In this section, we will talk about Newton's amazing conjecture, which led to the discovery of the law of universal gravitation.
Why does a stone released from the hands fall to the ground? Because it is attracted by the Earth, each of you will say. In fact, the stone falls to the Earth with free fall acceleration. 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 also acts on the Earth with the same modulus of force directed towards the stone. In other words, forces of mutual attraction act between the Earth and the stone.
Newton's guess
Newton was the first who first guessed, and then strictly proved, that the reason causing the fall of a stone to the Earth, the movement of the Moon around the Earth and the planets around the Sun, is one and the same. This is the gravitational force acting between any bodies of the Universe. Here is the course of his reasoning, given in Newton's main work "Mathematical Principles of Natural Philosophy": "A stone thrown horizontally will deviate
, \\
1
/ /
At
Rice. 3.2
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 with more speed, ! then it will fall further” (Fig. 3.2). Continuing these considerations, Newton \ comes to the conclusion that if it were not for air resistance, then the trajectory of a stone thrown from high mountain with a certain speed, it could become such that it would never reach the surface of the Earth at all, but would move around it “just as the planets describe their orbits in the sky”.
Now we have become so accustomed to 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 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 on the Earth or the movement of the planets in their orbits) is the force of universal gravitation. What does this force depend on?
The dependence of the force of gravity on the mass of bodies
In § 1.23 we talked about the free fall of bodies. Galileo's experiments were mentioned, which proved that the Earth communicates the same acceleration to all bodies in a given place, regardless of their mass. This is only possible if force of gravity to the Earth is directly proportional to the mass of the body. It is in this case that the acceleration of free fall, equal to the ratio of the force of gravity to the mass of the body, is a constant value.
Indeed, in this case, an increase in the mass m, for example, by a factor of two will lead to an increase in the modulus of the force F also by a factor of two, and the acceleration
F
rhenium, which is equal to the ratio - , will remain unchanged.
Generalizing this conclusion for the forces of gravity between any bodies, we conclude that the force of universal gravitation is directly proportional to the mass of the body on which this force acts. But at least two bodies participate in mutual attraction. Each of them, according to Newton's third law, is subject to the same modulus of gravitational forces. Therefore, each of these forces must be proportional both to the mass of one body and to the mass of the other body.
Therefore, the force of universal gravitation between two bodies is directly proportional to the product of their masses:
F - here2. (3.2.1)
What else determines the gravitational force acting on a given body from another body?
The dependence of the force of gravity on the distance between bodies
It can be assumed that the force of gravity should depend on the distance between the bodies. To test the correctness of this assumption and to find the dependence of the force of gravity on the distance between bodies, Newton turned to the motion of the Earth's satellite - the Moon. Its motion was studied in those days much more accurately than the motion of the planets.
The revolution 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. Therefore, the Earth imparts centripetal acceleration to the Moon. It is calculated by the formula
l 2
a \u003d - Tg
where B is the radius of the lunar orbit, equal to approximately 60 radii of the Earth, T \u003d 27 days 7 h 43 min \u003d 2.4 106 s is the period of the Moon's revolution around the Earth. Taking into account that the radius of the Earth R3 = 6.4 106 m, we obtain that the centripetal acceleration of the Moon is equal to:
2 6 4k 60 ¦ 6.4 ¦ 10
M „ „„ „. , O
a = 2 ~ 0.0027 m/s*.
(2.4 ¦ 106 s)
The found value of acceleration is less than the acceleration of free fall of bodies near the Earth's surface (9.8 m/s2) by approximately 3600 = 602 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 the earth's gravity, and, consequently, the force of gravity itself, by 602 times.
This leads to an important conclusion: the acceleration imparted to bodies by the force of attraction to the Earth decreases in inverse proportion to the square of the distance to the center of the Earth:
ci
a = -k, (3.2.2)
R
where Cj is a constant coefficient, the same for all bodies.
Kepler's laws
The study of the motion of the planets showed that this motion is caused by the force of gravity towards the Sun. Using careful long-term observations of the Danish astronomer Tycho Brahe, the German scientist Johannes Kepler at the beginning of the 17th century. established the kinematic laws of planetary motion - the so-called Kepler's laws.
Kepler's first law
All planets move in ellipses with the Sun at one of the foci.
An ellipse (Fig. 3.3) is a flat closed curve, the sum of the distances from any point of which to two fixed points, called foci, is constant. This sum of distances is equal to the length of the major axis AB of the ellipse, i.e.
FgP + F2P = 2b,
where Fl and F2 are the foci of the ellipse, and b = ^^ is its semi-major axis; O is the center of the ellipse. The point of the orbit closest to the Sun is called perihelion, and the point farthest from it is called p.
IN
Rice. 3.4
"2
B A A aphelion. If the Sun is in focus Fr (see Fig. 3.3), then point A is perihelion, and point B is aphelion.
Kepler's second law
The radius-vector of the planet for the same intervals of time describes equal areas. So, if the shaded sectors (Fig. 3.4) have the same area, then the paths si> s2> s3 will be traversed by the planet in equal time intervals. It can be seen from the figure that Sj > s2. Consequently, the linear velocity of the planet at different points of its orbit is not the same. At perihelion, the speed of the planet is greatest, at aphelion - the smallest.
Kepler's third law
The squares of the orbital periods of the planets around the Sun are related as the cubes of the semi-major axes of their orbits. Denoting the semi-major axis of the orbit and the period of revolution of one of the planets through bx and Tv and the other - through b2 and T2, Kepler's third law can be written as follows:
From this formula it can be seen that the farther the planet is from the Sun, the longer its period of revolution around the Sun.
Based on Kepler's laws, certain conclusions can be drawn about the accelerations imparted to the planets by the Sun. For simplicity, we will assume that the orbits are not elliptical, but circular. For the planets of the solar system, this replacement is not a very rough approximation.
Then the force of attraction from the side of the Sun in this approximation should be directed for all planets to the center of the Sun.
If through T we denote the periods of revolution of the planets, and through R the radii of their orbits, then, according to Kepler's third law, for two planets we can write
t\L? T2 R2
Normal acceleration when moving in a circle a = co2R. Therefore, the ratio of the accelerations of the planets
Q-i GlD.
7G=-2~- (3-2-5)
2t:r0
Using equation (3.2.4), we get
T2
Since Kepler's third law is valid for all planets, then the acceleration of each planet is inversely proportional to the square of its distance from the Sun:
Oh oh
a = -|. (3.2.6)
WT
The constant C2 is the same for all planets, but does not coincide with the constant C2 in the formula for the acceleration imparted to bodies the globe.
Expressions (3.2.2) and (3.2.6) show that the gravitational force in both cases (attraction to the Earth and attraction to the Sun) gives all bodies an acceleration that does not depend on their mass and decreases inversely with the square of the distance between them:
F~a~-2. (3.2.7)
R
Law of gravity
The existence of dependences (3.2.1) and (3.2.7) means that the force of universal gravitation 12
TP.L Sh
F~
R2? ТТТ-i ТПп
F=G
In 1667, Newton finally formulated the law of universal gravitation:
(3.2.8) R
The force of mutual attraction of 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. The proportionality factor G is called the gravitational constant.
Interaction of point and extended bodies
The law of universal gravitation (3.2.8) is valid only for such bodies, the dimensions of which are negligible compared to the distance between them. In other words, it is valid only for material points. In this case, the forces of gravitational interaction are directed along the line connecting these points (Fig. 3.5). Such forces are called central.
To find the gravitational force acting on a given body from another, in the case when the size 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. Adding up the gravitational forces acting on each element of a given body from all the elements of another body, we obtain the force acting on this element (Fig. 3.6). Having done such an operation for each element of a given body and adding the resulting forces, they find the total gravitational force acting on this body. This task is difficult.
There is, however, one practically important case when formula (3.2.8) is applicable to extended bodies. It is possible to prove
m^
Fig. 3.5 Fig. 3.6
It can be stated that spherical bodies, the density of which depends only on the distances to their centers, at distances between them that are greater than the sum of their radii, are attracted with forces whose modules are determined by formula (3.2.8). In this case, R is the distance between the centers of the balls.
And finally, since the dimensions of the bodies falling to the Earth are much smaller than the dimensions of the Earth, these bodies can be considered as point ones. Then under R in the formula (3.2.8) one should understand the distance from the 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.
? 1. The distance from Mars to the Sun is 52% greater than the distance from the Earth to the Sun. What is the length of a year on Mars? 2. How will the force of attraction between the balls change if the aluminum balls (Fig. 3.7) are replaced by steel balls of the same mass? the same volume?
Why does a stone released from the hands fall to the ground? Because it is attracted by the Earth, each of you will say. In fact, the stone falls to the Earth with free fall acceleration. 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 also acts on the Earth with the same modulus of force directed towards the stone. In other words, forces of mutual attraction act between the Earth and the stone.
Newton was the first who first guessed, and then strictly proved, that the reason causing the fall of a stone to the Earth, the movement of the Moon around the Earth and the planets around the Sun, is one and the same. This is the gravitational force acting between any bodies of 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 action 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, then it will fall further” (Fig. 1).
Continuing these reasoning, 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 Earth’s surface at all, but would move around it “like how the planets describe their orbits in celestial space.
Now we have become so accustomed to 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 on the Earth or the movement of the planets in their orbits) is the force of universal gravitation. What does this force depend on?
The dependence of the force of gravity 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 acceleration, according to Newton's second law, is inversely proportional to mass. How can one explain that the acceleration imparted to a body by the Earth's gravity is the same for all bodies? This is possible only if the force of attraction to the Earth is directly proportional to the mass of the body. In this case, an increase in the mass m, for example, by a factor of two will lead to an increase in the modulus of force F is also doubled, and the acceleration, which is equal to \(a = \frac (F)(m)\), will remain unchanged. Generalizing this conclusion for the forces of gravity between any bodies, we conclude that the force of universal gravitation is directly proportional to the mass of the body on which this force acts.
But at least two bodies participate in mutual attraction. Each of them, according to Newton's third law, is subject to the same modulus of gravitational forces. Therefore, each of these forces must be proportional both to the mass of one body and to the mass of the other body. Therefore, the force of universal gravitation between two bodies is directly proportional to the product of their masses:
\(F \sim m_1 \cdot m_2\)
The dependence of the force of gravity on the distance between bodies
It is well known from experience that the free fall acceleration is 9.8 m/s 2 and it is the same for bodies falling from a height of 1, 10 and 100 m, that is, 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 measured 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 free fall acceleration.
To find out how the distance between bodies affects the force of their mutual attraction, it would be necessary to find out what is the acceleration of bodies remote from the Earth at sufficiently large distances. 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 the natural satellite of the Earth - the Moon. If the force of attraction between the Earth and the Moon did not depend on the distance between them, then the centripetal acceleration of 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 revolution 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. Therefore, the Earth imparts centripetal acceleration to the Moon. It is calculated by the formula \(a = \frac (4 \pi^2 \cdot R)(T^2)\), where R- the radius of the lunar orbit, equal to approximately 60 radii of the Earth, T≈ 27 days 7 h 43 min ≈ 2.4∙10 6 s is the period of the Moon's revolution around the Earth. Given that the radius of the earth R h ≈ 6.4∙10 6 m, we get 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 value of acceleration is less than the acceleration of free fall of bodies near the surface of the Earth (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 the earth's gravity, and, consequently, the force of attraction itself by 60 2 times.
This leads to an important conclusion: the acceleration imparted to bodies by the force of attraction to 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 of 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 is valid only for bodies whose dimensions are negligibly small 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). Such forces are called central.
To find the gravitational force acting on a given body from the side of another, in the case when the size 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. Adding up the gravitational forces acting on each element of a given body from all the elements of another body, we obtain the force acting on this element (Fig. 3). Having done such an operation for each element of a given body and adding the resulting forces, they find the total gravitational force 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 proved that spherical bodies, the density of which depends only on the distances to their centers, at distances between them that are greater than the sum of their radii, attract with forces whose modules are determined by formula (1). In this case R is the distance between the centers of the balls.
And finally, since the dimensions of the bodies falling to the Earth are much smaller than the dimensions of the Earth, these bodies can be considered as point ones. 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.
The 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 the bodies is numerically equal to one ( R= 1 m) and the masses of the 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 with a distance between bodies equal to 1 m.
In SI, the gravitational constant is expressed as
.
Cavendish experience
The value of the gravitational constant G can only be found empirically. To do this, you need to measure the modulus of the gravitational force F, acting on the body mass m 1 side body weight m 2 at a known distance R between bodies.
The first measurements of the gravitational constant were made in the middle of the 18th century. Estimate, though very roughly, the value G at that time succeeded as a result of considering the attraction of the pendulum to the mountain, the mass of which was determined by geological methods.
Accurate measurements of the gravitational constant were first made in 1798 by the English physicist G. Cavendish using a device called a torsion balance. Schematically, the torsion balance is shown in Figure 4.
Cavendish fixed 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 arising in it when twisting through various angles were preliminarily determined. Two large lead balls (20 cm in diameter and weighing m 2 = 49.5 kg) could be brought close to small balls. Attractive forces from the large balls forced 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 twisting angle of the wire (or the 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 is only 1% different from the value of the gravitational constant accepted today:
G ≈ 6.67∙10 -11 (N∙m 2) / kg 2
Thus, the attraction forces of two bodies weighing 1 kg each, located at a distance of 1 m from each other, are only 6.67∙10 -11 N in modules. 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), the gravitational force becomes large. For example, the Earth pulls the Moon with force F≈ 2∙10 20 N.
Gravitational forces are the "weakest" of all the forces of nature. This is due to the fact that the gravitational constant is small. But with large masses of cosmic bodies, the forces of universal gravitation become very large. These forces keep all the planets near the Sun.
The meaning of the law of gravity
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 to come are determined with great accuracy and their trajectories are calculated. The law of universal gravitation is also applied in motion calculations artificial satellites Earth and interplanetary automatic vehicles.
Disturbances in the motion of the 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 if this planet alone revolved around the Sun. But in solar system There are many planets, all of them are attracted by both the Sun and each other. Therefore, there are disturbances in the motion of the planets. In the solar system, perturbations are small, because the attraction of the planet by the Sun is much stronger than the attraction of other planets. When calculating the apparent position of the planets, perturbations must be taken into account. When launching artificial celestial bodies and when calculating their trajectories, they use an approximate theory of the motion of celestial bodies - perturbation theory.
Discovery of Neptune. One of the clearest examples of the triumph of the law of universal gravitation is the discovery of the planet Neptune. In 1781, the English astronomer William Herschel discovered the planet Uranus. Its orbit was calculated and a table of the positions of this planet was compiled for many years to come. However, a check of this table, carried out in 1840, showed that its data differ from reality.
Scientists have suggested that the deviation in the motion of Uranus is caused by the attraction of an unknown planet, located even further from the Sun than Uranus. Knowing the deviations from the calculated trajectory (disturbances in the movement of Uranus), the Englishman Adams and the Frenchman Leverrier, using the law of universal gravitation, calculated the position of this planet in the sky. Adams completed the calculations earlier, but the observers to whom he reported his results were in no hurry to verify. Meanwhile, Leverrier, having completed his calculations, indicated to the German astronomer Halle the place where to look for an unknown planet. On the very first evening, September 28, 1846, Halle, pointing the telescope to the indicated place, discovered a new planet. They named her Neptune.
In the same way, on March 14, 1930, the planet Pluto was discovered. 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 the planets and their satellites; explain phenomena such as the ebb and flow of water in the oceans, and much more.
The forces of universal gravitation are the most universal of all the forces of nature. They act between any bodies that have mass, and all bodies have mass. There are no barriers to the forces of gravity. They act through any body.
Literature
- Kikoin I.K., Kikoin A.K. Physics: Proc. for 9 cells. avg. school - M.: Enlightenment, 1992. - 191 p.
- Physics: Mechanics. Grade 10: Proc. for in-depth study of physics / M.M. Balashov, A.I. Gomonova, A.B. Dolitsky and others; Ed. G.Ya. Myakishev. – M.: Bustard, 2002. – 496 p.
The fall of bodies to Earth in a vacuum is called free fall of bodies. When falling in a glass tube, from which air is pumped out with the help of a pump, a piece of lead, a cork and a light pen reach the bottom at the same time (Fig. 26). Therefore, in free fall, all bodies, regardless of their mass, move in the same way.
Free fall is uniformly accelerated motion.
The acceleration with which bodies fall to the Earth in a vacuum is called free fall acceleration. The gravitational acceleration is denoted by the letter g. At the surface of the globe, the modulus of free fall acceleration is approximately equal to
If the calculations do not require high accuracy, then it is assumed that the modulus of the free fall acceleration at the Earth's surface is equal to
The same value of the acceleration of freely falling bodies with different masses indicates that the force under which the body acquires the acceleration of free fall is proportional to the mass of the body. This force of attraction acting from the Earth on all bodies is called the force of gravity:
Gravity acts on any body near the surface of the Earth and at a distance from the surface, and at a distance of 10 km, where airplanes fly. And does gravity act at even greater distances from the Earth? Do gravity and gravitational acceleration depend on distance from the Earth? Many scientists thought about these questions, but for the first time he gave answers to them in the 17th century. the great English physicist Isaac Newton (1643-1727).
Dependence of gravity on distance.
Newton suggested that gravity acts at any distance from the Earth, but its value decreases inversely with the square of the distance from the center of the Earth. A test of this assumption could be the measurement of the force of attraction of some body located at a great distance from the Earth, and comparing it with the force of attraction of the same body at the surface of the Earth.
To determine the acceleration of a body under the action of gravity at a great distance from the Earth, Newton used the results of astronomical observations of the motion of the Moon.
He suggested that the force of attraction acting from the Earth to the Moon is the same force of gravity that acts on any bodies near the surface of the Earth. Therefore, the centripetal acceleration during the movement of the Moon in orbit around the Earth is the acceleration of the free fall of the Moon to the Earth.
The distance from the center of the earth to the center of the moon is km. This is about 60 times the distance from the center of the Earth to its surface.
If gravity decreases in inverse proportion to the square of the distance from the center of the Earth, then the acceleration of free fall in the orbit of the Moon should be one times less than the acceleration of free fall near the surface of the Earth
By known values the radius of the Moon's orbit and the period of its revolution around the Earth, Newton calculated the centripetal acceleration of the Moon. It turned out to be really equal.
The theoretically predicted value of the free fall acceleration coincided with the value obtained as a result of astronomical observations. This proved the validity of Newton's assumption that the force of gravity decreases inversely with the square of the distance from the center of the Earth:
The law of universal gravitation.
Just as the Moon revolves around the Earth, the Earth in turn revolves around the Sun. Mercury, Venus, Mars, Jupiter and other planets revolve around the Sun
solar system. Newton proved that the movement of the planets around the Sun occurs under the action of an attractive force directed towards the Sun and decreasing inversely with the square of the distance from it. The Earth attracts the Moon, and the Sun - the Earth, the Sun attracts Jupiter, and Jupiter - its satellites, etc. From this, Newton concluded that all bodies in the Universe mutually attract each other.
The force of mutual attraction acting between the Sun, planets, comets, stars and other bodies in the Universe, Newton called the force of universal gravitation.
The gravitational force acting on the Moon from the Earth is proportional to the mass of the Moon (see formula 9.1). It is obvious that the sleep of universal gravitation acting from the side of the Moon on the Earth is proportional to the mass of the Earth. These forces, according to Newton's third law, are equal to each other. Consequently, the universal gravitational force acting between the Moon and the Earth is proportional to the mass of the Earth and the mass of the Moon, that is, proportional to the product of their masses.
Having extended the established laws - the dependence of gravity on distance and on the masses of interacting bodies - to the interaction of all bodies in the Universe, Newton discovered in 1682 the law of universal gravitation: all bodies are attracted to each other, the force of universal gravitation is directly proportional to the product of the masses of bodies and inversely proportional the square of the distance between them:
The vectors of forces of universal gravitation are directed along the straight line connecting the bodies.
The law of universal gravitation in this form can be used to calculate the forces of interaction between bodies of any shape, if the dimensions of the bodies are much less than the distance between them. Newton proved that for homogeneous spherical bodies the law of universal gravitation in this form is applicable at any distance between the bodies. In this case, the distance between the centers of the balls is taken as the distance between the bodies.
The forces of universal gravitation are called gravitational forces, and the coefficient of proportionality in the law of universal gravitation is called the gravitational constant.
Gravitational constant.
If there is an attractive force between the globe and a piece of chalk, then there probably is an attractive force between half the globe and a piece of chalk. Continuing mentally this process of dividing the globe, we will come to the conclusion that gravitational forces must act between any bodies, ranging from stars and planets to molecules, atoms and elementary particles. This assumption was proved experimentally by the English physicist Henry Cavendish (1731-1810) in 1788.
Cavendish performed experiments to detect the gravitational interaction of small bodies
dimensions using a torsion balance. Two identical small lead balls about 5 cm in diameter were mounted on a rod about a length suspended on a thin copper wire. Against small balls, he installed large lead balls with a diameter of 20 cm each (Fig. 27). Experiments have shown that in this case the rod with small balls rotated, which indicates the presence of an attractive force between the lead balls.
The rotation of the rod is prevented by the elastic force that occurs when the suspension is twisted.
This force is proportional to the angle of rotation. The force of the gravitational interaction of the balls can be determined by the angle of rotation of the suspension.
The masses of the balls, the distance between them in the Cavendish experiment were known, the force of the gravitational interaction was measured directly; therefore, the experiment made it possible to determine the gravitational constant in the law of universal gravitation. According to modern data, it is equal to
The most important phenomenon constantly studied by physicists is motion. 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.
In contact with
Everything in the universe moves. Gravity is a familiar 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 up and down.
In this article, we will consider what Newton's universal attraction is - the classical theory of gravity. However, before moving on to formulas and examples, let's talk about the essence of the problem of attraction and give it a definition.
Perhaps the study of gravity was 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 gravity of bodies interested in ancient Greece.
Movement was understood as the essence of the sensual characteristics of the body, or rather, the body moved while the observer sees it. If we cannot measure, weigh, feel a phenomenon, does this mean that this phenomenon does not exist? Naturally, it doesn't. And since Aristotle understood this, reflections on the essence of gravity began.
As it turned out today, after many tens of centuries, gravity is the basis not only of the earth's attraction and the attraction of our planet to, but also the basis of the origin of the Universe and almost all existing elementary particles.
Movement task
Let's do a thought experiment. Take a small ball in your left hand. Let's take the same one on the right. Let's release the right ball, and it will start 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 must 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 that it has potential energy, where is it recorded in it?
This is the task set by Aristotle, Newton and Albert Einstein. And all three brilliant thinkers partly solved this problem for themselves, but today there are a number of issues that need to be resolved.
Newtonian gravity
In 1666, the greatest English physicist and mechanic I. Newton discovered a law capable of quantitatively calculating the force due to which all matter in the universe tends to each other. This phenomenon is called universal gravitation. When asked: "Formulate the law of universal gravitation", your answer should sound like this:
The force of gravitational interaction, which contributes to the attraction of two bodies, is in direct proportion to the masses of these bodies and inversely proportional 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 with radii r1 and r2 lie on top of each other, then the distance between their surfaces is zero, but there is an attractive force. The point is that the distance between their centers r1+r2 is nonzero. 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 is the force of attraction,
- - masses,
- r - distance,
- G is the gravitational constant, equal to 6.67 10−11 m³ / (kg s²).
What is weight, if we have just considered the force of attraction?
Force is a vector quantity, but 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 ratio should be understood 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 there is nothing surprising in the fact that we personally we feel the attraction of the sun is much weaker than the earth's. The massive Sun, although it has a large mass, is very far from us. also far from the Sun, but it is attracted to it, as it has a large mass. How to find the force of attraction of two bodies, namely, how to calculate the gravitational force of the Sun, the 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 free fall acceleration of the Earth (9.81 m/s 2).
Important! There are no two, three, ten kinds of forces of attraction. Gravity is the only force that quantifies attraction. Weight (P = mg) and gravitational force are one and the same.
If m is our mass, M is the mass of the globe, R is its radius, then the gravitational force acting on us is:
Thus, since F = mg:
.
The masses m cancel out, leaving the expression for the free fall acceleration:
As you can see, the acceleration of free fall is indeed a constant value, since its formula includes constant values - the radius, the mass of the Earth and the gravitational constant. Substituting the values of these constants, we will make sure that the acceleration of free fall is equal to 9.81 m / s 2.
At different latitudes, the radius of the planet is somewhat different, since the Earth is still not a perfect sphere. Because of this, the acceleration of free fall at different points on the globe is different.
Let's return to the attraction of the Earth and the Sun. Let's try to prove by example that the globe attracts us stronger 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 is 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 the Earth:
This result is fairly obvious from a simpler expression for the weight (P = mg).
The force of gravitational attraction between man and the Sun:
As you can see, our planet attracts us almost 2000 times stronger.
How to find the force of attraction between the Earth and the Sun? In the following way:
Now we see that the Sun pulls on our planet more than a billion billion times stronger than the planet pulls you and me.
first cosmic speed
After Isaac Newton discovered the law of universal gravitation, he became interested in how fast a body should be thrown so that it, having overcome the gravitational field, left the globe forever.
True, he imagined it a little differently, in his understanding it was not a vertically standing rocket directed into the sky, but a body that horizontally makes a jump from the top of a mountain. It was a logical illustration, since 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 there is so rarefied, the air particles are no longer as attached to gravity as those that "fell" to the surface.
Let's try to find out what cosmic speed is.
The first cosmic velocity v1 is the velocity 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 quantity for our planet.
Let's write Newton's second law for a body that revolves around the 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, the centrifugal acceleration acts on the body, thus:
.
The masses are reduced, we get:
,
This speed is called the first cosmic speed:
As you can see, the space velocity is absolutely independent of the mass of the body. Thus, any object accelerated to a speed of 7.9 km / s will leave our planet and enter its orbit.
first cosmic speed
Second space velocity
However, even having accelerated the body to the first cosmic speed, we will not be able to completely break its gravitational connection with the Earth. For this, the second cosmic velocity is needed. Upon reaching this speed, the body leaves the gravitational field of the planet and all possible closed orbits.
Important! By mistake, it is often believed that in order to get to the Moon, astronauts had to reach the second cosmic 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, we set the problem a little differently. Suppose a body flies from infinity to a planet. Question: what speed will be achieved on the surface upon landing (without taking into account the atmosphere, of course)? It is this speed and it will take the body to leave the planet.
The law of universal gravitation. Physics Grade 9
The law of universal gravitation.
Conclusion
We have learned that although gravity is the main force in the universe, many of the reasons for this phenomenon are still a mystery. We learned what Newton's universal gravitational force is, learned how to calculate it for various bodies, and also studied some useful consequences that follow from such a phenomenon as the universal law of gravitation.
The simplest arithmetic calculations convincingly show that the force of attraction of the Moon to the Sun is 2 times greater than that of the Moon to the Earth.This means that, according to the "Law of Universal Gravitation", the Moon must revolve around the Sun ...
The law of universal gravitation is not even science fiction, but just nonsense, bigger than the theory that the earth rests on turtles, elephants and whales...
Let us turn to another problem of scientific knowledge: is it always possible to establish the truth in principle - at least ever at all. No not always. Let's give an example based on the same "universal gravitation". As you know, the speed of light is finite, as a result, we see distant objects not where they are located at the moment, but we see them at the point where the ray of light we saw started from. Many stars, perhaps, do not exist at all, only their light comes on - a hackneyed topic. And here gravity- How fast does it spread? Even Laplace managed to establish that gravity from the Sun does not come from where we see it, but from another point. After analyzing the data accumulated by that time, Laplace found that "gravity" propagates faster than light, at least by seven orders! Modern measurements have pushed the speed of propagation of gravity even further - at least 11 orders of magnitude faster than the speed of light.
There are strong suspicions that "gravity" spreads in general instantly. But if this is actually the case, then how to establish it - after all, any measurements are theoretically impossible without some kind of error. So we will never know if this speed is finite or infinite. And the world in which it has a limit, and the world in which it is limitless - these are “two big differences”, and we will never know what kind of world we live in! Here is the limit that is set scientific knowledge. Accepting one point of view or another is a matter of faith, completely irrational, defying any logic. How defying any logic is faith in the “scientific picture of the world”, which is based on the “law of universal gravitation”, which exists only in zombie heads, and which is not detected in the world around us ...
Now let's leave the Newtonian law, and in conclusion we will give a clear example of the fact that the laws discovered on Earth do not exist at all. not universal to the rest of the universe.
Let's look at the same moon. Preferably on a full moon. Why does the Moon look like a disk - more like a pancake than a bun, the shape of which it has? After all, it is a ball, and the ball, if illuminated from the side of the photographer, looks something like this: in the center - a glare, then the illumination decreases, the image is darker towards the edges of the disk.
In the moon, the illumination in the sky is uniform - both in the center and along the edges, it is enough to look at the sky. You can use good binoculars or a camera with a strong optical "zoom", an example of such a photograph is given at the beginning of the article. It was taken with a 16x zoom. This image can be processed in any graphics editor, increasing the contrast to make sure that everything is true, moreover, the brightness at the edges of the disk at the top and bottom is even slightly higher than in the center, where it should theoretically be maximum.
Here we have an example of what the laws of optics on the moon and on earth are completely different! For some reason, the moon reflects all the incident light towards the Earth. We have no reason to extend the regularities revealed in the conditions of the Earth to the entire Universe. It is not a fact that physical "constants" are actually constants and do not change over time.
All of the above shows that the "theories" of "black holes", "Higgs bosons" and much more are not even science fiction, but just nonsense, bigger than the theory that the earth rests on turtles, elephants and whales...
Natural History: The Law of Gravity
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