The law of universal gravitation is a short summary. What is gravity for dummies: definition and theory in simple words

So, the movement of planets, for example the Moon around the Earth or the Earth around the Sun, is the same fall, but only a fall that lasts indefinitely (in any case, if we ignore the transition of energy into “non-mechanical” forms).

The conjecture about the unity of causes governing the movement of planets and the fall of earthly bodies was expressed by scientists long before Newton. Apparently, the first to clearly express this idea was the Greek philosopher Anaxagoras, a native of Asia Minor, who lived in Athens almost two thousand years ago. He said that the Moon, if it did not move, would fall to the Earth.

However, Anaxagoras’ brilliant guess, apparently, did not have any practical impact on the development of science. She was destined to be misunderstood by her contemporaries and forgotten by her descendants. Ancient and medieval thinkers, whose attention was attracted by the movement of the planets, were very far from the correct (and more often than not any) interpretation of the causes of this movement. After all, even the great Kepler, who, at the cost of enormous labor, was able to formulate the exact mathematical laws of planetary motion, believed that the cause of this motion was the rotation of the Sun.

According to Kepler's ideas, the Sun, rotating, constantly pushes the planets into rotation. True, it remained unclear why the time of revolution of the planets around the Sun differs from the period of revolution of the Sun around its own axis. Kepler wrote about this: “if the planets did not have natural resistance, then it would be impossible to give reasons why they should not follow exactly the rotation of the Sun. But although in reality all the planets move in the same direction in which the rotation of the Sun occurs, the speed of their movement is not the same. The fact is that they mix, in certain proportions, the inertia of their own mass with the speed of their movement.”

Kepler failed to understand that the coincidence of the directions of motion of the planets around the Sun with the direction of rotation of the Sun around its axis is not associated with the laws of planetary motion, but with the origin of our solar system. An artificial planet can be launched both in the direction of rotation of the Sun and against this rotation.

Robert Hooke came much closer than Kepler to the discovery of the law of attraction of bodies. Here are his actual words from a work entitled An Attempt to Study the Motion of the Earth, published in 1674: “I will develop a theory which is in every respect consistent with the generally accepted rules of mechanics. This theory is based on three assumptions: firstly, that all celestial bodies, without exception, have a gravity directed towards their center, due to which they attract not only their own parts, but also all celestial bodies within their sphere of action. According to the second assumption, all bodies moving in a rectilinear and uniform manner will move in a straight line until they are deflected by some force and begin to describe trajectories in a circle, an ellipse, or some other less simple curve. According to the third assumption, the forces of attraction act the more strongly, the closer to them the bodies on which they act are located. I have not yet been able to establish by experience what the different degrees of attraction are. But if we develop this idea further, astronomers will be able to determine the law according to which all celestial bodies move.”

Truly, one can only be amazed that Hooke himself did not want to engage in the development of these ideas, citing being busy with other work. But a scientist appeared who made a breakthrough in this area

The history of Newton's discovery of the law of universal gravitation is quite well known. 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 one and the same arose with Newton the student, that the first calculations did not give the correct results, since the data available at that time on the distance from the Earth to the Moon were inaccurate, that 16 years later new, corrected information about this distance appeared. To explain the laws of planetary motion, Newton applied the laws of dynamics he created and the law of universal gravitation that he himself established.

He named the Galilean principle of inertia as the first law of dynamics, including it in the system of basic laws-postulates of his theory.

At the same time, Newton had to eliminate the mistake of Galileo, who believed that uniform motion in a circle was motion by inertia. Newton pointed out (and this is the second law of dynamics) that the only way to change the motion of a body - the value or direction of the velocity - is to act on it with some force. In this case, the acceleration with which a body moves under the influence of a force is inversely proportional to the mass of the body.

According to Newton's third law of dynamics, “to every action there is always an equal and opposite reaction.”

Consistently applying the principles - the laws of dynamics, he first calculated the centripetal acceleration of the Moon as it moves in orbit around the Earth, and then was able to show that the ratio of this acceleration to the acceleration of free fall of bodies at the Earth's surface is equal to the ratio of the squares of the radii of the Earth and the lunar orbit. From this Newton concluded that the nature of gravity and the force that holds the Moon in orbit are the same. In other words, according to his conclusions, the Earth and the Moon are attracted to each other with a force inversely proportional to the square of the distance between their centers Fg ≈ 1∕r2.

Newton was able to show that the only explanation for the independence of the acceleration of free fall of bodies from their mass is the proportionality of the force of gravity to the mass.

Summarizing the findings, Newton wrote: “there can be no doubt that the nature of gravity on other planets is the same as on Earth. In fact, let us imagine that the earth's bodies are raised to the orbit of the Moon and sent together with the Moon, also devoid of any movement, to fall to the Earth. Based on what has already been proven (meaning the experiments of Galileo), there is no doubt that at the same times they will pass through the same spaces as the Moon, for their masses are related to the mass of the Moon in the same way as their weights are to its weight.” So Newton discovered and then formulated the law of universal gravitation, which is rightfully the property of science.

2. Properties of gravitational 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: universal. These forces, so to speak, are “the most universal” among all the forces of nature. Everything that has mass - and mass is inherent in any form, any kind of matter - must experience gravitational influences. Even light is no exception. If we visualize gravitational forces with the help of strings that stretch from one body to another, then an innumerable number of such strings would have to permeate space anywhere. At the same time, it is worth noting that it is impossible to break such a thread and protect yourself from gravitational forces. There are no barriers to universal gravity; their radius of action is unlimited (r = ∞). Gravitational forces are long-range forces. This is the “official name” of these forces in physics. Due to long-range action, gravity connects all bodies of 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 do not change their weight when transferred from one height to another (or, to be more precise, they change, but extremely insignificantly), precisely because with a relatively small change in distance - in this case from the center of the Earth - gravitational forces practically do not change.

By the way, it is for this reason that the law of measuring gravitational forces with distance was discovered “in the sky.” All the necessary data was drawn from astronomy. One should not, however, think that a decrease in gravity with height cannot be detected under terrestrial conditions. So, for example, a pendulum clock with an oscillation period of one second will fall behind a day by almost three seconds if it is raised from the basement to the top floor of Moscow University (200 meters) - and this is only due to a decrease in gravity.

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 distance is absolutely necessary.

Gravitational forces have another very interesting and unusual property, which will be discussed now.

For many centuries, medieval science accepted as an unshakable dogma Aristotle's statement that a body falls the faster the greater its weight. Even everyday experience confirms this: it is known that a piece of fluff falls slower than a stone. However, as Galileo was able to show for the first time, the whole point here is that air resistance, coming into play, radically distorts the picture that would be if only earthly gravity acted on all bodies. There is a remarkable experiment with the so-called Newton tube, which makes it possible to very easily evaluate the role of air resistance. Here is a short description of this experience. Imagine an ordinary glass tube (so that you can see what is happening inside) in which various objects are placed: pellets, pieces of cork, feathers or fluffs, etc. If you turn the tube over so that all this can fall, then the pellet will flash faster , followed by pieces of cork and, finally, the fluff will gradually fall. But let’s try to monitor the fall of the same objects when the air is pumped out of the tube. The fluff, having lost its former slowness, rushes along, keeping pace with the pellet and the cork. This means that its movement was delayed by air resistance, which had a lesser effect on the movement of the plug and even less on the movement of the pellet. Consequently, if it were not for air resistance, if only the forces of universal gravity acted on bodies - in a particular case, gravity - then all bodies would fall exactly the same, accelerating at the same pace.

But “there is nothing new under the sun.” Two thousand years ago, Lucretius Carus wrote in his famous poem “On the Nature of Things”:

everything that falls in rare air,

Should fall faster according to its own weight

Only because water or air is a subtle essence

I am not able to put obstacles in the way of things that are the same,

But it is more likely to yield to those with greater severity.

On the contrary, I am never capable of anything anywhere

The thing holds the emptiness and appears as some kind of support,

By nature, constantly giving in to everything.

Therefore, everything, rushing through the void without obstacles,

Have the same speed despite the difference in weight.

Of course, these wonderful words were a great guess. To turn this guess into a reliably established law, it took many experiments, starting with the famous experiments of Galileo, who studied the fall of balls of the same size, but made of different materials (marble, wood, lead, etc.) from the famous leaning Leaning Tower of Pisa, and ending with the most sophisticated modern measurements of the influence of gravity on light. And all this variety of experimental data persistently strengthens us in the belief that gravitational forces impart equal acceleration to all bodies; in particular, the acceleration of free fall caused by gravity is the same for all bodies and does not depend on the composition, structure, or mass of the bodies themselves.

This seemingly simple law expresses perhaps the most remarkable feature of gravitational forces. There are literally no other forces that accelerate all bodies equally, regardless of their mass.

So, this property of the forces of universal gravity can be compressed into one short statement: the gravitational force is proportional to the mass of bodies. Let us emphasize that here we are talking about the very mass that acts as a measure of inertia in Newton’s laws. It is even called inert mass.

The four words “gravitational force is proportional to mass” contain a surprisingly deep meaning. Large and small bodies, hot and cold, of very different chemical compositions, of any structure - they all experience the same gravitational interaction if their masses are equal.

Or maybe this law is really simple? After all, Galileo, for example, considered it almost self-evident. Here is his reasoning. Let two bodies of different weights fall. According to Aristotle, a heavy body should fall faster even in vacuum. Now let's connect the bodies. Then, on the one hand, the bodies should fall faster, since the total weight has increased. But, on the other hand, adding a part to a heavy body that falls more slowly should slow down this body. There is a contradiction that can be eliminated only if we assume that all bodies under the influence of gravity alone fall with the same acceleration. It's like everything is consistent! However, let us think again about the above reasoning. It is based on the common method of proof “by contradiction”: by assuming that a heavier body falls faster than a lighter one, we have arrived at a contradiction. And from the very beginning there was an assumption that the acceleration of free fall is determined by weight and only weight. (Strictly speaking, not by weight, but by mass.)

But this is not at all obvious in advance (i.e., before the experiment). What if this acceleration was determined by the volume of the bodies? Or temperature? Let's imagine that there is a gravitational charge, similar to an electric charge and, like the latter, completely unrelated directly to mass. The comparison with electric charge is very useful. Here are two specks of dust between the charged plates of a capacitor. Let these dust grains have equal charges, and the masses are in the ratio 1 to 2. Then the accelerations should differ by a factor of two: the forces determined by the charges are equal, and with equal forces, a body with twice the mass accelerates half as much. If you connect dust particles, then, obviously, the acceleration will have a new, intermediate value. No speculative approach without an experimental study of electrical forces can give anything here. The picture would be exactly the same if the gravitational charge were not associated with mass. But only experience can answer the question of whether such a connection exists. And we now understand that it was the experiments that proved the identical acceleration due to gravity for all bodies that essentially showed that the gravitational charge (gravitational or heavy mass) is equal to the inertial mass.

Experience and only experience can serve both as a basis for physical laws and as a criterion for their validity. Let us at least recall the record-breaking precision experiments conducted under the leadership of V.B. Braginsky at Moscow State University. These experiments, in which an accuracy of about 10-12 was obtained, once again confirmed the equality of heavy and inert mass.

It is on experience, on the wide testing of nature - from the modest scale of a small laboratory of a scientist to the grandiose cosmic scale - that the law of universal gravitation is based, which (to summarize everything said above) says:

The force of mutual attraction of any two bodies whose dimensions are much smaller than the distance between them is proportional to the product of the masses of these bodies and inversely proportional to the square of the distance between these bodies.

The proportionality coefficient is called the gravitational constant. If we measure length in meters, time in seconds, and mass in kilograms, the gravitational force will always be equal to 6.673*10-11, and its dimension will be m3/kg*s2 or N*m2/kg2, respectively.

G=6.673*10-11 N*m2/kg2

3. Gravitational waves.

Newton's law of universal gravitation does not say anything about the time of transmission of gravitational interaction. It is implicitly assumed that it occurs instantly, no matter how large the distances between the interacting bodies are. This view is generally typical of supporters of action at a distance. But from Einstein’s “special theory of relativity” it follows that gravity is transmitted from one body to another at the same speed as the light signal. If some body moves from its place, then the curvature of space and time caused by it does not change instantly. First, this will affect the immediate vicinity of the body, then the change will affect more and more distant areas, and, finally, a new distribution of curvature will be established throughout space, corresponding to the changed position of the body.

And here we come to the problem that has caused and continues to cause the greatest number of disputes and disagreements - the problem of gravitational radiation.

Can gravity exist if there is no mass creating it? According to Newton's law, definitely not. It makes no sense to even raise such a question there. However, as soon as we agreed that gravitational signals are transmitted, although at a very high, but still not infinite speed, everything changes radically. Indeed, imagine that at first the mass causing gravity, for example a ball, was at rest. All bodies around the ball will be affected by ordinary Newtonian forces. Now let’s remove the ball from its original place with great speed. At first, the surrounding bodies will not feel this. After all, gravitational forces do not change instantly. It takes time for changes in the curvature of space to spread in all directions. This means that the surrounding bodies will experience the same influence of the ball for some time, when the ball itself is no longer there (at least, in the same place).

It turns out that the curvatures of space acquire a certain independence, that it is possible to tear a body out of the area of ​​space where it caused the curvatures, and in such a way that these curvatures themselves, at least over large distances, will remain and develop according to their internal laws. Here is gravity without gravitating mass! We can go further. If you make the ball oscillate, then, as it turns out from Einstein’s theory, a kind of ripple is superimposed on the Newtonian picture of gravity - gravitational waves. To better imagine these waves, you need to use a model - a rubber film. If you not only press your finger on this film, but simultaneously make oscillatory movements with it, then these vibrations will begin to be transmitted along the stretched film in all directions. This is an analogue of gravitational waves. The further away from the source, the weaker such waves are.

And now at some point we will stop putting pressure on the film. The waves won't go away. They will exist independently, scattering further and further across the film, causing geometry to bend along the way.

In exactly the same way, waves of space curvature - gravitational waves - can exist independently. Many researchers draw this conclusion from Einstein’s theory.

Of course, all these effects are very weak. For example, the energy released when one match burns is many times greater than the energy of gravitational waves emitted by our entire solar system during the same time. But what is important here is not the quantitative, but the principled side of the matter.

Proponents of gravitational waves - and they seem to be in the majority now - predict another amazing phenomenon; the transformation of gravity into particles such as electrons and positrons (they must be born in pairs), protons, antitrons, etc. (Ivanenko, Wheeler, etc.).

It should look something like this. A wave of gravity reached a certain area of ​​space. At a certain moment, this gravity sharply, abruptly, decreases and at the same time, say, an electron-positron pair appears there. The same can be described as an abrupt decrease in the curvature of space with the simultaneous birth of a pair.

There are many attempts to translate this into quantum mechanical language. Particles are introduced into consideration - gravitons, which are compared to the non-quantum image of a gravitational wave. In the physical literature, the term “transmutation of gravitons into other particles” is in circulation, and these transmutations - mutual transformations - are possible between gravitons and, in principle, any other particles. After all, there are no particles that are insensitive to gravity.

Even though such transformations are unlikely, that is, they happen extremely rarely, on a cosmic scale they can turn out to be fundamental.

4. Curvature of space-time by gravity,

"Eddington's Parable"

A parable by the English physicist Eddington from the book “Space, Time and Gravity” (retelling):

“In an ocean that has only two dimensions, there once lived a breed of flat fish. It was observed that the fish generally swam in straight lines as long as they did not encounter obvious obstacles in their path. This behavior seemed quite natural. But there was a mysterious area in the ocean; when the fish fell into it, they seemed enchanted; some sailed through this area but changed the direction of their movement, others endlessly circled around this area. One fish (almost Descartes) proposed a theory of vortices; she said that in this area there are whirlpools that make everything that gets into them spin. Over time, a much more advanced theory was proposed (Newton's theory); they said that all fish are attracted to a very large fish - the sun fish, dormant in the middle of the region - and this explained the deviation of their paths. At first this theory seemed perhaps a little strange; but it was confirmed with amazing accuracy by a wide variety of observations. All fish have been found to have this attractive property, proportionate to their size; the law of attraction (analogous to the law of universal gravitation) was extremely simple, but despite this, it explained all movements with such precision that the accuracy of scientific research had never reached before. True, some fish, grumbling, declared that they did not understand how such an action at a distance was possible; but everyone agreed that this action was carried out by the ocean, and that it would be easier to understand when the nature of water was better studied. Therefore, almost every fish that wanted to explain gravity began by suggesting some mechanism by which it spread through water.

But there was a fish who looked at things differently. She noticed the fact that the big fish and the small ones always moved along the same paths, although it might seem that it would take a lot of force to deflect the big fish from its path. (The sunfish imparted equal accelerations to all bodies.) Therefore, instead of trying, she began to study in detail the paths of movement of fish and thus came to an astonishing solution to the problem. There was a high place in the world where the sunfish lay. The fish could not directly notice this because they were two-dimensional; but when the fish in its movement fell on the slope of this elevation, then although it tried to swim in a straight line, it involuntarily turned a little to the side. This was the secret of the mysterious attraction or curvature of paths that occurred in the mysterious area. »

This parable shows how the curvature of the world in which we live can give the illusion of gravity, and we see that an effect like gravity is the only way such curvature can manifest itself.

Briefly, this can be formulated as follows. Since gravity bends the paths of all bodies in the same way, we can think of gravity as the curvature of space-time.

5. Gravity on Earth.

If you think about the role that gravitational forces play in the life of our planet, entire oceans open up. And not only oceans of phenomena, but also oceans in the literal sense of the word. Oceans of water. Air ocean. Without gravity they would not exist.

A wave in the sea, the movement of every drop of water in the rivers that feed this 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 ship is sailing on the sea. What prevents him from drowning is known to everyone. This is the famous buoyant force of Archimedes. But it appears only because the water is compressed by gravity with a force that increases with increasing depth. Inside a spacecraft in flight, there is no buoyant force, and there is no weight either. 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.

Under the influence of long-acting pressure forces under these conditions, all substances that we are accustomed to consider solid behave like pitch or resin. Heavy materials sink to the bottom (if you can call the center of the Earth that way), and light materials float to the surface. This process has been going on for billions of years. It has not ended, as follows from Schmidt’s theory, even now. The concentration of heavy elements in the region of the Earth's center is slowly increasing.

Well, how does the attraction of the Sun and the closest celestial body of the Moon manifest itself on Earth? Only residents of the ocean coasts can observe this attraction without special instruments.

The sun acts in almost the same way on everything on and inside the Earth. The force with which the Sun attracts a person at noon, when he is closest to the Sun, is almost the same as the force acting on him at midnight. After all, the distance from the Earth to the Sun is ten thousand times greater than the Earth’s diameter, and an increase in the distance by one ten-thousandth when the Earth rotates half a turn around its axis practically does not change the force of gravity. Therefore, the Sun imparts almost identical accelerations to all parts of the globe and all bodies on its surface. Almost, but still not quite the same. Because of this difference, the ebb and flow of the ocean occurs.

On the section of the earth's surface facing the Sun, the force of gravity is somewhat greater than that necessary for the movement of this section along an elliptical orbit, and on the opposite side of the Earth it is somewhat less. As a result, according to Newton's laws of mechanics, the water in the ocean bulges slightly in the direction facing the Sun, and on the opposite side it recedes from the Earth's surface. Tidal forces, as they say, arise, stretching the globe and giving, roughly speaking, the surface of the oceans the shape of an ellipsoid.

The smaller the distances between interacting bodies, the greater the tidal forces. This is why the Moon has a greater influence on the shape of the world's oceans than the Sun. More precisely, tidal influence is determined by the ratio of the mass of a body to the cube of its distance from the Earth; this ratio for the Moon is approximately twice that for the Sun.

If there were no cohesion between the parts of the globe, then tidal forces would tear it apart.

Perhaps this happened to one of Saturn's satellites when it came close to this large planet. That fragmented ring that makes Saturn such a remarkable planet may be debris from the satellite.

So, the surface of the world's oceans is like an ellipsoid, the major axis of which faces the Moon. The earth rotates around its axis. Therefore, a tidal wave moves along the surface of the ocean towards the direction of rotation of the Earth. When it approaches the shore, the tide begins. In some places the water level rises to 18 meters. Then the tidal wave goes away and the tide begins to ebb. The water level in the ocean fluctuates, on average, with a period of 12 hours. 25min. (half a lunar day).

This simple picture is greatly distorted by the simultaneous tidal action of the Sun, water friction, continental resistance, the complexity of the configuration of ocean shores and bottom in coastal zones, and some other particular effects.

It is important that the tidal wave slows down the Earth's rotation.

True, the effect is very small. Over 100 years, the day increases by a thousandth of a second. But, acting for billions of years, the braking forces will lead to the fact that the Earth will always be turned to the Moon with one side, and the Earth’s day will become equal to the lunar month. This has already happened to Luna. The Moon is slowed down so much that it always faces the Earth with one side. To “look” at the far side of the Moon, it was necessary to send a spacecraft around it.

I. Newton was able to deduce from Kepler's laws one of the fundamental laws of nature - the law of universal gravitation. Newton knew that for all planets in the solar system, acceleration is inversely proportional to the square of the distance from the planet to the Sun and the coefficient of proportionality is the same for all planets.

From here it follows, first of all, that the force of attraction acting from the Sun on a planet must be proportional to the mass of this planet. In fact, if the acceleration of the planet is given by formula (123.5), then the force causing the acceleration

where is the mass of this planet. On the other hand, Newton knew the acceleration that the Earth imparts to the Moon; it was determined from observations of the movement of the Moon as it orbits the Earth. This acceleration is approximately one times less than the acceleration imparted by the Earth to bodies located near the Earth's surface. The distance from the Earth to the Moon is approximately equal to the Earth's radii. In other words, the Moon is several times farther from the center of the Earth than bodies located on the surface of the Earth, and its acceleration is several times less.

If we accept that the Moon moves under the influence of the Earth's gravity, then it follows that the force of the Earth's gravity, like the force of the Sun's gravity, decreases in inverse proportion to the square of the distance from the center of the Earth. Finally, the force of gravity of the Earth is directly proportional to the mass of the attracted body. Newton established this fact in experiments with pendulums. He discovered that the period of swing of a pendulum does not depend on its mass. This means that the Earth imparts the same acceleration to pendulums of different masses, and, consequently, the force of gravity of the Earth is proportional to the mass of the body on which it acts. The same, of course, follows from the same acceleration of gravity for bodies of different masses, but experiments with pendulums make it possible to verify this fact with greater accuracy.

These similar features of the gravitational forces of the Sun and the Earth led Newton to the conclusion that the nature of these forces is the same and that there are forces of universal gravity acting between all bodies and decreasing in inverse proportion to the square of the distance between the bodies. In this case, the gravitational force acting on a given body of mass must be proportional to the mass.

Based on these facts and considerations, Newton formulated the law of universal gravitation in this way: any two bodies are attracted to each other with a force that is directed along the line connecting them, directly proportional to the masses of both bodies and inversely proportional to the square of the distance between them, i.e. mutual gravitational force

where and are the masses of bodies, is the distance between them, and is the coefficient of proportionality, called the gravitational constant (the method of measuring it will be described below). Combining this formula with formula (123.4), we see that , where is the mass of the Sun. The forces of universal gravity satisfy Newton's third law. This was confirmed by all astronomical observations of the movement of celestial bodies.

In this formulation, the law of universal gravitation is applicable to bodies that can be considered material points, i.e., to bodies the distance between which is very large compared to their sizes, otherwise it would be necessary to take into account that different points of bodies are separated from each other at different distances . For homogeneous spherical bodies, the formula is valid for any distance between the bodies, if we take the distance between their centers as the value. In particular, in the case of attraction of a body by the Earth, the distance must be counted from the center of the Earth. This explains the fact that the force of gravity almost does not decrease as the height above the Earth increases (§ 54): since the radius of the Earth is approximately 6400, then when the position of the body above the Earth’s surface changes within even tens of kilometers, the force of gravity of the Earth remains practically unchanged.

The gravitational constant can be determined by measuring all other quantities included in the law of universal gravitation for any specific case.

It was possible for the first time to determine the value of the gravitational constant using torsion balances, the structure of which is schematically shown in Fig. 202. A light rocker, at the ends of which two identical balls of mass are attached, is hung on a long and thin thread. The rocker arm is equipped with a mirror, which allows optical measurement of small rotations of the rocker arm around the vertical axis. Two balls of significantly greater mass can be approached from different sides to the balls.

Rice. 202. Scheme of torsion balances for measuring the gravitational constant

The forces of attraction of small balls to large ones create a pair of forces that rotate the rocker clockwise (when viewed from above). By measuring the angle at which the rocker arm rotates when approaching the balls of the balls, and knowing the elastic properties of the thread on which the rocker arm is suspended, it is possible to determine the moment of the pair of forces with which the masses are attracted to the masses. Since the masses of the balls and the distance between their centers (at a given position of the rocker) are known, the value can be found from formula (124.1). It turned out to be equal

After the value was determined, it turned out to be possible to determine the mass of the Earth from the law of universal gravitation. Indeed, in accordance with this law, a body of mass located at the surface of the Earth is attracted to the Earth with a force

where is the mass of the Earth, and is its radius. On the other hand, we know that . Equating these quantities, we find

.

Thus, although the forces of universal gravity acting between bodies of different masses are equal, a body of small mass receives significant acceleration, and a body of large mass experiences low acceleration.

Since the total mass of all the planets of the Solar System is slightly more than the mass of the Sun, the acceleration that the Sun experiences as a result of the action of gravitational forces on it from the planets is negligible compared to the accelerations that the gravitational force of the Sun imparts to the planets. The gravitational forces acting between the planets are also relatively small. Therefore, when considering the laws of planetary motion (Kepler's laws), we did not take into account the motion of the Sun itself and approximately assumed that the trajectories of the planets were elliptical orbits, in one of the foci of which the Sun was located. However, in accurate calculations it is necessary to take into account those “perturbations” that gravitational forces from other planets introduce into the movement of the Sun itself or any planet.

124.1. How much will the force of gravity acting on a rocket projectile decrease when it rises 600 km above the Earth's surface? The radius of the Earth is taken to be 6400 km.

124.2. The mass of the Moon is 81 times less than the mass of the Earth, and the radius of the Moon is approximately 3.7 times less than the radius of the Earth. Find the weight of a person on the Moon if his weight on Earth is 600N.

124.3. The mass of the Moon is 81 times less than the mass of the Earth. Find on the line connecting the centers of the Earth and the Moon the point at which the gravitational forces of the Earth and the Moon acting on a body placed at this point are equal to each other.

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

Law of Gravity

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

Mathematically we can express this great law by the formula

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

Let's remember the acceleration of free fall. This is the effect of the law of attraction. Under the influence of force, a body changes speed the more slowly, the greater its mass. As a result, all bodies fall to Earth with the same acceleration.

What causes this invisible unique force? Today the existence of a gravitational field is known and proven. You can learn more about the nature of the gravitational field in the additional material on the topic.

Think about it, what is gravity? Where is it from? What is it? Surely it cannot be that the planet looks at the Sun, sees how far away it is, and calculates the inverse square of the distance in accordance with this law?

Direction of gravity

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

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

The main thing to remember

Some methods of geological exploration, tide prediction and, more recently, calculation of the movement of artificial satellites and interplanetary stations. Advance calculation of planetary positions.

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

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

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

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

The force here also varies inversely with the square of the distance, but the difference in the magnitude of the electrical and gravitational forces is striking. Trying to establish the general nature of gravity and electricity, we discover such a superiority of electrical forces over the forces of gravity that it is difficult to believe that both have the same source. How can you say that one is more powerful than the other? After all, everything depends on what the mass is and what the charge is. When discussing how strongly gravity acts, you have no right to say: “Let's take a mass of such and such a size,” because you choose it yourself. But if we take what Nature itself offers us (her own numbers and measures, which have nothing to do with our inches, years, with our measures), then we will be able to compare. We take an elementary charged particle, such as an electron. Two elementary particles, two electrons, due to an electric charge, repel each other with a force inversely proportional to the square of the distance between them, and due to gravity they are attracted to each other again with a force inversely proportional to the square of the distance.

Question: What is the ratio of gravitational force to electrical force? Gravity is to electrical repulsion as one is to a number with 42 zeros. This causes deepest bewilderment. Where could such a huge number come from?

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

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

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

When he came to a great result: the same cause causes phenomena of an amazingly wide range - from the fall of a thrown stone to the Earth to the movement of huge cosmic bodies. Newton found this reason and was able to accurately express it in the form of one formula - the law of universal gravitation.

Since the force of universal gravitation imparts the same acceleration to all bodies regardless of their mass, it must be proportional to the mass of the body on which it acts:

But since, for example, the Earth acts on the Moon with a force proportional to the mass of the Moon, then the Moon, according to Newton’s third law, must act on the Earth with the same force. Moreover, this force must be proportional to the mass of the Earth. If the force of gravity is truly universal, then from the side of a given body a force must act on any other body proportional to the mass of this other body. Consequently, the force of universal gravity must be proportional to the product of the masses of interacting bodies. This leads to the formulation law of universal gravitation.

Definition of the law of universal gravitation

The force of mutual attraction between two bodies is directly proportional to the product of the masses of these bodies and inversely proportional to the square of the distance between them:

Proportionality factor G called gravitational constant.

The gravitational constant is numerically equal to the force of attraction between two material points weighing 1 kg each, if the distance between them is 1 m. After all, when m 1 = m 2=1 kg and R=1 m we get G=F(numerically).

It must be borne in mind that the law of universal gravitation (4.5) as a universal law is valid for material points. In this case, the forces of gravitational interaction are directed along the line connecting these points ( Fig.4.2). This kind of force is called central.

It can be shown that homogeneous bodies shaped like a ball (even if they cannot be considered material points) also interact with the force determined by formula (4.5). In this case R- the distance between the centers of the balls. The forces of mutual attraction lie on a straight line passing through the centers of the balls. (Such forces are called central.) The bodies that we usually consider falling on the Earth have dimensions much smaller than the Earth’s radius ( R≈6400 km). Such bodies can, regardless of their shape, be considered as material points and determine the force of their attraction to the Earth using the law (4.5), keeping in mind that R is the distance from a given body to the center of the Earth.

Determination of the gravitational constant

Now let's find out how to find the gravitational constant. First of all, we note that G has a specific name. This is due to the fact that the units (and, accordingly, the names) of all quantities included in the law of universal gravitation have already been established earlier. The law of gravitation provides a new connection between known quantities with certain names of units. That is why the coefficient turns out to be a named quantity. Using the formula of the law of universal gravitation, it is easy to find the name of the SI unit of gravitational constant:

N m 2 / kg 2 = m 3 / (kg s 2).

For quantification G it is necessary to independently determine all the quantities included in the law of universal gravitation: both masses, force and distance between bodies. It is impossible to use astronomical observations for this, since the masses of the planets, the Sun, and the Earth can only be determined on the basis of the law of universal gravitation itself, if the value of the gravitational constant is known. The experiment must be carried out on Earth with bodies whose masses can be measured on a scale.

The difficulty is that the gravitational forces between bodies of small masses are extremely small. It is for this reason that we do not notice the attraction of our body to surrounding objects and the mutual attraction of objects to each other, although gravitational forces are the most universal of all forces in nature. Two people with masses of 60 kg at a distance of 1 m from each other are attracted with a force of only about 10 -9 N. Therefore, to measure the gravitational constant, fairly subtle experiments are needed.

The gravitational constant was first measured by the English physicist G. Cavendish in 1798 using an instrument called a torsion balance. The diagram of the torsion balance is shown in Figure 4.3. A light rocker with two identical weights at the ends is suspended from a thin elastic thread. Two heavy balls are fixed motionless nearby. Gravitational forces act between the weights and the stationary balls. Under the influence of these forces, the rocker turns and twists the thread. By the angle of twist you can determine the force of attraction. To do this, you only need to know the elastic properties of the thread. The masses of the bodies are known, and the distance between the centers of interacting bodies can be directly measured.

From these experiments the following value for the gravitational constant was obtained:

Only in the case when bodies of enormous mass interact (or at least the mass of one of the bodies is very large) does the gravitational force reach a large value. For example, the Earth and the Moon are attracted to each other with a force F≈2 10 20 H.

Dependence of the acceleration of free falling bodies on geographic latitude

One of the reasons for the increase in the acceleration of gravity when the point where the body is located moves from the equator to the poles is that the globe is somewhat flattened at the poles and the distance from the center of the Earth to its surface at the poles is less than at the equator. Another, more significant reason is the rotation of the Earth.

Equality of inertial and gravitational masses

The most striking property of gravitational forces is that they impart the same acceleration to all bodies, regardless of their masses. What would you say about a football player whose kick would be equally accelerated by an ordinary leather ball and a two-pound weight? Everyone will say that this is impossible. But the Earth is just such an “extraordinary football player” with the only difference that its effect on bodies is not of the nature of a short-term blow, but continues continuously for billions of years.

The extraordinary property of gravitational forces, as we have already said, is explained by the fact that these forces are proportional to the masses of both interacting bodies. This fact cannot but cause surprise if you think about it carefully. After all, the mass of a body, which is included in Newton’s second law, determines the inertial properties of the body, that is, its ability to acquire a certain acceleration under the influence of a given force. It is natural to call this mass inert mass and denote by m and.

It would seem, what relation can it have to the ability of bodies to attract each other? The mass that determines the ability of bodies to attract each other should be called gravitational mass m g.

It does not at all follow from Newtonian mechanics that the inertial and gravitational masses are the same, i.e. that

Equality (4.6) is a direct consequence of experiment. It means that we can simply talk about the mass of a body as a quantitative measure of both its inertial and gravitational properties.

The law of universal gravitation is one of the most universal laws of nature. It is valid for any bodies with mass.

The meaning of the law of universal gravitation

But if we approach this topic more radically, it turns out that the law of universal gravitation does not have the possibility of its application everywhere. This law has found its application for bodies that have the shape of a ball, it can be used for material points, and it is also acceptable for a ball having a large radius, where this ball can interact with bodies much smaller than its size.

As you may have guessed from the information provided in this lesson, the law of universal gravitation is the basis in the study of celestial mechanics. And as you know, celestial mechanics studies the movement of planets.

Thanks to this law of universal gravitation, it became possible to more accurately determine the location of celestial bodies and the ability to calculate their trajectory.

But for a body and an infinite plane, as well as for the interaction of an infinite rod and a ball, this formula cannot be applied.

With the help of this law, Newton was able to explain not only how the planets move, but also why sea tides arise. Over time, thanks to the work of Newton, astronomers managed to discover such planets of the solar system as Neptune and Pluto.

The importance of the discovery of the law of universal gravitation lies in the fact that with its help it became possible to make forecasts of solar and lunar eclipses and accurately calculate the movements of spacecraft.

The forces of universal gravity are the most universal of all the forces of nature. After all, their action extends to the interaction between any bodies that have mass. And as you know, any body has mass. The forces of gravity act through any body, since there are no barriers to the forces of gravity.

Task

And now, in order to consolidate knowledge about the law of universal gravitation, let's try to consider and solve an interesting problem. The rocket rose to a height h equal to 990 km. Determine how much the force of gravity acting on the rocket at a height h has decreased compared to the force of gravity mg acting on it at the surface of the Earth? The radius of the Earth is R = 6400 km. Let us denote by m the mass of the rocket, and by M the mass of the Earth.

At height h the force of gravity is:

From here we calculate:

Substituting the value will give the result:

The legend about how Newton discovered the law of universal gravitation after hitting the top of his head with an apple was invented by Voltaire. Moreover, Voltaire himself assured that this true story was told to him by Newton’s beloved niece Katherine Barton. It’s just strange that neither the niece herself nor her very close friend Jonathan Swift ever mentioned the fateful apple in their memoirs about Newton. By the way, Isaac Newton himself, writing in detail in his notebooks the results of experiments on the behavior of different bodies, noted only vessels filled with gold, silver, lead, sand, glass, water or wheat, not to mention an apple. However, this did not stop Newton’s descendants from taking tourists around the garden on the Woolstock estate and showing them that same apple tree before the storm destroyed it.

Yes, there was an apple tree, and apples probably fell from it, but how great was the merit of the apple in the discovery of the law of universal gravitation?

The debate about the apple has not subsided for 300 years, just like the debate about the law of universal gravitation itself or about who has the priority of discovery.uk

G.Ya.Myakishev, B.B.Bukhovtsev, N.N.Sotsky, Physics 10th grade

… Let mortals rejoice that such an adornment of the human race lived among them.

(Inscription on Isaac Newton's grave)

Every schoolchild knows the beautiful legend about how Isaac Newton discovered the law of universal gravitation: an apple fell on the great scientist’s head, and instead of getting angry, Isaac wondered why this happened? Why does the Earth attract everything, but what is thrown always falls down?

But most likely it was a beautiful legend invented later. In reality, Newton had to do difficult and painstaking work to discover his law. We want to tell you about how the great scientist discovered his famous law.

The principles of the natural scientist

Isaac Newton lived at the turn of the 17th and 18th centuries (1642-1727). Life at this time was completely different. Europe was rocked by wars, and in 1666, England, where Newton lived, was struck by a terrible epidemic called the “Black Death.” This event would later be called the “Great Plague of London.” Many of the sciences were just emerging; there were few educated people, as well as what they knew.

For example, a modern weekly newspaper contains more information than the average person at that time would learn in his entire life!

Despite all these difficulties, there were people who strived for knowledge, made discoveries and moved progress forward. One of them was the great English scientist Isaac Newton.

The principles that he called “rules of philosophizing” helped the scientist make his main discoveries.

Rule 1.“No other causes should be accepted in nature other than those that are true and sufficient to explain phenomena... nature does nothing in vain, and it would be in vain for many to do what can be done by fewer. Nature is simple and does not luxury with superfluous causes of things...”

The essence of this rule is that if we can exhaustively explain a new phenomenon by existing laws, then we should not introduce new ones. This rule in general form is called Occam's razor.

Rule 2.“In experimental physics, propositions derived from occurring phenomena using induction (that is, the method of induction), despite the possibility of assumptions contrary to them, should be revered as true, either exactly or approximately, until such phenomena are discovered by which they are further clarified or will be subject to exclusion.” This means that all laws of physics must be proven or disproved experimentally.

In his principles of philosophizing, Newton formulated the principles scientific method. Modern physics successfully explores and applies phenomena whose nature has not yet been clarified (for example, elementary particles). Since Newton, natural science has developed in the firm belief that the world can be known and that Nature is organized according to simple mathematical principles. This confidence became the philosophical basis for the tremendous progress of science and technology in human history.

Shoulders of Giants

You probably haven't heard of the Danish alchemist Quiet Brahe. However, it was he who was Kepler's teacher and the first to compile an accurate table of planetary movements based on his observations. It should be noted that these tables merely represented the coordinates of the planets in the sky. Quietly bequeathed them Johannes Kepler, to his student, who, after carefully studying these tables, realized that the movement of the planets is subject to a certain pattern. Kepler formulated them as follows:

1. All planets move around in an ellipse, with the Sun at one of the focuses.
2. The radius drawn from the Sun to the planet “sweeps” equal areas in equal periods of time.
3. The squares of the periods of two planets (T 1 and T 2) are related as the cubes of the semi-major axes of their orbits (R 1 and R 2):

What immediately strikes the eye is that the Sun plays a special role in these laws. But Kepler could not explain this role, just as he could not explain the reason for the movement of planets around the Sun.

Isaac Newton will once say that if he saw further than others, it was only because he stood on the shoulders of giants. He undertook to find the root cause of Kepler's laws.

World Law

Newton realized that in order to change the speed of a body, it is necessary to apply a force to it. Today every schoolchild knows this statement as Newton's first law: the change in the speed of a body per unit time (in other words, acceleration a) is directly proportional to the force (F), and inversely proportional to the mass of the body (m). The greater the mass of the body, the more effort we must expend to change its speed. Please note that Newton uses only one characteristic of a body - its mass, without considering its shape, what it is made of, what color it is, etc. This is an example of the use of Occam's razor. Newton believed that body mass is a necessary and sufficient “factor” to describe the interaction of bodies:

Newton imagined the planets as large bodies that move in a circle (or nearly a circle). In everyday life, he often observed a similar movement: children played with a ball to which a thread was tied, they twirled it over their heads. In this case, Newton saw the ball (planet) and that it was moving in a circle, but did not see the thread. Drawing a similar analogy and using his rules of philosophizing, Newton realized that it was necessary to look for a certain force - a “thread” that connects the planets and the Sun. Further reasoning was simplified after Newton applied his own laws of dynamics.

Newton, using his first law and Kepler's third law, obtained:

Thus, Newton determined that the Sun acts on the planets with force:

He also realized that all planets revolve around the Sun, and considered it natural that the mass of the Sun should be taken into account in the constant:

It was in this form that the law of universal gravitation corresponded to Kepler's observations and his laws of planetary motion. The value G = 6.67 x 10 (-11) H (m/kg) 2 was derived from observations of the planets. Thanks to this law, the movements of celestial bodies were described, and, moreover, we were able to predict the existence of objects invisible to us. In 1846, scientists calculated the orbit of a previously unknown planet, which by its existence influenced the movement of other planets in the solar system. It was .

Newton believed that simple principles and “mechanisms of interaction” underlie the most complex things. That is why he was able to discern a pattern in the observations of his predecessors and formulate it into the Law of Universal Gravitation.