As you know, any two bodies are attracted to each other. This property of bodies is due to their mass. Since other forms of matter (fields, radiation) also have mass, they also obey the law of gravity. The most famous manifestation of mass attraction is the existence of gravity, with which the Earth acts on all bodies.

Law of gravity

The force with which two bodies are attracted to each other is called the gravitational force (gravitational force). The magnitude of this force is determined by the law of universal gravitation, formulated by Newton.

Here:
F- the gravitational force with which two bodies are attracted to each other (Newton),
m1- mass of the first body (kg),
m2- mass of the second body (kg),
r- distance between the centers of mass of bodies (meter),
G ,

The mutual attraction of masses should not be confused with the forces of magnetic or electric attraction. These are forces of a completely different nature.

Gravitational forces cannot be repulsive. In addition, the gravitational interaction cannot be weakened or eliminated with the help of any screen.

Gravity

According to the gravitation formula, one can determine the force of gravity by substituting the mass of the Earth and the mass of the body in question into the numerator, and the distance into the denominator r bodies to the center of the earth:

Definition: Gravity decreases inversely with the square of the distance from the center of the earth.

Directly on the Earth's surface, gravity is calculated using a simplified formula.

The force of gravity Fgr does not vanish at finite distances r, it tends to zero only when the bodies are infinitely removed.

Acceleration of gravity

Acceleration of gravity at any distance from the Earth, as well as on other planets, can be determined by the formula for the force of earth's attraction. If you reduce by body weight, you can get:

The gravitational acceleration decreases inversely with the square of the distance from the center of the earth. The free fall acceleration formula is also valid for other celestial bodies.

Gravitational field, gravitational field

Each body (for example, the Earth) creates a force field around itself - a gravitational field. The intensity of this field at any point characterizes the force that acts on another body located at this point.

g- intensity of the gravitational field
F- gravitational force acting on a body of mass m
m- body mass in the gravitational field

Field strength g is a vector quantity whose direction is determined by the direction of the gravitational force F, and the numerical value - the formula for the acceleration of free fall.

The intensity of the gravitational field coincides in magnitude, direction and units of measurement with the acceleration of free fall, although in their physical meaning, these are completely different physical quantities. While the field strength characterizes the state of space at a given point, force and acceleration appear only when a test body is located at a given point.

From the graph of the function g=g(r) It is clearly seen that the intensity of the gravitational field g tends to zero when the distance r tends to infinity. Therefore, statements like "the satellite left the gravitational field of the Earth" are incorrect.

The gravitational fields of celestial bodies overlap. If we move along a straight line connecting the centers of the Earth and the Moon, then, starting from a certain place, the strength of the gravitational field of the Moon will prevail.

First space (orbital) speed

first cosmic speed- this is the speed that the body must have in order to rotate at a constant height above the surface of the planet.

Using the free fall acceleration formula, you can determine the speed of revolution of an artificial satellite of the Earth (and any other planet) at any height above its surface.

The force of gravity acting on the satellite is equal to the centrifugal force, i.e.

Here:
UK- first space (orbital) speed (m/s)
h
rEarth
m Earth- mass of the planet Earth (kg),
m- satellite mass (kg)
g- free fall acceleration at some distance from the Earth's surface (m/s?)
gEarth- free fall acceleration on the Earth's surface 9.81 (m/s?)
? - gravitational constant 6.67 10-11 (m3/(kg s2))

Formula (3) allows you to determine the speed of satellites in orbit. However, the final velocity of the launch vehicle at the moment the engines stop working must be greater in order to bring the satellite to the desired height.

These formulas are also valid for the case of the motion of the Moon around the Earth. They are also true in the case of the motion of the planets around the Sun, if the motion occurs along a trajectory slightly different from a circular one, i.e. along a path with a small eccentricity.

Second escape velocity (escape velocity)

Second space velocity- this is the minimum speed with which the body must move so that it can overcome the influence of the Earth's gravitational field without the cost of additional work, i.e. move an infinite distance from the earth.

If a:
m- body weight (kg)
M- mass of the planet Earth (kg)
h- satellite height above the planet's surface (m)
rEarth- initial distance between the centers of mass of bodies (Surface of planet Earth) (meter)
r- final distance between the centers of mass of bodies (meter)
G- gravitational constant 6.67 10-11 (m3/(kg s2))
U2k- second escape velocity (escape velocity) (m/s)

Then the kinetic energy of the body should be equal to the work to overcome the influence of the gravitational field:

After simplification and rearrangement, the second cosmic velocity will take the form:

In fact, the second cosmic speed for launching rockets from the surface of the planet, this is the speed that the body must have directly on the surface of the planet when h small, but the gravitational force is large. As you move away from the source of the gravitational force, the escape velocity decreases because the gravitational force decreases, and the kinetic energy required for escape decreases accordingly.

This law, called the law of universal gravitation, is written in mathematical form as follows:

where m 1 and m 2 are the masses of the bodies, R is the distance between them (see Fig. 11a), and G is the gravitational constant equal to 6.67.10-11 N.m 2 /kg2.

The law of universal gravitation was first formulated by I. Newton when he tried to explain one of I. Kepler's laws, which states that for all planets the ratio of the cube of their distance R to the Sun to the square of the period T of revolution around it is the same, i.e.

Let us derive the law of universal gravitation as Newton did, assuming that the planets move in circles. Then, according to Newton's second law, a planet with a mass mPl moving along a circle of radius R with a speed v and a centripetal acceleration v2/R must be acted upon by a force F directed towards the Sun (see Fig. 11b) and equal to:

The speed v of the planet can be expressed in terms of the radius R of the orbit and the period of revolution T:

Substituting (11.4) into (11.3) we obtain the following expression for F:

It follows from Kepler's law (11.2) that T2 = const.R3 . Therefore, (11.5) can be transformed into:

Thus, the Sun attracts the planet with a force directly proportional to the mass of the planet and inversely proportional to the square of the distance between them. Formula (11.6) is very similar to (11.1), only the mass of the Sun is missing in the numerator of the fraction on the right. However, if the force of attraction between the Sun and the planet depends on the mass of the planet, then this force must also depend on the mass of the Sun, which means that the constant on the right side of (11.6) contains the mass of the Sun as one of the factors. Therefore, Newton put forward his famous assumption that the gravitational force should depend on the product of the masses of the bodies and the law became the way we wrote it down in (11.1).

The law of universal gravitation and Newton's third law do not contradict each other. According to formula (11.1), the force with which body 1 attracts body 2 is equal to the force with which body 2 attracts body 1.

For bodies of ordinary size, gravitational forces are very small. So, two adjacent cars are attracted to each other with a force equal to the weight of a raindrop. Since G. Cavendish in 1798 determined the value of the gravitational constant, formula (11.1) has helped to make a lot of discoveries in the "world of huge masses and distances." For example, knowing the magnitude of the free fall acceleration (g=9.8 m/s2) and the radius of the Earth (R=6.4.106 m), we can calculate its mass mЗ as follows. Each body with a mass m1 near the surface of the Earth (i.e. at a distance R from its center) is affected by the gravitational force of its attraction equal to m1g, the substitution of which in (11.1) instead of F gives:

whence we obtain that m З = 6.1024 kg.

Review questions:

· Formulate the law of universal gravitation?

· What is the gravitational constant?

Rice. 11. (a) - to the formulation of the law of universal gravitation; (b) - to the derivation of the law of universal gravitation from Kepler's law.

§ 12. GRAVITY FORCE. THE WEIGHT. WEIGHTLESSNESS. FIRST SPACE VELOCITY.

Gravitational force is the force with which objects of a certain mass are attracted to each other, located at a certain distance from each other.

The English scientist Isaac Newton in 1867 discovered the law of universal gravitation. This is one of the fundamental laws of mechanics. The essence of this law is as follows:any two material particles are attracted to each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

The force of attraction is the first force that a person felt. This is the force with which the Earth acts on all bodies located on its surface. And any person feels this force as his own weight.

Law of gravity

There is a legend that Newton discovered the law of universal gravitation quite by accident, walking in the evening in the garden of his parents. Creative people are constantly in search, and scientific discoveries are not instantaneous insight, but the fruit of long-term mental work. Sitting under an apple tree, Newton was thinking about another idea, and suddenly an apple fell on his head. It was clear to Newton that the apple fell as a result of the Earth's gravity. “But why doesn’t the moon fall to the Earth? he thought. “It means that some other force is acting on it, keeping it in orbit.” This is how the famous law of gravity.

Scientists who had previously studied the rotation of celestial bodies believed that celestial bodies obey some completely different laws. That is, it was assumed that there are completely different laws of attraction on the surface of the Earth and in space.

Newton combined these supposed kinds of gravity. Analyzing Kepler's laws describing the motion of the planets, he came to the conclusion that the force of attraction arises between any bodies. That is, both the apple that fell in the garden and the planets in space are affected by forces that obey the same law - the law of universal gravitation.

Newton found that Kepler's laws only work if there is an attractive force between the planets. And this force is directly proportional to the masses of the planets and inversely proportional to the square of the distance between them.

The force of attraction is calculated by the formula F=G m 1 m 2 / r 2

m 1 is the mass of the first body;

m2is the mass of the second body;

r is the distance between the bodies;

G is the coefficient of proportionality, which is called gravitational constant or gravitational constant.

Its value was determined experimentally. G\u003d 6.67 10 -11 Nm 2 / kg 2

If two material points with a mass equal to a unit of mass are at a distance equal to a unit of distance, then they are attracted with a force equal to G.

The forces of attraction are the gravitational forces. They are also called gravity. They are subject to the law of universal gravitation and appear everywhere, since all bodies have mass.

Gravity

The gravitational force near the surface of the Earth is the force with which all bodies are attracted to the Earth. They call her gravity. It is considered constant if the distance of the body from the Earth's surface is small compared to the radius of the Earth.

Since gravity, which is the gravitational force, depends on the mass and radius of the planet, it will be different on different planets. Since the radius of the Moon is less than the radius of the Earth, then the force of attraction on the Moon is less than on the Earth by 6 times. And on Jupiter, on the contrary, gravity is 2.4 times greater than gravity on Earth. But body weight remains constant, no matter where it is measured.

Many people confuse the meaning of weight and gravity, believing that gravity is always equal to weight. But it is not.

The force with which the body presses on the support or stretches the suspension, this is the weight. If the support or suspension is removed, the body will begin to fall with the acceleration of free fall under the action of gravity. The force of gravity is proportional to the mass of the body. It is calculated according to the formulaF= m g , where m- body mass, g- acceleration of gravity.

Body weight can change, and sometimes disappear altogether. Imagine that we are in an elevator on the top floor. The elevator is worth it. At this moment, our weight P and the force of gravity F, with which the Earth pulls us, are equal. But as soon as the elevator began to move down with acceleration a , weight and gravity are no longer equal. According to Newton's second lawmg+ P = ma . P \u003d m g -ma.

It can be seen from the formula that our weight decreased as we moved down.

At the moment when the elevator picked up speed and began to move without acceleration, our weight is again equal to gravity. And when the elevator began to slow down its movement, acceleration a became negative and the weight increased. There is an overload.

And if the body moves down with the acceleration of free fall, then the weight will completely become equal to zero.

At a=g R=mg-ma= mg-mg=0

This is a state of weightlessness.

So, without exception, all material bodies in the Universe obey the law of universal gravitation. And the planets around the Sun, and all the bodies that are near the surface of the Earth.

In nature, only four basic fundamental forces are known (they are also called major interactions) - gravitational interaction, electromagnetic interaction, strong interaction and weak interaction.

Gravitational interaction is the weakest of all.Gravitational forcesbind together parts of the globe and the same interaction determines large-scale events in the universe.

Electromagnetic interaction holds electrons in atoms and binds atoms into molecules. Particular manifestations of these forces areCoulomb forcesacting between fixed electric charges.

Strong interaction binds nucleons in nuclei. This interaction is the strongest, but it acts only at very short distances.

Weak interaction acts between elementary particles and has a very short range. It manifests itself in beta decay.

4.1. Newton's law of universal gravitation

Between two material points there is a force of mutual attraction, which is directly proportional to the product of the masses of these points ( m and M ) and inversely proportional to the square of the distance between them ( r2 ) and directed along a straight line passing through the interacting bodiesF= (GmM/r 2) r o ,(1)

here r o - unit vector drawn in the direction of the force F(Fig. 1a).

This force is called gravitational force(or force of gravity). Gravitational forces are always attractive forces. The strength of the interaction between two bodies does not depend on the environment in which the bodies are located.

g 1 g 2

Fig.1a Fig.1b Fig.1c

The constant G is called gravitational constant. Its value is established empirically: G = 6.6720. 10 -11 N. m 2 / kg 2 - i.e. two point bodies weighing 1 kg each, located at a distance of 1 m from each other, are attracted with a force of 6.6720. 10 -11 N. The very small value of G just allows us to speak about the weakness of gravitational forces - they should be taken into account only in the case of large masses.

The masses included in equation (1) are called gravitational masses. This emphasizes that, in principle, the masses included in Newton's second law ( F=m in a) and into the law of universal gravitation ( F=(Gm gr M gr /r 2) r o) are of different nature. However, it has been established that the ratio m gr / m in for all bodies is the same with a relative error of up to 10 -10 .

4.2. Gravitational field (field of gravity) of a material point

It's believed that gravitational interaction is carried out with the help of gravitational field (gravitational field), which is generated by the bodies themselves. Two characteristics of this field are introduced: vector - and scalar - gravitational field potential.

4.2.1. Strength of the gravitational field

Let we have a material point with mass M. It is believed that a gravitational field arises around this mass. The force characteristic of such a field is gravity field strengthg, which is determined from the law of universal gravitation g= (GM/r2) r o ,(2)

where r o - a unit vector drawn from a material point in the direction of the gravitational force. Gravitational field strength gis a vector quantity and is the acceleration obtained by the point mass m, brought into the gravitational field, created by a point mass M. Indeed, comparing (1) and (2), we obtain for the case of equality of the gravitational and inertial masses F=m g.

We emphasize that the magnitude and direction of acceleration received by a body introduced into the gravitational field does not depend on the magnitude of the mass of the introduced body. Since the main task of dynamics is to determine the magnitude of the acceleration received by the body under the action of external forces, then, therefore, the strength of the gravitational field completely and unambiguously determines the force characteristics of the gravitational field. The dependence g(r) is shown in Fig. 2a.

Fig.2a Fig.2b Fig.2c

The field is called central, if at all points of the field the intensity vectors are directed along straight lines that intersect at one point, fixed with respect to any inertial frame of reference. In particular, the gravitational field of a material point is central: at all points of the field, the vectors gand F=m g, acting on a body brought into the gravitational field are directed radially from the mass M , which creates a field, to a point mass m (Fig. 1b).

The law of universal gravitation in the form (1) is established for bodies taken as material points, i.e. for such bodies, the dimensions of which are small compared to the distance between them. If the dimensions of the bodies cannot be neglected, then the bodies should be divided into point elements, according to formula (1), the forces of attraction between all elements taken in pairs should be calculated and then geometrically added. The strength of the gravitational field of a system consisting of material points with masses M 1 , M 2 , ..., M n , is equal to the sum of the field strengths from each of these masses separately ( principle of superposition of gravitational fields ): g=g i, where g i= (GM i /r i 2) r o i - field strength of one mass M i .

Graphical representation of the gravitational field using tension vectors g at different points of the field it is very inconvenient: for systems consisting of many material points, the intensity vectors are superimposed on each other and a very confusing picture is obtained. That's why for a graphical representation of the gravitational field, use lines of force (lines of tension), which are carried out in such a way that the tension vector is directed tangentially to the line of force. Tension lines are considered to be directed in the same way as the vector g(Fig. 1c), those. lines of force end at a material point. Since at each point in space the tension vector has only one direction, then tension lines never cross. For a material point, the lines of force are radial straight lines entering the point (Fig. 1b).

In order to be able to characterize not only the direction, but also the value of the field strength with the help of tension lines, these lines are drawn with a certain density: the number of tension lines penetrating a unit of surface area perpendicular to the tension lines must be equal to the modulus vector g.

Gravity, also known as attraction or gravitation, is a universal property of matter that all objects and bodies in the Universe possess. The essence of gravity is that all material bodies attract to themselves all other bodies that are around.

Gravity

If gravity is a general concept and quality that all objects in the Universe possess, then the earth's attraction is a special case of this all-encompassing phenomenon. The earth attracts to itself all the material objects that are on it. Thanks to this, people and animals can safely move around the earth, rivers, seas and oceans can remain within their shores, and air can not fly through the vast expanses of space, but form the atmosphere of our planet.

A fair question arises: if all objects have gravity, why does the Earth attract people and animals to itself, and not vice versa? Firstly, we also attract the Earth to ourselves, it's just that compared to its force of attraction, our gravity is negligible. Secondly, the force of gravity is directly proportional to the mass of the body: the smaller the mass of the body, the lower its gravitational forces.

The second indicator on which the force of attraction depends is the distance between objects: the greater the distance, the less the effect of gravity. Including due to this, the planets move in their orbits, and do not fall on each other.

It is noteworthy that the Earth, the Moon, the Sun and other planets owe their spherical shape precisely to the force of gravity. It acts in the direction of the center, pulling towards it the substance that makes up the "body" of the planet.

Earth's gravitational field

The gravitational field of the Earth is a force energy field that is formed around our planet due to the action of two forces:

• gravity;
• centrifugal force, which owes its appearance to the rotation of the Earth around its axis (daily rotation).

Since both gravity and centrifugal force act constantly, the gravitational field is also a constant phenomenon.

The gravitational forces of the Sun, the Moon and some other celestial bodies, as well as the atmospheric masses of the Earth, have an insignificant effect on the field.

Law of gravity and Sir Isaac Newton

The English physicist, Sir Isaac Newton, according to a well-known legend, once walking in the garden during the day, saw the moon in the sky. At the same time, an apple fell from the branch. Newton was then studying the law of motion and knew that an apple falls under the influence of a gravitational field, and the Moon revolves in an orbit around the Earth.

And then the idea came to the mind of a brilliant scientist, illuminated by insight, that perhaps the apple falls to the earth, obeying the same force due to which the Moon is in its orbit, and does not rush randomly throughout the galaxy. This is how the law of universal gravitation, also known as Newton's Third Law, was discovered.

In the language of mathematical formulas, this law looks like this:

F=GMm/D2 ,

where F- force of mutual gravitation between two bodies;

M- mass of the first body;

m- mass of the second body;

D2- distance between two bodies;

G- gravitational constant, equal to 6.67x10 -11.