The law of universal gravitation. The force of universal gravity: characteristics and practical significance

According to Newton's second law, the cause of a change in motion, that is, the cause of the acceleration of bodies, is force. In mechanics, various forces are considered physical nature. Many mechanical phenomena and processes are determined by the action of forces gravity.

Law universal gravity was discovered by Isaac Newton in 1682. As early as 1665, 23-year-old Newton suggested that the forces that keep the Moon in its orbit are of the same nature as the forces that cause an apple to fall to Earth. According to his hypothesis, attractive forces (gravitational forces) act between all bodies of the Universe, directed along the line connecting centers of mass(Fig. 1.10.1). The concept of the center of mass of a body will be strictly defined in 1.23.

For a homogeneous ball, the center of mass coincides with the center of the ball.

In subsequent years, Newton tried to find a physical explanation laws of planetary motion, discovered by astronomer Johannes Kepler at the beginning of the 17th century, and give a quantitative expression for gravitational forces. Knowing how the planets move, Newton wanted to determine what forces act on them. This path is called inverse problem of mechanics . If the main task of mechanics is to determine the coordinates of a body of known mass and its speed at any time based on known forces acting on the body and given initial conditions (direct problem of mechanics ), then when solving the inverse problem it is necessary to determine the forces acting on the body if it is known how it moves. The solution to this problem led Newton to the discovery of the law of universal gravitation.

All bodies are attracted to each other with a force directly proportional to their masses and inversely proportional to the square of the distance between them:

Proportionality factor G is the same for all bodies in nature. He is called gravitational constant

Many phenomena in nature are explained by the action of the forces of universal gravity. The movement of the planets in solar system, artificial earth satellites, flight paths ballistic missiles, the movement of bodies near the surface of the Earth - they all find an explanation based on the law of universal gravitation and the laws of dynamics.

One of the manifestations of the force of universal gravity is gravity . This is the common name for the force of attraction of bodies towards the Earth near its surface. If M- mass of the Earth, R- its radius, m is the mass of a given body, then the force of gravity is equal to

Where g - acceleration of gravity at the surface of the Earth:

The force of gravity is directed towards the center of the Earth. In the absence of other forces, the body falls freely to the Earth with the acceleration of gravity.

The average value of the acceleration due to gravity for various points on the Earth's surface is 9.81 m/s 2 . Knowing the acceleration of gravity and the radius of the Earth ( R= 6.38·10 6 m), we can calculate the mass of the Earth M:

As we move away from the Earth's surface, the force of gravity and the acceleration of gravity change in inverse proportion to the square of the distance r to the center of the Earth. Rice. 1.10.2 illustrates the change in the gravitational force acting on an astronaut in a spacecraft as it moves away from the Earth. The force with which an astronaut weighing 71.5 kg (Gagarin) is attracted to the Earth near its surface is 700 N.

An example of a system of two interacting bodies is the Earth-Moon system. The Moon is at a distance from the Earth r L = 3.84 10 6 m. This distance is approximately 60 times the radius of the Earth R H. Therefore, the acceleration of free fall a A, due to gravity, in the orbit of the Moon is

With such acceleration directed towards the center of the Earth, the Moon moves in orbit. Therefore, this acceleration is centripetal acceleration . It can be calculated using the kinematic formula for centripetal acceleration:

Where T= 27.3 days - the period of revolution of the Moon around the Earth. Coincidence of the results of calculations performed different ways, confirms Newton's assumption about the single nature of the force that holds the Moon in orbit and the force of gravity.

The Moon's own gravitational field determines the acceleration of gravity g L on its surface. The mass of the Moon is 81 times less than the mass of the Earth, and its radius is approximately 3.7 times less than the radius of the Earth. Therefore the acceleration g L is determined by the expression:

The astronauts who landed on the Moon found themselves in conditions of such weak gravity. A person in such conditions can make giant leaps. For example, if a person on Earth jumps to a height of 1 m, then on the Moon he could jump to a height of more than 6 m.

Let us now consider the question of artificial earth satellites. Artificial satellites move beyond earth's atmosphere, and they are affected only by gravitational forces from the Earth. Depending on the initial speed the trajectory of a cosmic body can be different. We will consider here only the case of an artificial satellite moving in a circular motion. near-Earth orbit. Such satellites fly at altitudes of the order of 200-300 km, and the distance to the center of the Earth can be approximately taken to be equal to its radius R H. Then the centripetal acceleration of the satellite imparted to it by gravitational forces is approximately equal to the acceleration of gravity g. Let us denote the speed of the satellite in low-Earth orbit as υ 1 . This speed is called first escape velocity . Using the kinematic formula for centripetal acceleration, we get:

Moving at such a speed, the satellite would circle the Earth in time

In fact, the period of revolution of a satellite in a circular orbit near the Earth's surface is slightly longer than the specified value due to the difference between the radius of the actual orbit and the radius of the Earth.

The motion of the satellite can be considered as free fall, similar to the movement of projectiles or ballistic missiles. The only difference is that the speed of the satellite is so high that the radius of curvature of its trajectory is equal to the radius of the Earth.

For satellites moving along circular trajectories at a considerable distance from the Earth, the Earth's gravity weakens in inverse proportion to the square of the radius r trajectories. The satellite speed υ is found from the condition

Thus, in high orbits, the speed of satellites is less than in low-Earth orbit.

Period T the revolution of such a satellite is equal to

Here T 1 - period of revolution of the satellite in low-Earth orbit. The satellite's orbital period increases with increasing orbital radius. It is easy to calculate that with a radius r orbit equal to approximately 6.6 R 3, the satellite's orbital period will be equal to 24 hours. A satellite with such an orbital period, launched in the equatorial plane, will hang motionless over a certain point earth's surface. Such satellites are used in space radio communication systems. Orbit with radius r = 6,6 R Z is called geostationary .

In 1667. Newton understood that in order for the Moon to revolve around the Earth, and the Earth and other planets around the Sun, there must be a force to keep them in a circular orbit. He suggested that the force of gravity acting on all bodies on Earth and the force that holds the planets in their circular orbits are one and the same force. This force is called force of universal gravity or gravitational force. This force is an attractive force and acts between all bodies. Newton formulated law of universal gravitation : two material points are attracted to each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

The proportionality coefficient G was unknown in Newton's time. It was first measured experimentally by the English scientist Cavendish. This coefficient is called gravitational constant. Her modern meaning equals . The gravitational constant is one of the most fundamental physical constants. The law of universal gravitation can be written in vector form. If the force acting on the second point from the first is equal F 21, and the radius vector of the second point relative to the first is equal to R 21, That:

The presented form of the law of universal gravitation is valid only for the gravitational interaction of material points. It cannot be used for bodies of arbitrary shape and size. Calculating the gravitational force in the general case is a very difficult task. However, there are bodies that are not material points, for which the gravitational force can be calculated using the given formula. These are bodies that have spherical symmetry, for example, having the shape of a ball. For such bodies, the above law is valid if by distance R we mean the distance between the centers of the bodies. In particular, the force of gravity acting on all bodies from the Earth can be calculated using this formula, since the Earth has the shape of a ball, and all other bodies can be considered material points compared to the radius of the Earth.

Since gravity is gravitational force, then we can write that the force of gravity acting on a body of mass m is equal to

Where MZ and RZ are the mass and radius of the Earth. On the other hand, the force of gravity is equal to mg, where g is the acceleration of gravity. So the acceleration of free fall is equal to

This is the formula for the acceleration of gravity on the surface of the Earth. If you move away from the surface of the Earth, the distance to the center of the Earth will increase, and the acceleration of gravity will correspondingly decrease. So at a height h above the Earth’s surface, the acceleration of gravity is equal to:

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, religious holidays. Observations indicated that in the center of the “world” there is a Luminary (Sun), around which they revolve in orbit celestial bodies. 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 of the discovery (1667) by Isaac Newton of the law of gravity in astronomy, it was finally established heliocentric system world of N. Copernicus. 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 2 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 free fall has different meanings 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 new science- physicists.

In this paragraph we will remind you about gravity, centripetal acceleration and body weight

Every body on the planet is affected by Earth's gravity. The force with which the Earth attracts each body is determined by the formula

The point of application is at the center of gravity of the body. Gravity always directed vertically downwards.

The force with which a body is attracted to the Earth under the influence of the Earth's gravitational field is called gravity. According to the law of universal gravitation, on the surface of the Earth (or near this surface), a body of mass m is acted upon by the force of gravity

F t =GMm/R 2

where M is the mass of the Earth; R is the radius of the Earth.
If only the force of gravity acts on a body, and all other forces are mutually balanced, the body undergoes free fall. According to Newton's second law and formula F t =GMm/R 2 the gravitational acceleration module g is found by the formula

g=F t /m=GM/R 2 .

From formula (2.29) it follows that the acceleration of free fall does not depend on the mass m of the falling body, i.e. for all bodies in a given place on the Earth it is the same. From formula (2.29) it follows that Ft = mg. In vector form

F t = mg

In § 5 it was noted that since the Earth is not a sphere, but an ellipsoid of revolution, its polar radius is less than the equatorial one. From the formula F t =GMm/R 2 it is clear that for this reason the force of gravity and the acceleration of gravity caused by it at the pole is greater than at the equator.

The force of gravity acts on all bodies located in the gravitational field of the Earth, but not all bodies fall to the Earth. This is explained by the fact that the movement of many bodies is impeded by other bodies, for example supports, suspension threads, etc. Bodies that limit the movement of other bodies are called connections. Under the influence of gravity, the bonds are deformed and the reaction force of the deformed connection, according to Newton’s third law, balances the force of gravity.

The acceleration of gravity is affected by the rotation of the Earth. This influence is explained as follows. The reference systems associated with the Earth's surface (except for the two associated with the Earth's poles) are not, strictly speaking, inertial reference systems - the Earth rotates around its axis, and together with it such reference systems move in circles with centripetal acceleration. This non-inertiality of reference systems is manifested, in particular, in the fact that the value of the acceleration of free fall turns out to be different in different places on the Earth and depends on geographical latitude the place where the reference frame associated with the Earth is located, relative to which the acceleration of gravity is determined.

Measurements carried out at different latitudes showed that numeric values free fall accelerations differ little from each other. Therefore, with not very accurate calculations, we can neglect the non-inertiality of the reference systems associated with the Earth’s surface, as well as the difference in the shape of the Earth from spherical, and assume that the acceleration of gravity anywhere on the Earth is the same and equal to 9.8 m/s 2 .

From the law of universal gravitation it follows that the force of gravity and the acceleration of gravity caused by it decrease with increasing distance from the Earth. At a height h from the Earth's surface, the gravitational acceleration modulus is determined by the formula

g=GM/(R+h) 2.

It has been established that at an altitude of 300 km above the Earth's surface, the acceleration of gravity is 1 m/s2 less than at the Earth's surface.
Consequently, near the Earth (up to heights of several kilometers) the force of gravity practically does not change, and therefore the free fall of bodies near the Earth is a uniformly accelerated motion.

Body weight. Weightlessness and overload

The force in which, due to attraction to the Earth, a body acts on its support or suspension is called body weight. Unlike gravity, which is a gravitational force applied to a body, weight is an elastic force applied to a support or suspension (i.e., a link).

Observations show that the weight of a body P, determined on a spring scale, is equal to the force of gravity F t acting on the body only if the scales with the body relative to the Earth are at rest or moving uniformly and rectilinearly; In this case

Р=F t=mg.

If the body moves at an accelerated rate, then its weight depends on the value of this acceleration and on its direction relative to the direction of the acceleration of gravity.

When a body is suspended on a spring scale, two forces act on it: the force of gravity F t =mg and the elastic force F yp of the spring. If in this case the body moves vertically up or down relative to the direction of acceleration of free fall, then the vector sum of the forces F t and F up gives a resultant, causing acceleration of the body, i.e.

F t + F up =ma.

According to the above definition of the concept of “weight”, we can write that P = -F yp. From the formula: F t + F up =ma. taking into account that F T =mg, it follows that mg-ma=-F yp . Therefore, P=m(g-a).

The forces Ft and Fup are directed along one vertical straight line. Therefore, if the acceleration of body a is directed downward (i.e., it coincides in direction with the acceleration of free fall g), then in modulus

P=m(g-a)

If the acceleration of the body is directed upward (i.e., opposite to the direction of the acceleration of free fall), then

P = m = m(g+a).

Consequently, the weight of a body whose acceleration coincides in direction with the acceleration of gravity, less weight body at rest, and the weight of the body, the acceleration of which is opposite to the direction of the acceleration of free fall, more weight body at rest. An increase in body weight caused by its accelerated movement is called overload.

In free fall a=g. From the formula: P=m(g-a)

it follows that in this case P = 0, i.e. there is no weight. Therefore, if bodies move only under the influence of gravity (i.e., fall freely), they are in a state weightlessness. A characteristic feature This state is the absence of deformations and internal stresses in freely falling bodies, which are caused by gravity in bodies at rest. The reason for the weightlessness of bodies is that the force of gravity imparts equal accelerations to a freely falling body and its support (or suspension).

In nature, there are various forces that characterize the interaction of bodies. Let us consider the forces that occur in mechanics.

Gravitational forces. Probably the very first force whose existence man realized was the force of gravity acting on bodies from the Earth.

And it took many centuries for people to understand that the force of gravity acts between any bodies. And it took many centuries for people to understand that the force of gravity acts between any bodies. The English physicist Newton was the first to understand this fact. Analyzing the laws that govern the motion of planets (Kepler's laws), he came to the conclusion that the observed laws of motion of planets can be fulfilled only if there is an attractive force between them, directly proportional to their masses and inversely proportional to the square of the distance between them.

Newton formulated law of universal gravitation. Any two bodies attract each other. The force of attraction between point bodies is directed along the straight line connecting them, is directly proportional to the masses of both and inversely proportional to the square of the distance between them:

In this case, point bodies are understood as bodies whose dimensions are many times smaller than the distance between them.

The forces of universal gravity are called gravitational forces. The proportionality coefficient G is called the gravitational constant. Its value was determined experimentally: G = 6.7 10¯¹¹ N m² / kg².

Gravity acting near the surface of the Earth is directed towards its center and is calculated by the formula:

where g is the acceleration of gravity (g = 9.8 m/s²).

The role of gravity in living nature is very significant, since the size, shape and proportions of living beings largely depend on its magnitude.

Body weight. Let's consider what happens when some load is placed on a horizontal plane (support). At the first moment after the load is lowered, it begins to move downward under the influence of gravity (Fig. 8).

The plane bends and an elastic force (support reaction) directed upward appears. After the elastic force (Fу) balances the force of gravity, the lowering of the body and the deflection of the support will stop.

The deflection of the support arose under the action of the body, therefore, a certain force (P) acts on the support from the side of the body, which is called the weight of the body (Fig. 8, b). According to Newton's third law, the weight of a body is equal in magnitude to the ground reaction force and is directed in the opposite direction.

P = - Fу = Fheavy.

Body weight is called the force P with which a body acts on a horizontal support that is motionless relative to it.

Since the force of gravity (weight) is applied to the support, it is deformed and, due to its elasticity, counteracts the force of gravity. The forces developed in this case from the side of the support are called support reaction forces, and the very phenomenon of the development of counteraction is called the support reaction. According to Newton's third law, the support reaction force is equal in magnitude to the force of gravity of the body and opposite in direction.

If a person on a support moves with the acceleration of the parts of his body directed from the support, then the reaction force of the support increases by the amount ma, where m is the mass of the person, and is the acceleration with which the parts of his body move. These dynamic effects can be recorded using strain gauge devices (dynamograms).

Weight should not be confused with body weight. The mass of a body characterizes its inert properties and does not depend either on the force of gravity or on the acceleration with which it moves.

The weight of a body characterizes the force with which it acts on the support and depends on both the force of gravity and the acceleration of movement.

For example, on the Moon the weight of a body is approximately 6 times less than the weight of a body on Earth. Mass in both cases is the same and is determined by the amount of matter in the body.

In everyday life, technology, and sports, weight is often indicated not in newtons (N), but in kilograms of force (kgf). The transition from one unit to another is carried out according to the formula: 1 kgf = 9.8 N.

When the support and the body are motionless, then the mass of the body is equal to the gravity of this body. When the support and the body move with some acceleration, then, depending on its direction, the body can experience either weightlessness or overload. When the acceleration coincides in direction and is equal to the acceleration of gravity, the weight of the body will be zero, therefore a state of weightlessness arises (ISS, high-speed elevator when lowering down). When the acceleration of the support movement is opposite to the acceleration of free fall, the person experiences an overload (a manned launch from the surface of the Earth spaceship, High-speed elevator going up).