Newton's law of gravity. The law and force of universal gravitation
In his declining years he spoke about how he discovered law of universal gravitation.
When young Isaac walked in the garden among the apple trees on his parents' estate, he saw the moon in the daytime sky. And next to him an apple fell to the ground, falling from its branch.
Since Newton was working on the laws of motion at that very time, he already knew that the apple fell under the influence of the Earth's gravitational field. And he knew that the Moon is not just in the sky, but revolves around the Earth in orbit, and, therefore, it is affected by some kind of force that keeps it from breaking out of orbit and flying in a straight line away into outer space. This is where the idea came to him that perhaps the same force makes the apple fall to the ground and the Moon remain in Earth orbit.
Before Newton, scientists believed that there were two types of gravity: terrestrial gravity (acting on Earth) and celestial gravity (acting in the heavens). This idea was firmly entrenched in the minds of people of that time.
Newton's insight was that he combined these two types of gravity in his mind. From this historical moment, the artificial and false separation of the Earth and the rest of the Universe ceased to exist.
This is how the law of universal gravitation was discovered, which is one of the universal laws of nature. According to the law, all material bodies attract each other, and the magnitude of the gravitational force does not depend on the chemical and physical properties of the bodies, on the state of their motion, on the properties of the environment where the bodies are located. Gravity on Earth is manifested, first of all, in the existence of gravity, which is the result of the attraction of any material body by the Earth. The term associated with this “gravity” (from Latin gravitas - heaviness) , equivalent to the term "gravity".
The law of gravity states that the force of gravitational attraction between two material points of mass m1 and m2, separated by a distance R, is proportional to both masses and inversely proportional to the square of the distance between them.
The very idea of the universal force of gravity was repeatedly expressed before Newton. Previously, Huygens, Roberval, Descartes, Borelli, Kepler, Gassendi, Epicurus and others thought about it.
According to Kepler's assumption, gravity is inversely proportional to the distance to the Sun and extends only in the ecliptic plane; Descartes considered it the result of vortices in the ether.
There were, however, guesses with a correct dependence on distance, but before Newton no one was able to clearly and mathematically conclusively connect the law of gravity (a force inversely proportional to the square of the distance) and the laws of planetary motion (Kepler's laws).
In his main work "Mathematical Principles of Natural Philosophy" (1687)
Isaac Newton derived the law of gravitation based on Kepler's empirical laws known at that time.
He showed that:
- the observed movements of the planets indicate the presence of a central force;
- conversely, the central force of attraction leads to elliptical (or hyperbolic) orbits.
Unlike the hypotheses of its predecessors, Newton's theory had a number of significant differences. Sir Isaac published not only the supposed formula of the law of universal gravitation, but actually proposed a complete mathematical model:
- law of gravitation;
- law of motion (Newton's second law);
- system of methods for mathematical research (mathematical analysis).
Taken together, this triad is sufficient for a complete study of the most complex movements of celestial bodies, thereby creating the foundations of celestial mechanics.
But Isaac Newton left open the question of the nature of gravity. The assumption about the instantaneous propagation of gravity in space (i.e., the assumption that with a change in the positions of bodies the gravitational force between them instantly changes), which is closely related to the nature of gravity, was also not explained. For more than two hundred years after Newton, physicists proposed various ways to improve Newton's theory of gravity. Only in 1915 these efforts were crowned with success by the creation Einstein's general theory of relativity , in which all these difficulties were overcome.
The law of universal gravitation was discovered by Newton in 1687 while studying the motion of the moon's satellite around the Earth. The English physicist clearly formulated a postulate characterizing the forces of attraction. In addition, by analyzing Kepler's laws, Newton calculated that gravitational forces must exist not only on our planet, but also in space.
Background
The law of universal gravitation was not born spontaneously. Since ancient times, people have studied the sky, mainly to compile agricultural calendars, calculate important dates, and religious holidays. Observations indicated that in the center of the “world” there is a Luminary (Sun), around which celestial bodies rotate in orbits. Subsequently, the dogmas of the church did not allow this to be considered, and people lost the knowledge accumulated over thousands of years.
In the 16th century, before the invention of telescopes, a galaxy of astronomers appeared who looked at the sky in a scientific way, discarding the prohibitions of the church. T. Brahe, having been observing space for many years, systematized the movements of the planets with special care. These highly accurate data helped I. Kepler subsequently discover his three laws.
By the time Isaac Newton discovered the law of gravitation (1667), the heliocentric system of the world of N. Copernicus was finally established in astronomy. According to it, each of the planets of the system rotates around the Sun in orbits that, with an approximation sufficient for many calculations, can be considered circular. At the beginning of the 17th century. I. Kepler, analyzing the works of T. Brahe, established kinematic laws characterizing the movements of the planets. The discovery became the foundation for elucidating the dynamics of planetary motion, that is, the forces that determine exactly this type of their motion.
Description of interaction
Unlike short-period weak and strong interactions, gravity and electromagnetic fields have long-range properties: their influence manifests itself over enormous distances. Mechanical phenomena in the macrocosm are affected by two forces: electromagnetic and gravitational. The influence of planets on satellites, the flight of an thrown or launched object, the floating of a body in a liquid - in each of these phenomena gravitational forces act. These objects are attracted by the planet and gravitate towards it, hence the name “law of universal gravitation”.
It has been proven that there is certainly a force of mutual attraction between physical bodies. Phenomena such as the fall of objects to the Earth, the rotation of the Moon and planets around the Sun, occurring under the influence of the forces of universal gravity, are called gravitational.
Law of universal gravitation: formula
Universal gravity is formulated as follows: any two material objects are attracted to each other with a certain force. The magnitude of this force is directly proportional to the product of the masses of these objects and inversely proportional to the square of the distance between them:
In the formula, m1 and m2 are the masses of the material objects being studied; r is the distance determined between the centers of mass of the calculated objects; G is a constant gravitational quantity expressing the force with which the mutual attraction of two objects weighing 1 kg each, located at a distance of 1 m, occurs.
What does the force of attraction depend on?
The law of gravity works differently depending on the region. Since the force of gravity depends on the values of latitude in a certain area, similarly, the acceleration of gravity has different values in different places. The force of gravity and, accordingly, the acceleration of free fall have a maximum value at the poles of the Earth - the force of gravity at these points is equal to the force of attraction. The minimum values will be at the equator.
The globe is slightly flattened, its polar radius is approximately 21.5 km less than the equatorial radius. However, this dependence is less significant compared to the daily rotation of the Earth. Calculations show that due to the oblateness of the Earth at the equator, the magnitude of the acceleration due to gravity is slightly less than its value at the pole by 0.18%, and after daily rotation - by 0.34%.
However, in the same place on Earth, the angle between the direction vectors is small, so the discrepancy between the force of attraction and the force of gravity is insignificant, and it can be neglected in calculations. That is, we can assume that the modules of these forces are the same - the acceleration of gravity near the Earth’s surface is the same everywhere and is approximately 9.8 m/s².
Conclusion
Isaac Newton was a scientist who made a scientific revolution, completely rebuilt the principles of dynamics and, on their basis, created a scientific picture of the world. His discovery influenced the development of science and the creation of material and spiritual culture. It fell to Newton's fate to revise the results of the idea of the world. In the 17th century Scientists have completed the grandiose work of building the foundation of a new science - physics.
Aristotle argued that massive objects fall to the ground faster than light ones.
Galileo showed at the beginning of the 17th century that all objects fall “equally.” And around the same time, Kepler wondered what made the planets move in their orbits. Maybe it's magnetism? Isaac Newton, working on "", reduced all these movements to the action of a single force called gravity, which obeys simple universal laws.
Galileo experimentally showed that the distance traveled by a body falling under the influence of gravity is proportional to the square of the time of fall: a ball falling within two seconds will travel four times as far as the same object within one second. Galileo also showed that speed is directly proportional to the time of fall, and from this he deduced that a cannonball flies along a parabolic trajectory - one of the types of conic sections, like the ellipses along which, according to Kepler, the planets move. But where does this connection come from?
When Cambridge University closed during the Great Plague in the mid-1660s, Newton returned to the family estate and formulated his law of gravity there, although he kept it secret for another 20 years. (The story of the falling apple was unheard of until eighty-year-old Newton told it after a large dinner party.)
He suggested that all objects in the Universe generate a gravitational force that attracts other objects (just as an apple is attracted to the Earth), and this same gravitational force determines the trajectories along which stars, planets and other celestial bodies move in space.
In his declining days, Isaac Newton told how this happened: he was walking through an apple orchard on his parents' estate and suddenly saw the moon in the daytime sky. And right there, before his eyes, an apple came off the branch and fell to the ground. Since Newton was working on the laws of motion at that very time, he already knew that the apple fell under the influence of the Earth's gravitational field. He also knew that the Moon does not just hang in the sky, but rotates in orbit around the Earth, and, therefore, it is affected by some kind of force that keeps it from breaking out of orbit and flying in a straight line away, into open space. Then it occurred to him that perhaps it was the same force that made both the apple fall to the ground and the Moon remain in orbit around the Earth.
Inverse square law
Newton was able to calculate the magnitude of the Moon’s acceleration under the influence of Earth’s gravity and found that it was thousands of times less than the acceleration of objects (the same apple) near the Earth. How can this be if they move under the same force?
Newton's explanation was that the force of gravity weakens with distance. An object on the Earth's surface is 60 times closer to the center of the planet than the Moon. The gravity around the Moon is 1/3600, or 1/602, that of an apple. Thus, the force of attraction between two objects - be it the Earth and an apple, the Earth and the Moon, or the Sun and a comet - is inversely proportional to the square of the distance separating them. Double the distance and the force decreases by a factor of four, triple it and the force becomes nine times less, etc. The force also depends on the mass of the objects - the greater the mass, the stronger the gravity.
The law of universal gravitation can be written as a formula:
F = G(Mm/r 2).
Where: the force of gravity is equal to the product of the larger mass M and less mass m divided by the square of the distance between them r 2 and multiplied by the gravitational constant, denoted by a capital letter G(lowercase g stands for gravity-induced acceleration).
This constant determines the attraction between any two masses anywhere in the Universe. In 1789 it was used to calculate the mass of the Earth (6·1024 kg). Newton's laws are excellent at predicting forces and motions in a system of two objects. But when you add a third, everything becomes significantly more complicated and leads (after 300 years) to the mathematics of chaos.
When he came to a great result: the same cause causes phenomena of an amazingly wide range - from the fall of a thrown stone to the Earth to the movement of huge cosmic bodies. Newton found this reason and was able to accurately express it in the form of one formula - the law of universal gravitation.
Since the force of universal gravitation imparts the same acceleration to all bodies regardless of their mass, it must be proportional to the mass of the body on which it acts:
But since, for example, the Earth acts on the Moon with a force proportional to the mass of the Moon, then the Moon, according to Newton’s third law, must act on the Earth with the same force. Moreover, this force must be proportional to the mass of the Earth. If the force of gravity is truly universal, then from the side of a given body a force must act on any other body proportional to the mass of this other body. Consequently, the force of universal gravity must be proportional to the product of the masses of interacting bodies. This leads to the formulation law of universal gravitation.
Definition of the law of universal gravitation
The force of mutual attraction between 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.
The gravitational constant is numerically equal to the force of attraction between two material points weighing 1 kg each, if the distance between them is 1 m. After all, when m 1 = m 2=1 kg and R=1 m we get G=F(numerically).
It must be borne in mind that the law of universal gravitation (4.5) as a universal law is valid for material points. In this case, the forces of gravitational interaction are directed along the line connecting these points ( Fig.4.2). This kind of force is called central.
It can be shown that homogeneous bodies shaped like a ball (even if they cannot be considered material points) also interact with the force determined by formula (4.5). In this case R- the distance between the centers of the balls. The forces of mutual attraction lie on a straight line passing through the centers of the balls. (Such forces are called central.) The bodies that we usually consider falling on the Earth have dimensions much smaller than the Earth’s radius ( R≈6400 km). Such bodies can, regardless of their shape, be considered as material points and determine the force of their attraction to the Earth using the law (4.5), keeping in mind that R is the distance from a given body to the center of the Earth.
Determination of the gravitational constant
Now let's find out how to find the gravitational constant. First of all, we note that G has a specific name. This is due to the fact that the units (and, accordingly, the names) of all quantities included in the law of universal gravitation have already been established earlier. The law of gravitation provides a new connection between known quantities with certain names of units. That is why the coefficient turns out to be a named quantity. Using the formula of the law of universal gravitation, it is easy to find the name of the SI unit of gravitational constant:
N m 2 / kg 2 = m 3 / (kg s 2).
For quantification G it is necessary to independently determine all the quantities included in the law of universal gravitation: both masses, force and distance between bodies. It is impossible to use astronomical observations for this, since the masses of the planets, the Sun, and the Earth can only be determined on the basis of the law of universal gravitation itself, if the value of the gravitational constant is known. The experiment must be carried out on Earth with bodies whose masses can be measured on a scale.
The difficulty is that the gravitational forces between bodies of small masses are extremely small. It is for this reason that we do not notice the attraction of our body to surrounding objects and the mutual attraction of objects to each other, although gravitational forces are the most universal of all forces in nature. Two people with masses of 60 kg at a distance of 1 m from each other are attracted with a force of only about 10 -9 N. Therefore, to measure the gravitational constant, fairly subtle experiments are needed.
The gravitational constant was first measured by the English physicist G. Cavendish in 1798 using an instrument called a torsion balance. The diagram of the torsion balance is shown in Figure 4.3. A light rocker with two identical weights at the ends is suspended from a thin elastic thread. Two heavy balls are fixed motionless nearby. Gravitational forces act between the weights and the stationary balls. Under the influence of these forces, the rocker turns and twists the thread. By the angle of twist you can determine the force of attraction. To do this, you only need to know the elastic properties of the thread. The masses of the bodies are known, and the distance between the centers of interacting bodies can be directly measured.
From these experiments the following value for the gravitational constant was obtained:
Only in the case when bodies of enormous mass interact (or at least the mass of one of the bodies is very large) does the gravitational force reach a large value. For example, the Earth and the Moon are attracted to each other with a force F≈2 10 20 H.
Dependence of the acceleration of free falling bodies on geographic latitude
One of the reasons for the increase in the acceleration of gravity when the point where the body is located moves from the equator to the poles is that the globe is somewhat flattened at the poles and the distance from the center of the Earth to its surface at the poles is less than at the equator. Another, more significant reason is the rotation of the Earth.
Equality of inertial and gravitational masses
The most striking property of gravitational forces is that they impart the same acceleration to all bodies, regardless of their masses. What would you say about a football player whose kick would be equally accelerated by an ordinary leather ball and a two-pound weight? Everyone will say that this is impossible. But the Earth is just such an “extraordinary football player” with the only difference that its effect on bodies is not of the nature of a short-term blow, but continues continuously for billions of years.
The extraordinary property of gravitational forces, as we have already said, is explained by the fact that these forces are proportional to the masses of both interacting bodies. This fact cannot but cause surprise if you think about it carefully. After all, the mass of a body, which is included in Newton’s second law, determines the inertial properties of the body, that is, its ability to acquire a certain acceleration under the influence of a given force. It is natural to call this mass inert mass and denote by m and.
It would seem, what relation can it have to the ability of bodies to attract each other? The mass that determines the ability of bodies to attract each other should be called gravitational mass m g.
It does not at all follow from Newtonian mechanics that the inertial and gravitational masses are the same, i.e. that
Equality (4.6) is a direct consequence of experiment. It means that we can simply talk about the mass of a body as a quantitative measure of both its inertial and gravitational properties.
The law of universal gravitation is one of the most universal laws of nature. It is valid for any bodies with mass.
The meaning of the law of universal gravitation
But if we approach this topic more radically, it turns out that the law of universal gravitation does not have the possibility of its application everywhere. This law has found its application for bodies that have the shape of a ball, it can be used for material points, and it is also acceptable for a ball having a large radius, where this ball can interact with bodies much smaller than its size.
As you may have guessed from the information provided in this lesson, the law of universal gravitation is the basis in the study of celestial mechanics. And as you know, celestial mechanics studies the movement of planets.
Thanks to this law of universal gravitation, it became possible to more accurately determine the location of celestial bodies and the ability to calculate their trajectory.
But for a body and an infinite plane, as well as for the interaction of an infinite rod and a ball, this formula cannot be applied.
With the help of this law, Newton was able to explain not only how the planets move, but also why sea tides arise. Over time, thanks to the work of Newton, astronomers managed to discover such planets of the solar system as Neptune and Pluto.
The importance of the discovery of the law of universal gravitation lies in the fact that with its help it became possible to make forecasts of solar and lunar eclipses and accurately calculate the movements of spacecraft.
The forces of universal gravity are the most universal of all the forces of nature. After all, their action extends to the interaction between any bodies that have mass. And as you know, any body has mass. The forces of gravity act through any body, since there are no barriers to the forces of gravity.
Task
And now, in order to consolidate knowledge about the law of universal gravitation, let's try to consider and solve an interesting problem. The rocket rose to a height h equal to 990 km. Determine how much the force of gravity acting on the rocket at a height h has decreased compared to the force of gravity mg acting on it at the surface of the Earth? The radius of the Earth is R = 6400 km. Let us denote by m the mass of the rocket, and by M the mass of the Earth.
At height h the force of gravity is:
From here we calculate:
Substituting the value will give the result:
The legend about how Newton discovered the law of universal gravitation after hitting the top of his head with an apple was invented by Voltaire. Moreover, Voltaire himself assured that this true story was told to him by Newton’s beloved niece Katherine Barton. It’s just strange that neither the niece herself nor her very close friend Jonathan Swift ever mentioned the fateful apple in their memoirs about Newton. By the way, Isaac Newton himself, writing in detail in his notebooks the results of experiments on the behavior of different bodies, noted only vessels filled with gold, silver, lead, sand, glass, water or wheat, not to mention an apple. However, this did not stop Newton’s descendants from taking tourists around the garden on the Woolstock estate and showing them that same apple tree before the storm destroyed it.
Yes, there was an apple tree, and apples probably fell from it, but how great was the merit of the apple in the discovery of the law of universal gravitation?
The debate about the apple has not subsided for 300 years, just like the debate about the law of universal gravitation itself or about who has the priority of discovery.uk
G.Ya.Myakishev, B.B.Bukhovtsev, N.N.Sotsky, Physics 10th grade
I. Newton was able to deduce from Kepler's laws one of the fundamental laws of nature - the law of universal gravitation. Newton knew that for all planets in the solar system, acceleration is inversely proportional to the square of the distance from the planet to the Sun and the coefficient of proportionality is the same for all planets.
From here it follows, first of all, that the force of attraction acting from the Sun on a planet must be proportional to the mass of this planet. In fact, if the acceleration of the planet is given by formula (123.5), then the force causing the acceleration
where is the mass of this planet. On the other hand, Newton knew the acceleration that the Earth imparts to the Moon; it was determined from observations of the movement of the Moon as it orbits the Earth. This acceleration is approximately one times less than the acceleration imparted by the Earth to bodies located near the Earth's surface. The distance from the Earth to the Moon is approximately equal to the Earth's radii. In other words, the Moon is several times farther from the center of the Earth than bodies located on the surface of the Earth, and its acceleration is several times less.
If we accept that the Moon moves under the influence of the Earth's gravity, then it follows that the force of the Earth's gravity, like the force of the Sun's gravity, decreases in inverse proportion to the square of the distance from the center of the Earth. Finally, the force of gravity of the Earth is directly proportional to the mass of the attracted body. Newton established this fact in experiments with pendulums. He discovered that the period of swing of a pendulum does not depend on its mass. This means that the Earth imparts the same acceleration to pendulums of different masses, and, consequently, the force of gravity of the Earth is proportional to the mass of the body on which it acts. The same, of course, follows from the same acceleration of gravity for bodies of different masses, but experiments with pendulums make it possible to verify this fact with greater accuracy.
These similar features of the gravitational forces of the Sun and the Earth led Newton to the conclusion that the nature of these forces is the same and that there are forces of universal gravity acting between all bodies and decreasing in inverse proportion to the square of the distance between the bodies. In this case, the gravitational force acting on a given body of mass must be proportional to the mass.
Based on these facts and considerations, Newton formulated the law of universal gravitation in this way: any two bodies are attracted to each other with a force that is directed along the line connecting them, directly proportional to the masses of both bodies and inversely proportional to the square of the distance between them, i.e. mutual gravitational force
where and are the masses of bodies, is the distance between them, and is the coefficient of proportionality, called the gravitational constant (the method of measuring it will be described below). Combining this formula with formula (123.4), we see that , where is the mass of the Sun. The forces of universal gravity satisfy Newton's third law. This was confirmed by all astronomical observations of the movement of celestial bodies.
In this formulation, the law of universal gravitation is applicable to bodies that can be considered material points, i.e., to bodies the distance between which is very large compared to their sizes, otherwise it would be necessary to take into account that different points of bodies are separated from each other at different distances . For homogeneous spherical bodies, the formula is valid for any distance between the bodies, if we take the distance between their centers as the value. In particular, in the case of attraction of a body by the Earth, the distance must be counted from the center of the Earth. This explains the fact that the force of gravity almost does not decrease as the height above the Earth increases (§ 54): since the radius of the Earth is approximately 6400, then when the position of the body above the Earth’s surface changes within even tens of kilometers, the force of gravity of the Earth remains practically unchanged.
The gravitational constant can be determined by measuring all other quantities included in the law of universal gravitation for any specific case.
It was possible for the first time to determine the value of the gravitational constant using torsion balances, the structure of which is schematically shown in Fig. 202. A light rocker, at the ends of which two identical balls of mass are attached, is hung on a long and thin thread. The rocker arm is equipped with a mirror, which allows optical measurement of small rotations of the rocker arm around the vertical axis. Two balls of significantly greater mass can be approached from different sides to the balls.
Rice. 202. Scheme of torsion balances for measuring the gravitational constant
The forces of attraction of small balls to large ones create a pair of forces that rotate the rocker clockwise (when viewed from above). By measuring the angle at which the rocker arm rotates when approaching the balls of the balls, and knowing the elastic properties of the thread on which the rocker arm is suspended, it is possible to determine the moment of the pair of forces with which the masses are attracted to the masses. Since the masses of the balls and the distance between their centers (at a given position of the rocker) are known, the value can be found from formula (124.1). It turned out to be equal
After the value was determined, it turned out to be possible to determine the mass of the Earth from the law of universal gravitation. Indeed, in accordance with this law, a body of mass located at the surface of the Earth is attracted to the Earth with a force
where is the mass of the Earth, and is its radius. On the other hand, we know that . Equating these quantities, we find
.
Thus, although the forces of universal gravity acting between bodies of different masses are equal, a body of small mass receives significant acceleration, and a body of large mass experiences low acceleration.
Since the total mass of all the planets of the Solar System is slightly more than the mass of the Sun, the acceleration that the Sun experiences as a result of the action of gravitational forces on it from the planets is negligible compared to the accelerations that the gravitational force of the Sun imparts to the planets. The gravitational forces acting between the planets are also relatively small. Therefore, when considering the laws of planetary motion (Kepler's laws), we did not take into account the motion of the Sun itself and approximately assumed that the trajectories of the planets were elliptical orbits, in one of the foci of which the Sun was located. However, in accurate calculations it is necessary to take into account those “perturbations” that gravitational forces from other planets introduce into the movement of the Sun itself or any planet.
124.1. How much will the force of gravity acting on a rocket projectile decrease when it rises 600 km above the Earth's surface? The radius of the Earth is taken to be 6400 km.
124.2. The mass of the Moon is 81 times less than the mass of the Earth, and the radius of the Moon is approximately 3.7 times less than the radius of the Earth. Find the weight of a person on the Moon if his weight on Earth is 600N.
124.3. The mass of the Moon is 81 times less than the mass of the Earth. Find on the line connecting the centers of the Earth and the Moon the point at which the gravitational forces of the Earth and the Moon acting on a body placed at this point are equal to each other.
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