much shorter (5–6 hours).

The same effect accelerates the Moon's orbital motion and causes it to slowly move away from the Earth.

The tides caused by the Earth on the Moon have slowed down its rotation, and it is now facing the Earth on one side.

Why doesn't the planets move exactly according to Kepler's laws?
How was the location of the planet Neptune determined?
Which of the planets causes the greatest perturbations in the motion of other bodies in the solar system and why?
Which bodies in the solar system experience the greatest perturbations and why?

2) Exercise 12 (p. 80)

1. Determine the mass of Jupiter, knowing that its satellite, which is 422,000 km away from Jupiter, has a period of revolution of 1.77 days.

For comparison, use data for the Earth system–Moon.

Law of gravity

Presentation for the lesson: "The law of universal gravitation."

Development content

KVVK on the topic "The law of universal gravitation"

1. The history of the discovery of the law of universal gravitation.

2. How to prove that the force of gravity is proportional to the mass of the body?

3. How to prove that the force of gravity is proportional to the mass of both interacting bodies?

4. How to prove that the force of gravity is inversely proportional to the square of the distance between the bodies?

5. The law of universal gravitation. mathematical expression. Formulation.

6. How was the value of the gravitational constant measured?

7. The value of the gravitational constant. Unit in SI.

8. Limits of applicability of the law of universal gravitation.

9. Discovery of the planets using the law of universal gravitation.

10. What is gravity? How is it different from gravity?

11. Two formulas for calculating gravity.

12. How is free fall acceleration measured? What does it equal?

13. What does the free fall acceleration depend on and what does it not depend on?

14. Center of gravity. What is the center of gravity of plane figures?

15. How to measure body weight?

16. How to measure the mass of the earth?

On the way to discovery

Polish astronomer, mathematician, mechanic,

The first thought belonged to the English scientist Gilbert. He suggested that the planets of the solar system are giant magnets, so the forces that bind them are of a magnetic nature.

24.05. 1544 — 30.11.1603

Rene Descartes suggested that the Universe is filled with whirlwinds of thin invisible matter. These whirlwinds drag the planets into a "circular revolution around the Sun. Each planet has its own vortex. Planets are similar to light bodies that have fallen into water funnels. The hypotheses of Hilbert and Descartes were based on analogy and had no experimental support.

31.03. 1596 — 11.02. 1650

Debate between Descartes (right) and Queen Christina, by Pierre-Louis Dumesnil

The history of the discovery of the law of universal gravitation.

Danish astronomer, astrologer and alchemist of the Renaissance. The first in Europe began to conduct systematic and highly accurate astronomical observations .

(27.12. 1571 - 15.11. 1630)

German mathematician, astronomer, mechanic, optician, discoverer laws of planetary motion solar system.

Kepler's first law(1609):

All planets move in elliptical orbits with the Sun at one of the foci.

Kepler's second law(1609):

the radius vector of the planet describes equal areas in equal time intervals.

Kepler's third law(1618):

the squares of the periods of the planets are related as the cubes of the semi-major axes of their orbits:

The law of inertia: the movement of a body that is not acted upon by external forces or their resultant is zero is uniform movement around the circumference

15. 02. 1564 - 08. 01. 1642

I shall present a system of the world, which differs in many particulars from all hitherto known systems, but which agrees in all respects with the ordinary mechanical laws.

28. 07. 1635 - 03. 03. 1703

Attractive forces act the more, the closer the body on which they act to the center of attraction.

Kepler's third law: the squares of the periods of the planets are related as the cubes of the semi-major axes of their orbits.

08. 11. 1656 - 25. 01. 1742

Falling bodies to the ground

moon around the earth

Planets around the sun

Ebb and flow

How to prove that the force of gravity is proportional to the mass of the body?

1) From Newton's second law

How to prove that the force of gravity is proportional to the mass of both interacting bodies?

2) According to Newton's third law

How to prove that the force of gravity is inversely proportional to the square of the distance between the bodies?

The law of universal gravitation. mathematical expression.

Law of gravity:

All bodies are attracted to each other with a force that is directly proportional to the mass of each of them and inversely proportional to the square of the distance between them.

How was the value of the gravitational constant measured?

The value of the gravitational constant. Unit in SI.

G - gravitational constant

10. 10. 1731 - 24. 02. 1810

Limits of applicability of the law of universal gravitation.

Discovery of the planets using the law of universal gravitation.

The difference between these forces is much smaller than either of them, and, therefore, they can be considered approximately equal.

What is gravity? How is it different from gravity? Two formulas for calculating gravity.

The difference between these forces is much smaller than each of them, and, therefore, they can be considered approximately equal.

Measuring free fall acceleration? What does it equal?

What does the free fall acceleration depend on and what does it not depend on?

1) from the height above the Earth

2) from the latitude of the place (Earth is a non-inertial frame of reference)

3) from the rocks of the earth's crust (gravitometry)

4) from the shape of the Earth, flattened at the poles (pole - 9.83 m / s 2, 9.78 m / s 2 - equator)

Hooray. I became 0.7 N lighter!

geometric point, invariably associated with solid body, through which passes the resultant of all gravity forces acting on the particles of this body at any position of the latter in space; it may not coincide with any of the points of a given body (for example, near a ring). If a free body is hung on threads attached sequentially to different points of the body, then the directions of these threads will intersect at the center of the body.

Center of gravity. What is the center of gravity of plane figures?

Center of gravity geometric point, invariably associated with a solid body, through which the resultant of all gravity forces acting on particles passes

this body at any position of the latter in space;

it may not coincide with any of the points of a given body (for example, near a ring). If a free body is hung on threads attached in series to different

points of the body, then the directions of these threads will intersect at the center of gravity of the body.

How to measure body weight? How to measure the mass of the earth?

Problem solution example

1. At what distance from the surface of the Earth is the free fall acceleration equal to 1 m / s 2? The radius of the Earth is 6400 km, the free fall acceleration at the Earth's surface is 9.8 m/s 2 .

Gravity is the force with which a body is attracted to the Earth due to the law of universal gravitation:

m - body mass, M - mass of the Earth,

In the condition of the problem, the mass of the Earth is not given. It can be found in the following way. The force of gravity of a body on the surface of the Earth (h = 0) can also be written as the force of gravity:

Examples of test tasks:

1. Between two celestial bodies of the same mass, located at a distance r from each other, there is an attractive force of magnitude F one . If the distance between the bodies is doubled, how will this force change?

2. The figure shows four pairs of spherically symmetrical bodies located relative to each other at different distances between the centers of these bodies.

Force of interaction of two bodies of identical masses M located at a distance R from each other, is F 0 . For which pair of bodies the force of gravitational interaction is 4 F 0 ?

§ § 15 - 16 (teach, retell, answer KVVK),

Law of gravity (page 1 of 3)

Almost everything in the solar system revolves around the sun. Some planets have satellites, but they, making their way around the planet, move along with it around the Sun. The Sun has a mass exceeding the mass of the entire population of the solar system by 750 times. Because of this, the Sun causes the planets and everything else to move in orbits around it. On a cosmic scale, mass is main characteristic bodies, because all celestial bodies obey the law of universal gravitation.

Based on the laws of planetary motion established by I. Kepler, the great English scientist Isaac Newton (1643-1727), at that time recognized by no one else, discovered the law of universal gravitation, with the help of which it was possible to calculate with great accuracy for that time the movement of the Moon, planets and comets, explain the ebb and flow of the ocean.

Man uses these laws not only for a deeper knowledge of nature (for example, to determine the masses of celestial bodies), but also to solve practical tasks(cosmonautics, astrodynamics).

The work consists of an introduction, the main part, a conclusion and a list of references.

To fully appreciate the brilliance of the discovery of the Law of Gravity, let's return to its background. There is a legend that while walking through the apple orchard on the estate of his parents, Newton saw the moon in the daytime sky, and right before his eyes an apple broke off from a branch and fell to the ground. Since Newton was working on the laws of motion at the same time, he already knew that the apple fell under the influence of the Earth's gravitational field. He also knew that the Moon does not just hang in the sky, but rotates in an orbit around the Earth, and, therefore, some kind of force acts on it, which keeps it from breaking out of orbit and flying away in a straight line, into outer space. Then it occurred to him that perhaps this is the same force that causes both the apple to fall to the earth and the moon to remain in orbit around the earth - the force of gravity that exists between all bodies.

The very idea of the universal force of gravity has been repeatedly expressed before: Epicurus, Gassendi, Kepler, Borelli, Descartes, Roberval, Huygens and others thought about it. Descartes considered it the result of vortices in the ether. The history of science shows that almost all the arguments concerning the motion of celestial bodies before Newton boiled down mainly to the fact that celestial bodies, being perfect, move in circular orbits due to their perfection, since the circle is an ideal geometric figure.

140). In the center of the universe, Ptolemy placed the Earth, around which the planets and stars moved in large and small circles, as in a round dance. The geocentric system of Ptolemy lasted more than 14 centuries and was replaced by the heliocentric system of Copernicus only in the middle of the 16th century.

At the beginning of the 17th century, on the basis of the Copernican system, the German astronomer I. Kepler formulated three empirical laws of the motion of the planets of the solar system, using the results of observations of the motion of the planets by the Danish astronomer T. Brahe.

Kepler's First Law (1609): "All the planets move in elliptical orbits with the Sun at one of the foci."

The elongation of the ellipse depends on the speed of the planet; the distance the planet is from the center of the ellipse. A change in the speed of a celestial body leads to the transformation of an elliptical orbit into a hyperbolic one, moving along which you can leave the solar system.

Figure 1 - Elliptical orbit of a planet with mass

m <

Almost all the planets of the solar system (except Pluto) move in orbits close to circular.

Kepler's second law (1609): "The radius vector of a planet describes equal areas in equal time intervals" (Fig. 2).

Figure 2 - The law of areas - the second law of Kepler

Kepler's second law shows the equality of the areas described by the radius vector of a celestial body in equal time intervals. In this case, the speed of the body varies depending on the distance to the Earth (this is especially noticeable if the body moves along a highly elongated elliptical orbit). The closer the body to the planet, the greater the speed of the body.

At R=a, the periods of circulation of bodies in these orbits are the same

Kepler's laws, which forever became the basis of theoretical astronomy, were explained in the mechanics of I. Newton, in particular in the law of universal gravitation.

Despite the fact that Kepler's laws were the most important stage in understanding the motion of the planets, they still remained only empirical rules obtained from astronomical observations; Kepler failed to find the reason that determines these patterns common to all planets. Kepler's laws needed a theoretical justification.

This is precisely what Newton's ideas differed from the conjectures of other scientists. Before Newton, no one was able to clearly and mathematically conclusively link the law of gravity (a force inversely proportional to the square of distance) and the laws of planetary motion (Kepler's laws).

The two greatest scientists, far ahead of their time, created a science called celestial mechanics, discovered the laws of motion of celestial bodies under the influence of gravitational forces, and even if their achievements were limited to this, they would still be included in the pantheon of the greats of this world.

But Newton tested his law of gravitation with Kepler's laws. All three of Kepler's laws are consequences of the law of gravity. And Newton discovered it. The results of Newtonian calculations are now called Newton's law of universal gravitation, which we will consider in the next chapter.

2 Law of gravity

Topic: Law of gravity

1 The laws of planetary motion - Kepler's laws

2 Law of gravity

2.1 Discovery of Isaac Newton

2.2 Movement of bodies under the influence of gravity

3 artificial earth satellites

Bibliography

Man, studying phenomena, comprehends their essence and discovers the laws of nature. Thus, a body raised above the Earth and left to itself will begin to fall. It changes its speed, therefore, it is affected by gravity. This phenomenon is observed everywhere on our planet: the Earth attracts all bodies to itself, including us. Is it only the Earth that has the property of acting on all bodies by the force of attraction?

The purpose of the work: to study the law of universal gravitation, to show its practical significance, to reveal the concept of the interaction of bodies using this law as an example.

1 The laws of planetary motion - Kepler's laws

So, when the great predecessors of Newton studied the uniformly accelerated motion of bodies falling on the surface of the Earth, they were sure that they were observing a phenomenon of a purely terrestrial nature - existing only not far from the surface of our planet. When other scientists, studying the movement of celestial bodies, believed that completely different laws of motion operate in the celestial spheres than the laws governing motion here on Earth.

Thus, in modern terms, it was believed that there are two types of gravity, and this idea was firmly entrenched in the minds of people of that time. Everyone believed that there is terrestrial gravity, acting on the imperfect Earth, and there is celestial gravity, acting on the perfect heavens. The study of the motion of the planets and the structure of the solar system led, ultimately, to the creation of the theory of gravity - the discovery of the law of universal gravitation.

The first attempt to create a model of the universe was made by Ptolemy (

On fig. 1 shows the elliptical orbit of the planet, the mass of which is much less than the mass of the Sun. The sun is at one of the foci of the ellipse. The point P of the trajectory closest to the Sun is called perihelion, point A, the most distant from the Sun, is called aphelion. The distance between aphelion and perihelion is the major axis of the ellipse.

m<

Kepler's third law (1619): "The squares of the periods of revolution of the planets are related as the cubes of the semi-major axes of their orbits":

Kepler's third law holds for all planets in the solar system with an accuracy better than 1%.

Figure 3 shows two orbits, one of which is circular with radius R, and the other is elliptical with semi-major axis a. The third law states that if R=a, then the periods of circulation of bodies in these orbits are the same.

Figure 3 - Circular and elliptical orbits

And only Newton made a private, but very important conclusion: there must be a connection between the centripetal acceleration of the Moon and the acceleration of free fall on the Earth. This relationship had to be established numerically and verified.

It so happened that they did not intersect in time. Only thirteen years after Kepler's death, Newton was born. Both of them were supporters of the heliocentric system of Copernicus.

Having studied the motion of Mars for many years, Kepler experimentally discovers three laws of planetary motion, more than fifty years before Newton's discovery of the law of universal gravitation. Still not understanding why the planets move this way and not otherwise. It was a brilliant vision.

2.1 Discovery of Isaac Newton

The law of universal gravitation was discovered by I. Newton in 1682. According to his hypothesis, attractive forces (gravitational forces) act between all bodies of the Universe, directed along the line connecting the centers of mass (Fig. 4). For a body in the form of a homogeneous ball, the center of mass coincides with the center of the ball.

Figure 4 - Gravitational forces of attraction between bodies,

In subsequent years, Newton tried to find a physical explanation for the laws of planetary motion discovered by I. Kepler at the beginning of the 17th century, and to give a quantitative expression for gravitational forces. So, knowing how the planets move, Newton wanted to determine what forces act on them. This path is called the inverse problem of mechanics.

If the main task of mechanics is to determine the coordinates of a body of known mass and its velocity at any moment of time from known forces acting on the body and given initial conditions (the direct problem of mechanics), then when solving the inverse problem, it is necessary to determine the forces acting on the body, if it is known how it moves.

The solution of this problem led Newton to the discovery of the law of universal gravitation: "All bodies are attracted to each other with a force directly proportional to their masses and inversely proportional to the square of the distance between them." Like all physical laws, it takes the form of a mathematical equation

The coefficient of proportionality G is the same for all bodies in nature. It's called the gravitational constant.

G = 6.67 10–11 N m2/kg2 (SI)

There are several important remarks to be made about this law.

Firstly, its action explicitly extends to all physical material bodies in the Universe without exception. In particular, for example, you and a book experience equal in magnitude and opposite in direction forces of mutual gravitational attraction. Of course, these forces are so small that even the most accurate of modern instruments cannot detect them, but they really exist and can be calculated.

In the same way, you experience mutual attraction with a distant quasar, tens of billions of light years away. Again, the forces of this attraction are too small to be instrumentally registered and measured.

The second point is that the force of gravity of the Earth at its surface equally affects all material bodies located anywhere on the globe. Right now, we are affected by the force of gravity, calculated according to the above formula, and we really feel it as our own weight. If we drop something, it, under the influence of the same force, will rush to the ground with uniform acceleration.

2.2 Movement of bodies under the influence of gravity

Many phenomena are explained by the action of forces of universal gravitation in nature: the movement of the planets in the solar system, artificial satellites of the Earth, the flight paths of ballistic missiles, the movement of bodies near the surface of the Earth - all of them are explained on the basis of the law of universal gravitation and the laws of dynamics.

The law of universal gravitation explains the mechanical structure of the solar system, and Kepler's laws describing the trajectories of the planets can be derived from it. For Kepler, his laws were purely descriptive - the scientist simply generalized his observations in mathematical form, without subsuming any theoretical foundations under the formulas. In the great system of the world order according to Newton, Kepler's laws become a direct consequence of the universal laws of mechanics and the law of universal gravitation. That is, we again observe how empirical conclusions obtained at one level turn into strictly substantiated logical conclusions when moving to the next step in deepening our knowledge of the world.

Newton was the first to suggest that gravitational forces determine 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 force of universal gravitation is the force of gravity - this is how it is customary to call the force of attraction of bodies to the Earth near its surface.

If M is the mass of the Earth, RЗ is its radius, m is the mass of the given body, then the force of gravity is equal to

where g is the free fall acceleration;

at the surface of the earth

The force of gravity is directed towards the center of the earth. In the absence of other forces, the body falls freely to the Earth with free fall acceleration.

The average value of the gravitational acceleration for various points on the Earth's surface is 9.81 m/s2. Knowing the acceleration of free fall and the radius of the Earth (RЗ = 6.38 106 m), we can calculate the mass of the Earth

The picture of the structure of the solar system, which follows from these equations and combines terrestrial and celestial gravity, can be understood by a simple example. Suppose we are standing at the edge of a sheer cliff, next to a cannon and a hill of cannonballs. If you simply drop the core from the edge of the cliff vertically, it will begin to fall down vertically and with uniform acceleration. Its motion will be described by Newton's laws for uniformly accelerated motion of a body with acceleration g. If you now release the core from the cannon in the direction of the horizon, it will fly - and will fall in an arc. And in this case, its movement will be described by Newton's laws, only now they are applied to a body moving under the influence of gravity and having a certain initial speed in a horizontal plane. Now, as you repeatedly load a heavier cannonball into the cannon and fire it, you will find that as each successive cannonball leaves the barrel at a higher initial velocity, the cannonballs fall farther and farther away from the base of the cliff.

Now let's imagine that we stuffed so much gunpowder into the cannon that the speed of the cannonball is enough to fly around the globe. Neglecting air resistance, the cannonball, having flown around the Earth, will return to its starting point at exactly the same speed with which it initially flew out of the cannon. What will happen next is clear: the core will not stop there and will continue to wind circle after circle around the planet.

In other words, we will get an artificial satellite orbiting the Earth, like a natural satellite - the Moon.

So, step by step, we moved from describing the motion of a body falling solely under the influence of “earthly” gravity (Newtonian apple) to describing the motion of a satellite (Moon) in orbit, without changing the nature of the gravitational influence from “earthly” to “heavenly”. It was this insight that allowed Newton to link together the two forces of gravitational attraction that were considered different in nature before him.

When moving away from the surface of the Earth, the force of gravity and the acceleration of free fall change inversely with the square of the distance r to the center of the Earth. An example of a system of two interacting bodies is the Earth–Moon system. The Moon is located at a distance rL = 3.84 106 m from the Earth. This distance is approximately 60 times greater than the radius of the Earth RЗ. Consequently, the acceleration of free fall aL, due to the Earth's gravity, in the orbit of the Moon is

With such an acceleration directed towards the center of the Earth, the Moon moves in an orbit. Therefore, this acceleration is centripetal acceleration. It can be calculated from the kinematic formula for centripetal acceleration

where T = 27.3 days is the period of revolution of the Moon around the Earth.

The coincidence of the results of calculations performed by different methods confirms Newton's assumption about the unified nature of the force holding the Moon in orbit and the force of gravity.

The Moon's own gravitational field determines the free fall acceleration gL on its surface. The mass of the Moon 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 gL is determined by the expression

Astronauts who landed on the moon found themselves in conditions of such weak gravity. A person in such conditions can make giant jumps. For example, if a person on Earth jumps to a height of 1 m, then on the Moon he could jump to a height of more than 6 m.

Consider the question of artificial earth satellites. Artificial Earth satellites move outside the Earth's atmosphere, and only gravitational forces from the Earth act on them.

Depending on the initial speed, the trajectory of a space body can be different. Consider the case of an artificial satellite moving in a circular near-Earth orbit. Such satellites fly at altitudes of the order of 200–300 km, and the distance to the center of the Earth can be approximately taken to be equal to its radius R3. Then the satellite's centripetal acceleration imparted to it by gravitational forces is approximately equal to the gravitational acceleration g. We denote the speed of the satellite in near-Earth orbit as υ1 - this speed is called the first cosmic speed. Using the kinematic formula for centripetal acceleration, we get

Moving at this speed, the satellite would circle the Earth in time

In fact, the period of revolution of the satellite in a circular orbit near the Earth's surface is somewhat larger than the specified value due to the difference between the radius of the real orbit and the radius of the Earth. The motion of a satellite can be thought of as a free fall, similar to the motion of projectiles or ballistic missiles. The only difference is that the speed of the satellite is so great that the radius of curvature of its trajectory is equal to the radius of the Earth.

For satellites moving along circular trajectories at a considerable distance from the Earth, the Earth's gravity weakens inversely with the square of the radius r of the trajectory. Thus, in high orbits, the speed of movement of satellites is less than in near-Earth orbit.

The orbital period of a satellite increases with increasing orbital radius. It is easy to calculate that with an orbit radius r equal to approximately 6.6 R3, the period of revolution of the satellite will be equal to 24 hours. A satellite with such a period of revolution, launched in the plane of the equator, will hang motionless over a certain point on the earth's surface. Such satellites are used in space radio communication systems. An orbit with radius r = 6.6 R3 is called geostationary.

The second cosmic speed is the minimum speed that needs to be reported to the spacecraft near the surface of the Earth, so that, having overcome the earth's gravity, it turns into an artificial satellite of the Sun (artificial planet). In this case, the ship will move away from the Earth along a parabolic trajectory.

Figure 5 illustrates space velocities. If the speed of the spacecraft is υ1 = 7.9 103 m/s and is directed parallel to the Earth's surface, then the spacecraft will move in a circular orbit at a low altitude above the Earth. At initial velocities exceeding υ1 but less than υ2 = 11.2 103 m/s, the ship's orbit will be elliptical. At an initial speed of υ2, the ship will move along a parabola, and at an even higher initial speed, along a hyperbola.

Figure 5 - Cosmic velocities

Velocities near the Earth's surface are indicated: 1) υ = υ1 – circular trajectory;

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

4) υ = υ2 is a parabolic trajectory; 5) υ > υ2 is a hyperbolic trajectory;

6) the trajectory of the moon

Thus, we found out that all movements in the solar system obey Newton's law of universal gravitation.

Based on the small mass of the planets, and even more so of other bodies of the solar system, we can approximately assume that the movements in the near-solar space obey Kepler's laws.

All bodies move around the Sun in elliptical orbits, in one of the focuses of which is the Sun. The closer a celestial body is to the Sun, the faster its orbital speed (the planet Pluto, the most distant known, moves 6 times slower than the Earth).

Bodies can also move along open orbits: parabola or hyperbola. This happens if the speed of the body is equal to or exceeds the value of the second cosmic velocity for the Sun at a given distance from the central luminary. If we are talking about a satellite of the planet, then the cosmic velocity must be calculated relative to the mass of the planet and the distance to its center.

3 Artificial earth satellites

On February 12, 1961, the automatic interplanetary station "Venera-1" went beyond the limits of gravity