The law of gravity of the earth. The law and force of universal gravitation

Obi-Wan Kenobi said that strength holds the galaxy together. The same can be said about gravity. Fact: Gravity allows us to walk on the Earth, the Earth to revolve around the Sun, and the Sun to move around the supermassive black hole at the center of our galaxy. How to understand gravity? This is discussed in our article.

Let us say right away that you will not find here a uniquely correct answer to the question “What is gravity.” Because it simply doesn't exist! Gravity is one of the most mysterious phenomena, over which scientists are puzzling and still cannot fully explain its nature.

There are many hypotheses and opinions. There are more than a dozen theories of gravity, alternative and classical. We will look at the most interesting, relevant and modern ones.

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Gravity is a physical fundamental interaction

There are 4 fundamental interactions in physics. Thanks to them, the world is exactly what it is. Gravity is one of these interactions.

Fundamental interactions:

• gravity;
• electromagnetism;
• strong interaction;
• weak interaction.

Currently, the current theory describing gravity is general relativity ( general theory relativity). It was proposed by Albert Einstein in 1915-1916.

However, we know that it is too early to talk about the ultimate truth. After all, several centuries before the appearance of general relativity in physics, Newton’s theory dominated to describe gravity, which was significantly expanded.

Within the framework of general relativity, it is currently impossible to explain and describe all issues related to gravity.

Before Newton, it was widely believed that gravity on earth and gravity in heaven were different things. It was believed that the planets move according to their own ideal laws, different from those on Earth.

Newton discovered the law of universal gravitation in 1667. Of course, this law existed even during the time of dinosaurs and much earlier.

Ancient philosophers thought about the existence of gravity. Galileo experimentally calculated the acceleration of gravity on Earth, discovering that it is the same for bodies of any mass. Kepler studied the laws of motion of celestial bodies.

Newton managed to formulate and generalize the results of his observations. Here's what he got:

Two bodies attract each other with a force called gravitational force or gravity.

Formula for the force of attraction between bodies:

G is the gravitational constant, m is the mass of bodies, r is the distance between the centers of mass of bodies.

What is the physical meaning of the gravitational constant? It is equal to the force with which bodies with masses of 1 kilogram each act on each other, being at a distance of 1 meter from each other.

According to Newton's theory, every object creates a gravitational field. The accuracy of Newton's law has been tested at distances less than one centimeter. Of course, for small masses these forces are insignificant and can be neglected.

Newton's formula is applicable both for calculating the force of attraction of planets to the sun and for small objects. We simply do not notice with what force, say, balls are attracted to pool table. Nevertheless, this force exists and can be calculated.

The force of attraction acts between any bodies in the Universe. Its effect extends to any distance.

Newton's law of universal gravitation does not explain the nature of the force of gravity, but establishes quantitative laws. Newton's theory does not contradict GTR. It is quite sufficient for solving practical problems on an Earth scale and for calculating the motion of celestial bodies.

Gravity in general relativity

Despite the fact that Newton's theory is quite applicable in practice, it has a number of disadvantages. The law of universal gravitation is mathematical description, but does not give an idea of ​​the fundamental physical nature of things.

According to Newton, the force of gravity acts at any distance. And it works instantly. Considering that the most high speed in the world - the speed of light, there is a discrepancy. How can gravity act instantly at any distance, when it takes light not an instant, but several seconds or even years to overcome them?

Within the framework of general relativity, gravity is considered not as a force that acts on bodies, but as a curvature of space and time under the influence of mass. Thus, gravity is not a force interaction.

What is the effect of gravity? Let's try to describe it using an analogy.

Let's imagine space in the form of an elastic sheet. If you place a light tennis ball on it, the surface will remain level. But if you place a heavy weight next to the ball, it will press a hole on the surface, and the ball will begin to roll towards the large, heavy weight. This is “gravity”.

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Discovery of gravitational waves

Gravitational waves were predicted by Albert Einstein back in 1916, but they were discovered only a hundred years later, in 2015.

What's happened gravitational waves? Let's draw an analogy again. If you throw a stone into calm water, circles will appear on the surface of the water from where it falls. Gravitational waves are the same ripples, disturbances. Just not on the water, but in world space-time.

Instead of water there is space-time, and instead of a stone, say, a black hole. Any accelerated movement of mass generates a gravitational wave. If the bodies are in a state of free fall, when a gravitational wave passes, the distance between them will change.

Since gravity is a very weak force, the detection of gravitational waves was associated with large technical difficulties. Modern technologies made it possible to detect a burst of gravitational waves only from supermassive sources.

A suitable event for detecting a gravitational wave is the merger of black holes. Unfortunately or fortunately, this happens quite rarely. Nevertheless, scientists managed to register a wave that literally rolled across the space of the Universe.

To record gravitational waves, a detector with a diameter of 4 kilometers was built. During the passage of the wave, vibrations of mirrors on suspensions in a vacuum and the interference of light reflected from them were recorded.

Gravitational waves confirmed the validity of general relativity.

Gravity and elementary particles

In the standard model, certain elementary particles are responsible for each interaction. We can say that particles are carriers of interactions.

The graviton, a hypothetical massless particle with energy, is responsible for gravity. By the way, in our separate material, read more about the Higgs boson, which has caused a lot of noise, and other elementary particles.

Finally, here are some interesting facts about gravity.

10 facts about gravity

1. To overcome the force of Earth's gravity, a body must have a speed of 7.91 km/s. This is the first escape velocity. It is enough for a body (for example, a space probe) to move in orbit around the planet.
2. To escape from the Earth's gravitational field, spaceship must have a speed of at least 11.2 km/s. This is the second escape velocity.
3. The objects with the strongest gravity are black holes. Their gravity is so strong that they even attract light (photons).
4. You will not find the force of gravity in any equation of quantum mechanics. The fact is that when you try to include gravity in the equations, they lose their relevance. This is one of the most important problems of modern physics.
5. The word gravity comes from the Latin “gravis”, which means “heavy”.
6. The more massive the object, the stronger the gravity. If a person who weighs 60 kilograms on Earth weighs himself on Jupiter, the scales will show 142 kilograms.
7. NASA scientists are trying to develop a gravity beam that will allow objects to be moved without contact, overcoming the force of gravity.
8. Astronauts in orbit also experience gravity. More precisely, microgravity. They seem to fall endlessly along with the ship they are in.
9. Gravity always attracts and never repels.
10. The black hole, the size of a tennis ball, attracts objects with the same force as our planet.

Now you know the definition of gravity and can tell what formula is used to calculate the force of attraction. If the granite of science is pressing you to the ground stronger than gravity, contact our student service. We will help you study easily under the heaviest loads!

This article will focus on the history of the discovery of the law of universal gravitation. Here we will get acquainted with biographical information from the life of the scientist who discovered this physical dogma, consider its main provisions, the relationship with quantum gravity, the course of development and much more.

Genius

Sir Isaac Newton is a scientist originally from England. At one time, he devoted a lot of attention and effort to such sciences as physics and mathematics, and also brought a lot of new things to mechanics and astronomy. He is rightfully considered one of the first founders of physics in its classical model. He is the author of the fundamental work “Mathematical Principles of Natural Philosophy,” where he presented information about the three laws of mechanics and the law of universal gravitation. Isaac Newton laid the foundations of classical mechanics with these works. He also developed an integral type, light theory. He also contributed huge contribution in physical optics and developed many other theories in physics and mathematics.

Law

The law of universal gravitation and the history of its discovery go back to the distant past. Its classical form is a law that describes gravitational-type interactions that do not go beyond the framework of mechanics.

Its essence was that the indicator of the force F of gravitational thrust arising between 2 bodies or points of matter m1 and m2, separated from each other by a certain distance r, maintains proportionality in relation to both indicators of mass and is inversely proportional to the square of the distance between the bodies:

F = G, where the symbol G denotes the gravitational constant equal to 6.67408(31).10 -11 m 3 /kgf 2.

Newton's gravity

Before considering the history of the discovery of the law of universal gravitation, let us familiarize ourselves in more detail with its general characteristics.

In the theory created by Newton, all bodies with large mass should generate a special field around themselves that attracts other objects to itself. It's called a gravitational field, and it has potential.

A body with spherical symmetry forms a field outside itself, similar to that created by a material point of the same mass located at the center of the body.

The direction of the trajectory of such a point in the gravitational field created by a body with a much larger mass obeys. Objects of the universe, such as, for example, a planet or a comet, also obey it, moving along an ellipse or hyperbola. The distortion that other massive bodies create is taken into account using the provisions of perturbation theory.

Analyzing accuracy

After Newton discovered the law of universal gravitation, it had to be tested and proven many times. For this purpose, a series of calculations and observations were made. Having come to agreement with its provisions and based on the accuracy of its indicator, the experimental form of evaluation serves as a clear confirmation of general relativity. Measuring the quadrupole interactions of a body that rotates, but its antennas remain stationary, shows us that the process of increasing δ depends on the potential r -(1+δ), at a distance of several meters and is in the limit (2.1±6.2) .10 -3 . A number of other practical confirmations allowed this law to establish itself and take a single form, without modifications. In 2007, this dogma was rechecked at a distance of less than a centimeter (55 microns-9.59 mm). Taking into account the errors of the experiment, scientists examined the distance range and found no obvious deviations in this law.

Observation of the Moon's orbit in relation to the Earth also confirmed its validity.

Euclidean space

Newton's classical theory of gravity is associated with Euclidean space. The actual equality with a fairly high accuracy (10 -9) of the indicators of the distance measure in the denominator of the equality discussed above shows us the Euclidean basis of the space of Newtonian mechanics, with a three-dimensional physical form. At such a point of matter, the area of ​​the spherical surface has exact proportionality with respect to the square of its radius.

Data from history

Let us consider a brief history of the discovery of the law of universal gravitation.

Ideas were put forward by other scientists who lived before Newton. Epicurus, Kepler, Descartes, Roberval, Gassendi, Huygens and others thought about it. Kepler hypothesized that the force of gravity is inversely proportional to the distance from the Sun and extends only in the ecliptic planes; according to Descartes, it was a consequence of the activity of vortices in the thickness of the ether. There were a number of guesses that reflected the correct guesses about the dependence on distance.

A letter from Newton to Halley contained information that the predecessors of Sir Isaac himself were Hooke, Wren and Buyot Ismael. However, before him, no one had been able to clearly, using mathematical methods, connect the law of gravity and planetary motion.

The history of the discovery of the law of universal gravitation is closely connected with the work “Mathematical Principles of Natural Philosophy” (1687). In this work, Newton was able to derive the law in question thanks to Kepler's empirical law, which was already known by that time. He shows us that:

• the form of movement of any visible planet indicates the presence of a central force;
• the force of attraction of the central type forms elliptical or hyperbolic orbits.

About Newton's theory

An examination of the brief history of the discovery of the law of universal gravitation can also point us to a number of differences that distinguished it from previous hypotheses. Newton not only published the proposed formula for the phenomenon under consideration, but also proposed a mathematical model in its entirety:

• position on the law of gravity;
• provision on the law of motion;
• systematics of methods of mathematical research.

This triad could fairly accurately study even the most complex movements of celestial objects, thus creating the basis for celestial mechanics. Until Einstein began his work, this model did not require a fundamental set of corrections. Only the mathematical apparatus had to be significantly improved.

Object for discussion

The discovered and proven law throughout the eighteenth century became famous subject active disputes and scrupulous checks. However, the century ended with general agreement with his postulates and statements. Using the calculations of the law, it was possible to accurately determine the paths of movement of bodies in the heavens. Direct verification was carried out in 1798. He did this using a torsion type balance with great sensitivity. In the history of the discovery of the universal law of gravity, it is necessary to give a special place to the interpretations introduced by Poisson. He developed the concept of gravitational potential and the Poisson equation, with which it was possible to calculate this potential. This type of model made it possible to study the gravitational field in the presence of an arbitrary distribution of matter.

Newton's theory had many difficulties. The main one could be considered the inexplicability of long-range action. It was impossible to accurately answer the question of how gravitational forces are sent through vacuum space at infinite speed.

"Evolution" of the law

Over the next two hundred years, and even more, many physicists attempted to propose various ways to improve Newton's theory. These efforts ended in triumph in 1915, namely the creation of the General Theory of Relativity, which was created by Einstein. He was able to overcome the whole range of difficulties. In accordance with the principle of correspondence, Newton's theory turned out to be an approach to the beginning of work on a theory in more general view, which can be used if certain conditions are met:

1. The potential of gravitational nature cannot be too large in the systems under study. The solar system is an example of compliance with all the rules for the movement of celestial bodies. The relativistic phenomenon finds itself in a noticeable manifestation of the perihelion shift.
2. The speed of movement in this group of systems is insignificant in comparison with the speed of light.

Proof that in a weak stationary gravitational field, general relativity calculations take the form of Newtonian ones is the presence of a scalar gravitational potential in a stationary field with weakly expressed force characteristics, which is capable of satisfying the conditions of the Poisson equation.

Quantum scale

However, in history scientific discovery the law of universal gravitation, nor the General Theory of Relativity could serve as the final gravitational theory, since both do not satisfactorily describe gravitational-type processes on the quantum scale. An attempt to create a quantum gravitational theory is one of the most important tasks of modern physics.

From the point of view of quantum gravity, interaction between objects is created through the exchange of virtual gravitons. In accordance with the uncertainty principle, the energy potential of virtual gravitons is inversely proportional to the period of time in which it existed, from the point of emission by one object to the moment in time at which it was absorbed by another point.

In view of this, it turns out that on a small distance scale the interaction of bodies entails the exchange of virtual-type gravitons. Thanks to these considerations, it is possible to conclude a statement about Newton’s law of potential and its dependence in accordance with the inverse proportionality index with respect to distance. The analogy between Coulomb's and Newton's laws is explained by the fact that the weight of gravitons is zero. The weight of photons has the same meaning.

Misconception

In the school curriculum, the answer to the question from history, how Newton discovered the law of universal gravitation, is the story of a falling apple fruit. According to this legend, it fell on the scientist’s head. However, this is a widespread misconception, and in reality everything was possible without such a case of possible head injury. Newton himself sometimes confirmed this myth, but in reality the law was not a spontaneous discovery and did not come in a fit of momentary insight. As was written above, it was developed over a long time and was first presented in the works on the “Mathematical Principles”, which were released to the public in 1687.

Law of Gravity

Gravity (universal gravitation, gravitation)(from Latin gravitas - “gravity”) - a 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, it imparts the same acceleration to all bodies without exception, regardless of their mass. Mainly gravity plays a decisive role on a cosmic scale. Term gravity also used as the name of the branch of physics that studies gravitational interaction. The most successful modern physical theory in classical physics that describes gravity is the general theory of relativity; the quantum theory of gravitational interaction has not yet been constructed.

Gravitational interaction

Gravitational interaction is one of the four fundamental interactions in our world. Within the framework of classical mechanics, gravitational interaction is described law of universal gravitation Newton, who states that the force of gravitational attraction between two material points of mass m 1 and m 2 separated by distance R, is 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, 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 ​​the sphere with increasing radius, which leads to a quadratic decrease in the contribution of any unit area to area of ​​the entire sphere.

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

As the number of interacting bodies increases, the task becomes dramatically more complicated. Thus, the already famous three-body problem (i.e. movement of three bodies with non-zero masses) cannot be solved analytically in a general form. With a numerical solution, instability of the solutions relative to the initial conditions occurs quite quickly. When applied to the Solar System, this instability makes it impossible to predict the motion of planets on scales larger than a hundred million years.

In some special cases, it is possible to find an approximate solution. The most important case is when the mass of one body is significantly greater than the mass of other bodies (examples: the solar system and the dynamics of the rings of Saturn). 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 may arise, such as resonances, attractors, chaos, etc. A clear example of such phenomena is the non-trivial structure of the rings of Saturn.

Despite attempts to describe the behavior of the system from 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 at relativistic speeds, the effects of general relativity begin to appear:

• deviation of the law of gravity from Newton's;
• delay of potentials 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 no longer holds true;
• 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: energy losses in the binary system with the pulsar PSR B1913+16 - the Hulse-Taylor pulsar - are in good agreement with a 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 gravitational radiation of most natural sources directional, which significantly complicates its detection. Gravity power l-field source is proportional (v / c) 2l + 2 , if the multipole is of electric type, and (v / c) 2l + 4 - if the multipole is of magnetic type, where v is the characteristic speed of movement of sources in the radiating system, and c- 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- quadrupole moment tensor of the mass distribution of the radiating system. Constant (1/W) allows us to estimate the order of magnitude of the radiation power.

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

Subtle effects of gravity

In addition to the classical effects of gravitational attraction and time dilation, the general theory of relativity predicts the existence of other manifestations of gravity, which under 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 entrainment of inertial frames of reference (or the Lense-Thirring effect) and the gravitomagnetic field. In 2005, NASA's unmanned Gravity Probe B conducted an unprecedented precision experiment to measure these effects near Earth, but its full results have not yet been 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 constructed. However, at low energies, in the spirit of quantum field theory, gravitational interaction can be represented as an exchange of gravitons - gauge bosons with spin 2.

Standard theories of gravity

Due to the fact that 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 vast majority of cases one can limit oneself to the classical description of gravitational interaction.

There is a modern canonical classical theory of gravity - general theory of relativity, and many clarifying hypotheses and theories varying degrees development, competing with each other (see the article Alternative theories of gravity). All of these theories make very similar predictions within the approximation in which experimental tests are currently carried out. The following are several basic, most well-developed or known theories of gravity.

• 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 flat Minkowski space, in which the laws of conservation of energy-momentum and angular momentum are unambiguously satisfied. 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, the graviton mass should be taken into account, and gauge conditions associated with the Minkowski space metric should be used. This does not allow the gravitational field to be destroyed even locally by choosing some suitable reference frame.

As in general relativity, in RTG matter refers to 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 by opponents of RTG, which boil down to the following points:

A similar thing occurs in RTG, where the second tensor equation is introduced to take into account the connection between non-Euclidean space and Minkowski space. Due to the presence of a dimensionless fitting 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

• Vizgin V. P. Relativistic theory of gravity (origins and formation, 1900-1915). M.: Nauka, 1981. - 352c.
• Vizgin V. P. 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. - 200 p.

see also

• Gravimeter

Links

• The law of universal gravitation or “Why doesn’t the Moon fall to Earth?” - Just about difficult things

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

In contact with

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

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

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

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

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

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

Movement task

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

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

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

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

Newton's gravity

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

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

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

For the law of gravity the formula is as follows:

,

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

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

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 relation should be perceived as a unit vector directed from one center to another:

.

Law of Gravitational Interaction

Weight and gravity

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

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

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

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

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

Thus, since F = mg:

.

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

As we can see, the acceleration of gravity is truly a constant value, since its formula includes constant quantities - 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 gravity is equal to 9.81 m/s 2.

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

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

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

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

Gravitational attraction between man and Earth:

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

The force of gravitational attraction between man and the Sun:

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

How to find the force of attraction between the Earth and the Sun? In the following way:

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

First escape velocity

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

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

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

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

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

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

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

,

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

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

.

The masses are reduced, we get:

,

This speed is called the first escape velocity:

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

First escape velocity

Second escape velocity

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

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

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

Second escape velocity

Let's write down the law of conservation of energy:

,

where on the right side of the equality is the work of gravity: A = Fs.

From this we obtain that the second escape velocity is equal to:

Thus, the second escape velocity is times greater than the first:

The law of universal gravitation. Physics 9th grade

Law of Universal Gravitation.

Conclusion

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

Newton's classical theory of gravity (Newton's Law of Universal Gravitation)- a law describing gravitational interaction within the framework of classical mechanics. This law was discovered by Newton around 1666. It says that strength F (\displaystyle F) gravitational attraction between two material points of mass m 1 (\displaystyle m_(1)) And m 2 (\displaystyle m_(2)), separated by distance R (\displaystyle R), is proportional to both masses and inversely proportional to the square of the distance between them - that is:

F = G ⋅ m 1 ⋅ m 2 R 2 (\displaystyle F=G\cdot (m_(1)\cdot m_(2) \over R^(2)))

Here G (\displaystyle G)- gravitational constant equal to 6.67408(31)·10 −11 m³/(kg·s²) :.

Encyclopedic YouTube

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✪ Introduction to Newton's law of universal gravitation

✪ Law of Gravity

✪ physics LAW OF UNIVERSAL GRAVITY 9th grade

✪ About Isaac Newton ( Short story)

✪ Lesson 60. The law of universal gravitation. Gravitational constant

Subtitles

Now let's learn a little about gravity, or gravitation. As you know, gravity, especially in a beginner or even in a fairly advanced course of physics, is a concept that can be calculated and the basic parameters that determine it, but in fact, gravity is not entirely understandable. Even if you are familiar with the general theory of relativity, if you are asked what gravity is, you can answer: it is the curvature of space-time and the like. However, it is still difficult to get an intuition as to why two objects, simply because they have so-called mass, are attracted to each other. At least for me it's mystical. Having noted this, let us begin to consider the concept of gravity. We will do this by studying Newton's law of universal gravitation, which is valid for most situations. This law states: the force of mutual gravitational attraction F between two material points with masses m₁ and m₂ is equal to the product of the gravitational constant G by the mass of the first object m₁ and the second object m₂, divided by the square of the distance d between them. This is a fairly simple formula. Let's try to transform it and see if we can get some results that are familiar to us. We use this formula to calculate the acceleration of gravity near the Earth's surface. Let's draw the Earth first. Just to understand what we are talking about. This is our Earth. Let's say we need to calculate the gravitational acceleration acting on Sal, that is, on me. Here I am. Let's try to apply this equation to calculate the magnitude of the acceleration of my fall towards the center of the Earth, or to the center Earth masses. The quantity indicated by the capital letter G is the universal gravitational constant. Once again: G is the universal gravitational constant. Although, as far as I know, although I am not an expert on this matter, it seems to me that its value can change, that is, it is not a real constant, and I assume that when different dimensions its magnitude varies. But for our purposes, as well as in most physics courses, it is a constant, a constant equal to 6.67 * 10^(−11) cubic meters divided by a kilogram per second squared. Yes, its dimension looks strange, but it is enough for you to understand that these are conventional units necessary to, as a result of multiplying by the masses of objects and dividing by the square of the distance, obtain the dimension of force - newton, or kilogram per meter divided by second squared. So there's no need to worry about these units: just know that we'll have to work with meters, seconds, and kilograms. Let's substitute this number into the formula for force: 6.67 * 10^(−11). Since we need to know the acceleration acting on Sal, m₁ is equal to the mass of Sal, that is, me. I wouldn’t like to expose how much I weigh in this story, so let’s leave this mass as a variable, denoting ms. The second mass in the equation is the mass of the Earth. Let's write down its meaning by looking at Wikipedia. So, the mass of the Earth is 5.97 * 10^24 kilograms. Yes, the Earth is more massive than Sal. By the way, weight and mass are different concepts. So, the force F is equal to the product of the gravitational constant G by the mass ms, then by the mass of the Earth, and divide all this by the square of the distance. You may object: what is the distance between the Earth and what stands on it? After all, if objects touch, the distance is zero. It is important to understand here: the distance between two objects in this formula is the distance between their centers of mass. In most cases, a person's center of mass is located about three feet above the surface of the Earth, unless the person is very tall. Anyway, my center of mass may be three feet above the ground. Where is the center of mass of the Earth? Obviously in the center of the Earth. What is the radius of the Earth? 6371 kilometers, or approximately 6 million meters. Since the height of my center of mass is about one millionth the distance to the center of mass of the Earth, it can be neglected in this case. Then the distance will be equal to 6 and so on, like all other quantities, you need to write it in standard form- 6.371 * 10^6, since 6000 km is 6 million meters, and a million is 10^6. We write, rounding all fractions to the second decimal place, the distance is 6.37 * 10^6 meters. The formula contains the square of the distance, so let's square everything. Let's try to simplify now. First, let's multiply the values ​​in the numerator and move forward the variable ms. Then the force F is equal to the entire mass of Sal top part, let's calculate it separately. So 6.67 times 5.97 equals 39.82. 39.82. This work significant parts, which should now be multiplied by 10 to the required degree. 10^(−11) and 10^24 have the same base, so to multiply them it is enough to add the exponents. Adding 24 and −11, we get 13, resulting in 10^13. Let's find the denominator. It is equal to 6.37 squared times 10^6 also squared. As you remember, if a number written as a power is raised to another power, then the exponents are multiplied, which means that 10^6 squared is equal to 10 to the power of 6 multiplied by 2, or 10^12. Next, we calculate the square of 6.37 using a calculator and get... Square 6.37. And it's 40.58. 40.58. All that remains is to divide 39.82 by 40.58. Divide 39.82 by 40.58, which equals 0.981. Then we divide 10^13 by 10^12, which is equal to 10^1, or just 10. And 0.981 times 10 is 9.81. After simplification and simple calculations, we found that the gravitational force near the Earth’s surface acting on Sal is equal to Sel’s mass multiplied by 9.81. What does this give us? Is it now possible to calculate gravitational acceleration? It is known that force is equal to the product of mass and acceleration, therefore the gravitational force is simply equal to the product of Sal’s mass and gravitational acceleration, which is usually denoted by the lowercase letter g. So, on the one hand, the force of gravity is equal to 9.81 times Sal's mass. On the other hand, it is equal to Sal’s mass per gravitational acceleration. Dividing both sides of the equation by Sal’s mass, we find that the coefficient 9.81 is the gravitational acceleration. And if we included in the calculations full entry units of dimension, then, having reduced kilograms, we would see that gravitational acceleration is measured in meters divided by a second squared, like any acceleration. You can also notice that the resulting value is very close to the one we used when solving problems about the motion of a thrown body: 9.8 meters per second squared. This is impressive. Let's do another quick gravity problem because we have a couple of minutes left. Let's say we have another planet called Baby Earth. Let the radius of the Baby rS be half the radius of the Earth rE, and its mass mS is also equal to half the mass of the Earth mE. What will be the force of gravity acting here on any object, and how much less is it less than the force of gravity? Although, let's leave the problem for next time, then I'll solve it. See you. Subtitles by the Amara.org community

Properties of Newtonian gravity

In Newtonian theory, each massive body generates a force field of attraction towards this body, which is called a gravitational field. This field is potential, and the function of gravitational potential for a material point with mass M (\displaystyle M) is determined by the formula:

In general, when the density of a substance ρ (\displaystyle \rho ) distributed randomly, satisfies the Poisson equation:

The solution to this equation is written as:

Where r (\displaystyle r) - distance between volume element d V (\displaystyle dV) and the point at which the potential is determined φ (\displaystyle \varphi ), C (\displaystyle C) - arbitrary constant.

The force of attraction acting in a gravitational field on a material point with mass m (\displaystyle m), is related to the potential by the formula:

A spherically symmetrical body creates the same field outside its boundaries as a material point of the same mass located in the center of the body.

The trajectory of a material point in a gravitational field created by a much larger material point obeys Kepler's laws. In particular, planets and comets in solar system move along ellipses or hyperbolas. The influence of other planets, which distorts this picture, can be taken into account using perturbation theory.

Accuracy of Newton's law of universal gravitation

An experimental assessment of the degree of accuracy of Newton's law of gravitation is one of the confirmations of the general theory of relativity. Experiments on measuring the quadrupole interaction of a rotating body and a stationary antenna showed that the increment δ (\displaystyle \delta ) in the expression for the dependence of the Newtonian potential r − (1 + δ) (\displaystyle r^(-(1+\delta))) at distances of several meters is within (2 , 1 ± 6 , 2) ∗ 10 − 3 (\displaystyle (2.1\pm 6.2)*10^(-3)). Other experiments also confirmed the absence of modifications in the law of universal gravitation.

Newton's law of universal gravitation in 2007 was also tested at distances smaller than one centimeter (from 55 microns to 9.53 mm). Taking into account the experimental errors, no deviations from Newton's law were found in the studied range of distances.

Precision laser ranging observations of the Moon's orbit confirm the law of universal gravitation at the distance from the Earth to the Moon with precision 3 ⋅ 10 − 11 (\displaystyle 3\cdot 10^(-11)).

Connection with the geometry of Euclidean space

The fact of equality with very high accuracy 10 − 9 (\displaystyle 10^(-9)) exponent of the distance in the denominator of the expression for the force of gravity to the number 2 (\displaystyle 2) reflects the Euclidean nature of the three-dimensional physical space of Newtonian mechanics. In three-dimensional Euclidean space, the surface area of ​​a sphere is exactly proportional to the square of its radius

Historical sketch

The very idea of ​​the universal force of gravity was repeatedly expressed before Newton. Previously, Epicurus, Gassendi, Kepler, Borelli, Descartes, Roberval, Huygens and others thought about it. Kepler believed that gravity is inversely proportional to the distance to the Sun and extends only in the ecliptic plane; Descartes considered it to be the result of vortices in the ether. There were, however, guesses with a correct dependence on distance; Newton, in a letter to Halley, mentions Bulliald, Wren and Hooke as his predecessors. But before Newton, no one was able to clearly and mathematically conclusively connect the law of gravity (a force inversely proportional to the square of the distance) and the laws of planetary motion (Kepler’s laws).

• law of gravitation;
• law of motion (Newton's second law);
• system of methods for mathematical research (mathematical analysis).

Taken together, this triad is sufficient for a complete study of the most complex movements of celestial bodies, thereby creating the foundations of celestial mechanics. Before Einstein, no fundamental amendments to this model were needed, although the mathematical apparatus turned out to be necessary to significantly develop.

Note that Newton's theory of gravity was no longer, strictly speaking, heliocentric. Already in the two-body problem, the planet rotates not around the Sun, but around a common center of gravity, since not only the Sun attracts the planet, but the planet also attracts the Sun. Finally, it became clear that it was necessary to take into account the influence of the planets on each other.

During the 18th century, the law of universal gravitation was the subject of active debate (it was opposed by supporters of the Descartes school) and careful testing. By the end of the century, it became generally accepted that the law of universal gravitation makes it possible to explain and predict the movements of celestial bodies with great accuracy. Henry Cavendish in 1798 carried out a direct test of the validity of the law of gravity in terrestrial conditions, using extremely sensitive torsion balances. An important step was the introduction by Poisson in 1813 of the concept of gravitational potential and the Poisson equation for this potential; this model made it possible to study the gravitational field with an arbitrary distribution of matter. After this, Newton's law began to be regarded as a fundamental law of nature.

At the same time, Newton's theory contained a number of difficulties. The main one is the inexplicable long-range action: the force of attraction was transmitted incomprehensibly through completely empty space, and infinitely quickly. Essentially, Newton's model was purely mathematical, without any physical content. In addition, if the Universe, as was then assumed, is Euclidean and infinite, and at the same time the average density of matter in it is non-zero, then a gravitational paradox arises. At the end of the 19th century, another problem emerged: the discrepancy between the theoretical and observed displacement of the perihelion of Mercury.

Further development

General theory of relativity

For more than two hundred years after Newton, physicists proposed various ways to improve Newton's theory of gravity. These efforts were crowned with success in 1915, with the creation of Einstein's general theory of relativity, in which all these difficulties were overcome. Newton's theory, in full agreement with the correspondence principle, turned out to be an approximation of a more general theory, applicable when two conditions are met:

In weak stationary gravitational fields, the equations of motion become Newtonian (gravitational potential). To prove this, we show that the scalar gravitational potential in weak stationary gravitational fields satisfies the Poisson equation

It is known (Gravitational potential) that in this case the gravitational potential has the form:

Let us find the component of the energy-momentum tensor from the gravitational field equations of the general theory of relativity:

Where R i k (\displaystyle R_(ik))- curvature tensor. For we can introduce the kinetic energy-momentum tensor ρ u i u k (\displaystyle \rho u_(i)u_(k)). Neglecting quantities of the order u/c (\displaystyle u/c), you can put all the components T i k (\displaystyle T_(ik)), except T 44 (\displaystyle T_(44)), equal to zero. Component T 44 (\displaystyle T_(44)) equal to T 44 = ρ c 2 (\displaystyle T_(44)=\rho c^(2)) and therefore T = g i k T i k = g 44 T 44 = − ρ c 2 (\displaystyle T=g^(ik)T_(ik)=g^(44)T_(44)=-\rho c^(2)). Thus, the gravitational field equations take the form R 44 = − 1 2 ϰ ρ c 2 (\displaystyle R_(44)=-(\frac (1)(2))\varkappa \rho c^(2)). Due to the formula

value of the curvature tensor component R 44 (\displaystyle R_(44)) can be taken equal R 44 = − ∂ Γ 44 α ∂ x α (\displaystyle R_(44)=-(\frac (\partial \Gamma _(44)^(\alpha ))(\partial x^(\alpha )))) and since Γ 44 α ≈ − 1 2 ∂ g 44 ∂ x α (\displaystyle \Gamma _(44)^(\alpha )\approx -(\frac (1)(2))(\frac (\partial g_(44) )(\partial x^(\alpha )))), R 44 = 1 2 ∑ α ∂ 2 g 44 ∂ x α 2 = 1 2 Δ g 44 = − Δ Φ c 2 (\displaystyle R_(44)=(\frac (1)(2))\sum _(\ alpha )(\frac (\partial ^(2)g_(44))(\partial x_(\alpha )^(2)))=(\frac (1)(2))\Delta g_(44)=- (\frac (\Delta \Phi )(c^(2)))). Thus, we arrive at the Poisson equation:

Quantum gravity

However, the general theory of relativity is not the final theory of gravity, since it unsatisfactorily describes gravitational processes on a quantum scale (at distances on the order of the Planck distance, about 1.6⋅10 −35). The construction of a consistent quantum theory of gravity is one of the most important unsolved problems of modern physics.

From the point of view of quantum gravity, gravitational interaction occurs through the exchange of virtual gravitons between interacting bodies. According to the uncertainty principle, the energy of a virtual graviton is inversely proportional to the time of its existence from the moment of emission by one body to the moment of absorption by another body. The lifetime is proportional to the distance between the bodies. Thus, at short distances, interacting bodies can exchange virtual gravitons with short and long lengths waves, and at large distances only by long-wave gravitons. From these considerations we can obtain the law of inverse proportionality of the Newtonian potential to distance. The analogy between Newton's law and Coulomb's law is explained by the fact that the graviton mass, like the mass