About Attractions - Earthly and Lunar.

13. Movement of celestial bodies under the influence of gravitational forces

1. Cosmic velocities and shape of orbits

Based on observations of the motion of the Moon and analyzing the laws of planetary motion discovered by Kepler, I. Newton (1643-1727) established the law gravity. According to this law, as you already know from the physics course, all bodies in the Universe 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:

here m 1 and m 2 are the masses of two bodies, r is the distance between them, and G is the coefficient of proportionality, called the gravitational constant. Its numerical value depends on the units in which force, mass and distance are expressed. The law of universal gravitation explains the movement of planets and comets around the Sun, the movement of satellites around planets, binary and multiple stars around their common center of mass.

Newton proved that under the influence of mutual gravitation, bodies can move relative to each other along ellipse(particularly for circle), on parabola and by hyperbole. Newton found that the type of orbit that a body describes depends on its speed at a given point in the orbit(Fig. 34).

At some speed the body describes circle near the center of attraction. This speed is called the first cosmic or circular speed, it is reported to bodies launched as artificial satellites of the Earth in circular orbits. (The derivation of the formula for calculating the first cosmic velocity is known from the course of physics.) First space velocity near the Earth's surface is about 8 km/s (7.9 km/s).

If the body is given a speed that is twice the circular speed (11.2 km / s), called the second cosmic or parabolic speed, then the body will forever move away from the Earth and can become a satellite of the Sun. In this case, the movement of the body will occur along parabola relative to the earth. At an even greater speed relative to the Earth, the body will fly along a hyperbola. Moving along a parabola or hyperbole, the body goes around the Sun only once and forever moves away from it.

The average speed of the Earth's orbit is 30 km/s. The Earth's orbit is close to a circle, therefore, the speed of the Earth's movement along the orbit is close to circular at the distance of the Earth from the Sun. parabolic speed at the distance of the Earth from the Sun is km/s≈42 km/s. At such a speed relative to the Sun, the body will leave the Earth's orbit solar system.

2. Disturbances in the motion of the planets

Kepler's laws are exactly observed only when we consider the motion of two isolated bodies under the influence of their mutual attraction. There are many planets in the solar system, all of them are not only attracted by the Sun, but also attract each other, so their movements do not exactly obey Kepler's laws.

Deviations from motion that would occur strictly according to Kepler's laws are called perturbations. In the solar system, perturbations are small, because the attraction of each planet by the Sun is much stronger than the attraction of other planets.

The biggest perturbation in the solar system is caused by the planet Jupiter, which is about 300 times more massive than the Earth. Jupiter has a special strong influence On the movement of asteroids and comets when they come close to it. In particular, if the directions of the comet's accelerations caused by the attraction of Jupiter and the Sun coincide, then the comet can develop such a high speed that, moving along a hyperbola, it will leave the solar system forever. There were cases when the attraction of Jupiter held back the comet, the eccentricity of its orbit became smaller and the period of revolution sharply decreased.

When calculating the apparent position of the planets, perturbations must be taken into account. Now high-speed electronic computers help to make such calculations. When running artificial celestial bodies and when calculating their trajectories, they use the theory of motion of celestial bodies, in particular, the theory of perturbations.

The ability to send automatic interplanetary stations along the desired, pre-calculated trajectories, to bring them to the goal, taking into account disturbances in movement - all these are vivid examples of the cognizability of the laws of nature. The sky, which according to the believers is the abode of the gods, has become an arena human activity just like the earth. Religion has always contrasted the Earth and the sky and declared the sky inaccessible. Now, among the planets, artificial celestial bodies are moving, created by man, which he can control by radio from great distances.

3. Discovery of Neptune

One of clear examples achievements of science, one of the evidence of the unlimited cognizability of nature was the discovery of the planet Neptune by calculations - "on the tip of a pen."

Uranus - the planet following Saturn, which for many centuries was considered the most distant of the planets, was discovered by V. Herschel at the end of the 18th century. Uranus is hardly visible to the naked eye. By the 40s of the XIX century. accurate observations have shown that Uranus deviates just barely from the path it should follow, given the perturbations from all the known planets. Thus the theory of motion of celestial bodies, so rigorous and precise, was put to the test.

Le Verrier (in France) and Adams (in England) suggested that if perturbations from the known planets do not explain the deviation in the motion of Uranus, it means that the attraction of an as yet unknown body acts on it. They almost simultaneously calculated where behind Uranus there should be an unknown body that produces these deviations by its attraction. They calculated the orbit of an unknown planet, its mass and indicated the place in the sky where in given time there must have been an unknown planet. This planet was found in a telescope at the place indicated by them in 1846. It was called Neptune. Neptune is not visible to the naked eye. Thus, the disagreement between theory and practice, which seemed to undermine the authority of materialistic science, led to its triumph.

4. Tides

Under the influence of mutual attraction of particles, the body tends to take the shape of a ball. The shape of the Sun, planets, their satellites and stars is therefore close to spherical. Rotation of bodies (as you know from physical experiments) leads to their flattening, to compression along the axis of rotation. Therefore, the globe is slightly compressed at the poles, and the rapidly rotating Jupiter and Saturn are most compressed.

But the shape of the planets can also change from the action of the forces of their mutual attraction. A spherical body (planet) moves as a whole under the influence of the gravitational attraction of another body as if the entire force of attraction were applied to its center. However, individual parts of the planet are located at different distances from the attracting body, so the gravitational acceleration in them is also different, which leads to the emergence of forces that tend to deform the planet. The difference in accelerations caused by the attraction of another body at a given point and in the center of the planet is called tidal acceleration.

Consider, for example, the Earth-Moon system. The same element of mass in the center of the Earth will be attracted by the Moon weaker than on the side facing the Moon, and stronger than on the opposite side. As a result, the Earth, and first of all water shell Earth, slightly extended in both directions along the line connecting it with the Moon. In Figure 35, the ocean is depicted covering the entire Earth for clarity. At points lying on the line Earth - Moon, the water level is highest - there are tides. Along the circle, the plane of which is perpendicular to the direction of the Earth-Moon line and passes through the center of the Earth, the water level is the lowest - there is a low tide. At daily rotation Earth in the tide band alternately enter different places on the Earth. It is easy to understand that there can be two high tides and two low tides in a day.

The sun also causes ebbs and flows on Earth, but due to the great distance of the Sun, they are smaller than the moon and less noticeable.

The tides move a huge amount of water. At present, they are beginning to use the enormous energy of water, which participates in the tides, on the shores of the oceans and open seas.

The axis of the tidal protrusions must always be directed towards the Moon. As the Earth rotates, it tends to turn the water tidal bulge. Since the Earth rotates around its axis much faster than the Moon revolves around the Earth, the Moon pulls the water hump towards itself. There is friction between the water and the solid bottom of the ocean. As a result, the so-called tidal friction. It slows down the rotation of the Earth, and the days become longer over time (once they were only 5-6 hours). The strong tides caused on Mercury and Venus by the Sun, apparently, were the reason for their extremely slow rotation around their axis. The tides caused by the Earth have slowed down the rotation of the Moon so much that it always faces the Earth on one side. So the tides are an important factor evolution of celestial bodies and the Earth.

5. Mass and density of the Earth

The law of universal gravitation also allows us to determine one of the the most important characteristics celestial bodies - the mass, in particular the mass of our planet. Indeed, based on the law of universal gravitation, the acceleration free fall

Therefore, if the values ​​of the acceleration of free fall, the gravitational constant and the radius of the Earth are known, then its mass can be determined.

Substituting in the specified formula value g \u003d 9.8 m / s 2, G \u003d 6.67 * 10 -11 N * m 2 / kg 2, R \u003d 6370 km, we find that the mass of the Earth is M \u003d 6 * 10 24 kg.

Knowing the mass and volume of the Earth, you can calculate it average density. It is equal to 5.5 * 10 3 kg / m 3. But the density of the Earth increases with depth, and, according to calculations, near the center, in the core of the Earth, it is equal to 1.1*10 4 kg/m 3 . The increase in density with depth occurs due to an increase in the content heavy elements as well as by increasing the pressure.

(WITH internal structure Earth, studied by astronomical and geophysical methods, you met in the course of physical geography.)

Exercise 12

1. What is the density of the Moon if its mass is 81 times, and the radius is 4 times less than that of the Earth?

2. What is the mass of the Earth if angular velocity The moon is 13.2° per day, and the average distance to it is 380,000 km?

6. Determination of the masses of celestial bodies

Newton proved that a more precise formula for Kepler's third law is:

where M 1 and M 2 are the masses of any celestial bodies, and m 1 and m 2 are the masses of their satellites, respectively. Thus, the planets are considered satellites of the Sun. We see that the refined formula of this law differs from the approximate one by the presence of a factor containing masses. If under M 1 \u003d M 2 \u003d M we understand the mass of the Sun, and under m 1 and m 2 - the masses of two different planets, then the ratio will differ little from unity, since m 1 and m 2 are very small compared to the mass of the Sun. In this case, the exact formula will not noticeably differ from the approximate one.

To compare the masses of the Earth and another planet, for example Jupiter, in the original formula, the index 1 must be attributed to the movement of the Moon around the Earth with a mass M 1, and 2 - to the movement of any satellite around Jupiter with a mass M 2.

The masses of planets that do not have satellites are determined by the perturbations that they produce by their attraction in the motion of their neighboring planets, as well as in the motion of comets, asteroids or spacecraft.

Exercise 13

1. Determine the mass of Jupiter by comparing the Jupiter system with a satellite with the Earth-Moon system, if the first satellite of Jupiter is 422,000 km away from it and has an orbital period of 1.77 days. The data for the moon should be known to you.

2. Calculate at what distance from the Earth on the Earth-Moon line are those points at which the attraction of the Earth and the Moon are the same, knowing that the distance between the Moon and the Earth is 60 radii of the Earth, and the mass of the Earth is 81 times the mass of the Moon.

If the Earth did not attract the Moon, then the latter would fly into world space in the direction of the point BUT. But due to the attraction of the Earth, the Moon deviates from rectilinear path and moves along some arc towards the point B.

not only the movement of the Moon, but also the movement of all celestial bodies in the solar system.

This research proceeded with Newton not entirely smoothly. Since the planets are gigantic spherical bodies, it was very difficult to determine how they are attracted to each other. In the end, Newton was able to prove that spherical bodies attract each other as if all their mass was concentrated at their centers.

But in order to find the ratio of distances from the center the globe to the bodies on earth's surface, and to the Moon, it was required to know exactly the length of the Earth's radius. The dimensions of the Earth were not yet precisely determined, and for his calculations, Newton used the inaccurate, as it turned out later, value of the radius of the globe given by the Dutch scientist Snellius. Having received an incorrect result, Newton bitterly postponed this work.

Many years later, the scientist again returned to his calculations. The reason for this was a message in the Royal Society of London 1 famous French astronomer Picard about more exact definition them the magnitude of the earth's radius. Using data

Picard, Newton did all the work again and proved the correctness of his assumption.

But even after that, Newton did not publish his outstanding discovery for a long time. He tried to verify it comprehensively, applying the law he derived to the motion of the planets around the Sun and to the motion of the satellites of Jupiter and Saturn. And everywhere the data of these observations coincided with the theory.

Newton applied this law to the motion of comets and proved that parabolic motions are theoretically possible. He suggested that comets move either along very elongated ellipses or along open curves - parabolas.

Based on the law of gravitation, Newton compared the masses of the Sun, Earth and planets and supplemented this law with a new provision: the gravitational force of two bodies depends not only on the distance between them, but also on their masses. He proved that the gravitational force of two bodies is directly proportional to their masses, i.e., it is the greater, the greater the mass of mutually attracting bodies.

Earthly bodies also mutually attract each other. This is revealed in very precise experiments.

People are attracted to each other. It is known that two people, separated by one meter, are mutually attracted with a force equal to approximately one fortieth of a milligram. The person who is

Comets move in orbits shaped like ellipses, parabolas and hyperbolas.

on the surface of the Earth, attracts it with a force equal to its weight.

Newton's discovery led to the creation of a new picture of the world, namely: planets move at tremendous speeds in the solar system, they are located at colossal distances from each other.

1 London Royal Society- English Academy of Sciences.

Student . The story is widely known that the discovery of Newton's law of universal gravitation was caused by the fall of an apple from a tree. How reliable this story is, we do not know, but it remains a fact that the question that we have gathered today to discuss: "Why does the moon not fall to the Earth?" interested Newton and led him to the discovery of the law of gravity. Newton argued that between the Earth and all material bodies there is a gravitational force, which is inversely proportional to the square of the distance.

Newton calculated the acceleration imparted to the Moon by the Earth. The acceleration of freely falling bodies near the Earth's surface is equal to g=9.8 m/s 2 . The Moon is removed from the Earth at a distance equal to about 60 Earth radii. Therefore, Newton reasoned, the acceleration at this distance will be: . The Moon, falling with such an acceleration, should approach the Earth in the first second by 0.0013 m. But the Moon, in addition, moves by inertia in the direction instantaneous speed, i.e., along a straight line, tangent at a given point to its orbit around the Earth (Fig. 25).

Moving by inertia, the Moon should move away from the Earth, as the calculation shows, in one second by 1.3 mm. Of course, such a motion, in which in the first second the Moon would move along the radius to the center of the Earth, and in the second second - tangentially, does not really exist. Both movements add up continuously. As a result, the Moon moves along a curved line close to a circle.

Let us conduct an experiment from which it is clear how the force of attraction acting on a body at right angles to the direction of its motion transforms rectilinear motion into curvilinear. A ball, having rolled down from an inclined chute, by inertia continues to move in a straight line. If, however, a magnet is placed on the side, then under the influence of the force of attraction to the magnet, the trajectory of the ball is curved (Fig. 26).

The moon revolves around the earth, held by the force of gravity. A steel rope that could keep the moon in orbit would have to have a diameter of about 600 km. But, despite such a huge force of attraction, the Moon does not fall to the Earth, because, having initial speed, moves by inertia.

Knowing the distance from the Earth to the Moon and the number of revolutions of the Moon around the Earth, Newton determined the centripetal acceleration of the Moon. We got a number already known to us: 0.0027 m/s2.
Stop the force of attraction of the Moon to the Earth - and the Moon will rush in a straight line into the abyss of outer space. So in the device shown in Figure 27, the ball will fly away tangentially if the thread holding the ball on the circle breaks. In the device you know on a centrifugal machine (Fig. 28), only the connection (thread) keeps the balls in a circular orbit.

When the thread breaks, the balls scatter along the tangents. It is difficult to catch their rectilinear movement with the eye when they are devoid of connection, but if we make a drawing (Fig. 29), it will be seen that the balls move in a rectilinear manner, tangentially to the circle.

Stop moving by inertia - and the moon would fall to the Earth. The fall would have lasted four days, nineteen hours, fifty-four minutes, fifty-seven seconds, Newton calculated.

The teacher present at the class. The report is over. Who has questions?

Question . With what force does the earth pull the moon?

Student . This can be determined by the formula expressing the law of gravity: , where G is the gravitational constant, M and m are the masses of the Earth and the Moon, r is the distance between them. I expected this question and did the calculation beforehand. The earth pulls the moon with a force of about 2 * 10 20 N.

Question . The law of universal gravitation applies to all bodies, which means that the Sun also attracts the Moon. I wonder with what strength?

Answer . The mass of the Sun is 300,000 times the mass of the Earth, but the distance between the Sun and the Moon is 400 times greater than the distance between the Earth and the Moon. Therefore, in the formula, the numerator will increase by 300,000 times, and the denominator - by 400 2, or 160,000 times. The gravitational force will be almost twice as large.

Question . Why doesn't the moon fall on the sun?

Answer . The moon falls on the sun in the same way as on the earth, that is, only enough to remain at about the same distance, revolving around the sun.

- Around the Earth!

- Wrong, not around the Earth, but around the Sun. The Earth revolves around the Sun together with its satellite - the Moon, which means that the Moon also revolves around the Sun.

Question . The moon does not fall to the Earth, because, having an initial speed, it moves by inertia. But according to Newton's third law, the forces with which two bodies act on each other are equal in absolute value and oppositely directed. Therefore, with what force the Earth attracts the Moon to itself, with the same force the Moon attracts the Earth. Why doesn't the Earth fall on the Moon? Or does it revolve around the moon?

Teacher . The fact is that both the Moon and the Earth revolve around a common center of mass. Recall the experience with the balls and the centrifugal machine. The mass of one of the balls is twice the mass of the other. In order for the balls connected by a thread to remain in equilibrium with respect to the axis of rotation during rotation, their distances from the axis, or center of rotation, must be inversely proportional to the masses. The point around which these balls revolve is called the center of mass of the two balls.

Newton's third law is not violated in the experiment with balls: the forces with which the balls pull each other towards a common center of mass are equal. The common center of mass of the Earth and the Moon revolves around the Sun.

Question . Can the force with which the Earth pulls on the Moon be called the weight of the Moon?

Student . No! We call the weight of the body the force caused by the attraction of the Earth, with which the body presses on some support, for example, a scale pan, or stretches the spring of a dynamometer. If you put a stand under the Moon (from the side facing the Earth), then the Moon will not put pressure on it. The moon would not stretch the spring of the dynamometer, if we could hang it. The entire action of the force of attraction of the Moon by the Earth is expressed only in keeping the Moon in orbit, in imparting centripetal acceleration to it. It can be said about the Moon that in relation to the Earth it is weightless in the same way as objects are weightless in satellite spacecraft when the engine stops working and only the force of gravity to the Earth acts on the ship.

Question . Where is the center of mass of the Earth-Moon system?

Answer . The distance from the Earth to the Moon is 384,000 km. The ratio of the mass of the Moon to the mass of the Earth is 1:81. The distances from the center of mass to the centers of the Moon and the Earth will be inversely proportional to these numbers. Dividing 384,000 km by 82, we get approximately 4,700 km. This means that the center of mass is located at a distance of 4700 km from the center of the Earth.

- Why is equal to the radius Earth?

– About 6400 km.

– Consequently, the center of mass of the Earth-Moon system lies inside the globe (Fig. 30, point O). Therefore, if you do not pursue accuracy, you can talk about the revolution of the Moon around the Earth.

Question . Which is easier: to fly from the Earth to the Moon or from the Moon to the Earth?

Answer . For the rocket to become artificial satellite Earth, it must be informed of an initial speed approximately equal to 8 km / s. In order for the rocket to leave the Earth's sphere of gravity, the so-called second cosmic velocity, equal to 11.2 km / s, is needed. To launch rockets from the moon lower speed: after all, the force of gravity on the moon is six times less than on earth.

Question . I don't understand why bodies inside a rocket have no weight. Maybe it's only at that point on the way to the Moon, at which the force of attraction to the Moon is balanced by the force of attraction to the Earth?

Teacher . No. The bodies inside the rocket become weightless from the moment when the engines stop working and the rocket begins free flight in orbit around the Earth, while being in the Earth's gravitational field. In free flight around the Earth, both the satellite and all objects in it relative to the center of mass of the Earth move with the same centripetal acceleration and therefore weightless.

1st question. How did balls not connected by a thread move on a centrifugal machine: along a radius or tangent to a circle?

The answer depends on the choice of the frame of reference, i.e., on the choice of the body with respect to which we are considering the motion of the balls. If we take the surface of the table as the reference system, then the balls move along tangents to the circles they describe. If we take the rotating device itself as the reference system, then the balls move along the radius. Without specifying the reference system, the question of the nature of the motion does not make sense. To move means to move relative to other bodies, and we must necessarily indicate with respect to which ones.

2nd question. What does the moon revolve around?

If we consider the movement relative to the Earth, then the Moon revolves around the Earth. If the Sun is taken as the reference body, then it is around the Sun. I will explain what was said with a picture from the book “ Entertaining astronomy» Perelman (Fig. 31). Say, with respect to which body the movement of celestial bodies is shown here.

- Relative to the Sun.

- Right. But it is easy to see that the Moon is constantly changing its position relative to the Earth.

Teacher . Of course they can't. At the position of the Earth or the Moon (note I say "or", not "and") at the point of intersection of the orbits shown, the distance between the Earth and the Moon is 380,000 km. To better understand this, draw a diagram of this for the next lesson. complex movement. Draw the Earth's orbit as an arc of a circle with a radius of 15 cm (the distance from the Earth to the Sun, as you know, is 150,000,000 km). On an arc equal to 1/12 of a circle (the monthly path of the Earth), mark on equal distances five points, counting and extreme. These points will be the centers of the lunar orbits relative to the Earth in consecutive quarters of the month. The radius of the lunar orbits cannot be plotted on the same scale as the Earth's orbit, as it would be too small. To draw lunar orbits, you need to increase the selected scale by about ten times, then the radius lunar orbit will be about 4 mm. Indicate the position of the Moon on each orbit, starting with the full moon, and connect the marked points with a smooth dotted line.

At the next lesson of the circle, one of the students showed the required diagram (Fig. 32).

The story of a student drawing a diagram: “I learned a lot while drawing this diagram. It was necessary to correctly determine the position of the Moon in its phases, to think about the direction of movement of the Moon and the Earth in their orbits. There are inaccuracies in the drawing. I will tell about them now. At the selected scale, the curvature of the lunar orbit is incorrectly depicted. It must always be concave with respect to the Sun, i.e., the center of curvature must be inside the orbit. In addition, there are not 12 lunar months in a year, but more. But one twelfth of a circle is easy to construct, so I conditionally accepted that there are 12 lunar months in a year. And finally, it is not the Earth itself that revolves around the Sun, but the common center of mass of the Earth-Moon system.

Briefly, his story is as follows. Even the ancients, observing the movement of the planets in the sky, guessed that all of them, together with the Earth, "walk" around the Sun. Later, when people forgot what they knew before, this discovery was rediscovered by Copernicus. And then arose new question: how exactly do the planets go around the sun, what is their movement? Do they go in a circle and the Sun is in the center, or do they move along some other curve? How fast are they moving? Etc.

It turned out not so soon. After Copernicus came again troubled times and great controversy flared up about whether the planets go with the Earth around the Sun or the Earth is at the center of the universe. Then a man named Tycho Brahe (Tycho Brahe (1546-1601) - Danish astronomer) figured out how to answer this question. He decided that he needed to watch very carefully where the planets appear in the sky, write it down exactly, and then already choose between two hostile theories. This was the beginning modern science, the key to a correct understanding of nature is to observe the object, write down all the details and hope that the information obtained in this way will serve as the basis for one or another theoretical interpretation. And so Tycho Brahe, a rich man who owned an island near Copenhagen, equipped his island with large bronze circles and special observation posts, and recorded night after night the positions of the planets. Only at the cost of such hard work we get any discovery.

When all this data was collected, it fell into the hands of Kepler. (Johannes Kepler (1571-1630) - German astronomer and mathematician, was Brahe's assistant), which tried to solve how the planets move around the sun. He searched for a solution by trial and error. Once it seemed to him that he had already received the answer: he decided that the planets move in a circle, but the Sun is not in the center. Then Kepler noticed that one of the planets, it seems Mars, deviates from the desired position by 8 arc minutes, and realized that the answer he received was incorrect, since Tycho Brahe could not allow such big mistake. Relying on the accuracy of his observations, he decided to revise his theory and eventually discovered three facts.

Laws of planetary motion around the sun

First, Kepler established that the planets move around the Sun in ellipses and the Sun is in one of the foci. An ellipse is a curve that all artists know about because it is a stretched circle. Children also know about it: they were told that if you thread a string into a ring, fasten its ends and insert a pencil into the ring, it will describe an ellipse.

The two points A and B are foci. The planet's orbit is an ellipse. The sun is in one of the foci. Another question arises: how does the planet move along the ellipse? Does it go faster when it is closer to the Sun? Does it slow down moving away from it? Kepler answered this question as well. He discovered that if you take two positions of the planet separated from each other by a certain period of time, say three weeks, then take another part of the orbit and there are also two positions of the planet separated by three weeks, and draw lines (scientists call them radius vectors) from the Sun to the planet, then the area enclosed between the orbit of the planet and a pair of lines that are separated from each other by three weeks is the same everywhere, in any part of the orbit. And for these areas to be the same, the planet must go faster when it is closer to the Sun, and slower when it is far from it.

A few years later, Kepler formulated the third rule, which concerned not the movement of one planet around the Sun, but connected the movements of various planets with each other. It said that time full turn planets around the Sun depends on the magnitude of the orbit and is proportional to square root from a cube of this size. And the size of the orbit is the diameter that intersects the most wide place ellipse.

So Kepler discovered three laws that can be reduced to one, if we say that the orbit of the planet is an ellipse - for equal periods of time, the radius vector of the planet describes equal areas and the time (period) of the planet's revolution around the Sun is proportional to the size of the orbit to the power of three second, i.e., to the square root of the cube of the size of the orbit. These three laws of Kepler completely describe the motion of the planets around the Sun.

Meanwhile, Galileo discovered the great principle of inertia. Then it was Newton's turn, who decided that a planet orbiting the Sun didn't need force to move forward; if there were no force, the planet would fly tangentially. But in fact, the planet does not fly in a straight line. She always finds herself not in the place where she would have fallen if she had flown freely, but closer to the Sun. In other words, its speed, its movement is deflected towards the Sun.

It became clear that the source of this force (gravitational force) is located somewhere near the Sun.

People looked at Jupiter through a telescope with satellites revolving around it, and it reminded them of a small solar system. Everything looked as if the satellites were attracted to Jupiter. The moon also revolves around the earth and is attracted to it in exactly the same way. Naturally, the idea arose that attraction acts everywhere. It only remained to generalize these observations and say that all bodies attract each other. This means that the Earth must attract the Moon in the same way as the Sun attracts the planets. But it is known that the Earth also attracts ordinary objects: for example, you sit firmly on a chair, although you might like to fly through the air. The gravitation of objects towards the Earth was a well-known phenomenon. Newton suggested that the Moon in orbit is kept by the same forces that attract objects to the Earth.

Why hot flashes happen

First, the tides. The tides are caused by the Moon itself pulling on the Earth and its oceans. So they thought before, but here's what turned out to be inexplicable: if the Moon attracts water and raises them above the near side of the Earth, then only one tide would occur per day - right under the Moon. In fact, as we know, the tides are repeated after about 12 hours, that is, twice a day. There was another school that held opposing views. Its adherents believed that the Moon attracts the Earth, and the water does not keep up with it. Newton was the first to understand what was really happening: the attraction of the Moon acts equally on the Earth and on water, if they are equally distant. But the water at point y is closer to the moon than the earth, and at point x it is farther away. In y, water is attracted to the Moon more strongly than the Earth, and in x it is weaker. Therefore, a combination of the two previous pictures is obtained, which gives a double tide.

In fact, the Earth does the same thing as the Moon - it moves in a circle. The force with which the Moon acts on the Earth is balanced - but with what? Just as the Moon moves in circles to balance the Earth's gravity, so does the Earth move in circles. Both of them revolve around a common center, and the forces on Earth are balanced in such a way that the water in x is attracted by the Moon weaker, in y - stronger, and in both places the water swells. So the hot flashes were explained and why they occur twice a day.

Discovery of the speed of light

With the development of science, measurements were made more and more accurately and the confirmation of Newton's laws became more and more convincing. The first accurate measurements concerned the satellites of Jupiter. It would seem that if you carefully observe their circulation, you can be sure that everything happens according to Newton. However, it turned out that this was not the case. Jupiter's satellites appeared at the calculated points either 8 minutes earlier or 8 minutes later than would have been expected according to Newton's laws. They were found to be ahead of schedule when Jupiter is approaching the Earth, and behind when Jupiter and Earth are moving apart, a very strange phenomenon.

Römer (Olaf Römer (1644-1710) - Danish astronomer), convinced of the correctness of the law of gravity, came to the interesting conclusion that in order to travel from the moons of Jupiter to the Earth, light needs certain time, and looking at the satellites of Jupiter, we see them not where they are now, but where they were a few minutes ago - as many minutes as it takes for light to reach us. When Jupiter is closer to us, the light comes faster, and when Jupiter is further away, the light goes longer; therefore Römer had to correct his observations for this difference in time, i.e. take into account that sometimes we make these observations earlier, and sometimes later. From this he was able to determine the speed of light. This was the first time it was established that light does not propagate instantaneously.

Discovery of the planet

Another problem arose: the planets should not move in ellipses, because, according to Newton's laws, they not only attract the Sun, but also attract each other - weakly, but still attract, and this slightly changes their movement. were already known major planets- Jupiter, Saturn, Uranus - and it was calculated how much they should deviate from their perfect Keplerian orbits-ellipses due to mutual attraction. When these calculations were completed and verified by observations, it was found that Jupiter and Saturn were moving in full accordance with the calculations, and something strange was happening with Uranus. It would seem that there is still reason to doubt Newton's laws; But most importantly, don't lose heart! Two people, John Couch Adams (1819-1892) - English mathematician and astronomer; Urbain Le Verrier (1811-1877) French astronomer, who performed these calculations independently and almost simultaneously, suggested that the motion of Uranus is influenced by an invisible planet. They sent letters to observatories suggesting, "Point your telescope this way and you will see an unknown planet." “What nonsense,” they said in one of the observatories, “some boy got a paper and a pencil in his hands, and he tells us where to look for new planet". In another observatory, the directorate was easier to climb - and Neptune was discovered there!

Ministry of Education of the Russian Federation

MOU "Secondary School with. Solodniki.

abstract

on the topic:

Why doesn't the moon fall to earth?

Completed by: Student 9 Cl,

Feklistov Andrey.

Checked:

Mikhailova E.A.

S. Solodniki 2006

1. Introduction

2. Law of gravity

3. Can the force with which the Earth attracts the Moon be called the weight of the Moon?

4. Is there a centrifugal force in the Earth-Moon system, what does it act on?

5. Around what does the moon revolve?

6. Can the Earth and the Moon collide? Their orbits around the Sun intersect, and not even once

7. Conclusion

8. Literature

Introduction

The starry sky has occupied the imagination of people at all times. Why do stars light up? How many of them shine at night? Are they far from us? Does the stellar universe have boundaries? Since ancient times, man has thought about these and many other questions, sought to understand and comprehend the structure of that big world in which we live. At the same time, the widest area was opened for the study of the Universe, where the forces of gravity play decisive role.

Among all the forces that exist in nature, the force of gravity differs, first of all, in that it manifests itself everywhere. All bodies have mass, which is defined as the ratio of the force applied to the body to the acceleration that the body acquires under the action of this force. The force of attraction acting between any two bodies depends on the masses of both bodies; it is proportional to the product of the masses of the considered bodies. In addition, the force of gravity is characterized by the fact that it obeys the law inversely proportional to the square of the distance. Other forces may depend on distance quite differently; many such forces are known.

All weighty bodies mutually experience gravity, this force determines the movement of the planets around the sun and satellites around the planets. The theory of gravity - the theory created by Newton, stood at the cradle of modern science. Another theory of gravity developed by Einstein is greatest achievement theoretical physics of the 20th century. During the centuries of the development of mankind, people observed the phenomenon of mutual attraction of bodies and measured its magnitude; they tried to put this phenomenon at their service, to surpass its influence, and, finally, to the very recent times calculate it with extreme accuracy during the first steps deep into the universe

The story is widely known that the discovery of Newton's law of universal gravitation was caused by the fall of an apple from a tree. We do not know how reliable this story is, but it remains a fact that the question: “why does the moon not fall to the earth?” interested Newton and led him to the discovery of the law of universal gravitation. The forces of universal gravitation are also called gravitational.

Law of gravity

Newton's merit lies not only in his brilliant conjecture about the mutual attraction of bodies, but also in the fact that he was able to find the law of their interaction, that is, a formula for calculating the gravitational force between two bodies.

The law of universal gravitation says: any two bodies are attracted to each other with a force directly proportional to the mass of each of them and inversely proportional to the square of the distance between them

Newton calculated the acceleration imparted to the Moon by the Earth. The acceleration of freely falling bodies at the earth's surface is 9.8 m/s 2. The Moon is removed from the Earth at a distance equal to about 60 Earth radii. Therefore, Newton reasoned, the acceleration at this distance will be: . The moon, falling with such an acceleration, should approach the Earth in the first second by 0.27 / 2 \u003d 0.13 cm

But the Moon, in addition, moves by inertia in the direction of the instantaneous velocity, i.e. along a straight line tangent at a given point to its orbit around the Earth (Fig. 1). Moving by inertia, the Moon should move away from the Earth, as the calculation shows, in one second by 1.3 mm. Of course, we do not observe such a movement, in which in the first second the Moon would move along the radius to the center of the Earth, and in the second second - tangentially. Both movements add up continuously. The moon moves along a curved line close to a circle.

Consider an experiment that shows how the force of attraction acting on a body at a right angle to the direction of motion by inertia transforms a rectilinear motion into a curvilinear one (Fig. 2). A ball, having rolled down from an inclined chute, by inertia continues to move in a straight line. If you put a magnet on the side, then under the influence of the force of attraction to the magnet, the trajectory of the ball is curved.

No matter how hard you try, you cannot throw a cork ball so that it describes circles in the air, but by tying a thread to it, you can make the ball rotate in a circle around your hand. Experiment (Fig. 3): a weight suspended from a thread passing through a glass tube pulls the thread. The force of the thread tension causes centripetal acceleration, which characterizes the change in linear velocity in the direction.

The moon revolves around the earth, held by the force of gravity. The steel rope that would replace this force should have a diameter of about 600 km. But, despite such a huge force of attraction, the Moon does not fall to the Earth, because it has an initial speed and, moreover, moves by inertia.

Knowing the distance from the Earth to the Moon and the number of revolutions of the Moon around the Earth, Newton determined the magnitude of the centripetal acceleration of the Moon.

It turned out the same number - 0.0027 m / s 2

Stop the force of attraction of the Moon to the Earth - and it will rush off in a straight line into the abyss of outer space. The ball will fly away tangentially (Fig. 3) if the thread holding the ball during rotation around the circle breaks. In the device in Fig. 4, on a centrifugal machine, only the connection (thread) keeps the balls in a circular orbit. When the thread breaks, the balls scatter along the tangents. It is difficult for the eye to catch their rectilinear movement when they are devoid of connection, but if we make such a drawing (Fig. 5), then it follows from it that the balls will move rectilinearly, tangentially to the circle.

Stop moving by inertia - and the moon would fall to the Earth. The fall would have lasted four days, nineteen hours, fifty-four minutes, fifty-seven seconds - Newton calculated so.

Using the formula of the law of universal gravitation, it is possible to determine with what force the Earth attracts the Moon: where G is the gravitational constant, t 1 and m 2 are the masses of the Earth and the Moon, r is the distance between them. Substituting specific data into the formula, we get the value of the force with which the Earth attracts the Moon and it is approximately 2 10 17 N

The law of universal gravitation applies to all bodies, which means that the Sun also attracts the Moon. Let's count with what force?

The mass of the Sun is 300,000 times the mass of the Earth, but the distance between the Sun and the Moon is 400 times greater than the distance between the Earth and the Moon. Therefore, in the formula, the numerator will increase by 300,000 times, and the denominator - by 400 2, or 160,000 times. The gravitational force will be almost twice as large.

But why doesn't the moon fall on the sun?

The moon falls on the sun in the same way as on the earth, i.e., only enough to remain at about the same distance, revolving around the sun.

The Earth revolves around the Sun together with its satellite - the Moon, which means that the Moon also revolves around the Sun.

The following question arises: the Moon does not fall to the Earth, because, having an initial speed, it moves by inertia. But according to Newton's third law, the forces with which two bodies act on each other are equal in magnitude and oppositely directed. Therefore, with what force the Earth attracts the Moon to itself, with the same force the Moon attracts the Earth. Why doesn't the Earth fall on the Moon? Or does it also revolve around the moon?

The fact is that both the Moon and the Earth revolve around a common center of mass, or, simplifying, we can say, around a common center of gravity. Recall the experience with the balls and the centrifugal machine. The mass of one of the balls is twice the mass of the other. In order for the balls connected by a thread to remain in equilibrium with respect to the axis of rotation during rotation, their distances from the axis, or center of rotation, must be inversely proportional to the masses. The point or center around which these balls revolve is called the center of mass of the two balls.

Newton's third law is not violated in the experiment with balls: the forces with which the balls pull each other towards the common center of mass are equal. In the Earth-Moon system, the common center of mass revolves around the Sun.

Can the force with which the Earth attracts Lu well, call the weight of the moon?

No. We call the weight of the body the force caused by the attraction of the Earth, with which the body presses on some support: a scale pan, for example, or stretches the spring of a dynamometer. If you put a stand under the Moon (from the side facing the Earth), then the Moon will not put pressure on it. The moon will not stretch the spring of the dynamometer, if they could hang it. The entire effect of the force of attraction of the Moon by the Earth is expressed only in keeping the Moon in orbit, in imparting centripetal acceleration to it. It can be said about the Moon that in relation to the Earth it is weightless in the same way as objects in a space ship-satellite are weightless when the engine stops working and only the force of attraction to the Earth acts on the ship, but this force cannot be called weight. All items released by the astronauts from their hands (pen, notepad) do not fall, but float freely inside the cabin. All bodies on the Moon, in relation to the Moon, of course, are weighty and will fall onto its surface if they are not held by something, but in relation to the Earth, these bodies will be weightless and cannot fall to the Earth.

Is there centrifugal force in the Earth-Moon system, what does it affect?

In the Earth-Moon system, the forces of mutual attraction of the Earth and the Moon are equal and oppositely directed, namely to the center of mass. Both of these forces are centripetal. There is no centrifugal force here.

The distance from the Earth to the Moon is approximately 384,000 km. The ratio of the mass of the Moon to the mass of the Earth is 1/81. Therefore, the distances from the center of mass to the centers of the Moon and the Earth will be inversely proportional to these numbers. Dividing 384,000 km by 81, we get approximately 4,700 km. So the center of mass is at a distance of 4700 km from the center of the earth.

The radius of the earth is about 6400 km. Consequently, the center of mass of the Earth-Moon system lies inside the globe. Therefore, if you do not pursue accuracy, you can talk about the revolution of the Moon around the Earth.

It is easier to fly from the Earth to the Moon or from the Moon to the Earth, because It is known that in order for a rocket to become an artificial satellite of the Earth, it must be given an initial velocity of ≈ 8 km/s. In order for the rocket to leave the sphere of gravity of the Earth, the so-called second cosmic velocity is needed, equal to 11.2 km/s To launch rockets from the moon, you need less speed. gravity on the Moon is six times less than on Earth.

The bodies inside the rocket become weightless from the moment when the engines stop working and the rocket will fly freely in orbit around the Earth, while being in the Earth's gravitational field. In free flight around the Earth, both the satellite and all objects in it relative to the center of mass of the Earth move with the same centripetal acceleration and therefore are weightless.

How did balls not connected by a thread move on a centrifugal machine: along a radius or tangent to a circle? The answer depends on the choice of the reference system, i.e., with respect to which reference body we will consider the movement of the balls. If we take the surface of the table as the reference system, then the balls move along tangents to the circles they describe. If we take the rotating device itself as the reference system, then the balls move along the radius. Without specifying the reference system, the question of motion does not make sense at all. To move means to move relative to other bodies, and we must necessarily indicate with respect to which ones.

What does the moon revolve around?

If we consider the movement relative to the Earth, then the Moon revolves around the Earth. If the Sun is taken as the reference body, then it is around the Sun.

Could the Earth and Moon collide? Their op bits around the sun intersect, and not even once .

Of course not. A collision is only possible if the Moon's orbit relative to the Earth intersects the Earth. With the position of the Earth or the Moon at the point of intersection of the orbits shown (relative to the Sun), the distance between the Earth and the Moon is on average 380,000 km. To better understand this, let's draw the following. The Earth's orbit was depicted as an arc of a circle with a radius of 15 cm (the distance from the Earth to the Sun is known to be 150,000,000 km). On an arc equal to part of a circle (the monthly path of the Earth), he noted five points at equal distances, counting the extreme ones. These points will be the centers of the lunar orbits relative to the Earth in consecutive quarters of the month. The radius of the lunar orbits cannot be plotted on the same scale as the Earth's orbit, as it would be too small. To draw lunar orbits, you need to increase the selected scale by about ten times, then the radius of the lunar orbit will be about 4 mm. After that indicated the position of the moon in each orbit, starting with the full moon, and connected the marked points with a smooth dotted line.

The main task was to divide the reference body. In the centrifugal machine experiment, both reference bodies are simultaneously projected onto the plane of the table, so it is very difficult to focus on one of them. This is how we solved our problem. A ruler made of thick paper (it can be replaced with a strip of tin, plexiglass, etc.) will serve as a rod along which a cardboard circle resembling a ball slides. The circle is double, glued along the circumference, but on two diametrically opposite sides there are slits through which a ruler is threaded. Holes are made along the axis of the ruler. The reference bodies are a ruler and a sheet of clean paper, which we attached with buttons to a sheet of plywood so as not to spoil the table. Having put the ruler on the pin, as if on an axis, they stuck the pin into the plywood (Fig. 6). When you turn the ruler to equal angles sequentially located holes turned out to be on one straight line. But when the ruler was turned, a cardboard circle slid along it, the successive positions of which had to be marked on paper. For this purpose, a hole was also made in the center of the circle.

With each turn of the ruler, the position of the center of the circle was marked on paper with the tip of a pencil. When the ruler passed through all the positions pre-planned for it, the ruler was removed. By connecting the marks on paper, we made sure that the center of the circle moved relative to the second reference body in a straight line, or rather, tangent to the initial circle.

But while working on the device, I made several interesting discoveries. Firstly, with a uniform rotation of the rod (ruler), the ball (circle) moves along it not uniformly, but accelerated. By inertia, the body must move uniformly and rectilinearly - this is the law of nature. But did our ball move only by inertia, that is, freely? Not! It was pushed by a rod and imparted acceleration to it. This will be clear to everyone if we turn to the drawing (Fig. 7). On a horizontal line (tangent) by dots 0, 1, 2, 3, 4 the positions of the ball are marked if it were moving completely freely. The corresponding positions of the radii with the same numerical designations show that the ball is moving with acceleration. The acceleration of the ball tells elastic force rod. In addition, the friction between the ball and the rod resists the movement. If we assume that the friction force is equal to the force that imparts acceleration to the ball, the movement of the ball along the rod must be uniform. As can be seen from Figure 8, the movement of the ball relative to the paper on the table is curvilinear. In drawing lessons, we were told that such a curve is called the “Archimedes spiral”. According to such a curve, the profile of the cams is drawn in some mechanisms when they want a uniform rotary motion turn into uniform translational motion. If two such curves are attached to each other, then the cam will receive a heart-shaped shape. With a uniform rotation of a part of this shape, the rod resting against it will perform a forward-return motion. I made a model of such a cam (Fig. 9) and a model of a mechanism for evenly winding threads on a bobbin (Fig. 10).

I did not make any discoveries during the assignment. But I learned a lot while making this diagram (Figure 11). It was necessary to correctly determine the position of the Moon in its phases, to think about the direction of movement of the Moon and the Earth in their orbits. There are inaccuracies in the drawing. I will tell about them now. At the selected scale, the curvature of the lunar orbit is incorrectly depicted. It must always be concave with respect to the Sun, i.e., the center of curvature must be inside the orbit. In addition, there are not 12 lunar months in a year, but more. But one twelfth of a circle is easy to construct, so I conditionally assumed that there are 12 lunar months in a year. And, finally, it is not the Earth itself that revolves around the Sun, but the common center of mass of the Earth-Moon system.

Conclusion

One of the clearest examples of the achievements of science, one of the evidence of the unlimited cognizability of nature was the discovery of the planet Neptune by calculations - "on the tip of a pen."

Uranus - the planet following Saturn, which for many centuries was considered the most distant of the planets, was discovered by V. Herschel at the end of the 18th century. Uranus is hardly visible to the naked eye. By the 40s of the XIX century. accurate observations have shown that Uranus deviates scarcely from the path it should follow, "taking into account the perturbations from all known planets. Thus, the theory of the motion of celestial bodies, so rigorous and accurate, was put to the test.

Le Verrier (in France) and Adams (in England) suggested that if perturbations from the known planets do not explain the deviation in the motion of Uranus, it means that the attraction of an as yet unknown body acts on it. They almost simultaneously calculated where behind Uranus there should be an unknown body that produces these deviations by its attraction. They calculated the orbit of the unknown planet, its mass and indicated the place in the sky where the unknown planet should have been at the given time. This planet was found in a telescope at the place indicated by them in 1846. It was called Neptune. Neptune is not visible to the naked eye. Thus, the disagreement between theory and practice, which seemed to undermine the authority of materialistic science, led to its triumph.

Bibliography:

1. M.I. Bludov - Conversations in Physics, part one, second edition, revised, Moscow "Enlightenment" 1972.

2. B.A. Vorontsov-velyamov - Astronomy! Grade 1, 19th edition, Moscow "Enlightenment" 1991.

3. A.A. Leonovich - I know the world, Physics, Moscow AST 1998.

4. A.V. Peryshkin, E.M. Gutnik - Physics Grade 9, Publishing House Bustard 1999.

5. Ya.I. Perelman - Entertaining physics, book 2, Edition 19, Nauka publishing house, Moscow 1976.

Tutoring

Need help learning a topic?

Our experts will advise or provide tutoring services on topics of interest to you.
Submit an application indicating the topic right now to find out about the possibility of obtaining a consultation.