8. The law of universal gravitation. Gravity and body weight.
The law of universal gravitation - two material points 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.
, WhereG – gravitational constant = 6.67*N
At the pole – mg== ,
At the equator – mg= –m
If the body is above the ground – mg== ,
Gravity is the force with which the planet acts on the body. The force of gravity is equal to the product of the mass of the body and the acceleration of free fall.
Weight is the force of a body acting on a support that prevents a fall, arising in the field of gravity.
9. Forces of dry and viscous friction. Movement on an inclined plane.
Friction forces arise when there is contact between m / y bodies.
Dry friction forces are the forces that arise when two solid bodies come into contact in the absence of a liquid or gaseous layer between them. Always directed tangentially to mating surfaces.
The static friction force is equal in magnitude to the external force and is directed in the opposite direction.
Ftr rest = -F
The force of sliding friction is always directed in the direction opposite to the direction of motion, depends on the relative speed of the bodies.
Viscous friction force - when a solid body moves in a liquid or gas.
With viscous friction, there is no static friction.
Depends on the speed of the body.
At low speeds
At high speeds
Movement on an inclined plane:
oy: 0=N-mgcosα, µ=tgα
10. Elastic body. Tensile forces and deformations. Relative extension. Voltage. Hooke's law.
When the body is deformed, a force arises that seeks to restore its previous dimensions and shape of the body - the force of elasticity.
1.Stretch x>0,Fy<0
2.Compression x<0,Fy>0
At small deformations (|x|< ε= – relative deformation. σ = =S - cross-sectional area of the deformed body - stress. ε=E– Young's modulus depends on material properties. Impulse
, or the amount of motion of a material point is a vector quantity equal to the product of the mass of a material point m and the speed of its movement v. - for a material point; – for a system of material points (through the impulses of these points); – for a system of material points (through the movement of the center of mass). Center of gravity of the system point C is called, the radius vector r C of which is equal to The equation of motion of the center of mass: The meaning of the equation is as follows: the product of the mass of the system and the acceleration of the center of mass is equal to the geometric sum of the external forces acting on the bodies of the system. As you can see, the law of motion of the center of mass resembles Newton's second law. If external forces do not act on the system or the sum of external forces is equal to zero, then the acceleration of the center of mass is equal to zero, and its speed is unchanged in time in absolute value and deposition, i.e. in this case, the center of mass moves uniformly and rectilinearly. In particular, this means that if the system is closed and its center of mass is motionless, then the internal forces of the system are not able to set the center of mass in motion. Rocket propulsion is based on this principle: in order to set a rocket in motion, it is necessary to throw exhaust gases and dust generated during the combustion of fuel in the opposite direction. Law of Conservation of Momentum To derive the law of conservation of momentum, consider some concepts. The set of material points (bodies) considered as a whole is called mechanical system. The forces of interaction between the material points of a mechanical system are called internal. The forces with which external bodies act on the material points of the system are called external. A mechanical system of bodies that is not affected by external force is called closed(or isolated). If we have a mechanical system consisting of many bodies, then, according to Newton's third law, the forces acting between these bodies will be equal and oppositely directed, i.e., the geometric sum of internal forces is zero. Consider a mechanical system consisting of n bodies whose mass and speed are respectively equal T 1 , m 2 ,
. ..,T n
And v 1 ,v 2 , .. .,v n. Let F" 1 ,F" 2 , ...,F" n - resultant internal forces acting on each of these bodies, a f 1 ,f 2 , ...,F n - resultant external forces. We write down Newton's second law for each of n bodies of the mechanical system: d/dt(m 1 v 1)= F" 1 +F 1 , d/dt(m 2 v 2)= F" 2 +F 2 , d/dt(m n v n)= F" n + F n. Adding these equations term by term, we get d/dt (m 1 v 1+m2 v 2+...+mn v n) = F" 1 +F" 2 +...+F" n +F 1 +F 2 +...+F n. But since the geometric sum of the internal forces of a mechanical system is equal to zero according to Newton's third law, then d/dt(m 1 v 1 + m 2 v 2 + ... + m n v n)= F 1
+ F 2 +...+
F n , or dp/dt= F 1 +
F 2 +...+
F n , (9.1) Where momentum of the system. Thus, the time derivative of the momentum of a mechanical system is equal to the geometric sum of the external forces acting on the system. In the absence of external forces (we consider a closed system) This expression is momentum conservation law:
the momentum of a closed system is conserved, i.e., does not change over time. The momentum conservation law is valid not only in classical physics, although it was obtained as a consequence of Newton's laws. Experiments prove that it is also true for closed systems of microparticles (they obey the laws of quantum mechanics). This law is universal, i.e. the law of conservation of momentum - fundamental law of nature. By what law are you going to hang me? The phenomenon of gravity is the law of universal gravitation. Two bodies act on each other with a force that is inversely proportional to the square of the distance between them and directly proportional to the product of their masses. Mathematically, we can express this great law by the formula Gravity acts over vast distances in the universe. But Newton argued that all objects are mutually attracted. Is it true that any two objects attract each other? Just imagine, it is known that the Earth attracts you sitting on a chair. But have you ever thought about the fact that a computer and a mouse attract each other? Or a pencil and pen on the table? In this case, we substitute the mass of the pen, the mass of the pencil into the formula, divide by the square of the distance between them, taking into account the gravitational constant, we obtain the force of their mutual attraction. But, it will come out so small (due to the small masses of the pen and pencil) that we do not feel its presence. Another thing is when it comes to the Earth and a chair, or the Sun and the Earth. The masses are significant, which means that we can already evaluate the effect of force. Let's think about free fall acceleration. This is the operation of the law of attraction. Under the action of a force, the body changes speed the slower, the greater the mass. As a result, all bodies fall to the Earth with the same acceleration. What is the cause of this invisible unique power? To date, the existence of a gravitational field is known and proven. You can learn more about the nature of the gravitational field in the additional material on the topic. Think about what gravity is. Where is it from? What does it represent? After all, it cannot be that the planet looks at the Sun, sees how far it is removed, calculates the inverse square of the distance in accordance with this law? There are two bodies, let's say body A and B. Body A attracts body B. The force with which body A acts begins on body B and is directed towards body A. That is, it "takes" body B and pulls it towards itself. Body B "does" the same thing with body A. Every body is attracted by the earth. The earth "takes" the body and pulls it towards its center. Therefore, this force will always be directed vertically downwards, and it is applied from the center of gravity of the body, it is called gravity. Some methods of geological exploration, tide prediction and, more recently, the calculation of the movement of artificial satellites and interplanetary stations. Early calculation of the position of the planets. Can we set up such an experiment ourselves, and not guess whether planets, objects are attracted? Such a direct experience made Cavendish (Henry Cavendish (1731-1810) - English physicist and chemist) using the device shown in the figure. The idea was to hang a rod with two balls on a very thin quartz thread and then bring two large lead balls to the side of them. The attraction of the balls will twist the thread slightly - slightly, because the forces of attraction between ordinary objects are very weak. With the help of such an instrument, Cavendish was able to directly measure the force, distance and magnitude of both masses and, thus, determine gravitational constant G. The unique discovery of the gravitational constant G, which characterizes the gravitational field in space, made it possible to determine the mass of the Earth, the Sun and other celestial bodies. Therefore, Cavendish called his experience "weighing the Earth." Interestingly, the various laws of physics have some common features. Let's turn to the laws of electricity (Coulomb force). Electric forces are also inversely proportional to the square of the distance, but already between the charges, and the thought involuntarily arises that this pattern has a deep meaning. Until now, no one has been able to present gravity and electricity as two different manifestations of the same essence. The force here also varies inversely with the square of the distance, but the difference in the magnitude of electric forces and gravitational forces is striking. In trying to establish the common nature of gravity and electricity, we find such a superiority of electric forces over gravitational forces that it is difficult to believe that both have the same source. How can you say that one is stronger than the other? After all, it all depends on what is the mass and what is the charge. Arguing about how strong gravity acts, you have no right to say: "Let's take a mass of such and such a size," because you choose it yourself. But if we take what Nature herself offers us (her own numbers and measures, which have nothing to do with our inches, years, our measures), then we can compare. We will take an elementary charged particle, such as, for example, an electron. Two elementary particles, two electrons, due to the electric charge repel each other with a force inversely proportional to the square of the distance between them, and due to gravity they are attracted to each other again with a force inversely proportional to the square of the distance. Question: What is the ratio of gravitational force to electrical force? Gravitation is related to electrical repulsion as one is to a number with 42 zeros. This is deeply puzzling. Where could such a huge number come from? People are looking for this huge factor in other natural phenomena. They go through all sorts of big numbers, and if you want a big number, why not take, say, the ratio of the diameter of the universe to the diameter of a proton - surprisingly, this is also a number with 42 zeros. And they say: maybe this coefficient is equal to the ratio of the diameter of the proton to the diameter of the universe? This is an interesting thought, but as the universe gradually expands, the constant of gravity must also change. Although this hypothesis has not yet been refuted, we do not have any evidence in its favor. On the contrary, some evidence suggests that the constant of gravity did not change in this way. This huge number remains a mystery to this day. Einstein had to modify the laws of gravity in accordance with the principles of relativity. The first of these principles says that the distance x cannot be overcome instantly, while according to Newton's theory, forces act instantly. Einstein had to change Newton's laws. These changes, refinements are very small. One of them is this: since light has energy, energy is equivalent to mass, and all masses attract, light also attracts and, therefore, passing by the Sun, must be deflected. This is how it actually happens. The force of gravity is also slightly modified in Einstein's theory. But this very slight change in the law of gravity is just enough to explain some of the apparent irregularities in Mercury's motion. Physical phenomena in the microcosm are subject to other laws than phenomena in the world of large scales. The question arises: how does gravity manifest itself in a world of small scales? The quantum theory of gravity will answer it. But there is no quantum theory of gravity yet. People have not yet been very successful in creating a theory of gravity that is fully consistent with quantum mechanical principles and with the uncertainty principle. 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 the relative position of the attracting 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 motion of the planets of the solar system discovered by Johannes Kepler on the basis of Tycho Brahe's astronomical observations. 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 motion of the Moon, artificial satellites of the Earth, or the free fall of bodies near the 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 the relativistic theory of gravitation created by Einstein, also called the general theory of 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? 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 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. 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. Many rightly call the 16th-17th centuries one of the most glorious periods in history. It was at this time that the foundations were largely laid, without which the further development of this science would be simply unthinkable. Copernicus, Galileo, Kepler have done a great job to declare physics as a science that can answer almost any question. Standing apart in a whole series of discoveries is the law of universal gravitation, the final formulation of which belongs to the outstanding English scientist Isaac Newton. The main significance of the work of this scientist was not in his discovery of the force of universal gravitation - both Galileo and Kepler spoke about the presence of this quantity even before Newton, but in the fact that he was the first to prove that the same forces act both on Earth and in outer space. same forces of interaction between bodies. Newton in practice confirmed and theoretically substantiated the fact that absolutely all bodies in the Universe, including those located on the Earth, interact with each other. This interaction is called gravitational, while the process of universal gravitation itself is called gravity. The force of universal gravitation, the definition and formulation of which he gave, is directly dependent on the product of the masses of interacting bodies, and inversely on the square of the distance between these objects. According to Newton, irrefutably confirmed by practical research, the force of universal gravitation is found by the following formula: In it, of particular importance belongs to the gravitational constant G, which is approximately equal to 6.67 * 10-11 (N * m2) / kg2. The gravitational force with which bodies are attracted to the Earth is a special case of Newton's law and is called gravity. In this case, the gravitational constant and the mass of the Earth itself can be neglected, so the formula for finding the force of gravity will look like this: Here g is nothing more than an acceleration whose numerical value is approximately equal to 9.8 m/s2. Newton's law explains not only the processes occurring directly on the Earth, it gives an answer to many questions related to the structure of the entire solar system. In particular, the force of universal gravitation between has a decisive influence on the motion of the planets in their orbits. The theoretical description of this movement was given by Kepler, but its justification became possible only after Newton formulated his famous law. Newton himself connected the phenomena of terrestrial and extraterrestrial gravitation using a simple example: when fired from it, it does not fly straight, but along an arcuate trajectory. At the same time, with an increase in the charge of gunpowder and the mass of the nucleus, the latter will fly farther and farther. Finally, if we assume that it is possible to obtain so much gunpowder and design such a cannon that the cannonball will fly around the globe, then, having made this movement, it will not stop, but will continue its circular (ellipsoidal) movement, turning into an artificial one. As a result, the force of the universal gravity is the same in nature both on Earth and in outer space.
where k is the stiffness of the body (N/m) depends on the shape and size of the body, as well as on the material.11. Impulse of the system of material points. The equation of motion of the center of mass. Impulse and its connection with force. Collisions and momentum of force. Law of conservation of momentum.
"
- And we hang everyone according to one law - the law of universal gravitation.Law of gravity
Direction of gravity
The main thing to remember
.
.
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
.
The motion of a body under the influence of gravity. Movement of artificial satellites. first cosmic speed
.
.
.
.
This interaction occurs between bodies because there is a special type of matter, unlike others, which in science is called the gravitational field. This field exists and acts around absolutely any object, while there is no protection against it, since it has an unparalleled ability to penetrate any materials.
