EntranceThe force of gravity between the earth and the sun is equal. What is the law of universal gravitation: the formula of the great discovery
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 acceleration free fall. 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 not for air resistance, then the trajectory of a stone thrown from high mountain at 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 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 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 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 the side of 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, the total gravitational force acting on this body is found. 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 a known distance R between bodies.
The first measurements of the gravitational constant were made in the middle of the 18th century. 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 mass 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 provisions of celestial bodies on the firmament for many decades in advance 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 there are many planets in the Solar System, 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 reported his results were in no hurry to check. Meanwhile, Leverrier, having completed his calculations, indicated 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
- Kikoin I.K., Kikoin A.K. Physics: Textbook. for 9th grade. avg. school – M.: Education, 1992. – 191 p.
- 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.
This means that, according to the “Law of Gravitation”, the Moon must revolve around the Sun...
The Law of Universal Gravity is not even science fiction, but just nonsense, greater 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. No not always. Let us give an example based on the same “universal gravity”. As you know, the speed of light is finite, as a result, we see distant objects not where they are located in this moment, but we see them at the point where the ray of light we saw started. Many stars may not exist at all, only their light comes through - a hackneyed topic. And here gravity- How fast does it spread? Laplace also managed to establish that gravity from the Sun does not come from where we see it, but from another point. Having analyzed the data accumulated by that time, Laplace established that “gravity” propagates faster than light, at least by seven orders of magnitude! Modern measurements pushed back the speed of gravity propagation even further - at least 11 orders of magnitude faster than the speed of light.
There are strong suspicions that “gravity” generally spreads instantly. But if this actually takes place, then how can this be established - after all, any measurements are theoretically impossible without some kind of error. So we will never know whether this speed is finite or infinite. And the world in which it has a limit, and the world in which it is unlimited, are “two big differences,” and we will never know what kind of world we live in! This is the limit that is set scientific knowledge. Accepting one point of view or another is a matter faith, completely irrational, defying any logic. How defying any logic is belief in “ scientific picture world", which is based on the "law of universal gravitation", which exists only in zombie heads, and which is in no way found in the surrounding world...
Now let's leave Newton's law, and in conclusion we will give a clear example of the fact that the laws discovered on Earth are completely not universal to the rest of the universe.
Let's look at the same Moon. Preferably during the 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, she is a ball, and the ball, if illuminated from the photographer’s side, looks something like this: in the center there is a glare, then the illumination drops, and the image is darker towards the edges of the disk.
The moon in the sky has uniform illumination - both in the center and at the edges, just 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 filmed at 16x zoom. This image can be processed in any graphics editor, increasing the contrast to make sure that everything is so, moreover, the brightness at the edges of the disk at the top and bottom is even slightly higher than in the center, where, according to theory, it should 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 falling light towards the Earth. We have no reason to extend the patterns identified 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, greater than the theory that the earth rests on turtles, elephants and whales...
Natural history: The law of universal gravitation
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The fall of bodies to the Earth in a vacuum is called the free fall of bodies. When falling in a glass tube from which air has been evacuated using a pump, a piece of lead, a cork and a light feather reach the bottom simultaneously (Fig. 26). Consequently, during free fall, all bodies, regardless of their mass, move the same way.
Free fall is a uniformly accelerated motion.
The acceleration with which bodies fall to Earth in a vacuum is called the acceleration of gravity. The acceleration due to gravity is symbolized by the letter g. At the surface of the globe, the gravitational acceleration modulus is approximately equal to
If high accuracy is not required in the calculations, then it is assumed that the module of gravity 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 the influence of which the body acquires the acceleration of free fall is proportional to the mass of the body. This attractive force acting on all bodies from the Earth is called gravity:
The force of gravity acts on any body near the surface of the Earth, both at a distance from the surface and at a distance of 10 km, where airplanes fly. Does gravity act at even greater distances from the Earth? Do the force of gravity and the acceleration of gravity depend on the distance to the Earth? Many scientists thought about these questions, but they were first answered in the 17th century. the great English physicist Isaac Newton (1643-1727).
Dependence of gravity on distance.
Newton proposed that gravity acts at any distance from the Earth, but its value decreases in inverse proportion to the square of the distance from the center of the Earth. A test of this assumption could be to measure the gravitational force of some body located at a great distance from the Earth and compare it with the gravitational force of the same body at the surface of the Earth.
To determine the acceleration of a body under the influence of gravity at a great distance from the Earth, Newton used the results of astronomical observations of the movement of the Moon.
He suggested that the force of gravity acting from the Earth on the Moon is the same force of gravity that acts on any bodies near the surface of the Earth. Therefore, the centripetal acceleration as the Moon moves in its orbit around the Earth is the acceleration of the Moon's free fall on the Earth.
The distance from the center of the Earth to the center of the Moon is km. This is approximately 60 times the distance from the center of the Earth to its surface.
If the force of gravity decreases in inverse proportion to the square of the distance from the center of the Earth, then the acceleration of gravity in the orbit of the Moon should be several times less than the acceleration of gravity at the surface of the Earth
By known values 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 acceleration due to gravity 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 in inverse proportion to the square of the distance from the center of the Earth:
The law of universal gravitation.
Just as the Moon moves around the Earth, the Earth in turn moves around the Sun. Mercury, Venus, Mars, Jupiter and other planets revolve around the Sun
solar system. Newton proved that the movement of planets around the Sun occurs under the influence of a force of gravity directed towards the Sun and decreasing in inverse proportion to the square of the distance from it. The Earth attracts the Moon, and the Sun attracts the Earth, the Sun attracts Jupiter, and Jupiter attracts its satellites, etc. From here Newton concluded that all bodies in the Universe mutually attract each other.
Newton called the force of mutual attraction acting between the Sun, planets, comets, stars and other bodies in the Universe the force of universal gravitation.
The force of universal gravity acting on the Moon from the Earth is proportional to the mass of the Moon (see formula 9.1). It is obvious that the force of universal gravitation acting from the Moon on the Earth is proportional to the mass of the Earth. According to Newton's third law, these forces are equal to each other. Consequently, the force of universal gravity 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 gravity: all bodies attract each other, the force of universal gravity is directly proportional to the product of the masses of bodies and inversely proportional square of the distance between them:
The vectors of universal gravitational forces 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 sizes of the bodies are significantly 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 proportionality coefficient in the law of universal gravitation is called the gravitational constant.
Gravitational constant.
If there is a force of attraction between the globe and a piece of chalk, then there is probably a force of attraction between half the globe and the 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, from stars and planets to molecules, atoms and elementary particles. This assumption was proven experimentally by the English physicist Henry Cavendish (1731-1810) in 1788.
Cavendish performed experiments to detect the gravitational interaction of small bodies
sizes using torsion balances. Two identical small lead balls with a diameter of approximately 5 cm were mounted on a rod about a length suspended on a thin copper wire. Against the small balls, he installed large lead balls with a diameter of 20 cm each (Fig. 27). Experiments showed 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 gravitational interaction between the balls can be determined by the angle of rotation of the suspension.
The masses of the balls and the distance between them in the Cavendish experiment were known, the force of gravitational interaction was measured directly; therefore, experience made it possible to determine the gravitational constant in the law of universal gravitation. According to modern data, it is equal
« Physics - 10th grade"
Why does the Moon move around the Earth?
What happens if the moon stops?
Why do planets revolve around the Sun?
Chapter 1 discussed in detail that Earth imparts to all bodies near the surface of the Earth the same acceleration - the acceleration of gravity. But if the globe imparts acceleration to a body, then, according to Newton’s second law, it acts on the body with some force. The force with which the Earth acts on a body is called gravity. First we will find this force, and then we will consider the force of universal gravity.
Acceleration in absolute value is determined from Newton's second law:
In general, it depends on the force acting on the body and its mass. Since the acceleration of gravity does not depend on mass, it is clear that the force of gravity must be proportional to mass:
The physical quantity is the acceleration of gravity, it is constant for all bodies.
Based on the formula F = mg, you can specify a simple and practically convenient method for measuring the mass of bodies by comparing the mass of a given body with a standard unit of mass. The ratio of the masses of two bodies is equal to the ratio of the forces of gravity acting on the bodies:
This means that the masses of bodies are the same if the forces of gravity acting on them are the same.
This is the basis for determining masses by weighing on spring or lever scales. By ensuring that the pressure force of the body on the scale pan, equal to the force of gravity applied to the body, is balanced by the pressure force of the weights on the other scale pan, equal strength gravity applied to the weights, we thereby determine the mass of the body.
The force of gravity acting on a given body near the Earth can be considered constant only at a certain latitude near the Earth's surface. If the body is lifted or moved to a place with a different latitude, then the acceleration of gravity, and therefore the force of gravity, will change.
The force of universal gravity.
Newton was the first to strictly prove that the cause of a stone falling to the Earth, the movement of the Moon around the Earth and the planets around the Sun are the same. This force of universal gravity, acting between any bodies in the Universe.
Newton came to the conclusion that if it were not for air resistance, then the trajectory of a stone thrown from a high mountain (Fig. 3.1) 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 the way the planets describe their orbits in celestial space.
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:
“Gravity exists for all bodies in general and is proportional to the mass of each of them... all planets gravitate towards each other...” I. Newton
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. From this follows the formulation of the law of universal gravitation.
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:
The proportionality factor G is 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. Indeed, with masses m 1 = m 2 = 1 kg and a distance r = 1 m, we obtain G = F (numerically).
It must be borne in mind that the law of universal gravitation (3.4) 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. 3.2, a).
It can be shown that homogeneous bodies shaped like a ball (even if they cannot be considered material points, Fig. 3.2, b) also interact with the force determined by formula (3.4). In this case, r is 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 to 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 (3.4), keeping in mind that r is the distance from a given body to the center of the Earth.
A stone thrown to the Earth 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." I. Newton
Determination of the gravitational constant.
Now let's find out how to find the gravitational constant. First of all, 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 gives 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 unit of gravitational constant in SI: N m 2 / kg 2 = m 3 / (kg s 2).
To quantify G, it is necessary to independently determine all the quantities included in the law of universal gravitation: both masses, force and distance between bodies.
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 3.3. A light rocker with two identical weights at the ends is suspended from a thin elastic thread. Two heavy balls are fixed nearby. Gravitational forces act between the weights and the stationary balls. Under the influence of these forces, the rocker turns and twists the thread until the resulting elastic force becomes equal to the gravitational force. 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:
G = 6.67 10 -11 N m 2 / kg 2.
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 of great importance. For example, the Earth and the Moon are attracted to each other with a force F ≈ 2 10 20 N.
Dependence of the acceleration of free fall of 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 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.
In Newton's theory, mass is the source of the gravitational field. We are in the Earth's gravitational field. At the same time, we are also sources of the gravitational field, but due to the fact that our mass is significantly less than the mass of the Earth, our field is much weaker and surrounding objects do not react to it.
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. The mass of a body, which is included in Newton’s second law, determines the inertial properties of the body, i.e. its ability to acquire a certain acceleration under the influence of a given force. This inert mass 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 is the gravitational mass m r.
It does not at all follow from Newtonian mechanics that the inertial and gravitational masses are the same, i.e. that
m and = m r . (3.5)
Equality (3.5) 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.
In this section we will talk about Newton's amazing guess, which led to the discovery of the law of universal gravitation.
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 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's conjecture
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 deflect
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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 at a higher speed, ! then it will fall further” (Fig. 3.2). Continuing these reasonings, Newton came 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 “just as 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
§ 1.23 talked about the free fall of bodies. Galileo's experiments were mentioned, which proved that the Earth imparts the same acceleration to all bodies in a given place, regardless of their mass. This is possible only if the force of gravity towards the Earth is directly proportional to the mass of the body. It is in this case that the acceleration of gravity, equal to the ratio of the force of gravity to the mass of the body, is a constant value.
Indeed, in this case, increasing the mass m, for example, by doubling will lead to an increase in the modulus of force F, also doubling, and accelerating
F
ratio, which is equal to the ratio -, 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 - here2. (3.2.1)
What else does the gravitational force acting on a given body from another body depend on?
Dependence of gravitational force on the distance between bodies
It can be assumed that the force of gravity should depend on the distance between the bodies. To check the correctness of this assumption and find the dependence of the gravitational force on the distance between bodies, Newton turned to the movement of the Earth's satellite, the Moon. Its movement was studied much more accurately in those days than the movement of the planets.
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 by the formula
l 2
a = - Tg
where B is the radius of the lunar orbit, equal to approximately 60 radii of the Earth, T = 27 days 7 hours 43 minutes = 2.4 106 s is the period of revolution of the Moon around the Earth. Considering 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 acceleration value is less than the acceleration of free fall of bodies at 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 gravity, and, consequently, the force of gravity itself by 602 times.
An important conclusion follows from this: 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:
ci
a = -k, (3.2.2)
R
where Cj - constant coefficient, the same for all bodies.
Kepler's laws
A study of the movement of planets showed that this movement is caused by the force of gravity towards the Sun. Using the careful observations of the Danish astronomer Tycho Brahe over many years, the German scientist Johannes Kepler early XVII V. 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 focus.
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 semimajor 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 at 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 describes in equal time intervals equal areas. So, if the shaded sectors (Fig. 3.4) have the same areas, then the paths si> s2> s3 will be traversed by the planet in equal periods of time. It is clear from the figure that Sj > s2. Consequently, the linear speed of motion of the planet at different points of its orbit is not the same. At perihelion the planet's speed is greatest, at aphelion it is least.
Kepler's third law
The squares of the periods of revolution of the planets around the Sun are related to the cubes of the semimajor axes of their orbits. Having designated the semimajor axis of the orbit and the period of revolution of one of the planets by bx and Tv and the other by b2 and T2, Kepler’s third law can be written as follows:
From this formula it is clear that the further a 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 consider the orbits not elliptical, but circular. For the planets of the Solar System, this replacement is not too rough an approximation.
Then the force of attraction from the Sun in this approximation should be directed for all planets towards the center of the Sun.
If we denote by T the periods of revolution of the planets, and by 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 is a = co2R. Therefore, the ratio of the accelerations of the planets
Q-i GD.
7G=-2~- (3-2-5)
2 t:r0
Using equation (3.2.4), we obtain
T2
Since Kepler's third law is valid for all planets, the acceleration of each planet is inversely proportional to the square of its distance from the Sun:
Oh oh
a = -|. (3.2.6)
VT
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 by the globe.
Expressions (3.2.2) and (3.2.6) show that the force of gravity in both cases (attraction to the Earth and attraction to the Sun) imparts to all bodies an acceleration that does not depend on their mass and decreases in inverse proportion to the square of the distance between them:
F~a~-2. (3.2.7)
R
Law of Gravity
The existence of dependencies (3.2.1) and (3.2.7) means that the force of universal gravity 12
TP.L Sh
F~
R2? TTT-i TPP
F=G
In 1667, Newton finally formulated the law of universal gravitation:
(3.2.8) R
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. The proportionality coefficient G is called the gravitational constant.
Interaction of point and extended bodies
The law of universal gravitation (3.2.8) is valid only for bodies whose dimensions 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). 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 elements so small 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. 3.6). Having performed such an operation for each element of a given body and adding up the resulting forces, the total gravitational force acting on this body is found. This task is difficult.
There is, however, one practically important case when formula (3.2.8) is applicable to extended bodies. Can you prove
m^
Fi Fig. 3.5 Fig. 3.6
It should be noted 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 (3.2.8). 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 R in formula (3.2.8) should be understood as 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 Earth to the Sun. How long is a year on Mars? 2. How will the force of attraction between the balls change if the aluminum balls (Fig. 3.7) are replaced with steel balls of the same mass? "Same volume?