Motion of a body under the action of gravity: definition, formulas. Law of gravity

The motion of a body under the action of gravity is one of the central topics in dynamic physics. Even an ordinary schoolboy knows that the dynamics section is based on three. Let's try to understand this topic thoroughly, and an article describing each example in detail will help us make studying the movement of a body under the influence of gravity as useful as possible.

A bit of history

People watched with curiosity various phenomena taking place in our lives. Mankind for a long time could not understand the principles and structure of many systems, but a long way of studying the world around us led our ancestors to a scientific revolution. Nowadays, when technology is developing at an incredible speed, people hardly think about how certain mechanisms work.

Meanwhile, our ancestors have always been interested in the mysteries of natural processes and the structure of the world, looking for answers to the most difficult questions and did not stop studying until they found answers to them. So, for example, the famous scientist Galileo Galilei, back in the 16th century, asked himself questions: "Why do bodies always fall down, what kind of force attracts them to the earth?" In 1589, he set up a series of experiments, the results of which proved to be very valuable. He studied in detail the patterns free fall various bodies, dropping objects from the famous tower in the city of Pisa. The laws that he deduced were improved and described in more detail by formulas by another famous English scientist - Sir Isaac Newton. It is he who owns the three laws on which almost all modern physics is based.

The fact that the laws of motion of bodies, described more than 500 years ago, are still relevant to this day, means that our planet is subject to immutable laws. Modern man it is necessary to at least superficially study the basic principles of arranging the world.

Basic and auxiliary concepts of dynamics

In order to fully understand the principles of such a movement, you must first familiarize yourself with some concepts. So, the most necessary theoretical terms:

• Interaction is the action of bodies on each other, in which there is a change or the beginning of their movement relative to each other. There are four types of interaction: electromagnetic, weak, strong and gravitational.
• Velocity is a physical quantity that denotes the speed with which a body is moving. Velocity is a vector, meaning it has not only a value, but also a direction.
• Acceleration is the value that shows us the rate of change in the speed of a body in a period of time. She is also
• The trajectory of the path is a curve, and sometimes a straight line, which the body outlines when moving. With uniform rectilinear motion, the trajectory may coincide with the displacement value.
• The path is the length of the trajectory, that is, exactly as much as the body has traveled in a certain amount of time.
• An inertial frame of reference is an environment in which Newton's first law is fulfilled, that is, the body retains its inertia, provided that all external forces are completely absent.

The above concepts are quite enough to correctly draw or imagine in your head a simulation of the movement of a body under the influence of gravity.

What does strength mean?

Let's move on to the main concept of our topic. So, force is a quantity, the meaning of which is the impact or influence of one body on another quantitatively. And gravity is the force that acts on absolutely every body located on the surface or near our planet. The question arises: where does this power come from? The answer lies in the law gravity.

What is gravity?

Any body from the side of the Earth is influenced by the gravitational force, which gives it some acceleration. Gravity always has a vertical downward direction, towards the center of the planet. In other words, gravity pulls objects towards the earth, which is why objects always fall down. It turns out that gravity is special case force of gravity. Newton deduced one of the main formulas for finding the force of attraction between two bodies. It looks like this: F \u003d G * (m 1 x m 2) / R 2.

What is the free fall acceleration?

A body released from a certain height always flies downwards under the influence of gravity. The movement of a body under the action of gravity vertically up and down can be described by equations, where the main constant will be the value of the acceleration "g". This value is due solely to the action of the force of attraction, and its value is approximately 9.8 m/s 2 . It turns out that a body thrown from a height without an initial speed will move down with an acceleration equal to the value "g".

Motion of a body under the action of gravity: formulas for solving problems

The basic formula for finding the force of gravity is as follows: F gravity \u003d m x g, where m is the mass of the body on which the force acts, and "g" is the acceleration of gravity (to simplify tasks, it is considered to be equal to 10 m / s 2) .

There are several more formulas used to find one or another unknown in the free movement of the body. So, for example, in order to calculate the path traveled by the body, it is necessary to substitute the known values ​​\u200b\u200binto this formula: S \u003d V 0 x t + a x t 2 / 2 (the path is equal to the sum of the products of the initial speed multiplied by time and acceleration by the square of time, divided on 2).

Equations for describing the vertical motion of a body

The movement of a body under the action of gravity along the vertical can be described by an equation that looks like this: x \u003d x 0 + v 0 x t + a x t 2 / 2. Using this expression, you can find the coordinates of the body at a known point in time. You just need to substitute the values ​​​​known in the problem: the initial location, the initial speed (if the body was not just released, but pushed with some force) and acceleration, in our case it will be equal to the acceleration g.

In the same way, you can find the speed of a body that moves under the influence of gravity. The expression for finding an unknown value at any time: v \u003d v 0 + g x t (the value of the initial speed can be equal to zero, then the speed will be equal to the product of the free fall acceleration by the time value for which the body moves).

Movement of bodies under the action of gravity: tasks and methods for their solutions

For many problems involving gravity, we recommend using the following plan:

1. To determine for yourself a convenient inertial frame of reference, it is usually customary to choose the Earth, because it meets many of the requirements for ISO.
2. Draw a small drawing or drawing that shows the main forces acting on the body. The motion of a body under the influence of gravity implies a sketch or diagram that indicates in which direction the body moves if it is subjected to an acceleration equal to g.
3. Then you should choose the direction for projecting the forces and resulting accelerations.
4. Write unknown quantities and determine their direction.
5. Finally, using the formulas above to solve problems, calculate all unknowns by substituting the data into the equations to find the acceleration or distance traveled.

Ready solution for an easy task

When it comes to such a phenomenon as the movement of a body under the influence of how it is more practical to solve the problem, it can be difficult. However, there are a few tricks, using which you can easily solve even the most difficult task. So, let's take a look at live examples of how to solve a particular problem. Let's start with an easy to understand problem.

A body was released from a height of 20 m without initial velocity. Determine how long it will take to reach the earth's surface.

Solution: we know the path traveled by the body, we know that the initial speed was equal to 0. We can also determine that only gravity acts on the body, it turns out that this is the movement of the body under the influence of gravity, and therefore we should use this formula: S = V 0 x t + a x t 2 /2. Since in our case a \u003d g, after some transformations we get the following equation: S \u003d g x t 2 / 2. Now it remains only to express the time through this formula, we get that t 2 \u003d 2S / g. Let us substitute the known values ​​​​(we assume that g \u003d 10 m / s 2) t 2 \u003d 2 x 20 / 10 \u003d 4. Therefore, t \u003d 2 s.

So our answer is: the body will fall to the ground in 2 seconds.

A trick that allows you to quickly solve the problem is as follows: you can notice that the described movement of the body in the above problem occurs in one direction (vertically down). It is very similar to uniformly accelerated motion, since no force acts on the body, except for gravity (we neglect the force of air resistance). Thanks to this, you can use an easy formula for finding a path with uniformly accelerated movement, bypassing the images of drawings with the arrangement of forces acting on the body.

An example of solving a more complex problem

And now let's see how it is better to solve problems for the movement of a body under the influence of gravity, if the body does not move vertically, but has a more complex nature of movement.

For example, the following task. An object of mass m is moving with unknown acceleration down an inclined plane whose coefficient of friction is k. Determine the value of the acceleration that is present when moving given body, if the slope angle α is known.

Solution: You should use the plan described above. First of all, draw a drawing of an inclined plane with the image of the body and all the forces acting on it. It turns out that three components act on it: the force of gravity, friction and the reaction force of the support. The general equation of the resultant forces looks like this: F friction + N + mg = ma.

The main highlight of the problem is the condition of inclination at an angle a. For ox and the oy axis, this condition must be taken into account, then we get the following expression: mg x sin α - Friction F = ma (for the ox axis) and N - mg x cos α = Friction F (for the oy axis).

F friction is easy to calculate by the formula for finding the friction force, it is equal to k x mg (friction coefficient multiplied by the product of body mass and free fall acceleration). After all the calculations, it remains only to substitute the found values ​​\u200b\u200bin the formula, a simplified equation will be obtained for calculating the acceleration with which the body moves along an inclined plane.

According to Newton's second law, the prerequisite for the configuration of motion, in other words, the prerequisite for the acceleration of bodies, is force. In mechanics, forces of various physical nature are considered. Many mechanical phenomena and processes are determined by the action of forces gravity. Law of global gravity was discovered by I. Newton in 1682. Back in 1665, the 23-year-old Newton suggested that the forces that keep the Moon in its orbit are of the same nature as the forces that make an apple fall to the Earth. According to his guess, attractive forces (gravitational forces) act between all the bodies of the Universe, directed along the strip connecting centers of mass(Fig. 1.10.1). For a body in the form of a homogeneous ball, the center of gravity coincides with the center of the ball.

In the following years, Newton tried to find a physical explanation for laws of planetary motion, discovered by the astrologer I. Kepler at the beginning of the 17th century, and give a quantitative expression for gravitational forces. Knowing how the planets move, Newton wanted to find out what forces act on them. This path is called reverse problem of mechanics. If the main task of mechanics is to determine the coordinates of a body of known mass and its speed at any moment of time according to the known forces acting on the body and given initial conditions ( smooth task of mechanics), then when solving the reverse problem, you need to find the forces acting on the body, if it is clear how it moves. The solution of this problem led Newton to the discovery of the law of global gravitation. All bodies are attracted to each other with a force directly proportional to their masses and inversely proportional to the square of the distance between them:

The proportionality coefficient G is similar for all bodies in nature. They call him gravitational constant

Many phenomena in nature are explained by the action of global gravitational forces. The movement of planets in the solar system, the movement of artificial satellites of the Earth, the lines of motion of ballistic missiles, the movement of bodies near the surface of the Earth - all these phenomena are explained on the basis of the law of global gravitation and the laws of dynamics. One of the manifestations of the force of global gravity is gravity. So it is customary to call the force of attraction of bodies to the Earth near its surface. If M is the mass of the Earth, RЗ is its radius, m is the mass of the given body, then the force of gravity is equal to

where g - acceleration of gravity at the earth's surface:

The force of gravity is oriented towards the center of the Earth. In the absence of other forces, the body falls freely to the Earth with free fall acceleration. The average value of the free fall acceleration for different points on the Earth's surface is 9.81 m/s2. Knowing the acceleration of free fall and the radius of the Earth (RЗ = 6.38 106 m), we can calculate the mass of the Earth M:

When moving away from the surface of the Earth, the force of gravity and the acceleration of gravity change back in proportion to the square of the distance r to the center of the Earth. Rice. 1.10.2 illustrates the change in the gravitational force acting on an astronaut in a cosmic ship as it moves away from the Earth. The force with which an astronaut is attracted to the Earth near its surface is assumed to be 700 N.

An example of a system of two interacting bodies is the Earth-Moon system. The Moon is located at a distance rL = 3.84 106 m from the Earth. This distance is approximately 60 times greater than the radius of the Earth RЗ. As follows, the acceleration of free fall aL, due to the Earth's gravity, in the orbit of the Moon is

With such an acceleration directed towards the center of the Earth, the Moon moves in an orbit. As follows, this acceleration is centripetal acceleration. It can be calculated using the kinematic formula for centripetal acceleration (see §1.6):

where T = 27.3 days is the period of the Moon's ascension around the Earth. The coincidence of the results of calculations performed by different methods confirms Newton's assumption about the unified nature of the force holding the Moon in orbit and the force of gravity. The Moon's own gravitational field determines the free fall acceleration gL on its surface. The mass of the Moon is 81 times less than the mass of the Earth, and its radius is about 3.7 times less than the radius of the Earth. Therefore, the acceleration gL will be determined by the expression:

The astronauts who landed on the moon found themselves in such conditions of weak gravity. A person in such conditions can make huge jumps. For example, if a person in terrestrial conditions jumps to a height of 1 m, then on the Moon he could jump to a height of more than 6 m. Let us now consider the question of artificial satellites of the Earth. Artificial satellites move outside earth's atmosphere, and only the gravitational forces from the Earth act on them. Depending on the initial speed, the line of motion of a galactic body can be different (see §1.24). We will consider here only the case of motion artificial satellite radial near-Earth orbit. Such satellites fly at altitudes of the order of 200-300 km, and the distance to the center of the Earth can be approximately taken to be equal to its radius R3. Then centripetal acceleration satellite, reported to him by the forces of gravity, is approximately equal to the acceleration of free fall g. Let us denote the speed of the satellite in near-Earth orbit as υ1. This speed is called first galactic speed. Using the kinematic formula for centripetal acceleration (see §1.6), we get:

Moving at such a speed, the satellite would circle the Earth in time. In fact, the period of the satellite's orbit in a radial orbit near the Earth's surface somewhat exceeds the indicated value due to the difference between the radius of the real orbit and the radius of the Earth. The motion of the satellite can be considered as free fall similar to the movement of projectiles or ballistic missiles. The difference lies solely in the fact that the speed of the satellite is so great that the radius of curvature of its line of motion is equal to the radius of the Earth. For satellites moving along radial trajectories at a significant distance from the Earth, the Earth's gravity weakens back in proportion to the square of the radius r of the line of motion. The satellite speed υ is found from the condition

Thus, in large orbits, the speed of movement of satellites is less than in near-Earth orbit. The period T of the appeal of such a satellite is

Here T1 is the orbital period of the satellite in near-Earth orbit. The orbital period of the satellite increases with increasing radius of the orbit. It is easy to calculate that with an orbit radius r equal to approximately 6.6 R3, the satellite's orbital period will be equal to 24 hours. A satellite with such an elevation period, launched in the plane of the equator, will hover motionlessly over a certain point earth's surface. Such satellites are used in galactic radio communication systems. An orbit with radius r = 6.6R3 is called geostationary.

(The terms gravity and gravitation are equivalent).

Acceleration, which experiences the body m 2 , located at a distance r from this body m 1 is equal to:
.
This value does not depend on the nature (composition) and mass of the body receiving acceleration. This ratio expresses an experimental fact, known even to Galilai, according to which all bodies fall into gravity. Earth's field with the same acceleration.

Newton established that acceleration and force are inversely proportional by comparing the acceleration of bodies falling near the surface of the Earth with the acceleration with which the Moon moves in its orbit. (The radius of the Earth and the approximate distance to the Moon were known by that time.) Further, it was shown that Kepler's laws follow from the law of universal gravitation, which were found by I. Kepler by processing numerous observations of the motions of the planets. This is how celestial mechanics arose. A brilliant confirmation of the Newtonian theory of T. was the prediction of the existence of a planet beyond Uranus (English astronomer J. Adams, French astronomer W. Le Verrier, 1843-45) and the discovery of this planet, which was called Neptune (German astronomer I. Galle , 1846).

The f-ly describing the motion of the planets includes the product G and the mass of the Sun, it is known with great accuracy. To define the same constant G laboratory experiments are required to measure the force of gravity. interaction of two bodies with a known mass. The first such experiment was set up by the English. scientist G. Cavendish (1798). Knowing G, it is possible to determine abs. the value of the mass of the Sun, the Earth and other celestial bodies.

The law of gravity in the form (1) is directly applicable to point bodies. It can be shown that it is also valid for extended bodies with a spherically symmetric mass distribution, and r is the distance between the centers of symmetry of bodies. For spherical bodies located far enough from each other, law (1) is valid approximately.

In the course of the development of the theory of thermodynamics, the idea of ​​the direct force interaction of bodies gradually gave way to the idea of ​​a field. Gravity field in Newton's theory is characterized by the potential , where x,y,z- coordinates, t- time, as well as field strength , i.e.
.
Gravity potential. field created by a set of resting masses does not depend on time. Gravity several potentials. bodies satisfy the principles of superposition, i.e. potential k.-l. the point of their common field is equal to the sum of the potentials of the bodies under consideration.

It is assumed that the gravitational the field is described in the inertial coordinate system, i.e. in a coordinate system, relative to which the body maintains a state of rest or uniform rectilinear motion if no forces act on it. In gravitational field the force acting on a particle of matter is equal to the product of its mass and the field strength at the location of the particle: F=mg. The acceleration of a particle relative to the inertial coordinate system (the so-called abs. acceleration) is, obviously, g.

Point body with mass dm creates gravity. potential
.
A continuous medium distributed in space with a density (may depend on time) creates gravity. potential equal to the sum of the potentials of all elements of the environment. In this case, the field strength is expressed as the vector sum of the strengths created by all particles.

Gravity the potential obeys the Poisson equation:
. (2)

It is clear that the potential of an isolated spherically symmetric body depends only on r. Outside such a body, the potential coincides with the potential of a point body located at the center of symmetry and having the same mass m. If at r>R, then at r>R. This justifies the approximation material points in celestial mechanics, where they usually deal with almost spherical. bodies that are, moreover, quite far apart. The exact Poissnoa equation, taking into account the real, asymmetric distribution of masses, is used, for example, in studying the structure of the Earth by gravimetric methods. T.'s law in the form of the Poisson equation is applied in the theoretical. study of the structure of stars. In stars, the force of gravity, which varies from point to point, is balanced by a pressure gradient; in rotating stars, centrifugal force is added to the pressure gradient.

Let us note some fundamental features of the classical theories T.
1) In the equation of motion of a material body - the second law of Newton's mechanics, ma=F(where F - acting force, a- the acceleration acquired by the body), and Newton's law of gravitation includes the same characteristic of the body - its mass. This implies that the inertial mass of the body and its gravitational mass are equal (for more details, see section 3).

2) Instantaneous value of gravity. potential is completely determined by the instantaneous mass distribution throughout space and the limiting conditions for the potential at infinity. For limited distributions of matter, the condition of vanishing at infinity (at ) is accepted. Adding a constant term to the potential violates the condition at infinity, but does not change the field strength g and does not change the ur-tion of the movement of material bodies in a given field.

3) Transition in accordance with Galilean transformations ( x"=x-vt, t"=t) from one inertial coordinate system to another, moving relative to the first at a constant speed v, does not change the Poisson equation and does not change the equation of movement of material bodies. In other words, mechanics, including Newton's theory of theory, is invariant under Galilean transformations.

4) Transition from an inertial coordinate system to an acceleration moving with acceleration a(t)(without rotation) does not change the Poisson equation, but leads to the appearance of an additional term that does not depend on the coordinates ma in ur-tions of movement. Exactly the same shuttle in the equations of motion occurs if in the inertial coordinate system to gravitational. to the potential, add a term that depends linearly on the coordinates, , i.e. add a homogeneous T. T. field, a homogeneous T. field can be compensated under conditions of accelerated movement.

2. Movement of bodies under the influence of gravitational forces

The most important task of Newtonian celestial mechanics is yavl. the problem of motion of two point material bodies interacting gravitationally. To solve it, using Newton's law of gravity, they make up the equations of motion of bodies. Holy Islands solutions of these ur-tions are known with exhaustive completeness. According to a well-known solution, it can be established that certain quantities characterizing the system remain constant in time. They are called integrals of motion. Main integrals of motion (conserved quantities) yavl. energy, momentum, angular momentum of the system. For a two-body system, a complete mechanical energy E, equal to the sum of the kinetic. energy ( T) and potential energy ( U), is saved:
E=T+U= const ,
where is the kinetic the energy of two bodies.

In the classic celestial mechanics potential energy due to gravity. phone interaction. For a pair of bodies, the gravitational (potential) energy is:
,
where is the gravitational mass potential m 2 at the location of the mass m 1 , a is the potential created by the mass m 1 at the location of the mass m 2. Zero value U have bodies spaced at an infinite distance. Because when the bodies approach each other, their kinetic energy increases and potential energy decreases, then, therefore, the sign U negative.

For stationary gravitating systems, cf. abs value gravity values. twice the energy cf. kinetic values. the energies of the particles that make up the system (see). So, for example, for a small mass m, revolving in a circular orbit around central body, the condition of equality of the centrifugal force mv2 /r gravitational force leads to , i.e. kinetic energy while . Hence, U=-2T and E=U+T=-T= const

In Newton's theory of gravity, a change in the position of a particle instantly leads to a change in the field throughout space (gravitational interaction occurs at infinite speed). In other words, in the classic theory T. the field serves the purpose of describing the instantaneous interaction at a distance, it does not have its own. degrees of freedom, cannot propagate and radiate. It is clear what the idea of ​​gravitation is. the field is valid only approximately for sufficiently slow motions of the sources. Accounting for the finite velocity of propagation of gravity. interaction is produced in the relativistic theory of thermodynamics (see below).

In the nonrelativistic theory of gravity, the total mechanical energy of a system of bodies (including the energy of gravitational interactions) must remain unchanged indefinitely. Newton's theory admits a systematic a decrease in this energy only in the presence of dissipation associated with the conversion of part of the energy into heat, for example. in inelastic collisions of bodies. If the bodies are viscous, then their deformations and oscillations when moving in gravity. the field also reduce the energy of the system of bodies due to the conversion of energy into heat.

3. Acceleration and gravity

inertial body mass ( m i) is a value that characterizes its ability to acquire one or another acceleration under the action of a given force. Inertial mass is included in Newton's second law of mechanics. Gravity weight ( m g) characterizes the body's ability to create one or another field T. Gravity. mass is included in the law of T.

From the experiments of Galileo, with the accuracy with which they were set, it followed that all bodies fall with the same acceleration, regardless of their nature and inertial mass. This means that the force with which the Earth acts on these bodies depends only on their inertial mass, and the force is proportional to the inertial mass of the body in question. But according to Newton's third law, the body under study acts on the Earth with exactly the same force with which the Earth acts on the body. Consequently, the force created by the falling body depends only on one of its characteristics - the inertial mass - and is proportional to it. At the same time, the falling body acts on the Earth with a force determined by gravity. body weight. Thus, for all bodies gravitational. mass is proportional to inertial. Counting m i and m g simply coinciding, find from experiments a specific numerical value of the constant G.

Proportionality of inertial and gravitational. masses in bodies of different nature was the subject of research in the experiments of Hung. physicist R. Eötvös (1922), Amer. physicist R. Dicke (1964) and Soviet physicist V.B. Braginsky (1971). It is tested in the laboratory with high accuracy (with an error

The high accuracy of these experiments makes it possible to estimate the influence on the mass of various types of bond energy between body particles (see ). Proportionality of inertial and gravitational. mass means that physical. interactions within the body are equally involved in the creation of its inertial and gravitational forces. wt.

Relative to a coordinate system moving with acceleration a, all free bodies acquire the same acceleration - a. Due to the equality of the inertial and gravitational. masses, they all acquire the same acceleration relative to the inertial coordinate system under the influence of gravitational forces. fields with intensity g=-a. That is why we can say that from the point of view of the laws of mechanics, homogeneous gravity. field is indistinguishable from the acceleration field. In an inhomogeneous gravitational field compensation of the field intensity by acceleration at once in all space is impossible. However, the field strength can be compensated by accelerating a specially selected coordinate system along the entire trajectory of a body freely moving under the action of forces T. Such a coordinate system is called. freely falling. The phenomenon of weightlessness takes place in it.

Space movement. spacecraft (satellite) in the Earth's T. field can be considered as the motion of a falling coordinate system. The acceleration of astronauts and all objects on the ship relative to the Earth is the same and equal to the acceleration of free fall, and relative to each other is practically zero, so they are in weightlessness.

In free fall in non-uniform gravity. Field compensation of the field strength by acceleration cannot be ubiquitous, since the acceleration of neighboring freely falling particles is not exactly the same, i.e. particles have relative acceleration. In space on a ship, relative accelerations are practically imperceptible, since in order of magnitude they are cm / s 2, where r is the distance from the ship to the center of the earth, is the mass of the earth, x- the size of the ship. These accelerations can be neglected and the gravitational force can be used. Earth's field in the distance r from its center homogeneous in volume with a characteristic size x. In any given volume of space, the inhomogeneity of gravitational field can be established by observations of sufficiently high accuracy, but for any given accuracy of observations, you can specify the amount of space in which the field will look homogeneous.

Relative accelerations manifest themselves, for example, on Earth in the form of ocean tides. The force with which the Moon pulls the Earth is different at different points on the Earth. The parts of the water surface closest to the Moon are attracted more strongly than the center of gravity of the Earth, and it, in turn, is stronger than the most distant parts of the oceans. Along the line connecting the Moon and the Earth, relative accelerations are directed from the center of the Earth, and in orthogonal directions - towards the center. As a result, the water shell of the Earth is deformed so that it is extended in the form of an ellipsoid along the Earth-Moon line. Due to the rotation of the Earth, tidal humps roll over the surface of the ocean twice a day. A similar but smaller tidal deformation is caused by gravity inhomogeneities. fields of the sun.

A. Einstein, based on the equivalence of homogeneous fields T. and accelerated coordinate systems in mechanics, suggested that such an equivalence applies in general to all, without exception, physical. phenomena. This postulate is called the principle of equivalence: all physical processes proceed in exactly the same way (under the same conditions) in an inertial frame of reference located in a uniform gravitational field, and in a frame of reference moving forward with acceleration in the absence of gravitational forces. fields. The equivalence principle played important role in the construction of Einstein's theory T.

4. Relativistic mechanics and field theory

The study of el.-mag. phenomena by M. Faraday and D. Maxwell in the second half of the 19th century. led to the creation of the theory of el.-magn. fields. The conclusions of this theory have been confirmed experimentally. Maxwell's equations are not invariant with respect to Galilean transformations, but are invariant with respect to Lorentz transformations, i.e. the laws of electromagnetism are formulated in the same way in all inertial coordinate systems connected by Lorentz transformations.

If the inertial coordinate system x", y", z", t" moves relative to the inertial coordinate system x, y, z, t at a constant speed v in axis direction x, then the Lorentz transformations have the form:
y"=y, z"=z, .
At low speeds () and neglecting the terms ( v/c) 2 and vx/c 2 these transformations go over to the Galilean transformations.

Logic analysis of the contradictions that arose when comparing the conclusions of the theory of el.-mag. phenomena from the classical ideas about space and time, led to the construction of a private (special) theory of relativity. A decisive step was taken by A. Einstein (1905), a huge role in its construction was played by the works of the Dutch physicist G. Lorenz and the French. mathematician A. Poincare. Particular relativity requires a revision of the classical ideas about space and time. In the classic In physics, the time interval between two events (for example, between two flashes of light), as well as the concept of the simultaneity of events, have an absolute meaning. They do not depend on the motion of the observer. In the theory of relativity, this is not so: judgments about time intervals between events and about segments of length depend on the motion of the observer (the coordinate system associated with him). These quantities turn out to be relative in approximately the same sense as they are relative, depending on the location of the observers, yavl. their judgments about the angle under which they see the same pair of objects. Invariant, absolute, independent of the coordinate system, yavl. only 4-dimensional interval ds between events, including as a period of time dt, and the distance element between them:
ds 2 =c 2 dt 2 -dx 2 -dy 2 -dz 2 . (3)
The transition from one inertial frame to another, preserving ds 2 unchanged, is carried out exactly in accordance with the Lorentz transformations.

Invariance ds 2 means that space and time are combined into a single 4-dimensional world - space-time. Expression (3) can also be written as:
, (4)
where the indices and run through the values ​​0, 1, 2, 3 and summation is performed over them, x 0 =ct, x 1 =x, x 2 =y, x 3 =z, , other quantities are equal to zero. The set of quantities is called the metric tensor of the flat space-time or the Minkowski world [in general relativity (GR) it was shown that space-time has curvature, see below].

In the term "metric tensor" the word "metric" indicates the role of these quantities in determining distances and time intervals. In general, metric tensor is a collection of ten functions depending on x 0 , x 1 , x 2 , x 3 in the selected coordinate system. Metric tensor (or simply metric) allows you to determine the distance and time interval between events separated by .

Specialist. the theory of relativity establishes the limiting speed of the movement of material bodies and the propagation of interactions in general. This speed coincides with the speed of light in vacuum. Together with a change in ideas about space and time, special. the theory of relativity clarified the concept of mass, momentum, force. In relativistic mechanics, i.e. in mechanics invariant under Lorentz transformations, the inertial mass of a body depends on the velocity: , where m 0 - bodies. The energy of the body and its momentum are combined into a 4-component energy-momentum vector. For continuum you can enter energy density, momentum density and momentum flux density. These quantities are combined into a 10-component quantity - the energy-momentum tensor. All components undergo a joint transformation when moving from one coordinate system to another. Relativistic theory of el.-magnet. fields (electrodynamics) are much richer than electrostatics, which is valid only in the limit of slow motions of charges. In electrodynamics, there is a combination of electric. and magnetic fields. Accounting for the finite rate of propagation of field changes and delay in the transfer of interaction leads to the concept of el.-magnet. waves, to-rye carry away energy from the radiating system.

Similarly, the relativistic theory of thermodynamics turned out to be more complicated than Newton's. Gravity the field of a moving body has a number of sv-in, similar to sv-you el.-mag. fields of a moving charged body in electrodynamics. Gravity the field at a great distance from the bodies depends on the position and movement of the bodies in the past, since gravitational. the field propagates at a finite speed. Radiation and distribution of gravitats becomes possible. waves (see). The relativistic theory of thermodynamics, as might be expected, turned out to be non-linear.

5. Curvature of space-time in general relativity

According to the principle of equivalence, no observations, using any laws of nature, can distinguish the acceleration created by a homogeneous field T. from the acceleration of a moving coordinate system. In homogeneous gravity. field, it is possible to achieve zero acceleration of all particles placed in a given region of space, if we consider them in a coordinate system freely falling along with the particles. Such a coordinate system is mentally represented in the form of a laboratory with rigid walls and clocks located in it. The situation is different in non-uniform gravity. field in which neighboring free particles have relative accelerations. They will move with acceleration, albeit small, relative to the center of the laboratory (coordinate system), and such a coordinate system should be recognized as only locally inertial. It is possible to consider the coordinate system as inertial only in the region where it is permissible to neglect the relative accelerations of particles. Therefore, in a non-uniform gravitational field only in a small region of space-time and with limited accuracy, space-time can be considered as flat and f-loy (3) can be used to determine the interval between events.

Impossibility to introduce an inertial coordinate system in non-uniform gravity. the field makes all conceivable coordinate systems more or less equal. Ur-niya gravitats. fields must be written in such a way that they are valid in all coordinate systems, without giving preference to c.-l. of them. Hence the name for the relativistic theory of thermodynamics - the general theory of relativity.

Gravity the fields produced by real bodies, such as the Sun or the Earth, are always non-uniform. They are called true or nonremovable fields. In such a gravity field, no local-inertial coordinate system can be extended to the entire space-time. This means that the interval ds 2 cannot be reduced to the form (3) in the entire space-time continuum, i.e. space-time cannot be flat. Einstein came up with the radical idea of ​​identifying non-uniform gravities. fields with space-time curvature. From these positions, gravitational the field of any body can be considered as a distortion of the geometry of space-time by this body.

Fundamentals of Mathematics. apparatus for the geometry of a space with curvature (non-Euclidean geometry) were laid down in the works of N.I. Lobachevsky, Hung. mathematics J. Bolyai, German. mathematicians K. Gauss and G. Riemann. In non-Euclidean geometry, curved space-time is characterized by metric. tensor included in the expression for the invariant interval:
, (5)
a special case of this expression yavl. formula (4). Having a set of functions , one can raise the question of the existence of such coordinate transformations, which would translate (5) into (3), i.e. would make it possible to check whether space-time is flat. The desired transformations are feasible if and only if a certain tensor, composed of f-tions, squares of their first derivatives and second derivatives, is equal to zero. This tensor is called the curvature tensor. In the general case, of course, it is not equal to zero.

A set of values ​​is used for an invariant, independent of the choice of coordinate system, description of the geometric. st-in curved space-time. With physical point of view of the curvature tensor, expressed in terms of the second derivatives of gravitational. potentials, describes tidal accelerations in non-uniform gravity. field.

The curvature tensor is a dimensional quantity, its dimension is the square of the reciprocal length. Curvature at each point of space-time corresponds to characteristic lengths - radii of curvature. In a small space-time region surrounding a given point, a curved space-time is indistinguishable from a flat one up to small terms , where l is the characteristic size of the region. In this sense, the curvature of the world has the same properties as, say, the curvature the globe: it is insignificant in small regions. The curvature tensor at a given point cannot be "destroyed" by any coordinate transformations. However, in a certain system of coordinates and with a predetermined accuracy, the field T. in a small region of space-time can be considered absent. In this area, all the laws of physics acquire the form that is consistent with the special. theory of relativity. This is how the principle of equivalence manifests itself, which was the basis of the theory of thermodynamics during its construction.

Metric the space-time tensor, and in particular the curvature of the world, are available for experimental determination. To prove the curvature of the globe, it is necessary to have a small "ideal" scale and use it to measure the distance between sufficiently distant points on the surface. A comparison of the measured distances will indicate the difference between the real geometry and the Euclidean one. Similarly, the geometry of space-time can be established by measurements made with "ideal" rulers and clocks. It is natural to assume, following Einstein, that the properties of a small "ideal" atom do not depend on where in the world it is placed. Therefore, having made, for example, a measurement of the shift in the frequency of light (by determining the gravitational redshift), it is possible in principle to determine the metric. space-time tensor and its curvature.

6. Einstein equations

By summing the curvature tensor with metric. tensor can form a symmetric tensor , which has the same number of components as the energy momentum tensor of matter, which serves as a source of gravity. fields.

Einstein suggested that the equations of gravity should establish a relationship between and . In addition, he took into account that in gravitational field, the continuity equation for matter must be fulfilled in the same way as the current continuity equation is performed in electrodynamics. Such ur-tions are performed automatically if the ur-niya gravitats. write fields like this:
. (6)
This is Einstein's equations, obtained by him in 1916. These equations also follow from the variations. principle that independently showed him. mathematician D. Hilbert.

Einstein's equations express the connection between the distribution and motion of matter, on the one hand, and geometric. St. you space-time - on the other.

In equations (6) on the left side are the components of the tensor describing the geometry of space-time, and on the right - the components of the energy-momentum tensor describing the physical. Holy Islands of matter and fields (sources of gravitational fields). Quantities are not just functions that describe the gravitational field, but at the same time are components of the metric space-time tensor.

Einstein wrote that most of his work (special relativity theory, quantum nature of light) was in line with the actual problems of his time. They would have been made by other scientists with a delay of no more than 2-3 years if these works had not been done Einstein made an exception for general relativity and wrote that the relativistic theory of thermodynamics might have been delayed by 50 years. methods of field theory, and another approach to the nonlinear theory of thermodynamics arose, starting from the concept of a field given in a flat space-time. interpretations of T.

It should be emphasized that it is precisely in astronomy and cosmology that questions are encountered in which geometrical approach yavl. preferred. An example is the cosmological the theory of a spatially closed universe, as well as the theory. Therefore, Einstein's theory, based on the geometric concept retains its full meaning.

In the geometric interpretation of the motion of a material point in gravity. the field is a movement along a 4-dimensional trajectory - geodetic. lines of space-time. In a world with curvature, the geodesic. line generalizes the concept of a straight line in Euclidean geometry. The ur-tions of the motion of matter contained in the ur-nies of Einstein are reduced to the ur-tions of the geodesic. lines for point bodies. Bodies (particles), which cannot be considered pointlike, deviate in their motion from the geodesic. lines and experience the action of tidal forces.

7. Weak gravitational fields and observed effects

Field T. Most astronomical. objects yavl. weak. An example is gravity. field of the earth. In order for the body to leave the Earth forever, it must be given a speed of 11.2 km / s near the Earth's surface, i.e. speed is small compared to the speed of light. In other words, gravitational the potential of the Earth is small compared to the square of the speed of light, which is yavl. criterion for the weakness of gravity. fields.

In the weak field approximation, the laws of the Newtonian theory of gravitation and mechanics follow from the equations of general relativity. The effects of general relativity under such conditions represent only minor corrections.

The simplest effect, although difficult to observe, yavl. slowing down the flow of time in gravity. field, or, in a more common formulation, the effect of shifting the frequency of light. If a light signal with a frequency is emitted at a point with a value of gravitational potential and is accepted with a frequency at a point with a potential value (where there is exactly the same transducer for frequency comparison), then the equality must hold. Gravity effect. frequency shift of light was predicted by Einstein back in 1911 on the basis of the law of conservation of photon energy in gravity. field. It is reliably established in the spectra of stars, measured with an accuracy of up to 1% in the laboratory and with an accuracy of up to 1% under space conditions. flight. In the most accurate experiment, a hydrogen-maser frequency standard was used; a rocket that has risen to a height of 10 thousand km. Another similar standard has been set on Earth. Their frequencies were compared at different heights. The results confirmed the predicted frequency change.

When passing near a gravitating body, an el.-mag. the signal experiences a relativistic delay in propagation time. According to its physical nature, this effect is similar to the previous one. According to radio observations of the planets and especially interplanetary space. ships, the delay effect coincides with the calculated value within 0.1% (see).

The most important from the point of view of verification of general relativity yavl. rotation of the orbit of a body revolving around a gravitating center (it is also called the perihelion shift effect). This effect makes it possible to reveal the nonlinear nature of the relativistic gravity. fields. According to Newtonian celestial mechanics, the motion of the planets around the Sun is described by the ellipse equation: , where p=a(1-e 2) - orbit parameter, a- big semi-axle, e- eccentricity (see). Taking into account relativistic corrections, the trajectory has the form:
.
For each revolution of the planet around the Sun, its major axis is elliptical. orbit is rotated in the direction of motion by an angle . For Mercury, the relativistic rotation angle is a century. The fact that the angle of rotation accumulates over time makes it easier to observe this effect. During one revolution, the angle of rotation of the major axis of the orbit is so insignificant ~ 0.1" that its detection is significantly complicated by the curvature of the light rays within solar system. Nevertheless, modern radar data confirm the relativistic effect of the shift of Mercury's perihelion with an accuracy of 1%.

The listed effects are classic. It is also possible to check other predictions of general relativity (for example, the precession of the gyroscope axis) in weak gravitation. field of the solar system. Relativistic effects are used not only to test the theory, but also to refine astrophysical parameters, for example, to determine the mass of the components of binary stars. For example, in a binary system including the pulsar PSR 1913+16, the perihelion shift effect is observed, which made it possible to determine the total mass of the system's components with an accuracy of 1%.

8. Gravity and quantum physics

Einstein's equations include classical gravity. field characterized by the components of the metric. tensor , and the matter energy-momentum enzor . To describe the motion of gravitating bodies, the quantum nature of matter, as a rule, is not important. This is because they usually deal with gravity. macroscopic interaction. bodies, consisting of a huge number of atoms and molecules. The quantum mechanical description of the motion of such bodies is practically indistinguishable from the classical one. Science does not yet have experimental data on gravity. interaction under conditions when quantum properties of particles interacting with gravity become essential. field, and quantum properties of the gravity itself. fields.

Quantum processes involving gravity. The fields are certainly important in space (see , ) and, possibly, will become available for study also in laboratory conditions. The unification of the theory of thermodynamics with quantum theory is one of the most important problems in physics, and the solution to it has already begun.

Under normal conditions, the influence of gravity. field on quantum systems is extremely small. To excite the atom ext. gravitational field, relative acceleration, created by gravity. field at a distance of "radius of the hydrogen atom" cm and equal to , should be comparable with the acceleration with which the electron moves in the atom, . (Here - the radius of curvature of the gravitational field of the Earth, equal to: see) In gravitational field of the Earth with a margin of 10 19 this ratio is not fulfilled, therefore, atoms under terrestrial conditions under the influence of gravity are not excited and do not experience energy shifts. levels.

Nevertheless, under certain conditions, the probability of transitions in a quantum system under the action of gravitational forces. margins may be noticeable. It is on this principle that some modern assumptions on the detection of gravity. waves.

In specially created (macroscopic) quantum systems, the transition between neighboring quantum levels can occur even under the influence of a very weak alternating gravitational field. waves. An example of such a system is an el.-magnet. field in a cavity with highly reflective walls. If initially the system had N field quanta (photons) (), then under the influence of gravity. waves, their number can change with a noticeable probability and become equal to N+2 or N-2. In other words, transitions with energy are possible. level , and they are, in principle, discoverable.

The role of intense gravitations is especially important. fields. Such fields probably existed at the beginning of the expansion of the Universe, near the cosmological singularities and can occur in the later stages of gravity. collapse. The high intensity of these fields is manifested in the fact that they are capable of leading to observable effects (the production of pairs of particles) even in the absence of atoms, real particles or photons. These fields have an effective impact on physical. vacuum - physical. fields in the lowest energy state. In vacuum, due to the fluctuations of quantized fields, the so-called. virtual, actually unobservable particles. If the intensity of the external gravitational field is so large that at distances characteristic of quantum fields and particles, it is able to produce work that exceeds the energy of a pair of particles, then as a result, a pair of particles can be born - their transformation from a virtual pair into a real one. Necessary condition this process should be comparability of the characteristic radius of curvature describing the intensity of gravity. field, with the Compton wavelength , associated with particles with a rest mass m. A similar condition must be satisfied for massless particles so that the process of production of a pair of quanta with energy is possible. In the above example of a cavity containing an el.-mag. field, this process is similar to a transition with a probability comparable to unity from a vacuum state N=0 to a state describing two quanta, N=2. In ordinary gravity fields, the probability of such processes is negligible. However, in space, they could lead to the birth of particles in the very early Universe, as well as to the so-called. quantum "evaporation" of black holes of small mass (according to) works of English. scientist S. Hawking).

Intense gravitational fields that can significantly affect the zero fluctuations of other physical. fields should equally effectively affect their own zero fluctuations. If the process of birth of quantum physical. fields, then with the same probability (and in some cases with even greater probability) the process of the birth of gravitational quanta itself should be possible. fields - gravions. A rigorous and exhaustive consideration of such processes is possible only on the basis of quantum theory T. Such a theory has not yet been created. Application to gravity. field of the same ideas and methods, which led to the successful construction of quantum electrodynamics, encounters serious difficulties. It is not yet clear what paths the development of the quantum theory of T will take. One thing is certain - the most important way to test such theories will be the search for the phenomena predicted by the theory in space.

Why does a stone released from the hands fall to the ground? Because it is attracted by the Earth, each of you will say. In fact, the stone falls to the Earth with free fall acceleration. Consequently, a force directed towards the Earth acts on the stone from the side of the Earth.

According to Newton's third law, the stone also acts on the Earth with the same modulus of force directed towards the stone. In other words, forces of mutual attraction act between the Earth and the stone.

Newton's guess

Newton was the first who first guessed, and then strictly proved, that the reason causing the fall of a stone to the Earth, the movement of the Moon around the Earth and the planets around the Sun, is one and the same. This is the gravitational force acting between any bodies of the Universe. Here is the course of his reasoning, given in Newton's main work "Mathematical Principles of Natural Philosophy": "A stone thrown horizontally will deviate from a straight path under the influence of gravity and, having described a curved trajectory, will finally fall to the Earth. If you throw it with greater speed, then it will fall further” (Fig. 3.2). Continuing these reasoning, Newton comes to the conclusion that if it were not for air resistance, then the trajectory of a stone thrown from a high mountain at a certain speed could become such that it would never reach the Earth’s surface at all, but would move around it “like how the planets describe their orbits in celestial space.

Rice. 3.2

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

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

The dependence of the force of gravity on the mass of bodies

In § 1.23 we talked about the free fall of bodies. Galileo's experiments were mentioned, which proved that the Earth communicates the same acceleration to all bodies in a given place, regardless of their mass. This is possible only if the force of attraction to the Earth is directly proportional to the mass of the body. It is in this case that the acceleration of free fall, equal to the ratio of the force of gravity to the mass of the body, is a constant value.

Indeed, in this case, an increase in the mass m, for example, by a factor of two will lead to an increase in the modulus of force also by a factor of two, and the acceleration, which is equal to the ratio, will remain unchanged.

Generalizing this conclusion for the forces of gravity between any bodies, we conclude that the force of universal gravitation is directly proportional to the mass of the body on which this force acts. But at least two bodies participate in mutual attraction. Each of them, according to Newton's third law, is subject to the same modulus of gravitational forces. Therefore, each of these forces must be proportional both to the mass of one body and to the mass of the other body.

So The gravitational force between two bodies is directly proportional to the product of their masses.:

What else determines the gravitational force acting on a given body from another body?

The dependence of the force of gravity on the distance between bodies

It can be assumed that the force of gravity should depend on the distance between the bodies. To test the correctness of this assumption and to find the dependence of the force of gravity on the distance between bodies, Newton turned to the motion of the Earth's satellite - the Moon. Its motion was studied in those days much more accurately than the motion of the planets.

The revolution of the Moon around the Earth occurs under the influence of the gravitational force between them. Approximately, the orbit of the Moon can be considered a circle. Therefore, the Earth imparts centripetal acceleration to the Moon. It is calculated by the formula

where R is the radius of the lunar orbit, equal to about 60 radii of the Earth, T \u003d 27 days 7 h 43 min \u003d 2.4 10 6 s is the period of the Moon's revolution around the Earth. Given that the radius of the Earth R 3 = 6.4 10 6 m, we get that the centripetal acceleration of the Moon is equal to:

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

Thus, an increase in the distance between the body and the Earth by 60 times led to a decrease in the acceleration imparted by the earth's gravity, and, consequently, the force of attraction itself by 60 2 times (1).

Hence follows important conclusion: the acceleration imparted to bodies by the force of attraction to the earth decreases in inverse proportion to the square of the distance to the center of the earth:

where C 1 - constant factor, the same for all bodies.

Kepler's laws

The study of the motion of the planets showed that this motion is caused by the force of gravity towards the Sun. Using careful long-term observations of the Danish astronomer Tycho Brahe, the German scientist Johannes Kepler in early XVII in. established the kinematic laws of planetary motion - the so-called Kepler's laws.

Kepler's first law

All planets move in ellipses with the Sun at one of the foci.

An ellipse (Fig. 3.3) is a flat closed curve, the sum of the distances from any point of which to two fixed points, called foci, is constant. This sum of distances is equal to the length of the major axis AB of the ellipse, i.e.

where F 1 and F 2 are the foci of the ellipse, and b = is its major axis; O is the center of the ellipse. The point of the orbit closest to the Sun is called perihelion, and the point farthest from it is called aphelion. If the Sun is in focus F 1 (see Fig. 3.3), then point A is perihelion, and point B is aphelion.

Rice. 3.3

Kepler's second law

The radius-vector of the planet for the same intervals of time describes equal areas. So, if the shaded sectors (Fig. 3.4) have the same area, then the paths s 1, s 2, s 3 will be traversed by the planet in equal time intervals. It can be seen from the figure that s 1 > s 2 . Consequently, the linear velocity of the planet at different points of its orbit is not the same. At perihelion, the speed of the planet is greatest, at aphelion - the smallest.

Rice. 3.4

Kepler's third law

The squares of the periods of revolution of the planets around the Sun are related as the cubes of the semi-major axes of their orbits. Denoting the semi-major axis of the orbit and the period of revolution of one of the planets through b 1 and T 1 and the other - through b 2 and T 2, Kepler's third law can be written as follows:

Based on Kepler's laws, certain conclusions can be drawn about the accelerations imparted to the planets by the Sun. For simplicity, we will assume that the orbits are not elliptical, but circular. For the planets of the solar system, this replacement is not a very rough approximation.

Then the force of attraction from the side of the Sun in this approximation should be directed for all planets to the center of the Sun.

If through T we denote the periods of revolution of the planets, and through R the radii of their orbits, then, according to Kepler's third law, for two planets we can write

Normal acceleration when moving in a circle a \u003d ω 2 R. Therefore, the ratio of the accelerations of the planets

Using equation (3.2.4), we get

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:

The constant C 2 is the same for all planets, but does not coincide with the constant C 1 in the formula for the acceleration imparted to bodies by the globe.

Expressions (3.2.2) and (3.2.6) show that the gravitational force in both cases (attraction to the Earth and attraction to the Sun) gives all bodies an acceleration that does not depend on their mass and decreases inversely with the square of the distance between them:

Law of gravity

The existence of dependences (3.2.1) and (3.2.7) means that the force of universal gravitation

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

The force of mutual attraction of two bodies is directly proportional to the product of the masses of these bodies and inversely proportional to the square of the distance between them. The coefficient of proportionality G is called the gravitational(2) constant.

Interaction of point and extended bodies

The law of universal gravitation (3.2.8) is valid only for such bodies, the dimensions of which are negligible compared to the distance between them. In other words, it is valid only for material points. In this case, the forces of gravitational interaction are directed along the line connecting these points (Fig. 3.5). Such forces are called central.

Rice. 3.5

To find the gravitational force acting on a given body from another, in the case when the size of the bodies cannot be neglected, proceed as follows. Both bodies are mentally divided into such small elements that each of them can be considered a point. Adding up the gravitational forces acting on each element of a given body from all the elements of another body, we obtain the force acting on this element (Fig. 3.6). Having done such an operation for each element of a given body and adding the resulting forces, they find the total gravitational force acting on this body. This task is difficult.

Rice. 3.6

There is, however, one practically important case when formula (3.2.8) is applicable to extended bodies. It can be proved that spherical bodies, the density of which depends only on the distances to their centers, at distances between them that are greater than the sum of their radii, attract with forces whose modules are determined by formula (3.2.8). In this case, R is the distance between the centers of the balls.

And finally, since the dimensions of the bodies falling to the Earth are much smaller than the dimensions of the Earth, these bodies can be considered as point ones. Then under R in the formula (3.2.8) one should understand the distance from the given body to the center of the Earth.

Questions for self-examination

1. The distance from Mars to the Sun is 52% greater than the distance from Earth to the Sun. What is the length of a year on Mars?
2. How will the force of attraction between the balls change if the aluminum balls (Fig. 3.7) are replaced by steel balls of the same mass? the same volume?

Rice. 3.7

(1) Interestingly, as a student, Newton realized that the Moon moves under the influence of gravity towards the Earth. But at that time the radius of the Earth was not known exactly, and the calculations did not lead to the correct result. Only 16 years later, new, corrected data appeared, and the law of universal gravitation was published.

(2) From the Latin word gravitas - heaviness.