According to Newton's laws, the motion of a body with acceleration is possible only under the action of a force. Because falling bodies move with an acceleration directed downwards, then they are affected by the force of attraction to the Earth. But not only the Earth has the property to act on all bodies by the force of attraction. Isaac Newton suggested that forces of attraction act between all bodies. These forces are called forces of gravity or gravitational forces.
Having extended the established laws - the dependence of the force of attraction of bodies to the Earth on the distances between the bodies and on the masses of interacting bodies, obtained as a result of observations - Newton discovered in 1682 law of gravity:All bodies are attracted to each other, the force of universal gravitation is directly proportional to the product of the masses of the bodies and inversely proportional to the square of the distance between them:
The vectors of forces of universal gravitation are directed along the straight line connecting the bodies. The proportionality factor G is called gravitational constant (universal gravitational constant) and equal to
.
gravity called the force of attraction acting from the Earth on all bodies:
.
Let 
is the mass of the earth, and 
is the radius of the earth. Consider the dependence of the acceleration of free fall on the height of the rise above the Earth's surface:
Body weight. Weightlessness
Body weight - the force with which a body presses on a support or suspension due to the attraction of this body to the ground. The weight of the body is applied to the support (suspension). The amount of body weight depends on how the body moves with support (suspension).
Body weight, i.e. the force with which the body acts on the support, and the elastic force with which the support acts on the body, in accordance with Newton's third law, are equal in absolute value and opposite in direction.
If the body is at rest on a horizontal support or moves uniformly, only the force of gravity and the elastic force from the side of the support act on it, therefore the weight of the body is equal to the force of gravity (but these forces are applied to different bodies):
.
With accelerated motion, the weight of the body will not be equal to the force of gravity. Consider the motion of a body with mass m under the action of gravity and elasticity with acceleration. According to Newton's 2nd law:
If the acceleration of the body is directed downward, then the weight of the body is less than the force of gravity; if the acceleration of the body is directed upwards, then all bodies are greater than the force of gravity.
The increase in body weight caused by the accelerated movement of the support or suspension is called overload.
If the body is freely falling, then from the formula * it follows that the weight of the body is zero. The disappearance of the weight during the movement of the support with the acceleration of free fall is called weightlessness.
The state of weightlessness is observed in an airplane or spacecraft when they move with the acceleration of free fall, regardless of the speed of their movement. Outside the earth's atmosphere, when the jet engines are turned off, only the force of universal gravitation acts on the spacecraft. Under the influence of this force, the spacecraft and all the bodies in it move with the same acceleration; therefore, the phenomenon of weightlessness is observed in the ship.
The motion of a body under the influence of gravity. Movement of artificial satellites. first cosmic speed
If the modulus of displacement of the body is much less than the distance to the center of the Earth, then the force of universal gravitation during the movement can be considered constant, and the movement of the body is uniformly accelerated. The simplest case of motion of a body under the action of gravity is free fall with zero initial velocity. In this case, the body moves with the acceleration of free fall towards the center of the Earth. If there is an initial velocity that is not directed vertically, then the body moves along a curved path (parabola, if air resistance is not taken into account).
At a certain initial velocity, a body thrown tangentially to the Earth's surface, under the action of gravity in the absence of an atmosphere, can move in a circle around the Earth without falling on it and without moving away from it. This speed is called first cosmic speed, and the body moving in this way - artificial earth satellite (AES).
Let's define the first cosmic velocity for the Earth. If a body under the influence of gravity moves around the Earth uniformly in a circle, then the acceleration of free fall is its centripetal acceleration:
.
Hence the first cosmic velocity is
.
The first cosmic velocity for any celestial body is determined in the same way. The free fall acceleration at a distance R from the center of a celestial body can be found using Newton's second law and the law of universal gravitation:
.
Therefore, the first cosmic velocity at a distance R from the center of a celestial body with mass M is equal to
.
To launch a satellite into near-Earth orbit, it must first be taken out of the atmosphere. Therefore, spaceships launch vertically. At an altitude of 200 - 300 km from the Earth's surface, where the atmosphere is rarefied and has almost no effect on the movement of the satellite, the rocket makes a turn and informs the satellite of the first cosmic velocity in the direction perpendicular to the vertical.
In nature, there are various forces that characterize the interaction of bodies. Consider those forces that occur in mechanics.
gravitational forces. Probably, the very first force, the existence of which was realized by a person, was the force of attraction acting on bodies from the side of the Earth.
And it took many centuries for people to understand that the force of gravity acts between any bodies. And it took many centuries for people to understand that the force of gravity acts between any bodies. The English physicist Newton was the first to understand this fact. Analyzing the laws that govern the motion of the planets (Kepler's laws), he came to the conclusion that the observed laws of planetary motion can only be fulfilled if there is an attractive force between them that is directly proportional to their masses and inversely proportional to the square of the distance between them.
Newton formulated law of gravity. Any two bodies are attracted to each other. The force of attraction between point bodies is directed along the straight line connecting them, is directly proportional to the masses of both and inversely proportional to the square of the distance between them:
In this case, point bodies are understood to mean bodies whose dimensions are many times smaller than the distance between them.
The forces of gravity are called gravitational forces. The coefficient of proportionality G is called the gravitational constant. Its value was determined experimentally: G = 6.7 10¯¹¹ N m² / kg².
gravity acting near the surface of the Earth, is directed towards its center and is calculated by the formula:
where g is the free fall acceleration (g = 9.8 m/s²).
The role of gravity in living nature is very significant, since the size, shape and proportions of living beings largely depend on its magnitude.
Body weight. Consider what happens when a load is placed on a horizontal plane (support). At the first moment after the load is lowered, it begins to move downward under the action of gravity (Fig. 8).
The plane bends and there is an elastic force (reaction of the support), directed upwards. After the elastic force (Fy) balances the force of gravity, the lowering of the body and the deflection of the support will stop.
The deflection of the support arose under the action of the body, therefore, a certain force (P) acts on the support from the side of the body, which is called the weight of the body (Fig. 8, b). According to Newton's third law, the weight of a body is equal in magnitude to the support reaction force and is directed in the opposite direction.
P \u003d - Fu \u003d F heavy.
body weight called the force P, with which the body acts on a horizontal support that is stationary relative to it.
Since gravity (weight) is applied to the support, it deforms and, due to elasticity, counteracts the force of gravity. The forces developed in this case from the side of the support are called the forces of the reaction of the support, and the very phenomenon of the development of counteraction is called the reaction of the support. According to Newton's third law, the reaction force of the support is equal in magnitude to the force of gravity of the body and opposite to it in direction.
If a person on a support moves with the acceleration of the links of his body directed away from the support, then the reaction force of the support increases by the value ma, where m is the mass of the person, and are the accelerations with which the links of his body move. These dynamic effects can be recorded using strain gauge devices (dynamograms).
Weight should not be confused with body mass. The mass of a body characterizes its inertial properties and does not depend on either the gravitational force or the acceleration with which it moves.
The weight of the body characterizes the force with which it acts on the support and depends both on the force of gravity and on the acceleration of movement.
For example, on the Moon, the weight of a body is about 6 times less than the weight of a body on Earth. The mass is the same in both cases and is determined by the amount of matter in the body.
In everyday life, technology, sports, weight is often indicated not in newtons (N), but in kilograms of force (kgf). The transition from one unit to another is carried out according to the formula: 1 kgf = 9.8 N.
When the support and the body are motionless, then the mass of the body is equal to the force of gravity of this body. When the support and the body move with some acceleration, then, depending on its direction, the body may experience either weightlessness or overload. When the acceleration coincides in direction and is equal to the acceleration of gravity, the weight of the body will be zero, so a state of weightlessness occurs (ISS, high-speed elevator when lowering down). When the acceleration of the movement of the support is opposite to the acceleration of free fall, the person experiences an overload (start from the surface of the Earth of a manned spacecraft, a high-speed elevator going up).
The interaction inherent in all bodies of the Universe and manifested in their mutual attraction to each other is called gravitational, and the very phenomenon of universal gravitation gravity .
Gravitational interaction carried out by means of a special type of matter called gravitational field.
Gravitational forces (gravitational forces) due to the mutual attraction of the bodies and directed along the line connecting the interacting points.
The expression for the force of gravity was given to Newton in 1666 when he was only 24 years old.
Law of gravity: two bodies are attracted to each other with forces that are directly proportional to the product of the masses of the bodies and inversely proportional to the square of the distance between them:
The law is valid provided that the dimensions of the bodies are negligibly small compared to the distances between them. Also, the formula can be used to calculate the forces of universal gravitation, for spherical bodies, for two bodies, one of which is a ball, the other is a material point.
The coefficient of proportionality G = 6.68 10 -11 is called gravitational constant.
physical meaning The gravitational constant is that it is numerically equal to the force with which two bodies weighing 1 kg each are attracted, located at a distance of 1 m from each other.
Gravity
The force with which the Earth attracts nearby bodies is called gravity , and the gravitational field of the Earth - gravity field .
The force of gravity is directed downward towards the center of the Earth. In the body, it passes through a point called center of gravity. The center of gravity of a homogeneous body with a center of symmetry (ball, rectangular or round plate, cylinder, etc.) is located at this center. Moreover, it may not coincide with any of the points of the given body (for example, near the ring).
In the general case, when it is required to find the center of gravity of any body of irregular shape, one should proceed from the following regularity: if the body is suspended on a thread attached sequentially to different points of the body, then the directions marked by the thread will intersect at one point, which is precisely the center the gravity of this body.
The modulus of gravity is found using the law of universal gravitation and is determined by the formula:
F t \u003d mg, (2.7)
where g is the free fall acceleration of the body (g=9.8 m/s 2 ≈10m/s 2).
Since the direction of free fall acceleration g coincides with the direction of gravity F t, the last equality can be rewritten as
It follows from (2.7) that, i.e., the ratio of the force acting on a body of mass m at any point in the field to the mass of the body determines the free fall acceleration at a given point in the field.
For points located at a height h from the Earth's surface, the free fall acceleration of a body is:
(2.8)
where R З is the radius of the Earth; MZ is the mass of the Earth; h is the distance from the center of gravity of the body to the surface of the Earth.
From this formula it follows that,
firstly, the free fall acceleration does not depend on the mass and dimensions of the body and,
Secondly, with increasing height above the Earth, the acceleration of free fall decreases. For example, at an altitude of 297 km, it turns out to be not 9.8 m/s 2 , but 9 m/s 2 .
The decrease in the acceleration of free fall means that the force of gravity also decreases as the height above the Earth increases. The farther the body is from the Earth, the weaker it attracts it.
From formula (1.73) it can be seen that g depends on the radius of the Earth R z.
But due to the oblateness of the Earth, it has a different meaning in different places: it decreases as you move from the equator to the pole. At the equator, for example, it is equal to 9.780m/s 2 , and at the pole - 9.832m/s 2 . In addition, local g values may differ from their average g cf values due to the heterogeneous structure of the earth's crust and subsoil, mountain ranges and depressions, as well as mineral deposits. The difference between the values of g and g cf is called gravitational anomalies:
Positive anomalies Δg >0 often indicate deposits of metal ores, and negative Δg<0– о залежах лёгких полезных ископаемых, например нефти и газа.
The method of determining mineral deposits by accurately measuring the acceleration of free fall is widely used in practice and is called gravimetric exploration.
An interesting feature of the gravitational field, which electromagnetic fields do not have, is its all-penetrating ability. If you can protect yourself from electric and magnetic fields with the help of special metal screens, then nothing can protect you from the gravitational field: it penetrates through any materials.
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's 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.
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. Take a small ball in your left hand. 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.