DEFINITION

The law of universal gravitation was discovered by I. Newton:

Two bodies are attracted to each other with , which is directly proportional to their product and inversely proportional to the square of the distance between them:

Description of the law of gravity

The coefficient is the gravitational constant. In the SI system, the gravitational constant has the value:

This constant, as can be seen, is very small, so the gravitational forces between bodies with small masses are also small and practically not felt. However, the motion of cosmic bodies is completely determined by gravity. The presence of universal gravitation or, in other words, gravitational interaction explains what the Earth and planets “hold” on, and why they move around the Sun along certain trajectories, and do not fly away from it. The law of universal gravitation allows us to determine many characteristics of celestial bodies - the masses of planets, stars, galaxies and even black holes. This law allows you to calculate the orbits of the planets with great accuracy and create mathematical model Universe.

With the help of the law of universal gravitation, it is also possible to calculate cosmic velocities. For example, the minimum speed at which a body moving horizontally above the Earth's surface will not fall on it, but will move in a circular orbit is 7.9 km / s (the first cosmic velocity). In order to leave the Earth, i.e. to overcome its gravitational attraction, the body must have a speed of 11.2 km / s, (the second cosmic velocity).

Gravity is one of the most amazing phenomena nature. In the absence of gravitational forces, the existence of the Universe would be impossible, the Universe could not even arise. Gravity is responsible for many processes in the Universe - its birth, the existence of order instead of chaos. The nature of gravity is still not fully understood. To date, no one has been able to develop a worthy mechanism and model of gravitational interaction.

Gravity

A special case of the manifestation of gravitational forces is gravity.

Gravity is always directed vertically downward (toward the center of the Earth).

If the force of gravity acts on the body, then the body performs. The type of movement depends on the direction and module of the initial speed.

We deal with the force of gravity every day. , after a while it is on the ground. The book, released from the hands, falls down. Having jumped, a person does not fly into outer space and descends to the ground.

Considering the free fall of a body near the Earth's surface as a result of the gravitational interaction of this body with the Earth, we can write:

whence the acceleration free fall:

The free fall acceleration does not depend on the mass of the body, but depends on the height of the body above the Earth. The globe is slightly flattened at the poles, so bodies near the poles are slightly closer to the center of the earth. In this regard, the acceleration of free fall depends on the latitude of the area: at the pole it is slightly greater than at the equator and other latitudes (at the equator m / s, at the North Pole equator m / s.

The same formula allows you to find the free fall acceleration on the surface of any planet with mass and radius .

Examples of problem solving

EXAMPLE 1 (the problem of "weighing" the Earth)

Exercise	The radius of the Earth is km, the acceleration of free fall on the surface of the planet is m/s. Using these data, estimate the approximate mass of the Earth.
Solution	Acceleration of free fall at the surface of the Earth: whence the mass of the Earth: In the C system, the radius of the Earth m. Substituting numerical values ​​into the formula physical quantities Let's estimate the mass of the Earth:
Answer	Mass of the Earth kg.

EXAMPLE 2

Exercise

An Earth satellite moves in a circular orbit at an altitude of 1000 km from the Earth's surface. How fast is the satellite moving? How long does it take for a satellite to make one complete revolution around the earth?

Solution

According to , the force acting on the satellite from the side of the Earth is equal to the product of the mass of the satellite and the acceleration with which it moves: From the side of the earth, the force of gravitational attraction acts on the satellite, which, according to the law of universal gravitation, is equal to: where and are the masses of the satellite and the Earth, respectively. Since the satellite is at a certain height above the surface of the Earth, the distance from it to the center of the Earth: where is the radius of the earth.

« Physics - Grade 10 "

Why does the moon move around the earth?
What happens if the moon stops?
Why do the planets revolve around the sun?

In Chapter 1, it was discussed in detail that the globe imparts the same acceleration to all bodies near the surface of the Earth - the acceleration of free fall. But if the globe imparts acceleration to the body, then, according to Newton's second law, it acts on the body with some force. The force with which the earth acts on the body is called gravity. First, let's find this force, and then consider the force of universal gravitation.

Modulo acceleration is determined from Newton's second law:

In the general case, it depends on the force acting on the body and its mass. Since the acceleration of free fall does not depend on the mass, it is clear that the force of gravity must be proportional to the mass:

The physical quantity is the free fall acceleration, it is constant for all bodies.

Based on the formula F = mg, you can specify a simple and practically convenient method for measuring the masses of bodies by comparing the mass of a given body with the standard unit of mass. The ratio of the masses of two bodies is equal to the ratio of the forces of gravity acting on the bodies:

This means that the masses of bodies are the same if the forces of gravity acting on them are the same.

This is the basis for the determination of masses by weighing on a spring or balance scale. By ensuring that the force of pressure of the body on the scales, equal to the force of gravity applied to the body, is balanced by the force of pressure of the weights on the other scales, equal to the force of gravity applied to the weights, we thereby determine the mass of the body.

The force of gravity acting on a given body near the Earth can be considered constant only at a certain latitude near the Earth's surface. If the body is lifted or moved to a place with a different latitude, then the acceleration of free fall, and hence the force of gravity, will change.

The force of gravity.

Newton was the first to rigorously prove that the reason that causes the fall of a stone to the Earth, the movement of the Moon around the Earth and the planets around the Sun, is the same. This gravitational force acting between any bodies of the Universe.

Newton came to the conclusion that if it were not for air resistance, then the trajectory of a stone thrown from a high mountain (Fig. 3.1) with 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 the sky.

Newton found this reason and was able to accurately express it in the form of one formula - the law of universal gravitation.

Since the force of universal gravitation imparts the same acceleration to all bodies, regardless of their mass, it must be proportional to the mass of the body on which it acts:

“Gravity exists for all bodies in general and is proportional to the mass of each of them ... all planets gravitate towards each other ...” I. Newton

But since, for example, the Earth acts on the Moon with a force proportional to the mass of the Moon, then the Moon, according to Newton's third law, must act on the Earth with the same force. Moreover, this force must be proportional to the mass of the Earth. If the gravitational force is truly universal, then from the side of a given body any other body must be acted upon by a force proportional to the mass of this other body. Consequently, the force of universal gravitation must be proportional to the product of the masses of the interacting bodies. From this follows the formulation of the law of universal gravitation.

Law of gravity:

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 proportionality factor G is called gravitational constant.

The gravitational constant is numerically equal to the force of attraction between two material points with a mass of 1 kg each, if the distance between them is 1 m. After all, with masses m 1 \u003d m 2 \u003d 1 kg and a distance r \u003d 1 m, we get G \u003d F (numerically).

It must be kept in mind that the law of universal gravitation (3.4) as a universal law is valid for material points. In this case, the forces of gravitational interaction are directed along the line connecting these points (Fig. 3.2, a).

It can be shown that homogeneous bodies having the shape of a ball (even if they cannot be considered material points, Fig. 3.2, b) also interact with the force defined by formula (3.4). In this case, r is the distance between the centers of the balls. The forces of mutual attraction lie on a straight line passing through the centers of the balls. Such forces are called central. The bodies whose fall to the Earth we usually consider are much smaller than the Earth's radius (R ≈ 6400 km).

Such bodies, regardless of their shape, can be considered as material points and the force of their attraction to the Earth can be determined using the law (3.4), bearing in mind that r is the distance from the given body to the center of the Earth.

A stone thrown to the Earth will deviate under the action of gravity from a straight path and, having described a curved trajectory, will finally fall to the Earth. If you throw it with more speed, it will fall further.” I. Newton

Definition of the gravitational constant.

Now let's find out how you can find the gravitational constant. First of all, note that G has a specific name. This is due to the fact that the units (and, accordingly, the names) of all quantities included in the law of universal gravitation have already been established earlier. The law of gravitation gives a new connection between known quantities with certain names of units. That is why the coefficient turns out to be a named value. Using the formula of the law of universal gravitation, it is easy to find the name of the unit of gravitational constant in SI: N m 2 / kg 2 \u003d m 3 / (kg s 2).

To quantify G, it is necessary to independently determine all the quantities included in the law of universal gravitation: both masses, force and distance between bodies.

The difficulty lies in the fact that the gravitational forces between bodies of small masses are extremely small. It is for this reason that we do not notice the attraction of our body to surrounding objects and the mutual attraction of objects to each other, although gravitational forces are the most universal of all forces in nature. Two people weighing 60 kg at a distance of 1 m from each other are attracted with a force of only about 10 -9 N. Therefore, to measure the gravitational constant, rather subtle experiments are needed.

The gravitational constant was first measured by the English physicist G. Cavendish in 1798 using a device called a torsion balance. The scheme of the torsion balance is shown in Figure 3.3. A light rocker with two identical weights at the ends is suspended on a thin elastic thread. Two heavy balls are motionlessly fixed nearby. Gravitational forces act between weights and motionless balls. Under the influence of these forces, the rocker turns and twists the thread until the resulting elastic force becomes equal to the gravitational force. The angle of twist can be used to determine the force of attraction. To do this, you only need to know the elastic properties of the thread. The masses of bodies are known, and the distance between the centers of interacting bodies can be directly measured.

From these experiments, the following value for the gravitational constant was obtained:

G \u003d 6.67 10 -11 N m 2 / kg 2.

Only in the case when bodies of enormous masses interact (or at least the mass of one of the bodies is very large), the gravitational force reaches of great importance. For example, the Earth and the Moon are attracted to each other with a force F ≈ 2 10 20 N.

Dependence of free fall acceleration of bodies on geographic latitude.

One of the reasons for the increase in the acceleration of free fall when moving the point where the body is located from the equator to the poles is that the globe is somewhat flattened at the poles and the distance from the center of the Earth to its surface at the poles is less than at the equator. Another reason is the rotation of the Earth.

Equality of inertial and gravitational masses.

The most striking property of gravitational forces is that they impart the same acceleration to all bodies, regardless of their masses. What would you say about a football player whose kick would equally accelerate an ordinary leather ball and a two-pound weight? Everyone will say that it is impossible. But the Earth is just such an “extraordinary football player”, with the only difference that its effect on bodies does not have the character of a short-term impact, but continues continuously for billions of years.

In Newton's theory, mass is the source of the gravitational field. We are in the Earth's gravitational field. At the same time, we are also sources of the gravitational field, but due to the fact that our mass is significantly less than the mass of the Earth, our field is much weaker and the surrounding objects do not react to it.

The unusual property of gravitational forces, as we have already said, is explained by the fact that these forces are proportional to the masses of both interacting bodies. The mass of the body, which is included in Newton's second law, determines the inertial properties of the body, i.e., its ability to acquire a certain acceleration under the action of a given force. This inertial mass m and.

It would seem, what relation can it have to the ability of bodies to attract each other? The mass that determines the ability of bodies to attract each other is the gravitational mass m r .

It does not follow at all from Newtonian mechanics that the inertial and gravitational masses are the same, i.e. that

m and = m r . (3.5)

Equality (3.5) is a direct consequence of experience. It means that one can simply speak of the mass of a body as a quantitative measure of both its inertial and gravitational properties.

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

In contact with

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

But, alas, the question is why and How do all bodies attract each other?, remains to this day not fully disclosed, although it has been studied up and down.

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

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

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

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

Movement task

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

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

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

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

Newtonian gravity

In 1666, the greatest English physicist and mechanic I. Newton discovered a law capable of quantitatively calculating the force due to which all matter in the universe tends to each other. This phenomenon is called universal gravitation. When asked: "Formulate the law of universal gravitation", your answer should sound like this:

The force of gravitational interaction, which contributes to the attraction of two bodies, is in direct proportion to the masses of these bodies and inversely proportional to the distance between them.

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

For the law of gravity, the formula is as follows:

,

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

What is weight, if we have just considered the force of attraction?

Force is a vector quantity, but in the law of universal gravitation it is traditionally written as a scalar. In a vector picture, the law will look like this:

.

But this does not mean that the force is inversely proportional to the cube of the distance between the centers. The ratio should be understood as a unit vector directed from one center to another:

.

Law of gravitational interaction

Weight and gravity

Having considered the law of gravity, one can understand that there is nothing surprising in the fact that we personally we feel the attraction of the sun is much weaker than the earth's. The massive Sun, although it has a large mass, is very far from us. also far from the Sun, but it is attracted to it, as it has a large mass. How to find the force of attraction of two bodies, namely, how to calculate the gravitational force of the Sun, the Earth and you and me - we will deal with this issue a little later.

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

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

Important! There are no two, three, ten kinds of forces of attraction. Gravity is the only force that quantifies attraction. Weight (P = mg) and gravitational force are one and the same.

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

Thus, since F = mg:

.

The masses m cancel out, leaving the expression for the free fall acceleration:

As you can see, the acceleration of free fall is indeed a constant value, since its formula includes constant values ​​- the radius, the mass of the Earth and the gravitational constant. Substituting the values ​​of these constants, we will make sure that the acceleration of free fall is equal to 9.81 m / s 2.

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

Let's return to the attraction of the Earth and the Sun. Let's try to prove by example that the globe attracts us stronger than the Sun.

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

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

Gravitational attraction between man and the Earth:

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

The force of gravitational attraction between man and the Sun:

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

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

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

first cosmic speed

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

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

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

Let's try to find out what cosmic speed is.

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

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

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

,

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

In orbit, centrifugal acceleration acts on the body, thus:

.

The masses are reduced, we get:

,

This speed is called the first cosmic speed:

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

first cosmic speed

Second space velocity

However, even having accelerated the body to the first cosmic speed, we will not be able to completely break its gravitational connection with the Earth. For this, the second cosmic velocity is needed. Upon reaching this speed, the body leaves the gravitational field of the planet and all possible closed orbits.

Important! By mistake, it is often believed that in order to get to the Moon, astronauts had to reach the second cosmic velocity, because they first had to "disconnect" from the gravitational field of the planet. This is not so: the Earth-Moon pair are in the Earth's gravitational field. Their common center of gravity is inside the globe.

In order to find this speed, we set the problem a little differently. Suppose a body flies from infinity to a planet. Question: what speed will be achieved on the surface upon landing (without taking into account the atmosphere, of course)? It is this speed and it will take the body to leave the planet.

The law of universal gravitation. Physics Grade 9

The law of universal gravitation.

Conclusion

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

Gravitational forces are described by the simplest quantitative laws. But despite this simplicity, the manifestations of gravitational forces can be very complex and diverse.

Gravitational interactions are described by the law of universal gravitation discovered by Newton:

Material points attract with a force proportional to the product of their masses and inversely proportional to the square of the distance between them:

Gravitational constant. The coefficient of proportionality is called the gravitational constant. This value characterizes the intensity of gravitational interaction and is one of the main physical constants. Its numerical value depends on the choice of the system of units and in SI units it is equal. From the formula it can be seen that the gravitational constant is numerically equal to the force of attraction of two turned masses of 1 kg located at a distance from each other. The value of the gravitational constant is so small that we do not notice the attraction between the bodies around us. Only because of the huge mass of the Earth, the attraction of surrounding bodies to the Earth decisively affects everything that happens around us.

Rice. 91. Gravitational interaction

Formula (1) gives only the modulus of the force of mutual attraction of point bodies. In fact, it is about two forces, since the force of gravity acts on each of the interacting bodies. These forces are equal in absolute value and opposite in direction in accordance with Newton's third law. They are directed along the straight line connecting the material points. Such forces are called central. A vector expression, for example, for the force with which a body of mass acts on a body of mass (Fig. 91), has the form

Although the radius vectors of material points depend on the choice of the origin of coordinates, their difference, and hence the force, depend only on relative position attracted bodies.

Kepler's laws. The well-known legend of the falling apple, which allegedly led Newton to the idea of ​​gravity, is hardly to be taken seriously. When establishing the law of universal gravitation, Newton proceeded from the laws of planetary motion discovered by Johannes Kepler on the basis of Tycho Brahe's astronomical observations. solar system. Kepler's three laws are:

1. The trajectories along which the planets move are ellipses, in one of the focuses of which is the Sun.

2. The radius vector of the planet, drawn from the Sun, sweeps the same areas in equal time intervals.

3. For all planets, the ratio of the square of the period of revolution to the cube of the semi-major axis of an elliptical orbit has the same value.

The orbits of most planets differ little from circular ones. For simplicity, we will assume that they are exactly circular. This does not contradict Kepler's first law, since the circle is a special case of an ellipse, in which both foci coincide. According to Kepler's second law, the motion of the planet along a circular trajectory occurs uniformly, i.e., with a constant modulo speed. At the same time, Kepler's third law says that the ratio of the square of the period of revolution T to the cube of the radius of a circular orbit is the same for all planets:

A planet moving in a circle at a constant speed has a centripetal acceleration equal to Let us use this to determine the force that imparts such an acceleration to the planet when condition (3) is met. According to Newton's second law, the acceleration of a planet is equal to the ratio of the force acting on it to the mass of the planet:

From here, taking into account Kepler's third law (3), it is easy to establish how the force depends on the mass of the planet and on the radius of its circular orbit. Multiplying both parts of (4) by we see that in the left part, according to (3), there is the same value for all planets. This means that the right side, which is equal, is the same for all planets. Therefore, i.e., the force of gravity is inversely proportional to the square of the distance from the Sun and directly proportional to the mass of the planet. But the sun and the planet appear in their gravitational

interaction as equal partners. They differ from each other only in masses. And since the force of attraction is proportional to the mass of the planet, then it must be proportional to the mass of the Sun M:

Introducing the coefficient of proportionality G into this formula, which should no longer depend either on the masses of the interacting bodies or on the distance between them, we arrive at the law of universal gravitation (1).

gravitational field. The gravitational interaction of bodies can be described using the concept of a gravitational field. The Newtonian formulation of the law of universal gravitation corresponds to the idea of ​​the direct action of bodies on each other at a distance, the so-called long-range action, without any participation of an intermediate medium. In modern physics, it is believed that the transfer of any interactions between bodies is carried out through the fields created by these bodies. One of the bodies does not directly affect the other, it endows the space surrounding it with certain properties - it creates a gravitational field, a special material environment, which affects the other body.

The idea of ​​a physical gravitational field performs both aesthetic and quite practical functions. The forces of gravity act at a distance, they pull where we can hardly see what is pulling. The force field is some kind of abstraction that replaces hooks, ropes or rubber bands for us. It is impossible to give any visual picture of the field, since the very concept of a physical field is one of the basic concepts that cannot be defined through other, simpler concepts. You can only describe its properties.

Considering the ability of the gravitational field to create a force, we believe that the field depends only on the body from which the force acts, and does not depend on the body on which it acts.

Note that within the framework of classical mechanics (Newtonian mechanics), both ideas - about long-range action and interaction through a gravitational field - lead to the same results and are equally admissible. The choice of one of these methods of description is determined solely by considerations of convenience.

The intensity of the gravitational field. The power characteristic of the gravitational field is its intensity measured by the force acting on a material point of a unit mass, i.e., the ratio

Obviously, the gravitational field created by the point mass M has spherical symmetry. This means that the intensity vector at any of its points is directed towards the mass M, which creates the field. The field strength modulus, as follows from the law of universal gravitation (1), is equal to

and depends only on the distance to the field source. The field strength of a point mass decreases with distance according to the inverse square law. In such fields, the motion of bodies occurs in accordance with Kepler's laws.

The principle of superposition. Experience shows that gravitational fields satisfy the principle of superposition. According to this principle, the gravitational field created by any mass does not depend on the presence of other masses. The strength of the field created by several bodies is equal to the vector sum of the field strengths created by these bodies separately.

The principle of superposition makes it possible to calculate the gravitational fields created by extended bodies. To do this, you need to mentally divide the body into separate elements, which can be considered material points, and find the vector sum of the field strengths created by these elements. Using the principle of superposition, it can be shown that the gravitational field created by a ball with a spherically symmetric mass distribution (in particular, a homogeneous ball) outside this ball is indistinguishable from the gravitational field of a material point of the same mass as the ball placed at the center of the ball. This means that the intensity of the gravitational field of the ball is given by the same formula (6). This simple result is given here without proof. It will be given for the case of electrostatic interaction when considering the field of a charged ball, where the force also decreases inversely with the square of the distance.

Attraction of spherical bodies. Using this result and invoking Newton's third law, it can be shown that two balls with a spherically symmetric distribution of masses each attract each other as if their masses were concentrated at their centers, i.e., just like point masses. We present the corresponding proof.

Let two balls with masses attract each other with forces (Fig. 92a). If we replace the first ball with a point mass (Fig. 92b), then the gravitational field created by it at the location of the second ball will not change and, therefore, the force acting on the second ball will not change. Based on the third

Newton's law from here we can conclude that the second ball acts with the same force both on the first ball and on the material point replacing it. This force is easy to find, given that the gravitational field created by the second ball in the place where the first ball is located , is indistinguishable from the field of a point mass placed at its center (Fig. 92c).

Rice. 92. Spherical bodies are attracted to each other as if their masses were concentrated at their centers

Thus, the force of attraction of the balls coincides with the force of attraction of two point masses, and the distance between them is equal to the distance between the centers of the balls.

From this example, the practical value of the concept of a gravitational field is clearly visible. Indeed, it would be very inconvenient to describe the force acting on one of the balls as the vector sum of the forces acting on its individual elements, given that each of these forces, in turn, is the vector sum of the interaction forces of this element with all the elements into which we must mentally break the second ball. Let us also pay attention to the fact that in the process of the above proof, we alternately considered either one ball or the other as the source of the gravitational field, depending on whether we were interested in the force acting on one or the other ball.

Now it is obvious that any body of mass located near the surface of the Earth, the linear dimensions of which are small compared to the radius of the Earth, is affected by the force of gravity, which, in accordance with (5), can be written as under M should be understood the mass of the globe, and instead of the radius of the Earth should be substituted

For formula (7) to be applicable, it is not necessary to consider the Earth as a homogeneous sphere, it is sufficient that the mass distribution be spherically symmetric.

Free fall. If a body near the surface of the Earth moves only under the action of gravity, i.e., falls freely, then its acceleration, according to Newton's second law, is equal to

But the right side of (8) gives the value of the intensity of the Earth's gravitational field near its surface. So, the intensity of the gravitational field and the acceleration of free fall in this field are one and the same. That is why we immediately designated these quantities with one letter

Weighing the Earth. Let us now dwell on the question of the experimental determination of the value of the gravitational constant. First of all, we note that it cannot be found from astronomical observations. Indeed, from observations of the motion of the planets, one can only find the product of the gravitational constant and the mass of the Sun. From observations of the movement of the Moon, artificial satellites of the Earth, or the free fall of bodies near earth's surface one can only find the product of the gravitational constant and the mass of the Earth. To determine it, it is necessary to be able to independently measure the mass of the source of the gravitational field. This can only be done in an experiment performed in the laboratory.

Rice. 93. Scheme of the Cavendish experiment

Such an experiment was first performed by Henry Cavendish using a torsion balance, to the ends of which small lead balls were attached (Fig. 93). Large heavy balls were fixed near them. Under the action of the forces of attraction of small balls to large balls, the yoke of the torsion balance turned slightly, and the force was measured by twisting the elastic suspension thread. To interpret this experiment, it is important to know that the balls interact in the same way as the corresponding material points of the same mass, because here, unlike the planets, the size of the balls cannot be considered small compared to the distance between them.

In his experiments, Cavendish obtained the value of the gravitational constant only differing from that accepted at the present time. In modern modifications of the Cavendish experiment, the accelerations imparted to small balls on the beam by the gravitational field of heavy balls are measured, which makes it possible to increase the accuracy of measurements. Knowledge of the gravitational constant makes it possible to determine the masses of the Earth, the Sun and other sources of gravity from observations of the motion of bodies in the gravitational fields they create. In this sense, the Cavendish experiment is sometimes figuratively called the weighing of the Earth.

Universal gravitation is described by a very simple law, which, as we have seen, is easily established on the basis of Kepler's laws. What is the greatness of Newton's discovery? It embodied the idea that the fall of an apple to the Earth and the movement of the Moon around the Earth, which is also in a certain sense a fall to the Earth, have a common cause. In those distant times, this was an amazing idea, since common wisdom said that celestial bodies move according to their "perfect" laws, and earthly objects obey "worldly" rules. Newton came to the conclusion that the uniform laws of nature are valid for the entire universe.

Enter such a unit of force that in the law of universal gravitation (1) the value of the gravitational constant C is equal to one. Compare this unit of force to the newton.

Are there deviations from Kepler's laws for the planets of the solar system? What are they due to?

How to establish the dependence of gravitational force on distance from Kepler's laws?

Why can't the gravitational constant be determined from astronomical observations?

What is a gravitational field? What are the advantages of describing the gravitational interaction using the concept of a field in comparison with the idea of ​​long-range action?

What is the principle of superposition for a gravitational field? What can be said about the gravitational field of a homogeneous sphere?

How are the strength of the gravitational field and the acceleration of free fall related?

Calculate the mass of the Earth M using the values ​​of the gravitational constant of the Earth's radius km and the acceleration due to gravity

Geometry and gravity. Several subtle points are connected with the simple formula of the law of universal gravitation (1), which deserve separate discussion. From Kepler's laws,

that the distance in the denominator of the expression for the force of gravity is included in the second degree. The whole set of astronomical observations leads to the conclusion that the value of the exponent is equal to two with very high accuracy, namely This fact is highly remarkable: the exact equality of the exponent to two reflects the Euclidean nature of three-dimensional physical space. This means that the position of bodies and the distance between them in space, the addition of displacements of bodies, etc., is described by Euclid's geometry. The exact equality of the exponent to two emphasizes the fact that in the three-dimensional Euclidean world the surface of a sphere is exactly proportional to the square of its radius.

Inertial and gravitational masses. It also follows from the above derivation of the law of gravitation that the force of the gravitational interaction of bodies is proportional to their masses, or rather, to the inertial masses that appear in Newton's second law and describe the inertial properties of bodies. But inertia and the ability to gravitational interactions are completely different properties of matter.

In determining mass based on inert properties, the law is used. Measurements of mass in accordance with this definition of it require a dynamic experiment - a known force is applied and acceleration is measured. This is how mass spectrometers are used to determine the masses of charged elementary particles and ions (and thus atoms).

In the definition of mass based on the phenomenon of gravitation, the law is used. Measurement of mass in accordance with such a definition is carried out using a static experiment - weighing. The bodies are placed motionless in a gravitational field (usually the field of the Earth) and the gravitational forces acting on them are compared. The mass defined in this way is called heavy or gravitational.

Will the inertial and gravitational masses be the same? After all, the quantitative measures of these properties, in principle, could be different. The first answer to this question was given by Galileo, although he apparently did not suspect it. In his experiments, he intended to prove that Aristotle's then prevailing assertions that heavy bodies fall faster than light ones were false.

To better follow the reasoning, we denote the inertial mass by and the gravitational mass by

where is the intensity of the gravitational field of the Earth, the same for all bodies. Now let's compare what happens if two bodies are simultaneously dropped from the same height. In accordance with Newton's second law, for each of the bodies one can write

But experience shows that the accelerations of both bodies are the same. Consequently, the relation will be the same for them. So, for all bodies

The gravitational masses of bodies are proportional to their inertial masses. By proper choice of units, they can be made simply equal.

The coincidence of the values ​​of the inertial and gravitational masses was confirmed many times with increasing accuracy in various experiments of scientists from different eras - Newton, Bessel, Eötvös, Dicke and, finally, Braginsky and Panov, who brought the relative measurement error to . To better imagine the sensitivity of instruments in such experiments, we note that this is equivalent to the ability to detect a change in the mass of a ship with a displacement of a thousand tons when one milligram is added to it.

In Newtonian mechanics, the coincidence of the values ​​of the inertial and gravitational masses has no physical reason and in this sense is random. This is simply an experimental fact established with very high accuracy. If this were not the case, Newtonian mechanics would not suffer in the slightest. In Einstein's relativistic theory of gravity, also called general theory relativity, the equality of the inertial and gravitational masses is of fundamental importance and was originally laid down in the basis of the theory. Einstein suggested that there is nothing surprising or accidental in this coincidence, because in reality the inertial and gravitational masses are one and the same physical quantity.

Why is the value of the exponent to which the distance between bodies is included in the law of universal gravitation related to the Euclidean nature of three-dimensional physical space?

How are inertial and gravitational masses determined in Newtonian mechanics? Why do some books not even mention these quantities, but just the mass of the body?

Suppose that in some world the gravitational mass of bodies is in no way related to their inertial mass. What could be observed with simultaneous free fall of different bodies?

What phenomena and experiments testify to the proportionality of the inertial and gravitational masses?

Between any bodies in nature there is a force of mutual attraction, called force of gravity(or gravity). was discovered by Isaac Newton in 1682. When he was still 23 years old, he 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.

Gravity (mg) is directed vertically strictly to the center of the earth; depending on the distance to the surface of the globe, the acceleration of free fall is different. At the surface of the Earth in middle latitudes, its value is about 9.8 m / s 2. as you move away from the earth's surface g decreases.

Body weight (weight force) – is the force with which the body acts onhorizontal support or stretches the suspension. It is assumed that the body stationary relative to the support or suspension. Let the body lie on a horizontal table that is motionless relative to the Earth. Denoted by letter R.

Body weight and gravity are different in nature: body weight is a manifestation of the action of intermolecular forces, and gravity has a gravitational nature.

If acceleration a = 0 , then the weight is equal to the force with which the body is attracted to the Earth, namely. [P] = H.

If the state is different, then the weight changes:

• if acceleration A not equal 0 , then the weight P \u003d mg - ma (down) or P = mg + ma (up);
• if the body falls freely or moves with free fall acceleration, i.e. a =g(Fig. 2), then the body weight is equal to 0 (P=0 ). The state of a body in which its weight is zero is called weightlessness.

IN weightlessness there are also astronauts. IN weightlessness momentarily you are, too, when you bounce while playing basketball or dancing.

Home experiment: A plastic bottle with a hole at the bottom is filled with water. We release from the hands from a certain height. As long as the bottle falls, water does not flow out of the hole.

The weight of a body moving with acceleration (in an elevator) The body in the elevator experiences overloads