Abstract. Universal gravity. What is the law of universal gravitation: the formula of the great discovery Determine the force of gravity between the earth and the sun

The most important phenomenon constantly studied by physicists is movement. Electromagnetic phenomena, laws of mechanics, thermodynamic and quantum processes - all this is a wide range of fragments of the universe studied by physics. And all these processes come down, one way or another, to one thing - to.

Everything in the Universe moves. Gravity is a common phenomenon for all people since childhood; we were born in the gravitational field of our planet; this physical phenomenon is perceived by us at the deepest intuitive level and, it would seem, does not even require study.

But, alas, the question is why and how do all bodies attract each other, remains to this day not fully disclosed, although it has been studied far and wide.

In this article we will look at what universal attraction is according to Newton - the classical theory of gravity. However, before moving on to formulas and examples, we will talk about the essence of the problem of attraction and give it a definition.

Perhaps the study of gravity became the beginning of natural philosophy (the science of understanding the essence of things), perhaps natural philosophy gave rise to the question of the essence of gravity, but, one way or another, the question of the gravitation of bodies became interested in ancient Greece.

Movement was understood as the essence of the sensory characteristic of the body, or rather, the body moved while the observer saw it. If we cannot measure, weigh, or feel a phenomenon, does this mean that this phenomenon does not exist? Naturally, it doesn't mean that. And since Aristotle understood this, reflections began on the essence of gravity.

As it turns out today, after many tens of centuries, gravity is the basis not only of gravity and the attraction of our planet to, but also the basis for the origin of the Universe and almost all existing elementary particles.

Movement task

Let's conduct a thought experiment. Let's take in left hand small ball. Let's take the same one on the right. Let's release the right ball and it will begin to fall down. The left one remains in the hand, it is still motionless.

Let's mentally stop the passage of time. The falling right ball “hangs” in the air, the left one still remains in the hand. The right ball is endowed with the “energy” of movement, the left one is not. But what is the deep, meaningful difference between them?

Where, in what part of the falling ball is it written that it should move? It has the same mass, the same volume. It has the same atoms, and they are no different from the atoms of a ball at rest. Ball has? Yes, this is the correct answer, but how does the ball know what has potential energy, where is it recorded in it?

This is precisely the task that Aristotle, Newton and Albert Einstein set themselves. And all three brilliant thinkers partly solved this problem for themselves, but today there are a number of issues that require resolution.

Newton's gravity

In 1666, the greatest English physicist and mechanic I. Newton discovered a law that can quantitatively calculate the force due to which all matter in the Universe tends to each other. This phenomenon is called universal gravity. When you are asked: “Formulate the law of universal gravitation,” your answer should sound like this:

The force of gravitational interaction contributing to the attraction of two bodies is located in direct proportion to the masses of these bodies and in inverse proportion to the distance between them.

Important! Newton's law of attraction uses the term "distance". This term should be understood not as the distance between the surfaces of bodies, but as the distance between their centers of gravity. For example, if two balls of radii r1 and r2 lie on top of each other, then the distance between their surfaces is zero, however gravity There is. The thing is that the distance between their centers r1+r2 is different from zero. On a cosmic scale, this clarification is not important, but for a satellite in orbit, this distance is equal to the height above the surface plus the radius of our planet. The distance between the Earth and the Moon is also measured as the distance between their centers, not their surfaces.

For the law of gravity the formula is as follows:

,

• F – force of attraction,
• – masses,
• r – distance,
• G – gravitational constant equal to 6.67·10−11 m³/(kg·s²).

What is weight, if we just looked at the force of gravity?

Strength is vector quantity, however, in the law of universal gravitation it is traditionally written as a scalar. In a vector picture, the law will look like this:

.

But this does not mean that the force is inversely proportional to the cube of the distance between the centers. The relation should be perceived as a unit vector directed from one center to another:

.

Law of Gravitational Interaction

Weight and gravity

Having considered the law of gravity, one can understand that it is not surprising that we personally we feel the Sun's gravity much weaker than the Earth's. Although the massive Sun has a large mass, it is very far from us. is also far from the Sun, but it is attracted to it, since it has a large mass. How to find the gravitational force of two bodies, namely, how to calculate the gravitational force of the Sun, Earth and you and me - we will deal with this issue a little later.

As far as we know, the force of gravity is:

where m is our mass, and g is the acceleration of free fall of the Earth (9.81 m/s 2).

Important! There are not two, three, ten types of attractive forces. Gravity is the only force that gives quantitative characteristics attraction. Weight (P = mg) and gravitational force are the same thing.

If m is our mass, M is the mass of the globe, R is its radius, then the gravitational force acting on us is equal to:

Thus, since F = mg:

.

The masses m are reduced, and the expression for the acceleration of free fall remains:

As we can see, the acceleration of free fall is indeed constant, since its formula includes constant quantities - radius, mass of the Earth and gravitational constant. Substituting the values ​​of these constants, we will make sure that the acceleration of gravity is equal to 9.81 m/s 2.

At different latitudes, the radius of the planet is slightly different, since the Earth is still not a perfect sphere. Because of this, the acceleration of free fall at individual points on the globe is different.

Let's return to the attraction of the Earth and the Sun. Let's try to prove with an example that the globe attracts you and me more strongly than the Sun.

For convenience, let’s take the mass of a person: m = 100 kg. Then:

• The distance between a person and the globe equal to the radius of the planet: R = 6.4∙10 6 m.
• The mass of the Earth is: M ≈ 6∙10 24 kg.
• The mass of the Sun is: Mc ≈ 2∙10 30 kg.
• Distance between our planet and the Sun (between the Sun and man): r=15∙10 10 m.

Gravitational attraction between man and Earth:

This result is quite obvious from the more simple expression for weight (P = mg).

The force of gravitational attraction between man and the Sun:

As we can see, our planet attracts us almost 2000 times stronger.

How to find the force of attraction between the Earth and the Sun? As follows:

Now we see that the Sun attracts our planet more than a billion billion times stronger than the planet attracts you and me.

First escape velocity

After Isaac Newton discovered the law of universal gravitation, he became interested in how fast a body must be thrown so that it, having overcome the gravitational field, leaves the globe forever.

True, he imagined it a little differently, in his understanding it was not a vertically standing rocket aimed at the sky, but a body that horizontally made a jump from the top of a mountain. This was a logical illustration because At the top of the mountain the force of gravity is slightly less.

So, at the top of Everest, the acceleration of gravity will not be the usual 9.8 m/s 2 , but almost m/s 2 . It is for this reason that the air there is so thin, the air particles are no longer as tied to gravity as those that “fell” to the surface.

Let's try to find out what escape velocity is.

The first escape velocity v1 is the speed at which the body leaves the surface of the Earth (or another planet) and enters a circular orbit.

Let's try to find out the numerical value of this value for our planet.

Let's write down Newton's second law for a body that rotates around a planet in a circular orbit:

,

where h is the height of the body above the surface, R is the radius of the Earth.

In orbit, a body is subject to centrifugal acceleration, thus:

.

The masses are reduced, we get:

,

This speed called the first escape velocity:

As you can see, escape velocity is absolutely independent of body mass. Thus, any object accelerated to a speed of 7.9 km/s will leave our planet and enter its orbit.

First escape velocity

Second escape velocity

However, even having accelerated the body to the first escape velocity, we will not be able to completely break its gravitational connection with the Earth. This is why we need a second escape velocity. When this speed is reached the body leaves the planet's gravitational field and all possible closed orbits.

Important! It is often mistakenly believed that in order to get to the Moon, astronauts had to reach the second escape velocity, because they first had to “disconnect” from the gravitational field of the planet. This is not so: the Earth-Moon pair are in the Earth’s gravitational field. Their common center of gravity is inside the globe.

In order to find this speed, let's pose the problem a little differently. Let's say a body flies from infinity to a planet. Question: what speed will be reached on the surface upon landing (without taking into account the atmosphere, of course)? This is exactly the speed the body will need to leave the planet.

The law of universal gravitation. Physics 9th grade

Law of Universal Gravitation.

Conclusion

We learned that although gravity is the main force in the Universe, many of the reasons for this phenomenon still remain a mystery. We learned what Newton's force of universal gravitation is, learned to calculate it for various bodies, and also studied some useful consequences that follow from such a phenomenon as universal law gravity.

The law of universal gravitation was discovered by Newton in 1687 while studying the motion of the moon's satellite around the Earth. The English physicist clearly formulated a postulate characterizing the forces of attraction. In addition, by analyzing Kepler's laws, Newton calculated that gravitational forces must exist not only on our planet, but also in space.

Background

The law of universal gravitation was not born spontaneously. Since ancient times, people have studied the sky, mainly to compile agricultural calendars, calculate important dates, religious holidays. Observations indicated that in the center of the “world” there is a Luminary (Sun), around which celestial bodies rotate in orbits. Subsequently, the dogmas of the church did not allow this to be considered, and people lost the knowledge accumulated over thousands of years.

In the 16th century, before the invention of telescopes, a galaxy of astronomers appeared who looked at the sky in a scientific way, discarding the prohibitions of the church. T. Brahe, having been observing space for many years, systematized the movements of the planets with special care. These highly accurate data helped I. Kepler subsequently discover his three laws.

By the time of the discovery (1667) by Isaac Newton of the law of gravity in astronomy, it was finally established heliocentric system world of N. Copernicus. According to it, each of the planets of the system rotates around the Sun in orbits that, with an approximation sufficient for many calculations, can be considered circular. IN early XVII V. I. Kepler, analyzing the works of T. Brahe, established kinematic laws characterizing the movements of the planets. The discovery became the foundation for elucidating the dynamics of planetary motion, that is, the forces that determine exactly this type of their motion.

Description of interaction

Unlike short-period weak and strong interactions, gravity and electromagnetic fields have long-range properties: their influence is manifested over gigantic distances. Mechanical phenomena in the macrocosm are affected by 2 forces: electromagnetic and gravitational. The influence of planets on satellites, the flight of an thrown or launched object, the floating of a body in a liquid - in each of these phenomena gravitational forces act. These objects are attracted by the planet and gravitate towards it, hence the name “law of universal gravitation”.

It has been proven that there is certainly a force of mutual attraction between physical bodies. Phenomena such as the fall of objects to the Earth, the rotation of the Moon and planets around the Sun, occurring under the influence of the forces of universal gravity, are called gravitational.

Law of universal gravitation: formula

Universal gravity is formulated as follows: any two material objects are attracted to each other with a certain force. The magnitude of this force is directly proportional to the product of the masses of these objects and inversely proportional to the square of the distance between them:

In the formula, m1 and m2 are the masses of the material objects being studied; r is the distance determined between the centers of mass of the calculated objects; G is a constant gravitational quantity expressing the force with which the mutual attraction of two objects weighing 1 kg each, located at a distance of 1 m, occurs.

What does the force of attraction depend on?

The law of gravity works differently depending on the region. Since the force of gravity depends on the values ​​of latitude in a certain area, similarly, the acceleration of free fall has different meanings in different places. The force of gravity and, accordingly, the acceleration of free fall have a maximum value at the Earth's poles - the force of gravity at these points is equal to the force of attraction. The minimum values ​​will be at the equator.

The globe is slightly flattened, its polar radius is approximately 21.5 km less than the equatorial radius. However, this dependence is less significant compared to the daily rotation of the Earth. Calculations show that due to the oblateness of the Earth at the equator, the magnitude of the acceleration due to gravity is slightly less than its value at the pole by 0.18%, and after daily rotation - by 0.34%.

However, in the same place on Earth, the angle between the direction vectors is small, so the discrepancy between the force of attraction and the force of gravity is insignificant, and it can be neglected in calculations. That is, we can assume that the modules of these forces are the same - the acceleration of gravity near the Earth’s surface is the same everywhere and is approximately 9.8 m/s².

Conclusion

Isaac Newton was a scientist who made a scientific revolution, completely rebuilt the principles of dynamics and based on them created scientific picture peace. His discovery influenced the development of science and the creation of material and spiritual culture. It fell to Newton's fate to revise the results of the idea of ​​the world. In the 17th century scientists have completed the grandiose work of building the foundation new science- physicists.

Why does a stone released from your hands fall to Earth? Because he is attracted by the Earth, each of you will say. In fact, the stone falls to the Earth with the acceleration of gravity. Consequently, a force directed towards the Earth acts on the stone from the side of the Earth. According to Newton's third law, the stone acts on the Earth with the same magnitude force directed towards the stone. In other words, forces of mutual attraction act between the Earth and the stone.

Newton was the first to first guess and then strictly prove that the reason that causes a stone to fall to the Earth, the movement of the Moon around the Earth and the planets around the Sun is the same. This is the force of gravity acting between any bodies in the Universe. Here is the course of his reasoning, given in Newton’s main work, “The Mathematical Principles of Natural Philosophy”:

“A stone thrown horizontally will deviate under the influence of gravity from a straight path and, having described a curved trajectory, will finally fall to the Earth. If you throw it at a higher speed, it will fall further” (Fig. 1).

Continuing these arguments, Newton comes to the conclusion that if it were not for air resistance, then the trajectory of a stone thrown from a high mountain at a certain speed could become such that it would never reach the surface of the Earth at all, but would move around it “like “how the planets describe their orbits in celestial space.”

Now we have become so familiar with the movement of satellites around the Earth that there is no need to explain Newton’s thought in more detail.

So, according to Newton, the movement of the Moon around the Earth or the planets around the Sun is also a free fall, but only a fall that lasts, without stopping, for billions of years. The reason for such a “fall” (whether we are really talking about the fall of an ordinary stone to the Earth or the movement of planets in their orbits) is the force of universal gravity. What does this force depend on?

Dependence of gravitational force on the mass of bodies

Galileo proved that during free fall the Earth imparts the same acceleration to all bodies in a given place, regardless of their mass. But according to Newton's second law, acceleration is inversely proportional to mass. How can we explain that the acceleration imparted to a body by the force of gravity of the Earth is the same for all bodies? This is possible only if the force of gravity towards the Earth is directly proportional to the mass of the body. In this case, increasing the mass m, for example, by doubling will lead to an increase in the force modulus F also doubled, and the acceleration, which is equal to \(a = \frac (F)(m)\), will remain unchanged. Generalizing this conclusion for gravitational forces between any bodies, we conclude that the force of universal gravity is directly proportional to the mass of the body on which this force acts.

But at least two bodies are involved in mutual attraction. Each of them, according to Newton’s third law, is acted upon by gravitational forces of equal magnitude. Therefore, each of these forces must be proportional to both the mass of one body and the mass of the other body. Therefore, the force of universal gravity between two bodies is directly proportional to the product of their masses:

\(F \sim m_1 \cdot m_2\)

Dependence of gravitational force on the distance between bodies

It is well known from experience that the acceleration of gravity is 9.8 m/s 2 and it is the same for bodies falling from a height of 1, 10 and 100 m, i.e. it does not depend on the distance between the body and the Earth. This seems to mean that force does not depend on distance. But Newton believed that distances should be counted not from the surface, but from the center of the Earth. But the radius of the Earth is 6400 km. It is clear that several tens, hundreds or even thousands of meters above the Earth’s surface cannot noticeably change the value of the acceleration of gravity.

To find out how the distance between bodies affects the strength of their mutual attraction, it would be necessary to find out what the acceleration of bodies distant from the Earth at sufficiently large distances is. However, it is difficult to observe and study the free fall of a body from a height of thousands of kilometers above the Earth. But nature itself came to the rescue here and made it possible to determine the acceleration of a body moving in a circle around the Earth and therefore possessing centripetal acceleration, caused, of course, by the same force of attraction to the Earth. Such a body is natural satellite Earth - Moon. If the force of attraction between the Earth and the Moon did not depend on the distance between them, then centripetal acceleration The Moon would be the same as the acceleration of a body freely falling near the surface of the Earth. In reality, the centripetal acceleration of the Moon is 0.0027 m/s 2 .

Let's prove it. The rotation of the Moon around the Earth occurs under the influence of the gravitational force between them. Approximately, the orbit of the Moon can be considered a circle. Consequently, the Earth imparts centripetal acceleration to the Moon. It is calculated using the formula \(a = \frac (4 \pi^2 \cdot R)(T^2)\), where R– radius of the lunar orbit, equal to approximately 60 Earth radii, T≈ 27 days 7 hours 43 minutes ≈ 2.4∙10 6 s – the period of the Moon’s revolution around the Earth. Considering that the radius of the Earth R z ≈ 6.4∙10 6 m, we find that the centripetal acceleration of the Moon is equal to:

\(a = \frac (4 \pi^2 \cdot 60 \cdot 6.4 \cdot 10^6)((2.4 \cdot 10^6)^2) \approx 0.0027\) m/s 2.

The found acceleration value is less than the acceleration of free fall of bodies at the Earth's surface (9.8 m/s 2) by approximately 3600 = 60 2 times.

Thus, an increase in the distance between the body and the Earth by 60 times led to a decrease in the acceleration imparted by gravity, and, consequently, the force of gravity itself by 60 2 times.

This leads to an important conclusion: the acceleration imparted to bodies by the force of gravity towards the Earth decreases in inverse proportion to the square of the distance to the center of the Earth

\(F \sim \frac (1)(R^2)\).

Law of Gravity

In 1667, Newton finally formulated the law of universal gravitation:

\(F = G \cdot \frac (m_1 \cdot m_2)(R^2).\quad (1)\)

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.

Law of Gravity valid only for bodies whose dimensions are negligible compared to the distance between them. In other words, it is only fair For material points . In this case, the forces of gravitational interaction are directed along the line connecting these points (Fig. 2). This kind of force is called central.

To find the gravitational force acting on a given body from another, in the case when the sizes of the bodies cannot be neglected, proceed as follows. Both bodies are mentally divided into such small elements that each of them can be considered a point. By adding up the gravitational forces acting on each element of a given body from all elements of another body, we obtain the force acting on this element (Fig. 3). Having performed such an operation for each element of a given body and adding up the resulting forces, they find full force gravity acting on this body. This task is difficult.

There is, however, one practically important case when formula (1) is applicable to extended bodies. It can be proven that spherical bodies, the density of which depends only on the distances to their centers, when the distances between them are greater than the sum of their radii, are attracted with forces whose moduli are determined by formula (1). In this case R is the distance between the centers of the balls.

And finally, since the sizes of bodies falling on the Earth are much smaller than the sizes of the Earth, these bodies can be considered as point bodies. Then under R in formula (1) one should understand the distance from a given body to the center of the Earth.

Between all bodies there are forces of mutual attraction, depending on the bodies themselves (their masses) and on the distance between them.

Physical meaning of the gravitational constant

From formula (1) we find

\(G = F \cdot \frac (R^2)(m_1 \cdot m_2)\).

It follows that if the distance between bodies is numerically equal to unity ( R= 1 m) and the masses of interacting bodies are also equal to unity ( m 1 = m 2 = 1 kg), then the gravitational constant is numerically equal to the force modulus F. Thus ( physical meaning ),

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

In SI, the gravitational constant is expressed as

.

Cavendish experience

The value of the gravitational constant G can only be found experimentally. To do this, you need to measure the gravitational force modulus F, acting on the body by mass m 1 from the side of a body of mass m 2 at known distance R between bodies.

The first measurements of the gravitational constant were made in mid-18th century V. Estimate, albeit very roughly, the value G at that time it was possible as a result of considering the attraction of a pendulum to a mountain, the mass of which was determined by geological methods.

Accurate measurements of the gravitational constant were first carried out in 1798 by the English physicist G. Cavendish using an instrument called a torsion balance. A torsion balance is shown schematically in Figure 4.

Cavendish secured two small lead balls (5 cm in diameter and weighing m 1 = 775 g each) at opposite ends of a two-meter rod. The rod was suspended on a thin wire. For this wire, the elastic forces that arise in it when twisted at various angles were previously determined. Two large lead balls (20 cm in diameter and weighing m 2 = 49.5 kg) could be brought close to the small balls. The attractive forces from the large balls caused the small balls to move towards them, while the stretched wire twisted a little. The degree of twist was a measure of the force acting between the balls. The angle of twist of the wire (or rotation of the rod with small balls) turned out to be so small that it had to be measured using an optical tube. The result obtained by Cavendish differs by only 1% from the value of the gravitational constant accepted today:

G ≈ 6.67∙10 -11 (N∙m 2)/kg 2

Thus, the attractive forces of two bodies weighing 1 kg each, located at a distance of 1 m from each other, are equal in modules to only 6.67∙10 -11 N. This is a very small force. Only in the case when bodies of enormous mass interact (or at least the mass of one of the bodies is large) does the gravitational force become large. For example, the Earth attracts the Moon with a force F≈ 2∙10 20 N.

Gravitational forces are the “weakest” of all natural forces. This is due to the fact that the gravitational constant is small. But with large masses of cosmic bodies, the forces of universal gravity become very large. These forces keep all the planets near the Sun.

The meaning of the law of universal gravitation

The law of universal gravitation underlies celestial mechanics - the science of planetary motion. With the help of this law, the positions of celestial bodies in the firmament for many decades in advance are determined with great accuracy and their trajectories are calculated. The law of universal gravitation is also used in motion calculations artificial satellites Earth and interplanetary automatic vehicles.

Disturbances in the motion of planets. Planets do not move strictly according to Kepler's laws. Kepler's laws would be strictly observed for the motion of a given planet only in the case when this one planet revolved around the Sun. But in solar system There are many planets, they are all attracted both by the Sun and by each other. Therefore, disturbances in the motion of the planets arise. In the Solar System, disturbances are small because the attraction of a planet by the Sun is much stronger than the attraction of other planets. When calculating the apparent positions of the planets, disturbances must be taken into account. When launching artificial celestial bodies and when calculating their trajectories, an approximate theory of the motion of celestial bodies is used - perturbation theory.

Discovery of Neptune. One of the striking examples of the triumph of the law of universal gravitation is the discovery of the planet Neptune. In 1781, the English astronomer William Herschel discovered the planet Uranus. Its orbit was calculated and a table of the positions of this planet was compiled for many years to come. However, a check of this table, carried out in 1840, showed that its data diverges from reality.

Scientists have suggested that the deviation in the movement of Uranus is caused by the attraction of an unknown planet located even further from the Sun than Uranus. Knowing the deviations from the calculated trajectory (disturbances in the movement of Uranus), the Englishman Adams and the Frenchman Leverrier, using the law of universal gravitation, calculated the position of this planet in the sky. Adams finished his calculations early, but the observers to whom he communicated his results were in no hurry to check. Meanwhile, Leverrier, having completed his calculations, pointed out to the German astronomer Halle the place where to look for the unknown planet. On the very first evening, September 28, 1846, Halle, pointing the telescope at the indicated location, discovered a new planet. She was named Neptune.

In the same way, the planet Pluto was discovered on March 14, 1930. Both discoveries are said to have been made "at the tip of a pen."

Using the law of universal gravitation, you can calculate the mass of planets and their satellites; explain phenomena such as the ebb and flow of water in the oceans, and much more.

The forces of universal gravity are the most universal of all the forces of nature. They act between any bodies that have mass, and all bodies have mass. There are no barriers to the forces of gravity. They act through any body.

Literature

1. Kikoin I.K., Kikoin A.K. Physics: Textbook. for 9th grade. avg. school – M.: Education, 1992. – 191 p.
2. Physics: Mechanics. 10th grade: Textbook. For in-depth study physicists / M.M. Balashov, A.I. Gomonova, A.B. Dolitsky and others; Ed. G.Ya. Myakisheva. – M.: Bustard, 2002. – 496 p.