You already know that there are attractive forces between all bodies, called forces of universal gravity.

Their action is manifested, for example, in the fact that bodies fall to the Earth, the Moon revolves around the Earth, and the planets revolve around the Sun. If gravitational forces disappeared, the Earth would fly away from the Sun (Fig. 14.1).

The law of universal gravitation was formulated in the second half of the 17th century by Isaac Newton.
Two material points of mass m 1 and m 2 located at a distance R are attracted with forces directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Modulus of each force

The proportionality factor G is called gravitational constant. (From the Latin “gravitas” - heaviness.) Measurements showed that

G = 6.67 * 10 -11 N * m 2 / kg 2. (2)

The law of universal gravitation reveals another important property of body mass: it is a measure of not only the inertia of the body, but also its gravitational properties.

1. What are the forces of attraction between two material points weighing 1 kg each, located at a distance of 1 m from each other? How many times is this force greater or less than the weight of a mosquito whose mass is 2.5 mg?

Such a small value of the gravitational constant explains why we do not notice the gravitational attraction between the objects around us.

Gravitational forces manifest themselves noticeably only when at least one of the interacting bodies has a huge mass - for example, it is a star or a planet.

3. How will the force of attraction between two material points change if the distance between them is increased by 3 times?

4. Two material points of mass m each are attracted with a force F. With what force are material points of mass 2m and 3m, located at the same distance, attracted?

2. The movement of planets around the Sun

The distance from the Sun to any planet is many times greater than the size of the Sun and the planet. Therefore, when considering the movement of planets, they can be considered material points. Therefore, the force of attraction of the planet to the Sun

where m is the mass of the planet, M С is the mass of the Sun, R is the distance from the Sun to the planet.

We will assume that the planet moves around the Sun uniformly in a circle. Then the speed of the planet’s movement can be found if we take into account that the acceleration of the planet a = v 2 /R is due to the action of the gravitational force F of the Sun and the fact that, according to Newton’s second law, F = ma.

5. Prove that the speed of the planet

the larger the orbital radius, the slower the planet's speed.

6. The radius of Saturn’s orbit is approximately 9 times larger than the radius of the Earth’s orbit. Find orally what is approximately the speed of Saturn if the Earth moves in its orbit at a speed of 30 km/s?

In a time equal to one revolution period T, the planet, moving with speed v, covers a path equal to the length of a circle of radius R.

7. Prove that the planet’s orbital period

From this formula it follows that the larger the orbital radius, the longer the planet's orbital period.

9. Prove that for all planets of the Solar system

Clue. Use formula (5).
From formula (6) it follows that For all planets in the Solar System, the ratio of the cube of the orbital radius to the square of the orbital period is the same. This pattern (it is called Kepler's third law) was discovered by the German scientist Johannes Kepler based on the results of many years of observations by the Danish astronomer Tycho Brahe.

3. Conditions for the applicability of the formula for the law of universal gravitation

Newton proved that the formula

F = G(m 1 m 2 /R 2)

For the force of attraction between two material points, you can also use:
– for homogeneous balls and spheres (R is the distance between the centers of balls or spheres, Fig. 14.2, a);

– for a homogeneous ball (sphere) and a material point (R is the distance from the center of the ball (sphere) to the material point, Fig. 14.2, b).

4. Gravity and the law of universal gravitation

The second of the above conditions means that using formula (1) you can find the force of attraction of a body of any shape to a homogeneous ball, which is much larger than this body. Therefore, using formula (1), it is possible to calculate the force of attraction to the Earth of a body located on its surface (Fig. 14.3, a). We get an expression for gravity:

(The Earth is not a homogeneous sphere, but it can be considered spherically symmetrical. This is sufficient for the possibility of applying formula (1).)

10. Prove that near the surface of the Earth

Where M Earth is the mass of the Earth, R Earth is its radius.
Clue. Use formula (7) and the fact that F t = mg.

Using formula (1), you can find the acceleration of gravity at a height h above the Earth's surface (Fig. 14.3, b).

11. Prove that

12. What is the acceleration of gravity at a height above the Earth’s surface equal to its radius?

13. How many times is the acceleration of gravity on the surface of the Moon less than on the surface of the Earth?
Clue. Use formula (8), in which you replace the mass and radius of the Earth with the mass and radius of the Moon.

14. The radius of a white dwarf star can be equal to the radius of the Earth, and its mass can be equal to the mass of the Sun. What is the weight of a kilogram weight on the surface of such a “dwarf”?

5. First escape velocity

Let's imagine that they installed a huge cannon on a very high mountain and fired from it in a horizontal direction (Fig. 14.4).

The greater the initial speed of the projectile, the further it will fall. It will not fall at all if its initial speed is selected so that it moves around the Earth in a circle. Flying in a circular orbit, the projectile will then become an artificial satellite of the Earth.

Let our satellite projectile move in low Earth orbit (this is the name for an orbit whose radius can be taken equal to the radius of the Earth R Earth).
With uniform motion in a circle, the satellite moves with centripetal acceleration a = v2/REarth, where v is the speed of the satellite. This acceleration is due to the action of gravity. Consequently, the satellite moves with gravitational acceleration directed towards the center of the Earth (Fig. 14.4). Therefore a = g.

15. Prove that when moving in low Earth orbit, the speed of the satellite

Clue. Use the formula a = v 2 /r for centripetal acceleration and the fact that when moving in an orbit of radius R Earth, the acceleration of the satellite is equal to the acceleration of gravity.

The speed v 1 that must be imparted to a body so that it moves under the influence of gravity in a circular orbit near the Earth's surface is called the first escape velocity. It is approximately equal to 8 km/s.

16. Express the first escape velocity in terms of the gravitational constant, mass and radius of the Earth.

Clue. In the formula obtained in the previous task, replace the mass and radius of the Earth with the mass and radius of the Moon.

In order for a body to leave the vicinity of the Earth forever, it must be given a speed of approximately 11.2 km/s. It is called the second escape velocity.

6. How the gravitational constant was measured

If we assume that the gravitational acceleration g near the Earth's surface, the mass and radius of the Earth are known, then the value of the gravitational constant G can be easily determined using formula (7). The problem, however, is that until the end of the 18th century the mass of the Earth could not be measured.

Therefore, in order to find the value of the gravitational constant G, it was necessary to measure the force of attraction of two bodies of known mass located at a certain distance from each other. At the end of the 18th century, the English scientist Henry Cavendish was able to carry out such an experiment.

He suspended a light horizontal rod with small metal balls a and b on a thin elastic thread and, using the angle of rotation of the thread, measured the attractive forces acting on these balls from large metal balls A and B (Fig. 14.5). The scientist measured small angles of rotation of the thread by the displacement of the “bunny” from the mirror attached to the thread.

Cavendish's experiment was figuratively called the "weighing of the Earth" because this experiment made it possible for the first time to measure the mass of the Earth.

18. Express the mass of the Earth in terms of G, g and R Earth.

Additional questions and tasks

19. Two ships weighing 6000 tons each are attracted by forces of 2 mN. What is the distance between the ships?

20. With what force does the Sun attract the Earth?

21. With what force does a person weighing 60 kg attract the Sun?

22. What is the acceleration of gravity at a distance from the Earth’s surface equal to its diameter?

23. How many times is the acceleration of the Moon, due to the Earth’s gravity, less than the acceleration of gravity on the Earth’s surface?

24. The acceleration of free fall on the surface of Mars is 2.65 times less than the acceleration of free fall on the surface of the Earth. The radius of Mars is approximately 3400 km. How many times is the mass of Mars less than the mass of Earth?

25. What is the orbital period of an artificial Earth satellite in low Earth orbit?

26. What is the first escape velocity for Mars? The mass of Mars is 6.4 * 10 23 kg, and the radius is 3400 km.

Every person in his life has come across this concept more than once, because gravity is the basis not only of modern physics, but also of a number of other related sciences.

Many scientists have been studying the attraction of bodies since ancient times, but the main discovery is attributed to Newton and is described as the well-known story of a fruit falling on one’s head.

What is gravity in simple words

Gravity is the attraction between several objects throughout the universe. The nature of the phenomenon varies, as it is determined by the mass of each of them and the extent between them, that is, the distance.

Newton's theory was based on the fact that both the falling fruit and the satellite of our planet are affected by the same force - gravity towards the Earth. But the satellite did not fall into earthly space precisely because of its mass and distance.

Gravity field

The gravitational field is the space within which the interaction of bodies occurs according to the laws of attraction.

Einstein's theory of relativity describes the field as a certain property of time and space, characteristically manifested when physical objects appear.

Gravity wave

These are certain types of field changes that are formed as a result of radiation from moving objects. They come off the object and spread in a wave effect.

Theories of gravity

The classical theory is Newtonian. However, it was imperfect and subsequently alternative options appeared.

These include:

• metric theories;
• non-metric;
• vector;
• Le Sage, who first described the phases;
• quantum gravity.

Today there are several dozen different theories, all of them either complement each other or look at phenomena from a different perspective.

It is worth noting: There is no ideal solution yet, but ongoing developments are opening up more possible answers regarding the attraction of bodies.

The force of gravitational attraction

The basic calculation is as follows - the gravitational force is proportional to the multiplication of the mass of the body by another, between which it is determined. This formula is expressed this way: force is inversely proportional to the distance between objects squared.

The gravitational field is potential, which means kinetic energy is conserved. This fact simplifies the solution of problems in which the force of attraction is measured.

Gravity in space

Despite the misconception of many, there is gravity in space. It is lower than on Earth, but still present.

As for the astronauts, who at first glance seem to be flying, they are actually in a state of slow decline. Visually, it seems that nothing attracts them, but in practice they experience gravity.

The strength of attraction depends on the distance, but no matter how large the distance between objects is, they will continue to be attracted to each other. Mutual attraction will never be zero.

Gravity in the Solar System

In the solar system, not only the Earth has gravity. Planets, as well as the Sun, attract objects to themselves.

Since the force is determined by the mass of the object, the Sun has the highest indicator. For example, if our planet has an indicator of one, then the luminary’s indicator will be almost twenty-eight.

Next in gravity after the Sun is Jupiter, so its gravitational force is three times higher than that of the Earth. Pluto has the smallest parameter.

For clarity, let’s denote this: in theory, on the Sun, the average person would weigh about two tons, but on the smallest planet of our system - only four kilograms.

What does the planet's gravity depend on?

Gravitational pull, as mentioned above, is the power with which the planet pulls toward itself objects located on its surface.

The force of gravity depends on the gravity of the object, the planet itself and the distance between them. If there are many kilometers, gravity is low, but it still keeps objects connected.

Several important and fascinating aspects related to gravity and its properties that are worth explaining to your child:

1. The phenomenon attracts everything, but never repels - this distinguishes it from other physical phenomena.
2. There is no such thing as zero. It is impossible to simulate a situation in which pressure does not apply, that is, gravity does not work.
3. The Earth is falling at an average speed of 11.2 kilometers per second; having reached this speed, you can leave the planet’s attraction well.
4. The existence of gravitational waves has not been scientifically proven, it is just a guess. If they ever become visible, then many mysteries of the cosmos related to the interaction of bodies will be revealed to humanity.

According to the theory of basic relativity of a scientist like Einstein, gravity is a curvature of the basic parameters of the existence of the material world, which represents the basis of the Universe.

Gravity is the mutual attraction of two objects. The strength of interaction depends on the gravity of the bodies and the distance between them. Not all the secrets of the phenomenon have been revealed yet, but today there are several dozen theories describing the concept and its properties.

The complexity of the objects being studied affects the research time. In most cases, the relationship between mass and distance is simply taken.

Gravity, also known as attraction or gravitation, is a universal property of matter that all objects and bodies in the Universe possess. The essence of gravity is that all material bodies attract all other bodies around them.

Earth gravity

If gravity is a general concept and quality that all objects in the Universe possess, then gravity is a special case of this comprehensive phenomenon. The earth attracts to itself all material objects located on it. Thanks to this, people and animals can safely move across the earth, rivers, seas and oceans can remain within their shores, and the air can not fly across the vast expanses of space, but form the atmosphere of our planet.

A fair question arises: if all objects have gravity, why does the Earth attract people and animals to itself, and not vice versa? Firstly, we also attract the Earth to us, it’s just that, compared to its force of attraction, our gravity is negligible. Secondly, the force of gravity depends directly on the mass of the body: the smaller the mass of the body, the lower its gravitational forces.

The second indicator on which the force of attraction depends is the distance between objects: the greater the distance, the less the effect of gravity. Thanks also to this, the planets move in their orbits and do not fall on each other.

It is noteworthy that the Earth, Moon, Sun and other planets owe their spherical shape precisely to the force of gravity. It acts in the direction of the center, pulling towards it the substance that makes up the “body” of the planet.

Earth's gravitational field

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

• gravity;
• centrifugal force, which owes its appearance to the rotation of the Earth around its axis (diurnal rotation).

Since both gravity and centrifugal force act constantly, the gravitational field is a constant phenomenon.

The field is slightly affected by the gravitational forces of the Sun, Moon and some other celestial bodies, as well as the atmospheric masses of the Earth.

The law of universal gravitation and Sir Isaac Newton

The English physicist, Sir Isaac Newton, according to a famous legend, one day while walking in the garden during the day, he saw the Moon in the sky. At the same time, an apple fell from the branch. Newton was then studying the law of motion and knew that an apple falls under the influence of a gravitational field, and the Moon rotates in orbit around the Earth.

And then the brilliant scientist, illuminated by insight, came up with the idea that perhaps the apple falls to the ground, obeying the same force thanks to which the Moon is in its orbit, and not rushing randomly throughout the galaxy. This is how the law of universal gravitation, also known as Newton’s Third Law, was discovered.

In the language of mathematical formulas, this law looks like this:

F=GMm/D 2 ,

Where F- the force of mutual gravity between two bodies;

M- mass of the first body;

m- mass of the second body;

D 2- the distance between two bodies;

G- gravitational constant equal to 6.67x10 -11.

We live on Earth, we move along its surface, as if along the edge of some rocky cliff that rises above a bottomless abyss. We stay on this edge of the abyss only thanks to what affects us Earth's gravitational force; we do not fall from the earth’s surface only because we have, as they say, some certain weight. We would instantly fly off this “cliff” and rapidly fly into the abyss of space if the gravity of our planet suddenly ceased to act. We would endlessly rush around in the abyss of world space, not knowing either the top or the bottom.

Movement on Earth

to his moving around the Earth we also owe it to gravity. We walk on the Earth and constantly overcome the resistance of this force, feeling its action like some heavy weight on our feet. This “load” especially makes itself felt when climbing uphill, when you have to drag it, like some kind of heavy weights hanging from your feet. It affects us no less sharply when going down the mountain, forcing us to speed up our steps. Overcoming gravity when moving around the Earth. These directions - “up” and “down” - are shown to us only by gravity. At all points on the earth's surface it is directed almost to the center of the earth. Therefore, the concepts of “bottom” and “top” will be diametrically opposed for the so-called antipodes, i.e. people living on diametrically opposite parts of the Earth’s surface. For example, the direction that shows “down” for those living in Moscow, shows “up” for residents of Tierra del Fuego. The directions showing "down" for people at the pole and at the equator are right angles; they are perpendicular to each other. Outside the Earth, with distance from it, the force of gravity decreases, as the force of gravity decreases (the force of attraction of the Earth, like any other world body, extends indefinitely far in space) and the centrifugal force increases, which reduces the force of gravity. Consequently, the higher we lift some cargo, for example, in a balloon, the less this cargo will weigh.

Earth's centrifugal force

Due to the daily rotation, centrifugal force of the earth. This force acts everywhere on the Earth's surface in a direction perpendicular to the Earth's axis and away from it. Centrifugal force small compared to gravity. At the equator it reaches its greatest value. But here, according to Newton’s calculations, the centrifugal force is only 1/289 of the attractive force. The further north you are from the equator, the less centrifugal force. At the pole itself it is zero.
The action of the centrifugal force of the Earth. At some height centrifugal force will increase so much that it will be equal to the force of attraction, and the force of gravity will first become zero, and then, with increasing distance from the Earth, it will take a negative value and will continuously increase, being directed in the opposite direction with respect to the Earth.

Gravity

The resultant force of Earth's gravity and centrifugal force is called gravity. The force of gravity at all points on the earth's surface would be the same if ours were a perfectly accurate and regular ball, if its mass were the same density everywhere and, finally, if there were no daily rotation around its axis. But, since our Earth is not a regular sphere, does not consist in all its parts of rocks of the same density and rotates all the time, then, consequently, the force of gravity at each point on the earth's surface is slightly different. Therefore, at every point on the earth’s surface the magnitude of gravity depends on the magnitude of the centrifugal force, which reduces the force of attraction, on the density of the earth's rocks and the distance from the center of the Earth. The greater this distance, the less gravity. The radii of the Earth, which at one end seem to rest against the Earth's equator, are the largest. Radii that end at the North or South Pole are the smallest. Therefore, all bodies at the equator have less gravity (less weight) than at the pole. It is known that at the pole the gravity is greater than at the equator by 1/289th. This difference in gravity of the same bodies at the equator and at the pole can be determined by weighing them using spring balances. If we weigh bodies on scales with weights, then we will not notice this difference. The scales will show the same weight both at the pole and at the equator; weights, like bodies that are weighed, will also, of course, change in weight.
Spring scales as a way to measure gravity at the equator and at the pole. Let’s assume that a ship with cargo weighs about 289 thousand tons in the polar regions, near the pole. Upon arrival at ports near the equator, the ship with cargo will weigh only about 288 thousand tons. Thus, at the equator the ship lost about a thousand tons in weight. All bodies are held on the earth's surface only due to the fact that gravity acts on them. In the morning, when you get out of bed, you are able to lower your feet to the floor only because this force pulls them down.

Gravity inside the Earth

Let's see how it changes gravity inside the earth. As we move deeper into the Earth, gravity continuously increases up to a certain depth. At a depth of about a thousand kilometers, gravity will have a maximum (greatest) value and will increase compared to its average value on the earth's surface (9.81 m/sec) by approximately five percent. With further deepening, the force of gravity will continuously decrease and at the center of the Earth will be equal to zero.

Assumptions regarding the Earth's rotation

Our The earth is spinning makes a full revolution around its axis in 24 hours. Centrifugal force, as is known, increases in proportion to the square of the angular velocity. Therefore, if the Earth accelerates its rotation around its axis by 17 times, then the centrifugal force will increase by 17 times squared, i.e. 289 times. Under normal conditions, as mentioned above, the centrifugal force at the equator is 1/289 of the gravitational force. When increasing 17 times the force of gravity and centrifugal force become equal. The force of gravity - the resultant of these two forces - with such an increase in the speed of the Earth's axial rotation will be equal to zero.
The value of centrifugal force during the rotation of the Earth. This speed of rotation of the Earth around its axis is called critical, since at such a speed of rotation of our planet, all bodies at the equator would lose their weight. The length of the day in this critical case will be approximately 1 hour 25 minutes. With further acceleration of the Earth's rotation, all bodies (primarily at the equator) will first lose their weight, and then will be thrown into space by centrifugal force, and the Earth itself will be torn into pieces by the same force. Our conclusion would be correct if the Earth were an absolutely rigid body and, when accelerating its rotational motion, would not change its shape, in other words, if the radius of the earth's equator retained its value. But it is known that as the Earth’s rotation accelerates, its surface will have to undergo some deformation: it will begin to compress towards the poles and expand towards the equator; it will take on an increasingly flattened appearance. The length of the radius of the earth's equator will begin to increase and thereby increase the centrifugal force. Thus, bodies at the equator will lose their weight before the Earth’s rotation speed increases 17 times, and a catastrophe with the Earth will occur before the day shortens its duration to 1 hour 25 minutes. In other words, the critical speed of the Earth's rotation will be somewhat lower, and the maximum length of the day will be slightly longer. Imagine mentally that the speed of rotation of the Earth, due to some unknown reasons, will approach critical. What will happen to the earth's inhabitants then? First of all, everywhere on Earth a day will be, for example, about two to three hours. Day and night will change kaleidoscopically quickly. The sun, like in a planetarium, will move very quickly across the sky, and as soon as you have time to wake up and wash yourself, it will disappear behind the horizon and night will come to replace it. People will no longer be able to accurately navigate time. No one will know what day of the month it is or what day of the week it is. Normal human life will be disorganized. The pendulum clock will slow down and then stop everywhere. They walk because gravity acts on them. After all, in our everyday life, when “walkers” begin to lag or hurry, it is necessary to shorten or lengthen their pendulum, or even hang some additional weight on the pendulum. Bodies at the equator will lose their weight. Under these imaginary conditions it will be possible to lift very heavy bodies easily. It won’t be difficult to put a horse, an elephant on your shoulders, or even lift a whole house. Birds will lose the ability to land. A flock of sparrows is circling over a trough of water. They chirp loudly, but are unable to come down. A handful of grain thrown by him would hang above the Earth in individual grains. Let us further assume that the Earth's rotation speed is getting closer and closer to critical. Our planet is greatly deformed and takes on an increasingly flattened appearance. It is likened to a rapidly rotating carousel and is about to throw off its inhabitants. The rivers will then stop flowing. They will be long standing swamps. Huge ocean ships will barely touch the water surface with their bottoms, submarines will not be able to dive into the depths of the sea, fish and marine animals will float on the surface of the seas and oceans, they will no longer be able to hide in the depths of the sea. Sailors will no longer be able to drop anchor, they will no longer control the rudders of their ships, large and small ships will stand motionless. Here is another imaginary picture. A passenger railway train stands at the station. The whistle has already been blown; the train must leave. The driver took all measures in his power. The fireman generously throws coal into the firebox. Large sparks fly from the chimney of the locomotive. The wheels are turning desperately. But the locomotive stands motionless. Its wheels do not touch the rails and there is no friction between them. There will come a time when people will not be able to go down to the floor; they will stick like flies to the ceiling. Let the speed of the Earth's rotation increase. The centrifugal force increasingly exceeds the force of gravity in its magnitude... Then people, animals, household items, houses, all objects on the Earth, its entire animal world will be thrown into cosmic space. The Australian continent will separate from the Earth and hang in space like a colossal black cloud. Africa will fly into the depths of the silent abyss, away from the Earth. The waters of the Indian Ocean will turn into a huge number of spherical drops and will also fly into boundless distances. The Mediterranean Sea, not yet having time to turn into giant accumulations of drops, with its entire thickness of water will be separated from the bottom, along which it will be possible to freely pass from Naples to Algeria. Finally, the speed of rotation will increase so much, the centrifugal force will increase so much, that the entire Earth will be torn apart. However, this cannot happen either. The speed of rotation of the Earth, as we said above, does not increase, but on the contrary, even decreases slightly - however, so little that, as we already know, over 50 thousand years the length of the day increases by only one second. In other words, the Earth now rotates at such a speed that is necessary for the animal and plant world of our planet to flourish under the calorific, life-giving rays of the Sun for many millennia.

Friction value

Now let's see what friction matters and what would happen if it were absent. Friction, as you know, has a harmful effect on our clothes: the sleeves of coats wear out first, and the soles of shoes wear out first, since sleeves and soles are most susceptible to friction. But imagine for a moment that the surface of our planet was as if well polished, completely smooth, and the possibility of friction would be excluded. Could we walk on such a surface? Of course not. Everyone knows that even on ice and a polished floor it is very difficult to walk and you have to be careful not to fall. But the surface of ice and polished floors still has some friction.
Friction force on ice. If the force of friction disappeared on the surface of the Earth, then indescribable chaos would reign on our planet forever. If there is no friction, the sea will rage forever and the storm will never subside. Sandstorms will not stop hanging over the Earth, and the wind will constantly blow. The melodic sounds of the piano, violin and the terrible roar of predatory animals will mix and endlessly spread in the air. In the absence of friction, a body that started to move would never stop. On an absolutely smooth earth's surface, various bodies and objects would forever be mixed in the most diverse directions. The world of the Earth would be ridiculous and tragic if there were no friction and attraction of the Earth.

Many thousands of years ago, people probably noticed that most objects fall faster and faster, and some fall evenly. But how exactly these objects fall was a question that interested no one. Where would primitive people have had the desire to find out how or why? If they pondered causes or explanations at all, superstitious awe immediately made them think of good and evil spirits. We can easily imagine that these people, with their dangerous lives, considered most ordinary phenomena to be “good” and most unusual phenomena to be “bad.”

All people in their development go through many stages of knowledge: from the nonsense of superstition to scientific thinking. At first, people performed experiments with two objects. For example, they took two stones and allowed them to fall freely, releasing them from their hands at the same time. Then they threw two stones again, but this time horizontally to the sides. Then they threw one stone to the side, and at the same moment they released the second one from their hands, but so that it simply fell vertically. People have learned a lot about nature from such experiments.

Fig.1

As humanity developed, it acquired not only knowledge, but also prejudices. Professional secrets and traditions of artisans gave way to organized knowledge of nature, which came from authorities and was preserved in recognized printed works.

This was the beginning of real science. People experimented on a daily basis, learning crafts or creating new machines. From experiments with falling bodies, people have established that small and large stones released from hands at the same time fall at the same speed. The same can be said about pieces of lead, gold, iron, glass, etc. of various sizes. From such experiments a simple general rule can be derived: the free fall of all bodies occurs in the same way, regardless of the size and material from which the bodies are made.

There was probably a long gap between observation of the causal relationships of phenomena and carefully executed experiments. Interest in the movement of freely falling and thrown bodies increased along with the improvement of weapons. The use of spears, arrows, catapults and even more sophisticated "instruments of war" made it possible to obtain primitive and vague information from the field of ballistics, but this took the form of working rules of artisans rather than scientific knowledge - they were not formulated ideas.

Two thousand years ago, the Greeks formulated the rules for the free fall of bodies and gave them explanations, but these rules and explanations were unfounded. Some ancient scientists apparently carried out quite reasonable experiments with falling bodies, but the use in the Middle Ages of ancient ideas proposed by Aristotle (about 340 BC) rather confused the issue. And this confusion lasted for many more centuries. The use of gunpowder greatly increased interest in the movement of bodies. But it was only Galileo (around 1600) who re-stated the fundamentals of ballistics in the form of clear rules consistent with practice.

The great Greek philosopher and scientist Aristotle apparently held the popular belief that heavy bodies fall faster than light ones. Aristotle and his followers sought to explain why certain phenomena occur, but did not always care to observe what was happening and how it was happening. Aristotle very simply explained the reasons for the fall of bodies: he said that bodies strive to find their natural place on the surface of the Earth. Describing how bodies fall, he made statements like the following: “... just as the downward movement of a piece of lead or gold or any other body endowed with weight occurs the faster, the larger its size...”, “. ..one body is heavier than another, having the same volume, but moving down faster...". Aristotle knew that stones fall faster than bird feathers, and pieces of wood fall faster than sawdust.

In the 14th century, a group of philosophers from Paris rebelled against Aristotle's theory and proposed a much more reasonable scheme, which was passed down from generation to generation and spread to Italy, influencing Galileo two centuries later. Parisian philosophers talked about accelerated movement and even about constant acceleration explaining these concepts in archaic language.

The great Italian scientist Galileo Galilei summarized the available information and ideas and critically analyzed them, and then described and began to disseminate what he considered to be true. Galileo understood that Aristotle's followers were confused by air resistance. He pointed out that dense objects, for which air resistance is insignificant, fall at almost the same speed. Galileo wrote: “... the difference in the speed of movement in the air of balls made of gold, lead, copper, porphyry and other heavy materials is so insignificant that a ball of gold in free fall at a distance of one hundred cubits would certainly outstrip a ball of copper by no more than four fingers. Having made this observation, I came to the conclusion that in a medium completely devoid of any resistance, all bodies would fall at the same speed." Having assumed what would happen if bodies fell freely in a vacuum, Galileo derived the following laws of falling bodies for the ideal case:

All bodies moving in the same way when falling: having started to fall at the same time, they move at the same speed

The movement occurs with “constant acceleration”; the rate of increase in the body's speed does not change, i.e. for each subsequent second the speed of the body increases by the same amount.

There is a legend that Galileo made a great demonstration of throwing light and heavy objects from the top of the Leaning Tower of Pisa (some say that he threw steel and wooden balls, while others claim that they were iron balls weighing 0.5 and 50 kg). There are no descriptions of such public experiences, and Galileo certainly did not demonstrate his rule in this way. Galileo knew that a wooden ball would fall much behind an iron ball, but he believed that a taller tower would be required to demonstrate the different falling speeds of two unequal iron balls.

So, small stones fall slightly behind large ones, and the difference becomes more noticeable the greater the distance the stones fly. And the point here is not just the size of the bodies: wooden and steel balls of the same size do not fall exactly the same. Galileo knew that a simple description of falling bodies was hampered by air resistance. Having discovered that as the size of bodies or the density of the material from which they are made increases, the movement of the bodies turns out to be more uniform, it is possible, based on some assumption, to formulate a rule for the ideal case. One could try to reduce air resistance by flowing around an object such as a sheet of paper, for example.

But Galileo could only reduce it and could not eliminate it completely. Therefore, he had to carry out the proof, moving from real observations of constantly decreasing air resistance to the ideal case where there is no air resistance. Later, with the benefit of hindsight, he was able to explain the differences in the actual experiments by attributing them to air resistance.

Soon after Galileo, air pumps were created, which made it possible to carry out experiments with free fall in a vacuum. To this end, Newton pumped air out of a long glass tube and dropped a bird's feather and a gold coin on top at the same time. Even bodies that differed greatly in density fell at the same speed. It was this experiment that provided a decisive test of Galileo's assumption. Galileo's experiments and reasoning led to a simple rule that was exactly valid in the case of free fall of bodies in a vacuum. This rule in the case of free fall of bodies in the air is fulfilled with limited accuracy. Therefore, one cannot believe in it as an ideal case. To fully study the free fall of bodies, it is necessary to know what changes in temperature, pressure, etc. occur during the fall, that is, to study other aspects of this phenomenon. But such studies would be confusing and complex, it would be difficult to notice their relationship, which is why so often in physics one has to be content only with the fact that the rule is a kind of simplification of a single law.

So, even the scientists of the Middle Ages and the Renaissance knew that without air resistance a body of any mass falls from the same height in the same time, Galileo not only tested it with experience and defended this statement, but also established the type of motion of a body falling vertically: “ ...they say that the natural motion of a falling body is continuously accelerating. However, in what respect this occurs has not yet been indicated; As far as I know, no one has yet proven that the spaces traversed by a falling body at equal intervals of time are related to each other like successive odd numbers.” So Galileo established the sign of uniformly accelerated motion:

S 1:S 2:S 3: ... = 1:2:3: ... (at V 0 = 0)

Thus, we can assume that free fall is uniformly accelerated motion. Since for uniformly accelerated motion the displacement is calculated by the formula

, then if we take three certain points 1,2,3 through which a body passes during a fall and write: (acceleration during free fall is the same for all bodies), it turns out that the ratio of displacements during uniformly accelerated motion is equal to:

S 1:S 2:S 3 = t 1 2:t 2 2:t 3 2

This is another important sign of uniformly accelerated motion, and therefore the free fall of bodies.

The acceleration of gravity can be measured. If we assume that the acceleration is constant, then it is quite easy to measure it by determining the period of time during which the body travels a known segment of the path and, again using the relation

. From here a=2S/t 2 . The constant acceleration due to gravity is symbolized by g. The acceleration of free fall is famous for the fact that it does not depend on the mass of the falling body. Indeed, if we recall the experience of the famous English scientist Newton with a bird feather and a gold coin, we can say that they fall with the same acceleration, although they have different masses.

The measurements give a g value of 9.8156 m/s 2 .

The acceleration vector of free fall is always directed vertically downward, along a plumb line at a given place on the Earth.

And yet: why do bodies fall? One might say, due to gravity or gravity. After all, the word “gravity” is of Latin origin and means “heavy” or “weighty.” We can say that bodies fall because they weigh. But then why do bodies weigh? And the answer can be this: because the Earth attracts them. And, indeed, everyone knows that the Earth attracts bodies because they fall. Yes, physics does not explain gravity; the Earth attracts bodies because nature works that way. However, physics can tell you a lot of interesting and useful things about gravity. Isaac Newton (1643-1727) studied the movement of celestial bodies - the planets and the Moon. He was more than once interested in the nature of the force that must act on the Moon so that, when moving around the earth, it is kept in an almost circular orbit. Newton also thought about the seemingly unrelated problem of gravity. Since falling bodies accelerate, Newton concluded that they are acted upon by a force that can be called the force of gravity or gravitation. But what causes this gravitational force? After all, if a force acts on a body, then it is caused by some other body. Any body on the surface of the Earth experiences the action of this gravitational force, and wherever the body is located, the force acting on it is directed towards the center of the Earth. Newton concluded that the Earth itself creates a gravitational force acting on bodies located on its surface.

The story of Newton's discovery of the law of universal gravitation is quite well known. According to legend, Newton was sitting in his garden and noticed an apple falling from a tree. He suddenly had a hunch that if the force of gravity acts at the top of a tree and even at the top of a mountain, then perhaps it acts at any distance. So the idea that it is the Earth’s gravity that holds the Moon in its orbit served as the basis for Newton to begin building his great theory of gravity.

For the first time, the idea that the nature of the forces that make a stone fall and determine the movement of celestial bodies is the same arose with Newton the student. But the first calculations did not give correct results because the data available at that time about the distance from the Earth to the Moon was inaccurate. 16 years later, new, corrected information about this distance appeared. After new calculations were carried out, covering the movement of the Moon, all the planets of the solar system discovered by that time, comets, ebbs and flows, the theory was published.

Many historians of science now believe that Newton made up this story in order to push the date of discovery back to the 1760s, while his correspondence and diaries indicate that he actually arrived at the law of universal gravitation only around 1685

Newton began by determining the magnitude of the gravitational force that the Earth exerts on the Moon by comparing it with the magnitude of the force acting on bodies on the surface of the Earth. On the surface of the Earth, the force of gravity imparts acceleration to bodies g = 9.8 m/s 2 . But what is the centripetal acceleration of the Moon? Since the Moon moves almost uniformly in a circle, its acceleration can be calculated using the formula:

a =g 2 /r

Through measurements, this acceleration can be found. It is equal

2.73*10 -3 m/s 2. If we express this acceleration in terms of the gravitational acceleration g near the Earth's surface, we obtain:

Thus, the acceleration of the Moon directed towards the Earth is 1/3600 of the acceleration of bodies near the Earth's surface. The Moon is 385,000 km away from the Earth, which is approximately 60 times the Earth's radius of 6,380 km. This means that the Moon is 60 times farther from the center of the Earth than bodies located on the surface of the Earth. But 60*60 = 3600! From this, Newton concluded that the force of gravity acting on any body from the Earth decreases in inverse proportion to the square of their distance from the center of the Earth:

Gravity~ 1/ r 2

The Moon, 60 Earth radii away, experiences a gravitational pull that is only 1/60 2 = 1/3600 of the force it would experience if it were on the Earth's surface. Any body placed at a distance of 385,000 km from the Earth, thanks to the Earth’s gravity, acquires the same acceleration as the Moon, namely 2.73 * 10 -3 m/s 2 .

Newton understood that the force of gravity depends not only on the distance to the attracted body, but also on its mass. Indeed, the force of gravity is directly proportional to the mass of the attracted body, according to Newton's second law. From Newton's third law it is clear that when the Earth acts with a gravitational force on another body (for example, the Moon), this body, in turn, acts on the Earth with an equal and opposite force:

Rice. 2

Thanks to this, Newton assumed that the magnitude of the gravitational force is proportional to both masses. Thus:

Where m 3 - mass of the Earth, m T- mass of another body, r- distance from the center of the Earth to the center of the body.

Continuing his study of gravity, Newton moved one step further. He determined that the force required to keep the various planets in their orbits around the Sun decreases in inverse proportion to the square of their distances from the Sun. This led him to the idea that the force acting between the Sun and each of the planets and keeping them in their orbits was also a gravitational force. He also suggested that the nature of the force that holds the planets in their orbits is identical to the nature of the force of gravity acting on all bodies near the earth's surface (we will talk about gravity later). The test confirmed the assumption of the unified nature of these forces. Then if gravitational influence exists between these bodies, then why shouldn’t it exist between all bodies? Thus Newton came to his famous The law of universal gravitation, which can be formulated as follows:

Every particle in the Universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line connecting the two particles.

The magnitude of this force can be written as:

where and are the masses of two particles, is the distance between them, and is the gravitational constant, which can be measured experimentally and has the same numerical value for all bodies.

This expression determines the magnitude of the gravitational force with which one particle acts on another, located at a distance from it. For two non-point, but homogeneous bodies, this expression correctly describes the interaction if is the distance between the centers of the bodies. In addition, if extended bodies are small compared to the distances between them, then we will not be much mistaken if we consider the bodies as point particles (as is the case for the Earth-Sun system).

If you need to consider the force of gravitational attraction acting on a given particle from two or more other particles, for example, the force acting on the Moon from the Earth and the Sun, then it is necessary for each pair of interacting particles to use the formula of the law of universal gravitation, and then add the forces vectorially, acting on the particle.

The value of the constant must be very small, since we do not notice any force acting between bodies of ordinary sizes. The force acting between two bodies of normal size was first measured in 1798. Henry Cavendish - 100 years after Newton published his law. To detect and measure such an incredibly small force, he used the setup shown in Fig. 3.

Two balls are attached to the ends of a light horizontal rod suspended from the middle to a thin thread. When the ball, labeled A, is brought close to one of the suspended balls, the force of gravitational attraction causes the ball attached to the rod to move, causing the thread to twist slightly. This slight displacement is measured using a narrow beam of light directed at a mirror mounted on a thread so that the reflected beam of light falls on the scale. Previous measurements of the twisting of the thread under the influence of known forces make it possible to determine the magnitude of the gravitational interaction force acting between two bodies. A device of this type is used in the design of a gravity meter, with the help of which very small changes in gravity can be measured near a rock that differs in density from neighboring rocks. This instrument is used by geologists to study the earth's crust and explore geological features that indicate an oil deposit. In one version of the Cavendish device, two balls are suspended at different heights. They will then be attracted differently by a deposit of dense rock close to the surface; therefore, the bar will rotate slightly when properly oriented relative to the deposit. Oil explorers are now replacing these gravity meters with instruments that directly measure small changes in the magnitude of the acceleration due to gravity, g, which will be discussed later.

Cavendish not only confirmed Newton's hypothesis that bodies attract each other and the formula correctly describes this force. Since Cavendish could measure quantities with good accuracy, he was also able to calculate the value of the constant. It is currently accepted that this constant is equal to

The diagram of one of the measurement experiments is shown in Fig. 4.

Two balls of equal mass are suspended from the ends of a balance beam. One of them is located above the lead plate, the other is below it. Lead (100 kg of lead was taken for the experiment) increases the weight of the right ball with its attraction and reduces the weight of the left one. The right ball outweighs the left one. The value is calculated based on the deviation of the balance beam.

The discovery of the law of universal gravitation is rightfully considered one of the greatest triumphs of science. And, associating this triumph with the name of Newton, one cannot help but want to ask why exactly this brilliant naturalist, and not Galileo, for example, who discovered the laws of free fall of bodies, not Robert Hooke or any of Newton’s other remarkable predecessors or contemporaries, managed to make this discovery?

This is not a matter of mere chance or falling apples. The main determining factor was that Newton had in his hands the laws he discovered that were applicable to the description of any movements. It was these laws, Newton's laws of mechanics, that made it absolutely clear that the basis that determines the features of movement are forces. Newton was the first who understood absolutely clearly what exactly needed to be looked for to explain the motion of the planets - it was necessary to look for forces and only forces. One of the most remarkable properties of the forces of universal gravitation, or, as they are often called, gravitational forces, is reflected in the very name given by Newton: worldwide. Everything that has mass - and mass is inherent in any form, any kind of matter - must experience gravitational interactions. At the same time, it is impossible to shield yourself from gravitational forces. There are no barriers to universal gravity. It is always possible to put up an insurmountable barrier to the electric and magnetic field. But gravitational interaction is freely transmitted through any body. Screens made of special substances impenetrable to gravity can only exist in the imagination of the authors of science fiction books.

So, gravitational forces are omnipresent and all-pervasive. Why don’t we feel the attraction of most bodies? If you calculate what proportion of the Earth’s gravity is, for example, the gravity of Everest, it turns out that it is only thousandths of a percent. The force of mutual attraction between two people of average weight with a distance of one meter between them does not exceed three hundredths of a milligram. The gravitational forces are so weak. The fact that gravitational forces, generally speaking, are much weaker than electrical forces, causes a peculiar division of the spheres of influence of these forces. For example, having calculated that in atoms the gravitational attraction of electrons to the nucleus is weaker than the electrical attraction by a factor, it is easy to understand that the processes inside the atom are determined practically by electrical forces alone. Gravitational forces become noticeable, and sometimes even colossal, when such huge masses as the masses of cosmic bodies: planets, stars, etc. appear in the interaction. Thus, the Earth and the Moon are attracted with a force of approximately 20,000,000,000,000,000 tons. Even stars so far from us, whose light travels from the Earth for years, are attracted to our planet with a force that is expressed by an impressive figure - hundreds of millions of tons.

The mutual attraction of two bodies decreases as they move away from each other. Let's mentally perform the following experiment: we will measure the force with which the Earth attracts a body, for example, a twenty-kilogram weight. Let the first experiment correspond to such conditions when the weight is placed at a very large distance from the Earth. Under these conditions, the force of attraction (which can be measured using the most ordinary spring scales) will be practically zero. As we approach the Earth, mutual attraction will appear and gradually increase, and finally, when the weight is on the surface of the Earth, the arrow of the spring scales will stop at the “20 kilograms” mark, since what we call weight, apart from the rotation of the earth, is nothing other than the force with which the Earth attracts bodies located on its surface (see below). If we continue the experiment and lower the weight into a deep shaft, this will reduce the force acting on the weight. This can be seen from the fact that if a weight is placed in the center of the earth, the attraction from all sides will be mutually balanced and the needle of the spring scale will stop exactly at zero.

So, one cannot simply say that gravitational forces decrease with increasing distance - one must always stipulate that these distances themselves, with this formulation, are taken to be much larger than the sizes of the bodies. It is in this case that the law formulated by Newton is correct that the forces of universal gravity decrease in inverse proportion to the square of the distance between attracting bodies. However, it remains unclear whether this is a rapid or not very rapid change with distance? Does such a law mean that interaction is practically felt only between the closest neighbors, or is it noticeable even at fairly large distances?

Let us compare the law of decreasing gravitational forces with distance with the law according to which illumination decreases with distance from the source. In both cases, the same law applies - inverse proportionality to the square of the distance. But we see stars located at such enormous distances from us that even a light beam, which has no rivals in speed, can travel only in billions of years. But if the light from these stars reaches us, then their attraction should be felt, at least very weakly. Consequently, the action of the forces of universal gravitation extends, necessarily decreasing, to almost unlimited distances. Their range of action is infinity. Gravitational forces are long-range forces. Due to long-range action, gravity binds all bodies in the universe.

The relative slowness of the decrease of forces with distance at each step is manifested in our earthly conditions: after all, all bodies, being moved from one height to another, change their weight extremely slightly. Precisely because with a relatively small change in distance - in this case to the center of the Earth - gravitational forces practically do not change.

The altitudes at which artificial satellites move are already comparable to the radius of the Earth, so to calculate their trajectory, taking into account the change in the force of gravity with increasing distance is absolutely necessary.

So, Galileo argued that all bodies released from a certain height near the surface of the Earth will fall with the same acceleration g (if we neglect air resistance). The force causing this acceleration is called gravity. Let us apply Newton's second law to gravity, considering as acceleration a acceleration of gravity g . Thus, the force of gravity acting on the body can be written as:

F g =mg

This force is directed downwards towards the center of the Earth.

Because in the SI system g = 9.8 , then the force of gravity acting on a body weighing 1 kg is.

Let us apply the formula of the law of universal gravitation to describe the force of gravity - the force of gravity between the earth and a body located on its surface. Then m 1 will be replaced by the mass of the Earth m 3, and r by the distance to the center of the Earth, i.e. by the Earth's radius r 3. Thus we get:

Where m is the mass of a body located on the surface of the Earth. From this equality it follows that:

In other words, the acceleration of free fall on the surface of the earth g determined by the quantities m 3 and r 3 .

On the Moon, on other planets, or in outer space, the force of gravity acting on a body of the same mass will be different. For example, on the Moon the magnitude g represents only one sixth g on Earth, and a body weighing 1 kg is subject to a force of gravity equal to only 1.7 N.

Until the gravitational constant G was measured, the mass of the Earth remained unknown. And only after G was measured, using the relationship it was possible to calculate the mass of the earth. This was first done by Henry Cavendish himself. Substituting the gravitational acceleration value g = 9.8 m/s and the radius of the earth r z = 6.38 10 6 into the formula, we obtain the following value for the mass of the Earth:

For the gravitational force acting on bodies located near the Earth's surface, you can simply use the expression mg. If it is necessary to calculate the gravitational force acting on a body located at some distance from the Earth, or the force caused by another celestial body (for example, the Moon or another planet), then the value of g should be used, calculated using the well-known formula in which r 3 and m 3 must be replaced by the corresponding distance and mass, you can also directly use the formula of the law of universal gravitation. There are several methods for determining the acceleration due to gravity very accurately. You can find g simply by weighing a standard weight on a spring balance. Geological scales must be amazing - their spring changes tension when adding less than a millionth of a gram of load. Torsional quartz balances give excellent results. Their design is, in principle, simple. A lever is welded to a horizontally stretched quartz thread, the weight of which slightly twists the thread:

A pendulum is also used for the same purposes. Until recently, pendulum methods for measuring g were the only ones, and only in the 60s - 70s. They began to be replaced by more convenient and accurate weighing methods. In any case, measuring the period of oscillation of a mathematical pendulum, according to the formula

you can find the value of g quite accurately. By measuring the value of g in different places on one instrument, one can judge the relative changes in gravity with an accuracy of parts per million.

The values ​​of the acceleration of gravity g at different points on the Earth are slightly different. From the formula g = Gm 3 you can see that the value of g should be smaller, for example, at the tops of mountains than at sea level, since the distance from the center of the Earth to the top of the mountain is somewhat greater. Indeed, this fact was established experimentally. However, the formula g=Gm 3 /r 3 2 does not give an exact value of g at all points, since the surface of the earth is not exactly spherical: not only do mountains and seas exist on its surface, but there is also a change in the radius of the earth at the equator; in addition, the mass of the earth is distributed non-uniformly; The rotation of the Earth also affects the change in g.

However, the properties of gravitational acceleration turned out to be more complex than Galileo assumed. Find out that the magnitude of acceleration depends on the latitude at which it is measured:

The magnitude of the acceleration due to gravity also changes with height above the Earth's surface:

The free fall acceleration vector is always directed vertically downward, and along a plumb line at a given place on the Earth.

Thus, at the same latitude and at the same altitude above sea level, the acceleration of gravity should be the same. Accurate measurements show that deviations from this norm—gravity anomalies—are very common. The reason for the anomalies is the non-uniform distribution of mass near the measurement site.

As already mentioned, the gravitational force on the part of a large body can be represented as the sum of forces acting on the part of individual particles of a large body. The attraction of a pendulum by the Earth is the result of the action of all the particles of the Earth on it. But it is clear that nearby particles make the greatest contribution to the total force - after all, attraction is inversely proportional to the square of the distance.

If heavy masses are concentrated near the measurement site, g will be greater than the norm; otherwise, g will be less than the norm.

If, for example, you measure g on a mountain or on a plane flying over the sea at the height of a mountain, then in the first case you will get a large number. The g value is also higher than normal on secluded ocean islands. It is clear that in both cases the increase in g is explained by the concentration of additional masses at the measurement site.

Not only the value of g, but also the direction of gravity can deviate from the norm. If you hang a weight on a thread, the elongated thread will show the vertical for this place. This vertical may deviate from the norm. The “normal” direction of the vertical is known to geologists from special maps on which the “ideal” figure of the Earth is constructed based on data on g values.

Let's perform an experiment with a plumb line at the foot of a large mountain. The plumb bob is pulled by the Earth to its center and by the mountain to the side. The plumb line must deviate under such conditions from the direction of the normal vertical. Since the mass of the Earth is much greater than the mass of the mountain, such deviations do not exceed a few arc seconds.

The “normal” vertical is determined by the stars, since for any geographic point it is calculated where the vertical of the “ideal” figure of the Earth “rests” in the sky at a given moment of the day and year.

Deviations of the plumb line sometimes lead to strange results. For example, in Florence, the influence of the Apennines leads not to attraction, but to repulsion of the plumb line. There can be only one explanation: there are huge voids in the mountains.

Remarkable results are obtained by measuring the acceleration of gravity on the scale of continents and oceans. Continents are much heavier than oceans, so it would seem that g values ​​over continents should be larger. Than over the oceans. In reality, the values ​​of g along the same latitude over oceans and continents are on average the same.

Again, there is only one explanation: the continents rest on lighter rocks, and the oceans on heavier rocks. And indeed, where direct research is possible, geologists establish that the oceans rest on heavy basaltic rocks, and the continents on light granites.

But the following question immediately arises: why do heavy and light rocks accurately compensate for the difference in the weights of continents and oceans? Such compensation cannot be a matter of chance; its reasons must be rooted in the structure of the Earth's shell.

Geologists believe that the upper parts of the earth's crust seem to float on an underlying plastic, that is, easily deformable mass. The pressure at depths of about 100 km should be the same everywhere, just as the pressure at the bottom of a vessel with water in which pieces of wood of different weights float is the same. Therefore, a column of matter with an area of ​​1 m 2 from the surface to a depth of 100 km should have the same weight both under the ocean and under the continents.

This equalization of pressures (it is called isostasy) leads to the fact that over the oceans and continents along the same latitude line the value of the gravitational acceleration g does not differ significantly. Local gravity anomalies serve geological exploration, the purpose of which is to find mineral deposits underground without digging holes or digging mines.

Heavy ore should be looked for in those places where g is greatest. In contrast, light salt deposits are detected by local underestimated g values. g can be measured with an accuracy of parts per million from 1 m/sec 2 .

Reconnaissance methods using pendulums and ultra-precise scales are called gravitational. They are of great practical importance, in particular for oil exploration. The fact is that with gravitational exploration methods it is easy to detect underground salt domes, and very often it turns out that where there is salt, there is oil. Moreover, oil lies in the depths, and salt is closer to the earth's surface. Oil was discovered using gravity exploration in Kazakhstan and other places.

Instead of pulling the cart with a spring, it can be accelerated by attaching a cord thrown over a pulley, from the opposite end of which a load is suspended. Then the force imparting acceleration will be due to weight this cargo. The acceleration of free fall is again imparted to the body by its weight.

In physics, weight is the official name of the force that is caused by the attraction of objects to the earth's surface - “the attraction of gravity.” The fact that bodies are attracted towards the center of the Earth makes this explanation reasonable.

No matter how you define it, weight is force. It is no different from any other force, except for two features: the weight is directed vertically and acts constantly, it cannot be eliminated.

To directly measure the weight of a body, we must use a spring scale, graduated in units of force. Since this is often inconvenient to do, we compare one weight with another using lever scales, i.e. we find the relation:

EARTH'S GRAVITY ACTING ON BODY X EARTH'S GRAVITY ACTING ON THE STANDARD OF MASS

Suppose that body X is attracted 3 times stronger than the mass standard. In this case, we say that the earth's gravity acting on body X is equal to 30 newtons of force, which means that it is 3 times greater than the earth's gravity, which acts on a kilogram of mass. The concepts of mass and weight are often confused, between which there is a significant difference. Mass is a property of the body itself (it is a measure of inertia or its “amount of matter”). Weight is the force with which the body acts on the support or stretches the suspension (weight is numerically equal to the force of gravity if the support or suspension has no acceleration).

If we use a spring scale to measure the weight of an object with very great accuracy, and then move the scale to another place, we will find that the weight of the object on the surface of the Earth varies somewhat from place to place. We know that far from the surface of the Earth, or in the depths of the globe, the weight should be much less.

Does the mass change? Scientists, reflecting on this issue, have long come to the conclusion that the mass should remain unchanged. Even at the center of the Earth, where gravity acting in all directions would produce zero net force, the body would still have the same mass.

Thus, the mass, measured by the difficulty we encounter when trying to accelerate the motion of a small cart, is the same everywhere: on the surface of the Earth, in the center of the Earth, on the Moon. Weight estimated by the elongation of the spring scales (and the feeling

in the muscles of the hand of a person holding a scale) will be significantly less on the Moon and practically equal to zero at the center of the Earth. (Fig.7)

How strong is the earth's gravity acting on different masses? How to compare the weights of two objects? Let's take two identical pieces of lead, say 1 kg each. The Earth attracts each of them with the same force, equal to a weight of 10 N. If you combine both pieces of 2 kg, then the vertical forces simply add up: The Earth attracts 2 kg twice as much as 1 kg. We will get exactly the same double attraction if we fuse both pieces into one or place them one on top of the other. The gravitational attractions of any homogeneous material simply add up, and there is no absorption or shielding of one piece of matter by another.

For any homogeneous material, weight is proportional to mass. Therefore, we believe that the Earth is the source of a “gravity field” emanating from its vertical center and capable of attracting any piece of matter. The gravity field acts equally on, say, every kilogram of lead. But what about the forces of attraction acting on equal masses of different materials, for example, 1 kg of lead and 1 kg of aluminum? The meaning of this question depends on what is meant by equal masses. The simplest way to compare masses, which is used in scientific research and in commercial practice, is the use of lever scales. They compare the forces that pull both loads. But having obtained equal masses of, say, lead and aluminum in this way, we can assume that equal weights have equal masses. But in fact, here we are talking about two completely different types of mass - inertial and gravitational mass.

The quantity in the formula represents the inert mass. In experiments with carts, which are accelerated by springs, the value acts as a characteristic of the “heaviness of the substance”, showing how difficult it is to impart acceleration to the body in question. A quantitative characteristic is a ratio. This mass is a measure of inertia, the tendency of mechanical systems to resist changes in state. Mass is a property that must be the same near the surface of the Earth, on the Moon, in deep space, and at the center of the Earth. What is its connection to gravity and what actually happens when weighed?

Completely independent of inertial mass, one can introduce the concept of gravitational mass as the amount of matter attracted by the Earth.

We believe that the Earth's gravitational field is the same for all objects in it, but we attribute it to different

We have different masses, which are proportional to the attraction of these objects by the field. This is gravitational mass. We say that different objects have different weights because they have different gravitational masses that are attracted by the gravitational field. Thus, gravitational masses are by definition proportional to weights as well as to gravity. Gravitational mass determines the force with which a body is attracted by the Earth. In this case, gravity is mutual: if the Earth attracts a stone, then the stone also attracts the Earth. This means that the gravitational mass of a body also determines how strongly it attracts another body, the Earth. Thus, gravitational mass measures the amount of matter that is affected by gravity, or the amount of matter that causes gravitational attractions between bodies.

The gravitational attraction on two identical pieces of lead is twice as strong as on one. The gravitational masses of the lead pieces must be proportional to the inertial masses, since the masses of both types are obviously proportional to the number of lead atoms. The same applies to pieces of any other material, say wax, but how do you compare a piece of lead to a piece of wax? The answer to this question is given by a symbolic experiment to study the fall of bodies of various sizes from the top of the leaning Leaning Tower of Pisa, the one that Galileo, according to legend, carried out. Let's drop two pieces of any material of any size. They fall with the same acceleration g. The force acting on a body and giving it acceleration6 is the Earth's gravity applied to this body. The force of attraction of bodies by the Earth is proportional to gravitational mass. But gravity imparts the same acceleration g to all bodies. Therefore, gravity, like weight, must be proportional to the inertial mass. Consequently, bodies of any shape contain equal proportions of both masses.

If we take 1 kg as the unit of both masses, then the gravitational and inertial masses will be the same for all bodies of any size from any material and in any place.

Here's how to prove it. Let us compare the standard kilogram made of platinum6 with a stone of unknown mass. Let's compare their inertial masses by moving each of the bodies in a horizontal direction under the influence of some force and measuring the acceleration. Let's assume that the mass of the stone is 5.31 kg. Earth's gravity is not involved in this comparison. Then we compare the gravitational masses of both bodies by measuring the gravitational attraction between each of them and some third body, most simply the Earth. This can be done by weighing both bodies. We will see that the gravitational mass of the stone is also 5.31 kg.

More than half a century before Newton proposed his law of universal gravitation, Johannes Kepler (1571-1630) discovered that “the intricate motion of the planets of the solar system could be described by three simple laws. Kepler's laws strengthened belief in the Copernican hypothesis that the planets revolve around the sun, a.

To assert at the beginning of the 17th century that the planets were around the Sun, and not around the Earth, was the greatest heresy. Giordano Bruno, who openly defended the Copernican system, was condemned as a heretic by the Holy Inquisition and burned at the stake. Even the great Galileo, despite his close friendship with the Pope, was imprisoned, condemned by the Inquisition and forced to publicly renounce his views.

In those days, the teachings of Aristotle and Ptolemy, which stated that the orbits of the planets arise as a result of complex movements along a system of circles, were considered sacred and inviolable. Thus, to describe the orbit of Mars, a dozen or so circles of varying diameters were required. Johannes Kepler set out to “prove” that Mars and the Earth must revolve around the Sun. He tried to find an orbit of the simplest geometric shape that would exactly correspond to the many dimensions of the planet's position. Years of tedious calculations passed before Kepler was able to formulate three simple laws that very accurately describe the motion of all planets:

First law:

one of the focuses of which is

Second law:

and the planet) describes at equal intervals

time equal areas

Third law:

distances from the Sun:

R 1 3 /T 1 2 = R 2 3 /T 2 2

The significance of Kepler's works is enormous. He discovered the laws, which Newton then connected with the law of universal gravitation. Of course, Kepler himself was not aware of what his discoveries would lead to. “He was engaged in tedious hints of empirical rules, which Newton was supposed to bring to a rational form in the future.” Kepler could not explain what caused the existence of elliptical orbits, but he admired the fact that they existed.

Based on Kepler's third law, Newton concluded that attractive forces should decrease with increasing distance and that attraction should vary as (distance) -2. Having discovered the law of universal gravitation, Newton transferred the simple idea of ​​\u200b\u200bthe movement of the Moon to the entire planetary system. He showed that attraction, according to the laws he derived, determines the movement of planets in elliptical orbits, and the Sun should be located at one of the foci of the ellipse. He was able to easily derive two other Kepler laws, which also follow from his hypothesis of universal gravitation. These laws are valid if only the attraction of the Sun is taken into account. But it is also necessary to take into account the effect of other planets on a moving planet, although in the solar system these attractions are small compared to the attraction of the Sun.

Kepler's second law follows from the arbitrary dependence of the force of gravity on distance if this force acts in a straight line connecting the centers of the planet and the Sun. But Kepler's first and third laws are satisfied only by the law of inverse proportionality of the forces of attraction to the square of the distance.

To obtain Kepler's third law, Newton simply combined the laws of motion with the law of gravity. For the case of circular orbits, one can reason as follows: let a planet whose mass is equal to m move with speed v in a circle of radius R around the Sun, whose mass is equal to M. This movement can only occur if the planet is acted upon by an external force F = mv 2 /R, creating centripetal acceleration v 2 /R. Let's assume that the attraction between the Sun and the planet creates the necessary force. Then:

GMm/r 2 = mv 2 /R

and the distance r between m and M is equal to the orbital radius R. But the speed

where T is the time during which the planet makes one revolution. Then

To obtain Kepler's third law, you need to transfer all R and T to one side of the equation, and all other quantities to the other:

R 3 /T 2 = GM/4p 2

If we now move to another planet with a different orbital radius and orbital period, then the new ratio will again be equal to GM/4p 2 ; this value will be the same for all planets, since G is a universal constant, and mass M is the same for all planets revolving around the Sun. Thus, the value of R 3 /T 2 will be the same for all planets in accordance with Kepler's third law. This calculation allows us to obtain the third law for elliptical orbits, but in this case R is the average value between the largest and smallest distance of the planet from the Sun.

Armed with powerful mathematical methods and guided by excellent intuition, Newton applied his theory to a large number of problems included in his PRINCIPLES, concerning the characteristics of the Moon, the Earth, other planets and their movement, as well as other celestial bodies: satellites, comets.

The moon experiences numerous disturbances that deviate it from uniform circular motion. First of all, it moves along a Keplerian ellipse, at one of the foci of which the Earth is located, like any satellite. But this orbit experiences slight variations due to the attraction of the Sun. At the new moon, the Moon is closer to the Sun than the full Moon, which appears two weeks later; this reason changes the attraction, which leads to the slowing down and speeding up of the Moon's movement during the month. This effect increases when the Sun is closer in winter, so that annual variations in the speed of the Moon are also observed. In addition, changes in the sun's gravity change the ellipticity of the lunar orbit; The lunar orbit tilts up and down, and the orbital plane rotates slowly. Thus, Newton showed that the noted irregularities in the movement of the Moon are caused by universal gravitation. He did not develop the question of solar gravity in all detail; the motion of the Moon remained a complex problem, which is being developed in ever increasing detail to this day.

Ocean tides have long remained a mystery, which it seemed could be explained by establishing their connection with the movement of the Moon. However, people believed that such a connection could not really exist, and even Galileo ridiculed this idea. Newton showed that the ebb and flow of the tides are caused by the uneven attraction of water in the ocean from the side of the Moon. The center of the lunar orbit does not coincide with the center of the Earth. The Moon and Earth rotate together around their common center of mass. This center of mass is located approximately 4800 km from the center of the Earth, only 1600 km from the surface of the Earth. When the Earth attracts the Moon, the Moon attracts the Earth with an equal and opposite force, resulting in a force Mv 2 /r, causing the Earth to move around the common center of mass with a period of one month. The part of the ocean closest to the Moon is attracted more strongly (it is closer), the water rises - and a tide arises. The part of the ocean located at a greater distance from the Moon is attracted less strongly than the land, and in this part of the ocean a hump of water also rises. Therefore, there are two tides in 24 hours. The sun also causes tides, although not so strong, because the large distance from the sun smoothes out the unevenness of attraction.

Newton revealed the nature of comets - these guests of the solar system, which have always aroused interest and even sacred horror. Newton showed that comets move in very elongated elliptical orbits, with the Sun at one focus. Their movement is determined, like the movement of planets, by gravity. But they are very small, so they can only be seen when they pass near the Sun. The comet's elliptical orbit can be measured and the time of its return to our region accurately predicted. Their regular return at the predicted times allows us to verify our observations and provides further confirmation of the law of universal gravitation.

In some cases, a comet experiences a strong gravitational disturbance while passing near large planets and moves to a new orbit with a different period. This is why we know that comets have little mass: planets influence their motion, but comets do not influence the motion of planets, although they act on them with the same force.

Comets move so fast and come so rarely that scientists are still waiting for the moment when they can apply modern means to study a large comet.

If you think about the role that gravitational forces play in the life of our planet, then entire oceans of phenomena open up, and even oceans in the literal sense of the word: oceans of water, oceans of air. Without gravity they would not exist.

A wave in the sea, all currents, all winds, clouds, the entire climate of the planet are determined by the play of two main factors: solar activity and gravity.

Gravity not only holds people, animals, water and air on Earth, but also compresses them. This compression at the Earth's surface is not so great, but its role is important.

The famous buoyant force of Archimedes appears only because it is compressed by gravity with a force that increases with depth.

The globe itself is compressed by gravitational forces to colossal pressures. At the center of the Earth, the pressure appears to exceed 3 million atmospheres.

As a creator of science, Newton created a new style that still retains its significance. As a scientific thinker, he is an outstanding founder of ideas. Newton came up with the remarkable idea of ​​universal gravitation. He left behind books on the laws of motion, gravity, astronomy and mathematics. Newton elevated astronomy; he gave it a completely new place in science and put it in order, using explanations based on the laws he created and tested.

The search for ways leading to an ever more complete and deep understanding of Universal Gravity continues. Solving great problems requires great work.

But no matter how the further development of our understanding of gravity goes, Newton’s brilliant creation of the twentieth century will always captivate with its unique daring and will always remain a great step on the path to understanding nature.

from original page N 17...

threw different masses, which are proportional to the attraction of these objects by the field. This is gravitational mass. We say that different objects have different weights because they have different gravitational masses that are attracted by the gravitational field. Thus, gravitational masses are by definition proportional to weights, as well as to the force of gravity. Gravitational mass determines the force with which a body is attracted by the Earth. In this case, gravity is mutual: if the Earth attracts a stone, then the stone also attracts the Earth. This means that the gravitational mass of a body also determines how strongly it attracts another body, the Earth. Thus, gravitational mass measures the amount of matter that is affected by gravity, or the amount of matter that causes gravitational attractions between bodies.

The gravitational attraction on two identical pieces of lead is twice as strong as on one. The gravitational masses of the lead pieces must be proportional to the inertial masses, since the masses of both types are obviously proportional to the number of lead atoms. The same applies to pieces of any other material, say wax, but how do you compare a piece of lead to a piece of wax? The answer to this question is given by a symbolic experiment to study the fall of bodies of various sizes from the top of the leaning Leaning Tower of Pisa, the one that, according to legend, was carried out by Galileo. Let's drop two pieces of any material of any size. They fall with the same acceleration g. The force acting on a body and giving it acceleration6 is the Earth's gravity applied to this body. The force of attraction of bodies by the Earth is proportional to gravitational mass. But gravity imparts the same acceleration g to all bodies. Therefore, gravity, like weight, must be proportional to the inertial mass. Consequently, bodies of any shape contain equal proportions of both masses.

If we take 1 kg as the unit of both masses, then the gravitational and inertial masses will be the same for all bodies of any size from any material and in any place.

Here's how to prove it. Let us compare the standard kilogram made of platinum6 with a stone of unknown mass. Let's compare their inertial masses by moving each of the bodies in a horizontal direction under the influence of some force and measuring the acceleration. Let's assume that the mass of the stone is 5.31 kg. Earth's gravity is not involved in this comparison. Then we compare the gravitational masses of both bodies by measuring the gravitational attraction between each of them and some third body, most simply the Earth. This can be done by weighing both bodies. We will see that the gravitational mass of the stone is also 5.31 kg.

More than half a century before Newton proposed his law of universal gravitation, Johannes Kepler (1571-1630) discovered that “the intricate motion of the planets of the solar system could be described by three simple laws. Kepler's laws strengthened belief in the Copernican hypothesis that the planets revolve around the sun, a.

To assert at the beginning of the 17th century that the planets were around the Sun, and not around the Earth, was the greatest heresy. Giordano Bruno, who openly defended the Copernican system, was condemned as a heretic by the Holy Inquisition and burned at the stake. Even the great Galileo, despite his close friendship with the Pope, was imprisoned, condemned by the Inquisition and forced to publicly renounce his views.

In those days, the teachings of Aristotle and Ptolemy, which stated that the orbits of the planets arise as a result of complex movements along a system of circles, were considered sacred and inviolable. Thus, to describe the orbit of Mars, a dozen or so circles of varying diameters were required. Johannes Kepler set out to “prove” that Mars and the Earth must revolve around the Sun. He tried to find an orbit of the simplest geometric shape that would exactly correspond to the many dimensions of the planet's position. Years of tedious calculations passed before Kepler was able to formulate three simple laws that very accurately describe the motion of all planets:

First law: Each planet moves in an ellipse, in

one of the focuses of which is

Second law: Radius vector (line connecting the Sun

and the planet) describes at equal intervals

time equal areas

Third law: Squares of planetary periods

are proportional to the cubes of their averages

distances from the Sun:

R 1 3 /T 1 2 = R 2 3 /T 2 2

The significance of Kepler's works is enormous. He discovered the laws, which Newton then connected with the law of universal gravitation. Of course, Kepler himself was not aware of what his discoveries would lead to. “He was engaged in tedious hints of empirical rules, which Newton was supposed to bring to a rational form in the future.” Kepler could not explain what caused the existence of elliptical orbits, but he admired the fact that they existed.

Based on Kepler's third law, Newton concluded that attractive forces should decrease with increasing distance and that attraction should vary as (distance) -2. Having discovered the law of universal gravitation, Newton transferred a simple idea of ​​​​the motion of the Moon to the entire planetary system. He showed that attraction, according to the laws he derived, determines the movement of planets in elliptical orbits, and the Sun should be located at one of the foci of the ellipse. He was able to easily derive two other Kepler laws, which also follow from his hypothesis of universal gravitation. These laws are valid if only the attraction of the Sun is taken into account. But it is also necessary to take into account the effect of other planets on a moving planet, although in the solar system these attractions are small compared to the attraction of the Sun.

Kepler's second law follows from the arbitrary dependence of the force of gravity on distance, if this force acts in a straight line connecting the centers of the planet and the Sun. But Kepler's first and third laws are satisfied only by the law of inverse proportionality of the forces of attraction to the square of the distance.

To obtain Kepler's third law, Newton simply combined the laws of motion with the law of gravity. For the case of circular orbits, one can reason as follows: let a planet whose mass is equal to m move with speed v in a circle of radius R around the Sun, whose mass is equal to M. This movement can only occur if the planet is acted upon by an external force F = mv 2 /R, creating centripetal acceleration v 2 /R. Let's assume that the attraction between the Sun and the planet creates the necessary force. Then:

GMm/r 2 = mv 2 /R

and the distance r between m and M is equal to the orbital radius R. But the speed

where T is the time during which the planet makes one revolution. Then

To obtain Kepler's third law, you need to transfer all R and T to one side of the equation, and all other quantities to the other:

R 3 /T 2 = GM/4p 2

If we now move to another planet with a different orbital radius and orbital period, then the new ratio will again be equal to GM/4p 2 ; this value will be the same for all planets, since G is a universal constant, and mass M is the same for all planets revolving around the Sun.