Based on the interpretation of Newton's second law, we can conclude that the change in motion occurs through force. Mechanics considers forces of different physical nature. Many of them are determined using the action of the forces of gravity.

In 1862, the law of universal gravitation was discovered by I. Newton. He suggested that the forces that hold the moon are of the same nature as the forces that cause an apple to fall to Earth. The meaning of the hypothesis is the presence of the action of forces of attraction directed along the line and connecting the centers of mass, as shown in Figure 1. 10 . one . A spherical body has a center of mass that coincides with the center of the ball.

Drawing 1 . 10 . 1 . Gravitational forces of attraction between bodies. F 1 → = - F 2 →.

Definition 1

With the known directions of motion of the planets, Newton tried to find out what forces act on them. This process was named inverse problem of mechanics.

The main task of mechanics is to determine the coordinates of a body of a known mass with its velocity at any moment of time using known forces acting on the body and a given condition (direct problem). The reverse is carried out with the determination of the forces acting on the body with its known direction. Such tasks led the scientist to the discovery of the definition of the law of universal gravitation.

All bodies are attracted to each other with a force that is directly proportional to their masses and inversely proportional to the square of the distance between them.

F = G m 1 m 2 r 2.

The value of G determines the coefficient of proportionality of all bodies in nature, called the gravitational constant and denoted by the formula G = 6, 67 · 10 - 11 N · m 2 / k g 2 (C And).

Most of the phenomena in nature are explained by the presence of the action of the force of universal gravity. The movement of planets, artificial satellites of the Earth, the flight path of ballistic missiles, the movement of bodies near the surface of the Earth - everything is explained by the law of gravity and dynamics.

Definition 3

The manifestation of the force of gravity is characterized by the presence gravity... This is the name of the force of attraction of bodies to the Earth and near its surface.

When M is denoted as the mass of the Earth, R Z is the radius, m is the mass of the body, then the gravity formula takes the form:

F = G M R З 2 m = m g.

Where g is acceleration free fall equal to g = G M R З 2.

The force of gravity is directed towards the center of the Earth, as shown in the Moon-Earth example. In the absence of the action of other forces, the body moves with the acceleration of free fall. Its average value is 9, 81 m / s 2. With a known G and radius R 3 = 6, 38 · 10 6 m, the Earth's mass M is calculated using the formula:

M = g R 3 2 G = 5, 98 10 24 k g.

If the body moves away from the surface of the Earth, then the action of the force of gravity and the acceleration of gravity change in inverse proportion to the square of the distance r to the center. Picture 1 . 10 . 2 shows how the gravitational force acting on the spacecraft astronaut changes with distance from the Earth. Obviously, the F of its attraction to the Earth is equal to 700 N.

Drawing 1 . 10 . 2 . Change in the force of gravity acting on the astronaut with distance from the Earth.

Example 1

The Earth-Moon is suitable as an example of the interaction of a system of two bodies.

The distance to the Moon is r Л = 3, 84 · 10 6 m. It is 60 times larger than the radius of the Earth R З. Hence, in the presence of gravity, the gravitational acceleration α Л of the Moon's orbit will be α Л = g R З r Л 2 = 9.81 m / s 2 60 2 = 0.0027 m / s 2.

It is directed towards the center of the Earth and is called centripetal. The calculation is made according to the formula a L = υ 2 r L = 4 π 2 r L T 2 = 0, 0027 m / s 2, where T = 27, 3 days is the period of the Moon's revolution around the Earth. The results and calculations performed in different ways suggest that Newton was right in his assumption of the same nature of the force that keeps the moon in orbit and the force of gravity.

The moon has its own gravitational field, which determines the acceleration of gravity g Л on the surface. The mass of the Moon is 81 times less than the mass of the Earth, and the radius is 3.7 times. Hence it is clear that the acceleration g Л should be determined from the expression:

g L = G M L R L 2 = G M Z 3, 7 2 T 3 2 = 0, 17 g = 1, 66 m / s 2.

This weak gravity is typical for astronauts on the moon. Therefore, you can make huge jumps and steps. A jump up one meter on Earth corresponds to a seven meter jump on the Moon.

The movement of artificial satellites is recorded outside the earth's atmosphere, therefore, the forces of gravity of the Earth act on them. The trajectory of a space body can change depending on the initial velocity. The movement of an artificial satellite in a near-earth orbit is approximately taken as the distance to the center of the Earth, equal to the radius R З. They fly at altitudes of 200 - 300 k m.

Definition 4

Hence it follows that the centripetal acceleration of the satellite, which is imparted by the forces of gravity, is equal to the acceleration of gravity g. The satellite speed will take the designation υ 1. They call her first space speed.

Applying the kinematic formula for centripetal acceleration, we obtain

a n = υ 1 2 R З = g, υ 1 = g R З = 7, 91 · 10 3 m / s.

At this speed, the satellite was able to fly around the Earth in a time equal to T 1 = 2 πR З υ 1 = 84 m and 12 s.

But the period of rotation of the satellite in a circular orbit near the Earth is much longer than indicated above, since there is a difference between the radius of the real orbit and the radius of the Earth.

The satellite moves according to the principle of free fall, remotely similar to the trajectory of a projectile or a ballistic missile. The difference lies in the high speed of the satellite, and the radius of curvature of its trajectory reaches the length of the radius of the Earth.

Satellites that move along circular trajectories at long distances have a weakened gravity, inversely proportional to the square of the radius r of the trajectory. Then finding the speed of the satellite follows the condition:

υ 2 к = g R 3 2 r 2, υ = g R 3 R З r = υ 1 R 3 r.

Therefore, the presence of satellites in high orbits indicates a lower speed of their movement than from a near-earth orbit. The formula for the circulation period is:

T = 2 πr υ = 2 πr υ 1 r R З = 2 πR З υ 1 r R 3 3/2 = T 1 2 π R З.

T 1 takes the value of the period of revolution of the satellite in the near-earth orbit. T increases with the size of the orbital radius. If r has a value of 6, 6 R 3 then the satellite T is equal to 24 hours. When it is launched in the equatorial plane, it will be observed as hanging over some point on the earth's surface. The use of such satellites is known in the space radio communication system. An orbit with a radius of r = 6.6 R 3 is called geostationary.

Drawing 1 . 10 . 3 . Model of the movement of satellites.

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In your 7th grade physics course, you studied the phenomenon of universal gravitation. It consists in the fact that forces of attraction act between all bodies in the Universe.

Newton came to the conclusion about the existence of gravitational forces (they are also called gravitational forces) as a result of studying the motion of the Moon around the Earth and planets around the Sun.

Newton's merit lies not only in his ingenious guess about the mutual attraction of bodies, but also in the fact that he was able to find the law of their interaction, that is, a formula for calculating the gravitational force between two bodies.

The law of gravitation says:

• any two bodies are attracted to each other with a force directly proportional to the mass of each of them and inversely proportional to the square of the distance between them

where F is the modulus of the vector of the force of gravitational attraction between bodies of masses m 1 and m 2, g is the distance between the bodies (their centers); G is a coefficient called gravitational constant.

If m 1 = m 2 = 1 kg and g = 1 m, then, as can be seen from the formula, the gravitational constant G is numerically equal to the force F. In other words, the gravitational constant is numerically equal to the force F of attraction of two bodies with a mass of 1 kg, located at a distance 1 m apart. Measurements show that

G = 6.67 10 -11 Nm 2 / kg 2.

The formula gives exact result when calculating the force of universal gravitation in three cases: 1) if the dimensions of the bodies are negligible compared to the distance between them (Fig. 32, a); 2) if both bodies are homogeneous and have a spherical shape (Fig. 32, b); 3) if one of the interacting bodies is a ball, the dimensions and mass of which are much larger than that of the second body (of any shape) located on the surface of this ball or near it (Fig. 32, c).

Rice. 32. Conditions defining the limits of applicability of the law of universal gravitation

The third of the considered cases is the basis for calculating the force of attraction to the Earth of any of the bodies on it using the above formula. In this case, the radius of the Earth should be taken as the distance between the bodies, since the dimensions of all bodies located on its surface or near it are negligible in comparison with the Earth's radius.

According to Newton's third law, an apple hanging on a branch or falling from it with the acceleration of free fall attracts the Earth to itself with the same modulus of force with which the Earth attracts it. But the acceleration of the Earth, caused by the force of its attraction to the apple, is close to zero, since the mass of the Earth is incommensurably greater than the mass of the apple.

Questions

1. What was called universal gravitation?
2. How else are the forces of gravity called?
3. Who and in what century discovered the law of universal gravitation?
4. Formulate the law of universal gravitation. Write down the formula that expresses this law.
5. When should the law of gravitation be applied to calculate gravitational forces?
6. Is the Earth attracted to an apple hanging from a branch?

Exercise 15

1. Give examples of the manifestation of the force of gravity.
2. The space station flies from the Earth to the Moon. How does the modulus of the vector of the force of its attraction to the Earth change in this case; to the moon? With the same or different in magnitude forces, the station is attracted to the Earth and the Moon when it is in the middle between them? If the forces are different, which one is greater and how many times? Justify all answers. (The mass of the Earth is known to be about 81 times the mass of the Moon.)
3. It is known that the mass of the Sun is 330,000 times the mass of the Earth. Is it true that the sun pulls the earth 330,000 times more than the earth pulls the sun? Explain the answer.
4. The ball thrown by the boy moved upward for some time. At the same time, its speed decreased all the time until it became equal to zero. Then the ball began to fall downward with increasing speed. Explain: a) whether the ball was gravitated towards the earth as it moved up; down; b) what caused the decrease in the speed of the ball when it moves up; increasing its speed when moving down; c) why, when the ball moved up, its speed decreased, and when it moved down, it increased.
5. Is a person on Earth attracted to the Moon? If so, what is it more attracted to - to the Moon or to the Earth? Is the moon attracted to this person? Justify the answers.

We all walk on the Earth because it attracts us. If the Earth did not attract all the bodies on its surface, then we, pushing off from it, would fly into space. But this does not happen, and everyone knows about the existence of gravity.

Are we pulling the Earth? The moon attracts!

Do we attract the Earth to ourselves? Funny question, isn't it? But let's figure it out. Do you know what the ebb and flow of the seas and oceans is? Every day the water leaves the shores, it is not known where it hangs for several hours, and then, as if nothing had happened, it returns back.

So the water at this time is not unknown where, but approximately in the middle of the ocean. There, something like a mountain is formed from water. Incredible, right? Water, which tends to spread, does not just flow itself down, but also forms mountains. And in these mountains a huge mass of water is concentrated.

Just estimate the entire volume of water that leaves the coast during low tides, and you will understand that we are talking about gigantic quantities. But since this happens, there must be some reason. And there is a reason. The reason lies in the fact that this water is attracted to the Moon.

Orbiting the Earth, the Moon passes over the oceans and attracts the oceanic waters. The moon revolves around the earth because it is attracted by the earth. But, it turns out that she herself at the same time attracts the Earth to herself. The land, however, is too big for her, but its influence is sufficient to move water in the oceans.

Force and the law of universal gravitation: concept and formula

Now let's go ahead and think: if two huge bodies, being nearby, both attract each other, is it not logical to assume that smaller bodies will also attract each other? Are they just much smaller and the force of their attraction will be small?

It turns out that this assumption is absolutely correct. Absolutely between all bodies in the Universe, there are forces of attraction, or, in other words, the forces of universal gravity.

Isaac Newton was the first to discover and formulate such a phenomenon in the form of a law. The law of universal gravitation says: all bodies are attracted to each other, while the force of their attraction is directly proportional to the mass of each of the bodies and inversely proportional to the square of the distance between them:

F = G * (m_1 * m_2) / r ^ 2,

where F is the magnitude of the vector of the force of attraction between the bodies, m_1 and m_2 are the masses of these bodies, r is the distance between the bodies, G is the gravitational constant.

The gravitational constant is numerically equal to the force that exists between bodies of mass 1 kg, located at a distance of 1 meter. This value was found experimentally: G = 6.67 * 〖10〗 ^ (- 11) N * m ^ 2⁄ 〖kg〗 ^ 2.

Returning to our original question: "are we pulling the Earth?", We can confidently answer: "yes." According to Newton's third law, we attract the Earth with exactly the same force with which the Earth attracts us. This force can be calculated from the law of universal gravitation.

And according to Newton's second law, the action of bodies on each other by any force is expressed in the form of the acceleration they impart to each other. But the acceleration imparted depends on the body weight.

The mass of the Earth is great, and it gives us the acceleration of free fall. And our mass is negligible compared to the Earth, and therefore the acceleration that we give to the Earth is practically zero. That is why we are attracted to the Earth and walk on it, and not vice versa.

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

History of the issue

The law of universal gravitation was not born spontaneously. Since ancient times, people have studied the sky, mainly for compiling agricultural calendars, calculating important dates, religious holidays. Observations indicated that in the center of the "world" there is a Luminary (the Sun), around which they revolve in orbits celestial bodies... Subsequently, the dogmas of the church did not allow to think so, and people lost the knowledge accumulated over thousands of years.

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

By the time of the discovery (1667) by Isaac Newton of the law of gravitation in astronomy, the heliocentric system of the world of N. Copernicus was finally established. According to her, each of the planets of the system revolves around the Luminary in orbits, which, with an approximation sufficient for many calculations, can be considered circular. At the beginning of the 17th century. I. Kepler, analyzing the work of T. Brahe, established the kinematic laws that characterize the motion of the planets. The discovery became the foundation for elucidating the dynamics of planetary motion, that is, the forces that determine precisely this type of their motion.

Description of interaction

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

It has been proven that the force of mutual attraction unconditionally acts between physical bodies. Such phenomena as the fall of objects on the Earth, the rotation of the Moon, planets around the Sun, occurring under the influence of the forces of universal attraction, are called gravitational.

The law of universal gravitation: formula

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

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

What determines the force of attraction

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

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

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

Conclusion

Isaac Newton was a scientist who made a scientific revolution, completely rebuilt the principles of dynamics and, on their basis, created scientific picture the world. His discovery influenced the development of science, the creation of material and spiritual culture. The fate of Newton fell to the task of revising the results of the concept of the world. In the XVII century. Scientists have completed the grandiose work of building the foundation new science- physics.

The law of universal gravitation

Gravity (universal gravity, gravity)(from Lat. gravitas - "heaviness") - long-range fundamental interaction in nature, to which all material bodies are subject. According to modern data, it is a universal interaction in the sense that, unlike any other forces, all bodies without exception, regardless of their mass, are given the same acceleration. Mainly gravity plays a decisive role on a cosmic scale. Term gravity is also used as the name of a branch of physics that studies gravitational interaction... The most successful modern physical theory in classical physics describing gravity is general relativity; the quantum theory of gravitational interaction has not yet been built.

Gravitational interaction

Gravitational interaction is one of four fundamental interactions in our world. Within the framework of classical mechanics, the gravitational interaction is described the law of gravity Newton, who states that the force of gravitational attraction between two material points of mass m 1 and m 2 separated by distance R, proportional to both masses and inversely proportional to the square of the distance - that is

Here G- gravitational constant equal to approximately m³ / (kg s²). The minus sign means that the force acting on the body is always equal in direction to the radius vector directed to the body, that is, the gravitational interaction always leads to the attraction of any bodies.

The law of universal gravitation is one of the applications of the inverse square law, which also occurs in the study of radiation (see, for example, Light pressure), and is a direct consequence of the quadratic increase in the area of ​​a sphere with increasing radius, which leads to a quadratic decrease in the contribution of any unit area to the area of ​​the entire sphere.

The simplest problem of celestial mechanics is the gravitational interaction of two bodies in empty space. This task is solved analytically to the end; the result of its solution is often formulated in the form of three Kepler's laws.

With an increase in the number of interacting bodies, the task becomes much more complicated. So, the already famous three-body problem (that is, the motion of three bodies with non-zero masses) cannot be solved analytically in a general form. With a numerical solution, the instability of the solutions relatively quickly sets in. initial conditions... Applied to the solar system, this instability makes it impossible to predict the motion of the planets on scales exceeding a hundred million years.

In some special cases, it is possible to find an approximate solution. The most important is the case when the mass of one body is significantly greater than the mass of other bodies (examples: solar system and the dynamics of Saturn's rings). In this case, as a first approximation, we can assume that light bodies do not interact with each other and move along Keplerian trajectories around the massive body. The interactions between them can be taken into account within the framework of perturbation theory and averaged over time. In this case, non-trivial phenomena such as resonances, attractors, chaos, etc. can arise. An illustrative example of such phenomena is the nontrivial structure of Saturn's rings.

Despite attempts to describe the behavior of the system from a large number attracting bodies of approximately the same mass, this cannot be done due to the phenomenon of dynamic chaos.

Strong gravitational fields

In strong gravitational fields, when moving with relativistic speeds, the effects of the general theory of relativity begin to manifest themselves:

• deviation of the law of gravitation from Newtonian;
• potential lag associated with the finite speed of propagation of gravitational disturbances; the appearance of gravitational waves;
• nonlinearity effects: gravitational waves tend to interact with each other, so the principle of superposition of waves in strong fields is no longer fulfilled;
• changing the geometry of space-time;
• the emergence of black holes;

Gravitational radiation

One of the important predictions of general relativity is gravitational radiation, the presence of which has not yet been confirmed by direct observations. However, there is indirect observational evidence in favor of its existence, namely: the energy losses in the binary system with the PSR B1913 + 16 pulsar - the Huls-Taylor pulsar - are in good agreement with the model in which this energy is carried away by gravitational radiation.

Gravitational radiation can only be generated by systems with variable quadrupole or higher multipole moments, this fact suggests that the gravitational radiation of most natural sources is directional, which significantly complicates its detection. Gravitational power l-the field source is proportional to (v / c) 2l + 2 if the multipole is of electrical type, and (v / c) 2l + 4 - if the multipole is magnetic type, where v is the characteristic speed of movement of sources in the emitting system, and c is the speed of light. Thus, the dominant moment will be the quadrupole moment of the electric type, and the power of the corresponding radiation is equal to:

where Q ij is the tensor of the quadrupole moment of the mass distribution of the emitting system. Constant (1 / W) allows you to estimate the order of magnitude of the radiation power.

From 1969 (Weber's experiments) to the present day (February 2007), attempts have been made to directly detect gravitational radiation. In the USA, Europe and Japan at the moment there are several operating ground-based detectors (GEO 600), as well as the project of the space gravitational detector of the Republic of Tatarstan.

Subtle effects of gravity

In addition to the classical effects of gravitational attraction and time dilation, general relativity predicts the existence of other manifestations of gravity, which in terrestrial conditions are very weak and their detection and experimental verification are therefore very difficult. Until recently, overcoming these difficulties seemed beyond the capabilities of experimenters.

Among them, in particular, we can name the dragging of inertial frames of reference (or the Lense-Thirring effect) and the gravitomagnetic field. In 2005, NASA's robotic Gravity Probe B conducted an unprecedentedly accurate experiment to measure these effects near Earth, but the full results have yet to be published.

Quantum theory of gravity

Despite more than half a century of attempts, gravity is the only fundamental interaction for which a consistent renormalizable quantum theory has not yet been built. However, at low energies, in the spirit of quantum field theory, the gravitational interaction can be represented as an exchange of gravitons - gauge bosons with spin 2.

Standard theories of gravity

Due to the fact that the quantum effects of gravity are extremely small even under the most extreme experimental and observational conditions, there are still no reliable observations of them. Theoretical estimates show that in the overwhelming majority of cases one can restrict oneself to the classical description of the gravitational interaction.

There is a modern canonical classical theory of gravity - the general theory of relativity, and many hypotheses that refine it and theories of varying degrees of elaboration, competing with each other (see the article Alternative theories of gravity). All these theories give very similar predictions within the framework of the approximation in which experimental tests are currently being carried out. Several of the main, most well-developed or known theories of gravity are described below.

• Gravity is not a geometric field, but a real physical force field described by a tensor.

• Gravitational phenomena should be considered within the framework of the flat Minkowski space, in which the laws of conservation of energy-momentum and angular momentum are unambiguously fulfilled. Then the motion of bodies in Minkowski space is equivalent to the motion of these bodies in effective Riemannian space.

• In tensor equations to determine the metric, one should take into account the graviton mass, and also use the gauge conditions associated with the metric of the Minkowski space. This does not allow annihilating the gravitational field even locally by choosing some suitable frame of reference.

As in general relativity, in RTG, matter is understood as all forms of matter (including the electromagnetic field), with the exception of the gravitational field itself. The consequences of the RTG theory are as follows: black holes as physical objects predicted in general relativity do not exist; The universe is flat, homogeneous, isotropic, stationary and Euclidean.

On the other hand, there are no less convincing arguments from the opponents of RTG, which boil down to the following provisions:

A similar situation takes place in the RTG, where the second tensor equation is introduced to take into account the connection between the non-Euclidean space and the Minkowski space. Due to the presence of a dimensionless adjustable parameter in the Jordan - Brans - Dicke theory, it becomes possible to choose it so that the results of the theory coincide with the results of gravitational experiments.

Sources and Notes

Literature

• V.P. Vizgin Relativistic theory of gravitation (origins and formation, 1900-1915). M .: Nauka, 1981 .-- 352c.
• V.P. Vizgin Unified theories in the 1st third of the twentieth century. M .: Nauka, 1985 .-- 304c.
• Ivanenko D. D., Sardanashvili G. A. Gravity, 3rd ed. M.: URSS, 2008 .-- 200p.

see also

• Gravimeter

Links

• The law of universal gravitation or "Why does the moon not fall to the earth?" - Just about the difficult