Gravity (universal gravitation, gravitation)(from lat. 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 gives the same acceleration to all bodies without exception, regardless of their mass. Primarily gravity plays a decisive role on a cosmic scale. Term gravity also used as the name of a branch of physics that studies the gravitational interaction. The most successful modern physical theory in classical physics, describing gravity, is the general theory of relativity, the quantum theory of gravitational interaction has not yet been built.
Gravitational interaction
Gravitational interaction is one of four fundamental interactions in our world. Within classical mechanics, the gravitational interaction is described by law gravity Newton who says that force gravitational attraction between two material points masses m 1 and m 2 separated by distance R, is proportional to both masses and inversely proportional to the square of the distance - i.e.
.Here G- gravitational constant, equal to approximately 
m³/(kg s²). The minus sign means that the force acting on the body is always equal in direction to the radius vector directed to the body, that is, the gravitational interaction always leads to the attraction of any bodies.
The law of universal gravitation is one of the applications of the inverse square law, which is also encountered in the study of radiation (see, for example, Light Pressure), and which 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 the area of the entire sphere.
The easiest task 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 three Kepler's laws.
As the number of interacting bodies increases, the problem becomes much more complicated. So, the already famous three-body problem (that is, the movement three bodies with nonzero masses) cannot be solved analytically in general view. With a numerical solution, the instability of solutions with respect to the initial conditions sets in rather quickly. When applied to the solar system, this instability makes it impossible to predict the motion of the planets on scales exceeding a hundred million years.
In some special cases, it is possible to find an approximate solution. The most important is the case when the mass of one body is significantly more mass other bodies (examples: solar system and the dynamics of Saturn's rings). In this case, in the first approximation, we can assume that light bodies do not interact with each other and move along Keplerian trajectories around a massive body. Interactions between them can be taken into account in the framework of perturbation theory, and averaged over time. In this case, non-trivial phenomena can arise, such as resonances, attractors, randomness, etc. illustrative example such phenomena - non-trivial structure of the rings of Saturn.
Despite attempts to describe the behavior of the system from a large number attracting bodies of approximately the same mass, this cannot be done due to the phenomenon of dynamic chaos.
Strong gravitational fields
In strong gravitational fields, when moving with relativistic speeds, the effects of general relativity begin to appear:
- deviation of the law of gravity from Newtonian;
- potential delay associated with the finite propagation velocity of gravitational perturbations; the appearance of gravitational waves;
- non-linear effects: gravitational waves tend to interact with each other, so the principle of superposition of waves in strong fields no longer performed;
- change in 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 a binary system with the PSR B1913+16 pulsar - the Hulse-Taylor pulsar - are in good agreement with the model in which this energy is carried away by gravitational radiation.
Gravitational radiation can only be generated by systems with variable quadrupole moments or higher multipole moments, this fact suggests that the gravitational radiation of most natural sources directional, which significantly complicates its detection. Gravity power l-poly source is proportional (v / c) 2l + 2 , if the multipole is of electric type, and (v / c) 2l + 4 - if multipole magnetic type, where v is the characteristic velocity of sources in the radiating system, and c is the speed of light. Thus, the dominant moment will be the quadrupole moment of the electric type, and the power of the corresponding radiation is equal to:
where Q ij is the tensor of the quadrupole moment of the mass distribution of the radiating system. Constant 
(1/W) makes it possible to estimate the order of magnitude of the radiation power.
Since 1969 (Weber's experiments (English)) and up to the present (February 2007), attempts have been made to directly detect gravitational radiation. In the USA, Europe and Japan in this moment there are several active 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, general relativity predicts the existence of other manifestations of gravity, which in earthly 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, one can name the drag of inertial reference frames (or the Lense-Thirring effect) and the gravitomagnetic field. In 2005 automatic apparatus NASA's Gravity Probe B has conducted an experiment of unprecedented accuracy to measure these effects near Earth, but the full results have yet to be published.
quantum theory of gravity
Despite more than half a century of attempts, gravity is the only fundamental interaction for which a consistent renormalizable quantum theory has not yet been built. However, at low energies, in the spirit of quantum field theory, the gravitational interaction can be represented as an exchange of gravitons - gauge bosons with spin 2.
Standard Theories of Gravity
Due to the fact that quantum effects gravitations 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 it is possible to restrict classic description gravitational interaction.
There is a modern canonical classical theory gravity - the general theory of relativity, and many hypotheses and theories that refine it varying degrees development, competing with each other (see the article Alternative theories of gravity). All of these theories give very similar predictions within the approximation in which experimental tests are currently being carried out. The following are some of the main, most well developed or famous theories 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 the flat Minkowski space, in which the laws of conservation of energy-momentum and angular momentum are unambiguously fulfilled. Then the motion of bodies in the Minkowski space is equivalent to the motion of these bodies in the effective Riemannian space.
- In tensor equations, to determine the metric, one should take into account the mass of the graviton, and also use the gauge conditions associated with the metric of the Minkowski space. This does not allow destroying the gravitational field even locally by choosing some suitable frame of reference.
As in general relativity, in RTG, matter refers to all forms of matter (including the electromagnetic field), with the exception of the gravitational field. 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, immobile and Euclidean.
On the other hand, there are at least convincing arguments opponents of the RTG, which boil down to the following provisions:
A similar thing happens in RTG, where the second tensor equation is introduced to take into account the connection between the non-Euclidean space and the Minkowski space. Due to the presence of a dimensionless 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.
| Theories of gravity | ||||||||
|
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. - 200p.
see also
- gravimeter
Links
- The law of universal gravitation or "Why does the moon not fall to the Earth?" - Just about the complex
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Sokol-Kutylovsky O.L.
On the forces of gravitational interaction
If you ask any student or professor of the physics or mechanics and mathematics departments of any university about the forces of gravitational interaction, it would seem that the most studied of all known force interactions, then all they can do is write formulas for Newton's force and for centrifugal force, which they will remember the incomprehensible Coriolis force and the existence of some mysterious gyroscopic forces. And all this despite the fact that all gravitational forces can be obtained from general principles classical physics.
1. What is known about gravitational forces
1.1. It is known that the force that arises between bodies in gravitational interaction, directly proportional to the mass of these bodies and inversely proportional to the square of the distance between them (the law of universal gravitation or Newton's law):
| , | (1) |
where G" 6.6720Ch 10 -11 LF m 2Ch kg -2 - gravitational constant, m, M- masses of interacting bodies and r- the shortest distance between the centers of mass of interacting bodies. Assuming that the body mass M on distance r creates a gravitational acceleration field directed towards its center of mass,
force (1) acting on a body of mass m, are also presented in the form:
where w is the angular velocity of rotation of the body around an axis that does not pass through the center of mass of the body, v
is the speed of the rectilinear motion of the body and r
is the radial vector connecting the axis of rotation with the particle or with the center of mass of the rotating body. The first term corresponds to the gravitational force of gravity (1), the second term in formula (3) is called the Coriolis force, and the third term is centrifugal force. The Coriolis force and the centrifugal force are considered fictitious, depending on the frame of reference, which absolutely does not correspond to experience and elementary common sense. How can a force be considered fictitious if it can perform real job? It is obvious that these are not fictitious physical forces, but the currently available knowledge and understanding of these forces.
The origin of the numerical coefficient "2" in the Coriolis force is doubtful, since this coefficient was obtained for the case when the instantaneous speed of the points of the body in a rotating frame of reference coincides with the speed of the moving body or is directed against it, that is, with the radial direction of the Coriolis force. The second case, when the speed of the body is orthogonal instantaneous speed points of the rotating frame of reference, in not considered. According to the method described in , the magnitude of the Coriolis force in the second case turns out to be zero, while at given angular and linear velocities it should be the same.
1.3. The angular velocity is an axial vector, that is, it is characterized by a certain value and is directed along a single selected axis. direction sign angular velocity determined by the right screw rule. The angular velocity of rotation is defined as the change in the angle of rotation per unit time, ω( t)=¶
φ/¶
t. In this definition φ( t) – periodic function time with a period of 2π radians. At the same time, the angular velocity is inverse function time. This follows, in particular, from its dimension. For these reasons, the derivative of the angular velocity with respect to time: ¶ ω /¶ t=-ω 2 .
The time derivative of the angular velocity corresponds to the axial vector of the angular acceleration. According to the conditional definition given in the physical encyclopedic dictionary, the axial vector of angular acceleration is directed along the axis of rotation, and in the same direction as the angular velocity, if the rotation is accelerated, and against the angular velocity, if the rotation is slow.
2. Gravitational forces acting on the center of mass of the body
Gravity and mechanical forces differ from each other in the nature of interaction: with the "contact" interaction of bodies, mechanical forces arise, and with remote gravitational interaction of bodies - gravitational forces.
2.1. Let us define all gravitational forces acting on the center of mass of a material body. Rotation of the body around own axis, passing through its center of mass, will not be considered yet. From the general principles of mechanics, it is known that force arises when the instantaneous momentum of a body changes. Let's act In a similar way as in determining the forces associated with rectilinear movement body, and in determining the forces associated with its rotation relative to the external axis:
or in expanded form:
where r
=r·[ cos(ω t)· x
+ sin(ω t)· y
], x
and y
are unit vectors in the direction of the corresponding coordinate axes, r is the modulus of the radial vector r
, r
1
=r
/r is the unit vector in the direction of the radial vector r
, t is time, and the coordinate axis z
coincides with the axis of rotation. Unit vector derivative value r
1
by time, ¶ r
1 /¶ t=ω· r
1^ , where r
1^ is the unit vector lying in the plane of rotation and orthogonal to the radial vector r
(Fig. 1).
Pay attention to possible changes radial vector, in accordance with equation (7), formula (6) takes the form:
| . | (8) |
Rice. one. Mutual arrangement radial vector r
, angular velocity ω
and instantaneous speed v
m body mass m,
in the coordinate system ( x, y, z) with the axis of rotation directed along the axis z. Unit vector r
1 =r
/r is orthogonal to the unit vector r
1^
.
2.2. All forces included in equation (8) are equal and are added according to the vector addition rule. The sum of forces (8) can be represented as four terms:
F G= F a+F ω1 + F ω2 + F ω3 .
Force F
a occurs in a straight line fast motion body or in the gravitational static interaction of a body with another body. Force F
ω1 corresponds to the Coriolis force for the case when a material body moves in a rotating system in the radial direction (along the radius of rotation). This force is directed towards the instantaneous velocity of the body or against it. Force F
ω2 is the force acting on any point of the rotating body. It is called the centrifugal force, but the same force is called the Coriolis force if the body in the rotating system moves in the direction of the instantaneous speed without changing the radius of rotation. Force F
ω2 is always directed radially. Considering equality ¶ r
1 /¶ t=ω· r
1^ , and the direction of the resulting vector in vector product, we obtain that during the rotation of each point of the body with an angular velocity ω
force acts on it F
ω2 = mω 2 r
, which coincides with the centrifugal force in formula (3).
Force F
ω3 is the force of inertia rotary motion. The inertia force of the rotational motion arises when the angular velocity of the rotating system and the bodies associated with it changes and is directed along the instantaneous velocity vector of the body at dw/dt<0 и против вектора мгновенной скорости тела при dw/dt>0. It occurs only during transient processes, and with a uniform rotation of the body, this force is absent. Direction gravitational force rotational inertia
 |
(9) |
shown in Fig. 2. Here r is the radial vector connecting the shortest way axis of rotation with the center of mass of the rotating body, ω is the axial vector of the angular velocity.
Rice. 2. The direction of the gravitational force of inertia of rotational motion, F ω3,
when moving the body from point 1 to point 2 when dw /
dt<0; r
is the radial vector ,
connecting the axis of rotation with the center of mass of the moving body; F T
- the force of attraction or the tensile force of the rope. The centrifugal force is not shown.
Vector sum of forces F
ω1 and F
ω2 creates the resulting force (the Coriolis force F
K) when a body moves in an arbitrary direction in a rotating system:
3.
Gravitational and mechanical forces arising from the rotation of the axis of rotation of the body
To determine all gravitational forces acting not only on the center of mass, but also on any other point of a material body, including those arising when the axis of rotation of this body rotates around another axis, it is necessary to return to formula (5).
The general formula for all gravitational and mechanical forces obtained earlier remains valid, but until now all the forces obtained were considered to be applied to the center of mass of the body. The influence of the rotation of its own axis of rotation on individual points of the body that do not coincide with the center of mass was not taken into account. Nevertheless, the formula (5) obtained earlier from the general principles of mechanics contains all the forces acting on any point of a rotating body, including the forces arising from the spatial rotation of the own axis of rotation of this body. Therefore, from formula (5), one can explicitly derive an equation for the force acting on an arbitrary point of a rotating material body when its own axis of rotation rotates through a certain angle in space. To do this, we represent equation (5) in the following form:
| (12) |
where S rґ w S
is the vector modulus rw w
, a ( rw w
) 1 is a unit vector directed along the vector rw w
. As shown, the time derivative of the vector rw w
when the value of this vector changes, it gives the gravitational and mechanical forces of rotation, from which the centrifugal force, the Coriolis force, and the inertia force of rotational motion are obtained:
where the fifth term is the force, or rather, it is the set of forces arising from the spatial rotation of the axis of rotation of the body at all points of this body, and the force arising at each point depends on the location of this point. In short notation, it is convenient to represent the total sum of all gravitational forces as:
 ,
|
(15) |
where Fa
is the Newton force with the gravitational acceleration vector a
, fw
1 – fw
3 - forces of rotational motion with the gravitational vector of the angular velocity w and e Fw W i
is the set of forces arising from the rotation of the axis of rotation of the body in all n points into which the body is evenly divided.
Let us represent the fifth term in expanded form. By definition, the radial vector r
is orthogonal to the angular velocity vector w, so the modulus of the vector rw w
is equal to the product of the modules of its constituent vectors:
The time derivative of a unit vector ( rw w
) 1 when changing it towards the angle j gives another unit vector, r 1 , located parallel to the plane of rotation S ( x, z) and orthogonal to the vector rw w
(Fig. 3). Moreover, as a factor, he has a coefficient numerically equal to the time derivative of the angle of rotation, W =¶ j /¶ t:
 .
|
(16) |
Since when the rotation axis is rotated, the movement of points of the material body is three-dimensional, and the rotation of the axis occurs in some plane S ( x, z), then the modulus of the unit vector relative to the plane of rotation is not constant, and during rotation it varies from zero to one. Therefore, when differentiating such a unit vector, its value relative to the plane in which this unit vector rotates must be taken into account. The length of the unit vector ( rw w
) 1 with respect to the plane of rotation S ( x, z) is the projection of this unit vector onto the plane of rotation. Unit vector derivative ( rw w
) 1 in the plane of rotation S ( x, z) can be represented as follows:
 ,
|
(17) |
where a is the angle between the vector rw w
and the plane of rotation S ( x, z).
The force acting on any point of a rotating body when turning its axis of rotation is applied not to the center of mass of this body, but directly to each given point. Therefore, the body must be divided into many points, and consider that each such point has a mass m i. Under the weight of a given point of the body, m i, means the mass concentrated in a volume small in relation to the whole body Vi so:
With a uniform density of the body r mass, and the point of application of force is the center of mass of a given volume Vi occupied by a part of a material body with a mass m i. Force acting on i-th point of a rotating body when turning its axis of rotation, takes the following form:
| , | (18) |
where m i is the mass of a given point of the body, r i is the shortest distance from a given point (at which the force is determined) to the axis of rotation of the body, w is the angular velocity of rotation of the body, W is the modulus of the angular velocity of rotation of the axis of rotation, a is the angle between the vector rw w and the plane of rotation S ( x, z), and r 1 is a unit vector directed parallel to the plane of rotation and orthogonal to the instantaneous velocity vector rw w .
Rice. 3. Force direction Fw W
, which arises when the axis of rotation of the body rotates in the plane S (x, z) with angular velocity W . At the point a with a radius vector emanating from a point with axis of rotation, force Fw W
=0; at the point b with a radius vector emanating from the center of the body, the force Fw W
has a maximum value.
The sum of all forces (18) acting on everything n points into which the body is evenly divided,
 |
(19) |
creates a moment of forces that rotate the body in the Y plane ( y, z), orthogonal to the plane of rotation S ( x, z) (Fig. 4).
From experiments with rotating bodies, the very presence of forces (19) is known, but they have not been clearly defined. In particular, in the theory of the gyroscope, the forces acting on the bearings of the gyroscope are called "gyroscopic" forces, but the origin of these physical forces is not disclosed. In a gyroscope, when its axis of rotation is rotated, force (18) acts on each point of the body, obtained here from the general principles of classical physics and expressed quantitatively in the form of a specific equation.
From the property of symmetry it follows that each point of the body corresponds to another point located symmetrically with respect to the axis of rotation, in which the force of the same magnitude, but having the opposite direction, acts (18). The joint action of such symmetrical pairs of forces during the rotation of the axis of a rotating body creates a moment of forces that rotates this body in the third plane Y ( y, z), which is orthogonal to the plane of rotation S ( x, z) and planes L (x, y), in which the body points rotate:
| . | (20) |
Rice. 4. The emergence of a moment of forces under the action of pairs of forces at points of the body located symmetrically with respect to the center of mass. 1 and 2 are two symmetrical points of a body rotating with an angular velocity w, in which, when the axis of rotation of the body rotates with an angular velocity W, equal forces arise Fw W
1 and Fw W
2 , respectively.
In this case, for unit vectors of angular velocities characterizing their direction, at any of the points of the body that do not coincide with the center of symmetry (center of mass), the vector identity is fulfilled:
| , | (21) |
where Q 1 is the unit axial vector of the angular velocity that occurs at the moment of action of the force (18), w 1 is the unit axial vector of the angular velocity of the body rotation and W 1 is the unit axial vector of the angular velocity of rotation of the axis of rotation (Fig. 2). Since the axis of rotation, coinciding with the vector of the angular velocity of rotation W, is always orthogonal to the axis of rotation, coinciding with the vector of the angular velocity of rotation of the body, w, then the angular velocity vector Q is always orthogonal to the vectors w and W : .
By rotating the coordinate system in space, the problem of finding the force (18) can always be reduced to a case similar to that considered in Fig. 3. Only the direction of the axial vector of the angular velocity w and the direction of the axial vector of the speed of rotation of the axis of rotation, W, can change, and, as a result of their change, it can change to the opposite direction of the force Fw W
.
The relationship between the absolute values of the angular velocities during free rotation of the body along three mutually orthogonal axes can be found by applying the law of conservation of energy of rotational motion. In the simplest case, for a homogeneous body with mass m in the form of a sphere with a radius r we have:
,
from where we get:
.
4. The total sum of the primary gravitational and mechanical forces acting on the body
4.1. Taking into account the forces (19) that arise when the rotation axis of the body rotates, the complete equation for the sum of all gravitational forces acting on any point of the material body participating in rectilinear and rotational motion, including with a spatial rotation of its own axis of rotation, has the following form :
 |
(22) |
where a
is the rectilinear acceleration vector of a body with mass m, r
is the radial vector connecting the axis of rotation of the body with the point of application of the force, r is the modulus of the radial vector r
,r
1 - unit vector, coinciding in direction with the radius vector r
, w is the angular velocity of rotation of the body, S rґ w S
is the module of the instantaneous velocity vector rw w
, (rw w
) 1 is a unit vector coinciding in direction with the vector rw w
, r
1^ is a unit vector located in the plane of rotation and orthogonal to the vector r
1 , W is the module of the angular velocity of rotation of the axis of rotation, r 1 is a unit vector directed parallel to the plane of rotation and orthogonal to the instantaneous velocity vector rw w
, a is the angle between the vector rw w
and the plane of rotation m i- weight i- that point of the body, concentrated in a small volume of the body Vi, whose center is the point of application of the force, and n is the number of points into which the body is divided. In formula (22) for the second, third and fourth forces, the sign can be taken positive, since these forces in the general formula are under the sign of the absolute value. The signs of forces are determined taking into account the direction of each specific force. With the help of the forces included in formula (22), it is possible to describe the mechanical motion of any point of a material body when it moves along an arbitrary trajectory, including the spatial rotation of its axis of rotation.
4.2. So, in the gravitational interaction there are only five different physical forces acting on the center of mass and on each of the points of the material body during the translational and rotational motion of this body, and only one of these forces (Newton's force) can act on a stationary body from the side of another body . Knowledge of all the forces of gravitational interaction makes it possible to understand the reason for the stability of dynamic mechanical systems (for example, planetary ones), and, taking into account electromagnetic forces, to explain the stability of the atom.
Literature:
1. L. D. Landau, A. I. Akhiezer, and E. M. Lifshits, Course of General Physics. Mechanics and molecular physics. — M.: Nauka, 1969.
2. Saveliev I.V. Course of general physics. T.1. Mechanics. Molecular physics. 3rd ed., rev. — M.: Nauka, 1987.
3. Sokol-Kutylovsky O.L. Gravitational and electromagnetic forces. Yekaterinburg, 2005
Sokol-Kutylovsky O.L., On the forces of gravitational interaction // "Academy of Trinitarianism", M., El No. 77-6567, publ. 13569, 18.07.2006
Gravity force
FORCE
The basis of mechanics is Newton's second law. When a law is written mathematically, the cause is written on the right, and the effect on the left. The cause is force, and the effect of force is acceleration. So the second law is written like this:
The acceleration of a body is proportional to the resulting force acting on the body and inversely proportional to the mass of the body. Directed acceleration in the direction of the resulting force. The resulting force is equal to the vector sum of all forces acting on the body: .
Real forces characterize the measure of interaction between two bodies. In the future, we will consider several types of interactions - gravitational, electrical, molecular. Each type of interaction has its own strength. If there are no interactions, then there are no forces. Therefore, first of all, it is necessary to find out which bodies interact with each other.
Gravity force
The body is thrown and flies over the Earth (Fig. 1.1). Available only
Rice. 1.1. Forces acting on a thrown stone ( a), stone acceleration ( b) and its speed ( in)
the interaction of the body with the Earth, which is characterized by the gravitational force of attraction (gravitation). According to the law of universal gravitation, the gravitational force is directed towards the center of the Earth and is equal to
where M is the mass of the earth, t- body mass, r is the distance from the center of the earth to the body, γ is the gravitational constant. There are no other interactions, so there are no other forces.
To find the acceleration of the stone, the gravitational force from formula 1.2 is substituted into formula 1.1 of Newton's second law. Obviously, the acceleration of the stone is always directed downward (Fig. 1.1, b). At the same time, the speed of the flying stone changes and at each point of the trajectory is directed tangentially to this trajectory (Fig. 1.1, in).
Newton's second law relates vector quantities - acceleration a and the resulting force. Any vector is given by magnitude (modulus) and direction. You can specify a vector with three projections on the coordinate axes, that is, three numbers. In this case, the choice of axes is determined by convenience. On fig. 1.1 axle X can be directed downward. Then the acceleration projections will be equal to a x, 0, 0. If the axis X point upwards, then the acceleration projections will become equal - a x,0,0. In what follows, we will choose the direction of the axis X so that it coincides in direction with the acceleration and for simplicity we will write not the quantity a x, but just a. So, the acceleration created by the gravitational force is
(1.3)
For bodies close to the earth's surface, r» R(earth radius R= 6400 km), so
m/s 2 (1.4)
Therefore, in the vertical direction, the thrown body moves with uniform acceleration.
From formula 1.3 it follows that the acceleration of free fall does not depend on the mass of the flying (falling) body and is determined only by the mass of the planet M and the distance of the body from the center of the planet r. The farther from the center of the planet is the body, the less acceleration of free fall.
Gravitational interaction− the weakest of four fundamental interactions. According to Newton's law of universal gravitation, the force of gravitational interaction F g of two point masses m 1 and m 2 is
G \u003d 6.67 10 -11 m 3 kg -1 cm -2 - gravitational constant, r - distance between the interacting masses m 1 and m 2. The ratio of the strength of the gravitational interaction between two protons to the strength of the Coulomb electrostatic interaction between them is 10 -36 .
The quantity G 1/2 m is called the gravitational charge. The gravitational charge is proportional to the mass of the body. Therefore, for the nonrelativistic case, according to Newton's law, the acceleration caused by the force of the gravitational interaction F g does not depend on the mass of the accelerated body. This statement is equivalence principle
.
The fundamental property of the gravitational field is that it determines the geometry of the space-time in which matter moves. According to modern concepts, the interaction between particles occurs through the exchange of particles between them - the carriers of interaction. It is believed that the carrier of the gravitational interaction is the graviton - a particle with spin J = 2. The graviton has not been detected experimentally. The quantum theory of gravity has not yet been created.
Consider the gravitational interaction between a homogeneous sphere of radius R, and the masses M and material point of mass m located at a distance r from the center of the sphere (Fig. 116).
In accordance with the above method for calculating forces, it is necessary to divide the sphere into small sections and sum up the forces acting on a material point from all sections of the sphere. Such a summation was first carried out by I. Newton. Without going into the mathematical subtleties of the calculation, we present the final result: the resulting force is directed towards the center of the ball (which is quite obvious), and the magnitude of this force is determined by the formula
In other words, the force of interaction turned out to be the same as the force of interaction of two point bodies, one of which is placed in the center of the sphere and its mass is equal to the mass of the sphere. The fact that the force of gravitational interaction is inversely proportional to the square of the distance between point bodies turned out to be essential in this calculation; for any other dependence of the force on the distance, the given result of the calculation would be incorrect.
The obtained conclusion can be generalized in an obvious way to the interaction of a point charge and a homogeneous ball. To prove it, it suffices to break the ball into thin spherical layers.
Similarly, it can be shown that the force of gravitational interaction between two spherically symmetric bodies is equal to the force of interaction between material points of the same masses located at the centers of the bodies. That is, when calculating the gravitational interaction, spherically symmetrical bodies can be considered material points located at the centers of these bodies, regardless of the size of the bodies themselves and the distance between them (Fig. 117).
Let us apply the results obtained to the force acting on all bodies located near the surface of the Earth. Let the body mass m is on top h above the surface of the earth. With good accuracy, the shape of the Earth can be considered spherical, so the force acting on the body from the side of the Earth is directed towards its center, and the modulus of this force is expressed by the formula
Where M is the mass of the Earth, R is its radius. It is known that the average radius of the Earth is equal to: R ≈ 6350 km. If the body is at small heights compared to the radius of the Earth, then the height of the body can be neglected, and in this case the force of attraction is equal to:
Where indicated
The gravitational force acting on all bodies near the surface of the Earth is called gravity. The acceleration vectors of free fall at different points are not parallel, as they are directed towards the center of the Earth. However, if we consider points that are at a small height compared to the radius of the Earth, then we can neglect the difference in the directions of free fall acceleration and assume that at all points of the region under consideration near the Earth's surface, the acceleration vector is constant both in magnitude and in direction ( Fig. 118).
In the framework of this approximation, we will call the force of gravity homogeneous.
6.7 Potential energy of gravitational attraction.
All bodies with mass are attracted to each other with a force that obeys the law of universal gravitation by I. Newton. Therefore, attracting bodies have an interaction energy.
We will show that the work of gravitational forces does not depend on the shape of the trajectory, that is, gravitational forces are also potential. To do this, consider the motion of a small body with mass m interacting with another massive body of mass M, which we will assume to be fixed (Fig. 90). As follows from Newton's law, the force \(~\vec F\) acting between the bodies is directed along the line connecting these bodies. Therefore, when the body moves m along an arc of a circle centered at the point where the body is located M, the work of the gravitational force is zero, since the force and displacement vectors remain mutually perpendicular all the time. When moving along a segment directed to the center of the body M, the displacement and force vectors are parallel, therefore, in this case, when the bodies approach each other, the work of the gravitational force is positive, and when the bodies move away, it is negative. Further, we note that during radial motion, the work of the attractive force depends only on the initial and final distances between the bodies. So when moving along segments (see Fig. 91) DE and D 1 E 1 perfect works are equal, since the laws of change of forces from distance on both segments are the same. Finally, an arbitrary body trajectory m can be divided into a set of arc and radial sections (for example, a broken line ABCDE). When moving along arcs, the work is equal to zero, when moving along radial segments, the work does not depend on the position of this segment - therefore, the work of the gravitational force depends only on the initial and final distances between the bodies, which was required to be proved.
Note that in proving potentiality, we used only the fact that the gravitational forces are central, that is, directed along the straight line connecting the bodies, and did not mention the specific form of the dependence of the force on distance. Hence, all central forces are potential.
We have proved the potentiality of the force of gravitational interaction between two point bodies. But for gravitational interactions, the principle of superposition is valid - the force acting on the body from the side of a system of point bodies is equal to the sum of the forces of pair interactions, each of which is potential, therefore, their sum is also potential. Indeed, if the work of each force of pair interaction does not depend on the trajectory, then their sum also does not depend on the shape of the trajectory. Thus, all gravitational forces are potential.
It remains for us to obtain a concrete expression for the potential energy of the gravitational interaction.
To calculate the work of the attractive force between two point bodies, it is enough to calculate this work when moving along a radial segment with a change in distance from r 1 to r 2 (Fig. 92).
Once again, we will use the graphical method, for which we plot the dependence of the attractive force \(~F = G\frac(mM)(r^2)\) on the distance r between the bodies, then the area under the graph of this dependence within the indicated limits will be equal to the desired work (Fig. 93). Calculating this area is not a very difficult task, however, it requires certain mathematical knowledge and skills. Without going into the details of this calculation, we present the final result, for a given dependence of the force on the distance, the area under the graph, or the work of the attractive force, is determined by the formula
\(~A_(12) = GmM \left(\frac(1)(r_2) - \frac(1)(r_1) \right)\) .
Since we have proved that gravitational forces are potential, this work is equal to the decrease in the potential energy of interaction, that is
\(~A_(12) = GmM \left(\frac(1)(r_2) - \frac(1)(r_1) \right) = -\Delta U = -(U_2 - U_1)\) .
From this expression, one can determine the expression for the potential energy of the gravitational interaction
\(~U(r) = - G \frac(mM)(r)\) . (one)
With this definition, the potential energy is negative and tends to zero at an infinite distance between the bodies \(~U(\infty) = 0\) . Formula (1) determines the work that the force of gravitational attraction will do with increasing distance from r to infinity, since with such a movement the vectors of force and displacement are directed in opposite directions, then this work is negative. With the opposite movement, when the bodies approach from an infinite distance to a distance, the work of the force of attraction will be positive. This work can be calculated by the definition of potential energy \(~A_(\infty \to r)U(r) = - (U(\infty)- U(r)) = G \frac(mM)(r)\) .
We emphasize that the potential energy is a characteristic of the interaction of at least two bodies. It is impossible to say that the energy of interaction "belongs" to one of the bodies, or how to "divide this energy between the bodies." Therefore, when we talk about a change in potential energy, we mean a change in the energy of a system of interacting bodies. However, in some cases it is still permissible to speak of a change in the potential energy of one body. So, when describing the motion of a small, compared to the Earth, body in the Earth's gravity field, we talk about the force acting on the body from the Earth, as a rule, without mentioning and not taking into account the equal force acting from the body on the Earth. The fact is that with the enormous mass of the Earth, the change in its speed is vanishingly small. Therefore, a change in the potential energy of interaction leads to a noticeable change in the kinetic energy of the body and an infinitesimal change in the kinetic energy of the Earth. In such a situation, it is permissible to speak about the potential energy of a body near the Earth's surface, that is, to "attribute" all the energy of the gravitational interaction to a small body. In the general case, one can speak of the potential energy of an individual body if the other interacting bodies are motionless.
We have repeatedly emphasized that the point at which the potential energy is assumed to be zero is chosen arbitrarily. In this case, such a point turned out to be a point at infinity. In a certain sense, this unusual conclusion can be recognized as reasonable: indeed, interaction disappears at an infinite distance - potential energy also disappears. From this point of view, the sign of potential energy also looks logical. Indeed, in order to separate two attracting bodies, external forces must do positive work, therefore, in such a process, the potential energy of the system must increase: here it increases, increases and ... becomes equal to zero! If the attracting bodies are in contact, then the force of attraction cannot do positive work, but if the bodies are separated, then such work can be done when the bodies approach each other. Therefore, it is often said that attracting bodies have negative energy, while repelling bodies have positive energy. This statement is true only if the zero level of potential energy is chosen at infinity.
So if two bodies are connected by a spring, then with an increase in the distance between the bodies, an attractive force will act between them, however, the energy of their interaction is positive. Do not forget that the zero level of potential energy corresponds to the state of an undeformed spring (and not infinity).
