8. Law gravity. 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 relative speed tel.
The force of viscous friction - when moving solid body in 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 momentum of a material point is a vector quantity equal to the product of the mass of the material point m and the speed of its movement v. - for a material point; - for the system 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 equal to 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 be 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. Topics of the USE codifier: forces in mechanics, the law of universal gravitation, gravity, free fall acceleration, body weight, weightlessness, artificial satellites of the Earth. Any two bodies are attracted to each other - for the sole reason that they have mass. This attractive force is called gravity or gravitational force. The gravitational interaction of any two bodies in the Universe obeys a fairly simple law. The law of universal gravitation.
Two material points with masses and are attracted to each other with a force directly proportional to their masses and inversely proportional to the square of the distance between them: (1)
The proportionality factor is called gravitational constant. This is a fundamental constant, and its numerical value was determined based on an experiment by Henry Cavendish: The order of magnitude of the gravitational constant explains why we do not notice the mutual attraction of the objects around us: the gravitational forces are too small for small masses of bodies. We observe only the attraction of objects to the Earth, the mass of which is approximately kg. Formula (1), being valid for material points, ceases to be true if the dimensions of the bodies cannot be neglected. There are, however, two practical exceptions. 1. Formula (1) is valid if the bodies are homogeneous balls. Then - the distance between their centers. The force of attraction is directed along the straight line connecting the centers of the balls. 2. Formula (1) is valid if one of the bodies is a homogeneous ball, and the other is a material point outside the ball. Then the distance from the point to the center of the ball. The force of attraction is directed along the straight line connecting the point with the center of the ball. The second case is especially important, since it allows one to apply formula (1) for the force of attraction of a body (for example, an artificial satellite) to the planet. Let's assume that the body is near some planet. Gravity is the force of gravitational attraction acting on the body from the side of the planet. In the vast majority of cases, gravity is the force of attraction towards the Earth. Let the body of mass lie on the surface of the Earth. The force of gravity acts on the body, where is the acceleration of free fall near the surface of the Earth. On the other hand, considering the Earth as a homogeneous ball, we can express the force of gravity according to the law of universal gravitation: where is the mass of the Earth, km is the radius of the Earth. From here we obtain the formula for the acceleration of free fall on the surface of the Earth: . (2)
The same formula, of course, allows you to find the acceleration of free fall on the surface of any planet of mass and radius . If the body is at a height above the surface of the planet, then for gravity we get: Here, is the free fall acceleration at height: In the last equality, we have used the relation which follows from formula (2) . Consider a body in a gravitational field. Suppose that there is a support or suspension that prevents the free fall of the body. Body weight
is the force with which a body acts on a support or suspension. We emphasize that the weight is applied not to the body, but to the support (suspension). On fig. 1 shows a body on a support. From the side of the Earth, gravity acts on the body (in the case of a homogeneous body of a simple shape, gravity is applied at the center of symmetry of the body). From the side of support, an elastic force acts on the body (the so-called support reaction). A force acts on the support from the side of the body - the weight of the body. According to Newton's third law, the forces and are equal in absolute value and opposite in direction. Let us assume that the body is at rest. Then the resultant of the forces applied to the body is zero. We have: Taking into account equality, we get . Therefore, if the body is at rest, then its weight is equal in modulus to the force of gravity. Task. The body of mass, together with the support, moves with acceleration directed vertically upwards. Find the weight of the body. Decision. Let's direct the axis vertically upwards (Fig. 2). Let's write Newton's second law: Let's move on to the projections on the axis: From here. Therefore, body weight As you can see, the weight of the body is greater than the force of gravity. Such a state is called overload. Task. The body of mass, together with the support, moves with acceleration directed vertically downwards. Find the weight of the body. Decision. Let's direct the axis vertically down (Fig. 3). The solution scheme is the same. Let's start with Newton's second law: Let's move on to the projections on the axis: Hence c. Therefore, body weight In this case, the weight of the body is less than the force of gravity. When (free fall of a body with support), the weight of the body vanishes. This is the state In order for an artificial satellite to make orbital motion around the planet, it needs to be told a certain speed. Find the speed of the circular motion of the satellite at a height above the surface of the planet. Mass of the planet, its radius (Fig. 4) The satellite will move under the action of a single force - the force of universal gravitation, directed towards the center of the planet. The acceleration of the satellite is also directed there - centripetal acceleration Denoting through the mass of the satellite, we write down Newton's second law in the projection on the axis directed to the center of the planet: , or From here we get the expression for the speed: first cosmic speed is the maximum speed of the circular movement of the satellite, corresponding to the altitude. For the first cosmic velocity we have or, taking into account the formula ( 2 ), For the Earth we have approximately. Why does a stone released from the hands fall to the ground? Because it is attracted by the Earth, each of you will say. In fact, the stone falls to the Earth with free fall acceleration. Consequently, a force directed towards the Earth acts on the stone from the side of the Earth. According to Newton's third law, the stone also acts on the Earth with the same modulus of force directed towards the stone. In other words, forces of mutual attraction act between the Earth and the stone. Newton was the first who first guessed, and then strictly proved, that the reason causing the fall of a stone to the Earth, the movement of the Moon around the Earth and the planets around the Sun, is one and the same. This is the gravitational force acting between any bodies of the Universe. Here is the course of his reasoning given in Newton's main work "The Mathematical Principles of Natural Philosophy": “A stone thrown horizontally 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 at a higher speed, then it will fall further” (Fig. 1). Continuing these reasoning, Newton comes to the conclusion that if it were not for air resistance, then the trajectory of a stone thrown from a high mountain at 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 celestial space. Now we have become so accustomed to the movement of satellites around the Earth that there is no need to explain Newton's idea in more detail. So, according to Newton, the movement of the Moon around the Earth or the planets around the Sun is also a free fall, but only a fall that lasts without stopping for billions of years. The reason for such a “fall” (whether it is really about the fall of an ordinary stone on the Earth or about the movement of the planets in their orbits) is the force of universal gravitation. What does this force depend on? Galileo proved that during free fall, the Earth imparts the same acceleration to all bodies in a given place, regardless of their mass. But acceleration, according to Newton's second law, is inversely proportional to mass. How can one explain that the acceleration imparted to a body by the Earth's gravity is the same for all bodies? This is possible only if the force of attraction to the Earth is directly proportional to the mass of the body. In this case, an increase in the mass m, for example, by a factor of two will lead to an increase in the modulus of force F is also doubled, and the acceleration, which is equal to \(a = \frac (F)(m)\), will remain unchanged. Generalizing this conclusion for the forces of gravity between any bodies, we conclude that the force of universal gravitation is directly proportional to the mass of the body on which this force acts. But at least two bodies participate in mutual attraction. Each of them, according to Newton's third law, is subject to the same modulus of gravitational forces. Therefore, each of these forces must be proportional both to the mass of one body and to the mass of the other body. Therefore, the force of universal gravitation between two bodies is directly proportional to the product of their masses: \(F \sim m_1 \cdot m_2\) It is well known from experience that the free fall acceleration is 9.8 m/s 2 and it is the same for bodies falling from a height of 1, 10 and 100 m, that is, it does not depend on the distance between the body and the Earth. This seems to mean that force does not depend on distance. But Newton believed that distances should be measured not from the surface, but from the center of the Earth. But the radius of the Earth is 6400 km. It is clear that several tens, hundreds or even thousands of meters above the Earth's surface cannot noticeably change the value of the free fall acceleration. To find out how the distance between bodies affects the force of their mutual attraction, it would be necessary to find out what is the acceleration of bodies remote from the Earth at sufficiently large distances. However, it is difficult to observe and study the free fall of a body from a height of thousands of kilometers above the Earth. But nature itself came to the rescue here and made it possible to determine the acceleration of a body moving in a circle around the Earth and therefore possessing centripetal acceleration, caused, of course, by the same force of attraction to the Earth. Such a body is the natural satellite of the Earth - the Moon. If the force of attraction between the Earth and the Moon did not depend on the distance between them, then the centripetal acceleration of the Moon would be the same as the acceleration of a body freely falling near the surface of the Earth. In reality, the centripetal acceleration of the Moon is 0.0027 m/s 2 . Let's prove it. The revolution of the Moon around the Earth occurs under the influence of the gravitational force between them. Approximately, the orbit of the Moon can be considered a circle. Therefore, the Earth imparts centripetal acceleration to the Moon. It is calculated by the formula \(a = \frac (4 \pi^2 \cdot R)(T^2)\), where R- the radius of the lunar orbit, equal to approximately 60 radii of the Earth, T≈ 27 days 7 h 43 min ≈ 2.4∙10 6 s is the period of the Moon's revolution around the Earth. Given that the radius of the earth R h ≈ 6.4∙10 6 m, we get that the centripetal acceleration of the Moon is equal to: \(a = \frac (4 \pi^2 \cdot 60 \cdot 6.4 \cdot 10^6)((2.4 \cdot 10^6)^2) \approx 0.0027\) m/s 2. The found value of acceleration is less than the acceleration of free fall of bodies near the surface of the Earth (9.8 m/s 2) by approximately 3600 = 60 2 times. Thus, an increase in the distance between the body and the Earth by 60 times led to a decrease in the acceleration imparted by the earth's gravity, and, consequently, the force of gravity itself, by 60 2 times. This leads to an important conclusion: the acceleration imparted to bodies by the force of attraction to the earth decreases in inverse proportion to the square of the distance to the center of the earth \(F \sim \frac (1)(R^2)\). In 1667, Newton finally formulated the law of universal gravitation: \(F = G \cdot \frac (m_1 \cdot m_2)(R^2).\quad (1)\) 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. Proportionality factor G called gravitational constant. Law of gravity is valid only for bodies whose dimensions are negligibly small compared to the distance between them. In other words, it is only fair for material points. In this case, the forces of gravitational interaction are directed along the line connecting these points (Fig. 2). Such forces are called central. To find the gravitational force acting on a given body from the side of another, in the case when the size of the bodies cannot be neglected, proceed as follows. Both bodies are mentally divided into such small elements that each of them can be considered a point. Adding up the gravitational forces acting on each element of a given body from all the elements of another body, we obtain the force acting on this element (Fig. 3). Having performed such an operation for each element of a given body and summing up the resulting forces, they find the total gravitational force acting on this body. This task is difficult. There is, however, one practically important case when formula (1) is applicable to extended bodies. It can be proved that spherical bodies, the density of which depends only on the distances to their centers, at distances between them that are greater than the sum of their radii, attract with forces whose modules are determined by formula (1). In this case R is the distance between the centers of the balls. And finally, since the dimensions of the bodies falling to the Earth are much smaller than the dimensions of the Earth, these bodies can be considered as point ones. Then under R in formula (1) one should understand the distance from a given body to the center of the Earth. Between all bodies there are forces of mutual attraction, depending on the bodies themselves (their masses) and on the distance between them. From formula (1) we find \(G = F \cdot \frac (R^2)(m_1 \cdot m_2)\). It follows that if the distance between the bodies is numerically equal to one ( R= 1 m) and the masses of the interacting bodies are also equal to unity ( m 1 = m 2 = 1 kg), then the gravitational constant is numerically equal to the force modulus F. Thus ( physical meaning
), the gravitational constant is numerically equal to the modulus of the gravitational force acting on a body with a mass of 1 kg from another body of the same mass with a distance between the bodies equal to 1 m. In SI, the gravitational constant is expressed as The value of the gravitational constant G can only be found empirically. To do this, you need to measure the modulus of the gravitational force F, acting on the body mass m 1 side body weight m 2 at a known distance R between bodies. The first measurements of the gravitational constant were made in the middle of the 18th century. Estimate, though very roughly, the value G at that time succeeded as a result of considering the attraction of the pendulum to the mountain, the mass of which was determined by geological methods. Accurate measurements of the gravitational constant were first made in 1798 by the English physicist G. Cavendish using a device called a torsion balance. Schematically, the torsion balance is shown in Figure 4. Cavendish fixed two small lead balls (5 cm in diameter and weighing m 1 = 775 g each) at opposite ends of a two meter rod. The rod was suspended on a thin wire. For this wire, the elastic forces arising in it when twisting through various angles were preliminarily determined. Two large lead balls (20 cm in diameter and weighing m 2 = 49.5 kg) could be brought close to small balls. Attractive forces from the large balls forced the small balls to move towards them, while the stretched wire twisted a little. The degree of twist was a measure of the force acting between the balls. The twisting angle of the wire (or the rotation of the rod with small balls) turned out to be so small that it had to be measured using an optical tube. The result obtained by Cavendish is only 1% different from the value of the gravitational constant accepted today: G ≈ 6.67∙10 -11 (N∙m 2) / kg 2 Thus, the attraction forces of two bodies weighing 1 kg each, located at a distance of 1 m from each other, are only 6.67∙10 -11 N in modules. This is a very small force. Only in the case when bodies of enormous mass interact (or at least the mass of one of the bodies is large), the gravitational force becomes large. For example, the Earth pulls the Moon with force F≈ 2∙10 20 N. Gravitational forces are the "weakest" of all the forces of nature. This is due to the fact that the gravitational constant is small. But with large masses of cosmic bodies, the forces of universal gravitation become very large. These forces keep all the planets near the Sun. The law of universal gravitation underlies celestial mechanics - the science of planetary motion. With the help of this law, the positions of celestial bodies in the firmament for many decades to come are determined with great accuracy and their trajectories are calculated. The law of universal gravitation is also used in calculations of the motion of artificial earth satellites and interplanetary automatic vehicles. Disturbances in the motion of the planets. Planets do not move strictly according to Kepler's laws. Kepler's laws would be strictly observed for the motion of a given planet only if this planet alone revolved around the Sun. But there are many planets in the solar system, all of them are attracted by both the Sun and each other. Therefore, there are disturbances in the motion of the planets. In the solar system, perturbations are small, because the attraction of the planet by the Sun is much stronger than the attraction of other planets. When calculating the apparent position of the planets, perturbations must be taken into account. When launching artificial celestial bodies and when calculating their trajectories, they use an approximate theory of the motion of celestial bodies - perturbation theory. Discovery of Neptune. One of the clearest examples of the triumph of the law of universal gravitation is the discovery of the planet Neptune. In 1781, the English astronomer William Herschel discovered the planet Uranus. Its orbit was calculated and a table of the positions of this planet was compiled for many years to come. However, a check of this table, carried out in 1840, showed that its data differ from reality. Scientists have suggested that the deviation in the motion of Uranus is caused by the attraction of an unknown planet, located even further from the Sun than Uranus. Knowing the deviations from the calculated trajectory (disturbances in the movement of Uranus), the Englishman Adams and the Frenchman Leverrier, using the law of universal gravitation, calculated the position of this planet in the sky. Adams completed the calculations earlier, but the observers to whom he reported his results were in no hurry to verify. Meanwhile, Leverrier, having completed his calculations, indicated to the German astronomer Halle the place where to look for an unknown planet. On the very first evening, September 28, 1846, Halle, pointing the telescope to the indicated place, discovered a new planet. They named her Neptune. In the same way, on March 14, 1930, the planet Pluto was discovered. Both discoveries are said to have been made "at the tip of a pen". Using the law of universal gravitation, you can calculate the mass of the planets and their satellites; explain phenomena such as the ebb and flow of water in the oceans, and much more. The forces of universal gravitation are the most universal of all the forces of nature. They act between any bodies that have mass, and all bodies have mass. There are no barriers to the forces of gravity. They act through any body. 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: 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, First of all, 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. On February 11, 2016, the experimental discovery of gravitational waves was announced, the existence of which was predicted in the last century by Albert Einstein. A gravitational wave is the propagation of a variable gravitational field in space. This wave is radiated by a moving mass and can break away from its source (as an electromagnetic wave breaks off from a charged particle moving with acceleration). It is believed that the study of gravitational waves will help shed light on the history of the universe and not only... They say that I. Newton himself told how he discovered the law of universal gravitation. Once a scientist was walking in the garden and saw the moon in the daytime sky. At that moment, an apple fell from a branch in front of his eyes. It was then that Newton thought that perhaps it was the same force that caused the apple to fall to the ground and the moon to remain in Earth orbit. We study the gravitational interaction Without exception, all physical bodies in the Universe are attracted to each other - this phenomenon is called universal gravitation or gravity (from Latin gravitas - weight). gravitational interaction - the interaction inherent in all bodies in the universe and manifested in their mutual attraction to each other. For example, now you and the textbook interact with the forces of gravitational attraction. But in this case, the forces are so small that even the most accurate instruments cannot detect them. The forces of gravitational attraction become noticeable only when at least one of the bodies has a mass comparable to the mass of celestial bodies (stars, planets, their satellites, etc.). Gravitational interaction is carried out due to a special kind of matter - the gravitational field that exists around any body - a star, planet, person, book, molecule, atom, etc. Discovering the law of gravity The first statements about gravity are found in ancient authors. Thus, the ancient Greek thinker Plutarch (c. 46 - c. 127) wrote: "The moon would fall to the Earth like a stone, as soon as the power of its flight would disappear." In the XVI-XVII centuries. European scientists again turned to the theory of the existence of mutual attraction of bodies. First of all, discoveries in astronomy served as the impetus: Nicolaus Copernicus (Fig. 33.1) proved that at the center of the Solar the system is the Sun, and all the planets revolve around it; Johannes Kepler (1571-1630) discovered the laws of planetary motion around the Sun; Galileo Galilei created a telescope and used it to see the moons of Jupiter. But why do the planets revolve around the Sun, and the satellites around the planets, what force keeps the cosmic bodies in their orbits? One of the first to understand this was the English scientist Robert Hooke (1635-1703). He wrote: "All celestial bodies have an attraction to their center, as a result of which they not only attract their own parts and do not allow them to scatter, but also attract all other celestial bodies that are in their sphere of action." It was R. Hooke who suggested that the force of attraction of two bodies is directly proportional to the masses of these bodies and inversely proportional to the square of the distance between them. However, this was proved by I. Newton, who formulated the law of universal gravitation: Rice. 33.2. According to Newton's third law, the forces of gravitational attraction of bodies are equal in absolute value and opposite in direction Rice. 33.3. Henry Cavendish (1731-1810), English physicist and chemist. Determined the gravitational constant, mass and average density of the Earth; a few years before S. Coulomb discovered the law of interaction of electric charges Between any two bodies there are forces of gravitational attraction (Fig. 33.2), which are directly proportional to the product of the masses of these bodies and inversely proportional to the square of the distance between them: The mathematical record of what law reminds you of the record of the law of universal gravitation? Write down the formula. The gravitational constant was first measured by the English scientist Henry Cavendish (Fig. 33.3) in 1798 using a torsion balance: The gravitational constant is numerically equal to the force with which two material points weighing 1 kg each interact at a distance of 1 m from each other (if m 1 \u003d m 2 \u003d 1 kg, and r \u003d 1 m, then F \u003d 6.67 10 -11 N). The law of universal gravitation makes it possible to describe a large number of phenomena, including the movement of natural and artificial bodies in the solar system, the movement of double stars, star clusters, etc. In astronomy, based on this law, they calculate the masses of celestial bodies, find out the nature of their movement, structure, evolution. gives an accurate result in the following cases: We find out the limits of applicability of the law of universal gravitation Rice. 33.5. The force of gravity is directed vertically downwards and is applied to a point called the center of gravity of the body. The center of gravity of a homogeneous symmetrical body is located at the center of symmetry; may be outside the body (c) Rice. 33.6. The distance r from the center of the Earth to the body is equal to the sum of the radius of the Earth R З and the height h at which the body is located 1) if the dimensions of the bodies are negligible compared to the distance between them (the bodies can be considered material points); 2) if both bodies have a spherical shape and a spherical distribution of matter; 3) if one of the bodies is a ball, the dimensions and mass of which are significantly greater than the dimensions and mass of another body located on the surface of this ball or at a distance from it. Note! The law of universal gravitation, like most laws of classical mechanics, is applied only in cases where the relative speed of the bodies is much less than the speed of light. In the general case, gravitation is described by the general theory of relativity created by A. Einstein. Why can one use the law of universal gravitation when calculating the force of attraction of the Earth to the Sun? Moon to Earth? man to the Earth (see Fig. 33.4)? determine the force of gravity Gravity P heavy - the force with which the Earth (or another astronomical body) attracts bodies located on its surface or near it (Fig. 33.5) *. According to the law of universal gravitation, the modulus of gravity ^ heavy acting on a body near the Earth can be calculated by the formula: where G is the gravitational constant; m is body weight; МЗ is the mass of the Earth; r \u003d R З + h is the distance from the center of the Earth to the body (Fig. 33.6). What is free fall acceleration The fall of bodies was first studied by Galileo Galilei, who experimentally proved: the reason that light bodies fall with less acceleration is air resistance; in the absence of air, all bodies - regardless of their mass, volume, shape - fall to the Earth with the same acceleration. More accurate experiments were carried out by Isaac Newton, who made a special device for this - Newton's tube. Experiments have shown: in a vacuum, a lead shot, a cork and a bird's feather fell the same way (a), in the air the feather was hopelessly lagging behind (b). The movement of a body only under the influence of gravity is called free fall. In free fall, the force of gravity acting on the body is not compensated by any force, therefore, according to Newton's second law, the body moves with acceleration. This acceleration is called the free fall acceleration and is denoted by the symbol g: Like gravity, the acceleration due to gravity is always directed vertically downwards. no matter which direction the body is moving. From the formula g \u003d - ^ heavy / ^ : So, we have two formulas for determining the modulus of gravity: From here we get the formula for calculating the acceleration of free fall: Analysis of the last formula shows: 1. The acceleration of free fall does not depend on the mass of the body (Galileo proved). 2. The gravitational acceleration decreases with an increase in the height h at which the body is above the Earth's surface, and a noticeable change occurs if h is tens and hundreds of kilometers (at a height h = 100 km, the gravitational acceleration will decrease by only 0.3 m / from 2). 3. If the body is on the surface of the Earth (h = 0) or at an altitude of several kilometers Rice. 33.7. The gravitational acceleration modulus at the equator is slightly less than at the pole g< g^ Note that due to the rotation of the Earth, and also due to the fact that the shape of the Earth is a geoid (the equatorial radius of the Earth is 21 km more than the polar one), the acceleration of free fall depends on the geographic latitude of the area (Fig. 33.7). From the 7th grade physics course, you know that g ~ 10 N / kg. Prove that 1 N/kg = 1 m/s 2 . Summing up The interaction inherent in all bodies in the Universe and manifested in their mutual attraction to each other is called gravitational. Gravitational interaction is carried out with the help of a special kind of matter - the gravitational field. The law of universal gravitation: between any two bodies there is a force of gravitational attraction, which is directly proportional to the product of the masses of these bodies and inversely proportional to the square of the distance gravitational constant. The force with which the Earth attracts bodies located on its surface or near it is called gravity. The force of gravity is directed vertically downwards, applied to the center of gravity of the body, and its modulus calculated according to the formulas: between them: The movement of bodies only under the action of gravity is called free fall, and the acceleration with which the bodies move in this case is called free fall acceleration g. This acceleration is always directed vertically downwards and does not depend on the mass of the body. On the Earth's surface g ~ 9.8 m/s 2 . test questions 1. What interaction is called gravitational? Give examples. 2. Formulate and write down the law of universal gravitation. 3. What is the physical meaning of the gravitational constant? What is it equal to? 4. What are the limits of applicability of the law of universal gravitation? 5. Define gravity. By what formulas is it calculated and how is it directed? 6. On what factors does free fall acceleration depend? Exercise number 33 1. Determine the mass of a body if a gravity force of 7.52 N acts on it on the surface of the Moon. What force of gravity will act on this body on the surface of the Earth? The free fall acceleration on the Moon is 1.6 m/s 2 . 2. Is it possible, using the law of universal gravitation, to calculate the force of attraction of two ocean liners (see figure)? 3. How will the force of gravitational attraction between two balls change if one of them is replaced by another with twice the mass? 4. Having measured the gravitational constant, G. Cavendish was able to determine the mass of the Earth, after which he proudly declared: "I weighed the Earth." Determine the mass of the Earth, knowing its radius (R З "6400 km), the acceleration of free fall on its surface and the gravitational constant. 5. Determine the acceleration of free fall at a height that is equal to three radii of the Earth. 6. Determine the gravitational acceleration on the surface of the planet, the mass and radius of which is twice as large as the mass and radius of the Earth. 7. Use additional sources of information and learn about the acceleration of free fall on the surface of the planets of the solar system. On which planet will you weigh less? Will your mass be less? 8. Equation of body motion: χ = -5ί + 5ί 2 . What is the initial speed and acceleration of the body? After what time interval will the body change its direction of motion? Experimental task The center of gravity of an irregularly shaped body can be determined by hanging it alternately from any two extreme points (see figure). Cut out a free-form figurine from thick paper or cardboard and determine its center of gravity. Place the figurine with the center of gravity on the tip of the needle or fountain pen. Make sure the figurine is in balance. Write down the plan for the experiment. Physics and technology in Ukraine Odessa National Polytechnic University, founded in 1918, today is one of the leading technical educational institutions in Ukraine. The names of such scientists as the Nobel Prize winner I. E. Tamm, academicians L. I. Mandelstam, N. D. Papaleksi, A. G. Amelin, M. A. Aganin, professors K. S. Zavriev, C. D. Clark, I. Yu. Timchenko and others. Outstanding engineers, designers, scientists, inventors studied and worked at the Odessa Polytechnic University: V. I. Atroshchenko, G. K. Boreskov, A. A. Ennan, A. E. Nudelman, A. F. Dashchenko, L. I. Gutenmakher, G. K. Suslov, V. V. Azhogin, L. I. Panov, B. S. Priester, A. V. Usov, A. V. Yakimov, etc. The main directions of scientific research and training of personnel of the Odessa Polytechnic are mechanical engineering, energy, chemical technologies, computer-integrated control systems, radio electronics, electromechanics, information technologies, telecommunications. Since 2010, the Rector of the University is Gennady Alexandrovich Oborsky, Doctor of Technical Sciences, Professor, a well-known specialist in the field of dynamics and reliability of technological systems. This is textbook material.
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.
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The law of universal gravitation.
Gravity.
Body weight. Weightlessness.
weightlessness
, in which the body does not press on the support at all.artificial satellites.
Rice. 4. Satellite in a circular orbit.
The dependence of the force of gravity on the mass of bodies
The dependence of the force of gravity on the distance between bodies
Law of gravity
The physical meaning of the gravitational constant
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Cavendish experience
The meaning of the law of gravity
Literature
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