Bodies under the influence of gravity. The movement of bodies under the influence of gravity. Body motion under the influence of gravity: formulas for solving problems

Based on the interpretation of Newton's second law, we can conclude that a change in motion occurs through force. Mechanics considers forces of various physical natures. Many of them are determined using the action of gravitational forces.

In 1862, the law of universal gravitation was discovered by I. Newton. He suggested that the forces that hold up the Moon are of the same nature as the forces that cause an apple to fall to the Earth. The meaning of the hypothesis is the presence of attractive forces directed along a line and connecting the centers of mass, as shown in Figure 1. 10 . 1 . A spherical body has a center of mass coinciding with the center of the ball.

Drawing 1 . 10 . 1 . Gravitational forces of attraction between bodies. F 1 → = - F 2 → .

Definition 1

Given the known directions of motion of the planets, Newton tried to find out what forces act on them. This process is called inverse problem of mechanics.

The main task of mechanics is to determine the coordinates of a body of known mass with its speed at any time using known forces acting on the body and a given condition (direct problem). The reverse is performed by determining the acting forces on a body with its known direction. Such problems led the scientist to the discovery of the definition of the law of universal gravitation.

Definition 2

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.

F = G m 1 m 2 r 2 .

The value of G determines the coefficient of proportionality of all bodies in nature, called the gravitational constant and denoted by the formula G = 6.67 · 10 - 11 N · m 2 / k g 2 (CI).

Most phenomena in nature are explained by the presence of the force of universal gravity. The movement of planets, artificial satellites of the Earth, the flight paths of ballistic missiles, the movement of bodies near the surface of the Earth - everything is explained by the law of gravity and dynamics.

Definition 3

The manifestation of gravity is characterized by the presence gravity. This is the name given to the force of attraction of bodies towards the Earth and near its surface.

When M is denoted as the mass of the Earth, RZ is the radius, m is the mass of the body, then the formula for gravity takes the form:

F = G M R З 2 m = m g .

Where g is the acceleration of gravity, equal to g = G M R 3 2.

Gravity is directed towards the center of the Earth, as shown in the Moon-Earth example. In the absence of other forces, the body moves with the acceleration of gravity. Its average value is 9.81 m/s2. With a known G and radius R 3 = 6.38 · 10 6 m, the mass of the Earth M is calculated using the formula:

M = g R 3 2 G = 5.98 10 24 k g.

If a body moves away from the Earth's surface, then the effect of gravity and acceleration due to gravity change in inverse proportion to the square of the distance r to the center. Picture 1 . 10 . 2 shows how the gravitational force acting on the astronaut of the ship changes with distance from the Earth. Obviously, the F of its attraction to the Earth is equal to 700 N.

Drawing 1 . 10 . 2 . Changes in the gravitational force acting on an astronaut as he moves away from the Earth.

Example 1

The Earth-Moon is a suitable example of the interaction of a two-body system.

The distance to the Moon is r L = 3.84 · 10 6 m. It is 60 times greater than the radius of the Earth R Z. This means that in the presence of gravity, the gravitational acceleration α L of the Moon’s orbit will be α L = g R Z r L 2 = 9.81 m/s 2 60 2 = 0.0027 m/s 2.

It is directed towards the center of the Earth and is called centripetal. The calculation is made according to the formula a L = υ 2 r L = 4 π 2 r L T 2 = 0.0027 m / s 2, where T = 27.3 days is the period of revolution of the Moon around the Earth. The results and calculations performed in different ways indicate that Newton was right in his assumption of the same nature of the force that keeps the Moon in orbit and the force of gravity.

The Moon has its own gravitational field, which determines the acceleration of gravity g L on the surface. The mass of the Moon is 81 times less than the mass of the Earth, and its radius is 3.7 times. This shows that the acceleration g L should be determined from the expression:

g L = G M L R L 2 = G M Z 3, 7 2 T 3 2 = 0, 17 g = 1, 66 m / s 2.

Such weak gravity is typical for astronauts on the Moon. Therefore, you can make huge jumps and steps. A one-meter jump on Earth corresponds to seven meters on the Moon.

The movement of artificial satellites is recorded outside the Earth's atmosphere, so they are affected by the Earth's gravitational forces. The trajectory of a cosmic body can vary depending on the initial speed. The movement of an artificial satellite in near-Earth orbit is approximately taken as the distance to the center of the Earth, equal to the radius R Z. They fly at altitudes of 200 - 300 km.

Definition 4

It follows that the centripetal acceleration of the satellite, which is imparted by gravitational forces, is equal to the acceleration of gravity g. The speed of the satellite will take the designation υ 1. They call her first escape velocity.

Applying the kinematic formula for centripetal acceleration, we obtain

a n = υ 1 2 R З = g, υ 1 = g R З = 7.91 · 10 3 m/s.

At this speed, the satellite was able to fly around the Earth in a time equal to T 1 = 2 πR З υ 1 = 84 min 12 s.

But the period of revolution of a satellite in a circular orbit near the Earth is much longer than indicated above, since there is a difference between the radius of the actual orbit and the radius of the Earth.

The satellite moves according to the principle of free fall, vaguely similar to the trajectory of a projectile or ballistic missile. The difference lies in the high speed of the satellite, and the radius of curvature of its trajectory reaches the length of the Earth's radius.

Satellites that move along circular trajectories over large distances have a weakened gravity, inversely proportional to the square of the radius r of the trajectory. Then finding the speed of the satellite follows the condition:

υ 2 к = g R 3 2 r 2, υ = g R 3 R З r = υ 1 R 3 r.

Therefore, the presence of satellites in high orbits indicates a lower speed of their movement than from near-Earth orbit. The formula for the circulation period is:

T = 2 πr υ = 2 πr υ 1 r R З = 2 πR З υ 1 r R 3 3 / 2 = T 1 2 π R З.

T 1 takes the value of the satellite's orbital period in low-Earth orbit. T increases with the size of the orbital radius. If r has a value of 6, 6 R 3 then the T of the satellite is 24 hours. When it is launched in the equatorial plane, it will be observed to hang above a certain point on the earth's surface. The use of such satellites is known in the space radio communication system. An orbit with a radius r = 6.6 RЗ is called geostationary.

Drawing 1 . 10 . 3 . Model of satellite motion.

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The action of universal gravitational forces in nature explains many phenomena: the movement of planets in the solar system, artificial satellites of the Earth, the flight paths of ballistic missiles, the movement of bodies near the surface of the Earth - all of them are explained on the basis of the law of universal gravitation and the laws of dynamics.

The law of gravity explains the mechanical structure of the solar system, and Kepler's laws describing the trajectories of planetary motion can be derived from it. For Kepler, his laws were purely descriptive - the scientist simply summarized his observations in mathematical form, without providing any theoretical foundations for the formulas. In the great system of the world order according to Newton, Kepler’s laws become a direct consequence of the universal laws of mechanics and the law of universal gravitation. That is, we again observe how empirical conclusions obtained at one level turn into strictly substantiated logical conclusions when moving to the next stage of deepening our knowledge about the world.

Newton was the first to express the idea that gravitational forces determine not only the movement of the planets of the solar system; they act between any bodies in the Universe. One of the manifestations of the force of universal gravitation is the force of gravity - this is the common name for the force of attraction of bodies towards the Earth near its surface.

If M is the mass of the Earth, RЗ is its radius, m is the mass of a given body, then the force of gravity is equal to

where g is the acceleration of free fall;

near 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/s2. Knowing the acceleration of gravity and the radius of the Earth (RЗ = 6.38·106 m), we can calculate the mass of the Earth

The picture of the structure of the solar system that follows from these equations and combines terrestrial and celestial gravity can be understood using a simple example. Suppose we are standing at the edge of a sheer cliff, next to a cannon and a pile of cannonballs. If you simply drop a cannonball vertically from the edge of a cliff, it will begin to fall down vertically and uniformly accelerated. Its motion will be described by Newton's laws for uniformly accelerated motion of a body with acceleration g. If you now fire a cannonball towards the horizon, it will fly and fall in an arc. And in this case, its movement will be described by Newton’s laws, only now they are applied to a body moving under the influence of gravity and having a certain initial speed in the horizontal plane. Now, as you load the cannon with increasingly heavier cannonballs and fire over and over again, you will find that as each successive cannonball leaves the barrel with a higher initial velocity, the cannonballs fall further and further from the base of the cliff.

Now imagine that we have packed so much gunpowder into a cannon that the speed of the cannonball is enough to fly around the globe. If we neglect air resistance, the cannonball, having flown around the Earth, will return to its starting point at exactly the same speed with which it initially flew out of the cannon. What will happen next is clear: the core will not stop there and will continue to wind circle after circle around the planet.

In other words, we will get an artificial satellite orbiting around the Earth, like a natural satellite - the Moon.

So, step by step, we moved from describing the motion of a body falling solely under the influence of “earthly” gravity (Newton’s apple) to describing the motion of a satellite (the Moon) in orbit, without changing the nature of the gravitational influence from “earthly” to “heavenly.” It was this insight that allowed Newton to connect together the two forces of gravitational attraction that were considered different in nature before him.

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. An example of a system of two interacting bodies is the Earth–Moon system. The Moon is located at a distance from the Earth rL = 3.84·106 m. This distance is approximately 60 times the Earth's radius RЗ. Consequently, the acceleration of free fall aL, 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 is the period of revolution of the Moon around the Earth.

The coincidence of the results of calculations performed in 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 gL 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 gЛ will be 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's consider the issue of artificial Earth satellites. Artificial satellites of the Earth move outside the 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. Let us consider the case of an artificial satellite moving in a circular 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 RЗ. 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 by υ1 - this speed is called the first cosmic speed. Using the kinematic formula for centripetal acceleration, we obtain

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 a satellite can be thought of as a free fall, similar to the motion 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 of the trajectory. Thus, in high orbits the speed of satellites is less than in low-Earth orbit.

The satellite's orbital period increases with increasing orbital radius. It is easy to calculate that with an orbital radius r equal to approximately 6.6 RЗ, 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 on the earth's surface. Such satellites are used in space radio communication systems. An orbit with a radius r = 6.6 RЗ is called geostationary.

The second cosmic speed is the minimum speed that must be imparted to a spacecraft at the surface of the Earth so that it, having overcome gravity, turns into an artificial satellite of the Sun (artificial planet). In this case, the ship will move away from the Earth along a parabolic trajectory.

Figure 5 illustrates escape velocities. If the speed of the spacecraft is υ1 = 7.9·103 m/s and is directed parallel to the Earth’s surface, then the ship will move in a circular orbit at a low altitude above the Earth. At initial velocities exceeding υ1, but less than υ2 = 11.2·103 m/s, the ship’s orbit will be elliptical. At an initial speed of υ2, the ship will move along a parabola, and at an even higher initial speed, along a hyperbola.

Cosmic speeds

The velocities near the Earth's surface are indicated: 1) υ = υ1 – circular trajectory;

2) υ1< υ < υ2 – эллиптическая траектория; 3) υ = 11,1·103 м/с – сильно вытянутый эллипс;

4) υ = υ2 – parabolic trajectory; 5) υ > υ2 – hyperbolic trajectory;

6) Moon trajectory

Thus, we found out that all movements in the solar system obey Newton’s law of universal gravitation.

Based on the small mass of the planets, and especially other bodies of the Solar System, we can approximately assume that movements in the circumsolar space obey Kepler’s laws.

All bodies move around the Sun in elliptical orbits, with the Sun at one of the focuses. The closer a celestial body is to the Sun, the faster its orbital speed (the planet Pluto, the most distant known, moves 6 times slower than the Earth).

Bodies can also move in open orbits: parabola or hyperbola. This happens if the speed of the body is equal to or exceeds the value of the second cosmic velocity for the Sun at a given distance from the central body. If we are talking about a satellite of a planet, then the escape velocity must be calculated relative to the mass of the planet and the distance to its center.

The motion of a body under the influence of gravity is one of the central topics in dynamic physics. Even an ordinary school student knows that the dynamics section is based on three. Let's try to analyze this topic thoroughly, and an article describing each example in detail will help us make the study of the movement of a body under the influence of gravity as useful as possible.

A little history

People watched with curiosity various phenomena occurring in our lives. For a long time, humanity could not understand the principles and structure of many systems, but a long journey of studying the world around us led our ancestors to a scientific revolution. Nowadays, when technology is developing at an incredible speed, people hardly think about how certain mechanisms work.

Meanwhile, our ancestors were always interested in the mysteries of natural processes and the structure of the world, looked for answers to the most complex questions and did not stop studying until they found answers to them. For example, the famous scientist Galileo Galilei asked the questions back in the 16th century: “Why do bodies always fall down, what force attracts them to the ground?” In 1589, he carried out a series of experiments, the results of which turned out to be very valuable. He studied in detail the patterns of free fall of various bodies, dropping objects from the famous tower in the city of Pisa. The laws he derived were improved and described in more detail by formulas by another famous English scientist, Sir Isaac Newton. It is he who owns the three laws on which almost all modern physics is based.

The fact that the patterns of body movement described more than 500 years ago are still relevant today means that our planet is subject to unchanging laws. Modern man needs to at least superficially study the basic principles of the world.

Basic and auxiliary concepts of dynamics

In order to fully understand the principles of such a movement, you should first become familiar with some concepts. So, the most necessary theoretical terms:

Interaction is the influence of bodies on each other, during which a change occurs or the beginning of their movement relative to each other. There are four types of interaction: electromagnetic, weak, strong and gravitational.
Speed is a physical quantity that indicates the speed with which a body moves. Speed is a vector, meaning it not only has a value, but also a direction.
Acceleration is the quantity that shows us the rate of change in the speed of a body over a period of time. She is also
The trajectory of the path is a curve, and sometimes a straight line, which the body outlines when moving. With uniform rectilinear motion, the trajectory can coincide with the displacement value.
The path is the length of the trajectory, that is, exactly as much as the body has traveled in a certain amount of time.
An inertial reference frame is a medium in which Newton's first law is satisfied, that is, the body retains its inertia, provided that all external forces are completely absent.

The above concepts are quite enough to correctly draw or imagine in your head a simulation of the movement of a body under the influence of gravity.

What does strength mean?

Let's move on to the main concept of our topic. So, force is a quantity, the meaning of which is the impact or influence of one body on another quantitatively. And gravity is the force that acts on absolutely every body located on the surface or near our planet. The question arises: where does this very power come from? The answer lies in the law of universal gravitation.

What is gravity?

Any body from the Earth is influenced by the gravitational force, which imparts some acceleration to it. The force of gravity always has a vertical direction downwards, towards the center of the planet. In other words, gravity pulls objects toward the Earth, which is why objects always fall down. It turns out that gravity is a special case of the force of universal gravitation. Newton derived one of the main formulas for finding the force of attraction between two bodies. It looks like this: F = G * (m 1 x m 2) / R 2.

What is the acceleration due to gravity?

A body that is released from a certain height always flies down under the influence of gravity. The movement of a body under the influence of gravity vertically up and down can be described by equations, where the main constant will be the acceleration value "g". This value is due solely to the force of gravity, and its value is approximately 9.8 m/s 2 . It turns out that a body thrown from a height without an initial speed will move down with an acceleration equal to the “g” value.

Body motion under the influence of gravity: formulas for solving problems

The basic formula for finding the force of gravity is as follows: F gravity = m x g, where m is the mass of the body on which the force acts, and “g” is the acceleration of gravity (to simplify problems, it is usually considered equal to 10 m/s 2) .

There are several more formulas used to find one or another unknown when a body moves freely. So, for example, in order to calculate the path traveled by a body, it is necessary to substitute known values into this formula: S = V 0 x t + a x t 2 / 2 (the path is equal to the sum of the products of the initial speed multiplied by time and acceleration by the square of time divided on 2).

Equations for describing the vertical motion of a body

The vertical movement of a body under the influence of gravity can be described by an equation that looks like this: x = x 0 + v 0 x t + a x t 2 / 2. Using this expression, you can find the coordinates of the body at a known moment in time. You just need to substitute the quantities known in the problem: initial location, initial speed (if the body was not just released, but pushed with some force) and acceleration, in our case it will be equal to acceleration g.

In the same way, you can find the speed of a body that moves under the influence of gravity. The expression for finding an unknown quantity at any moment of time: v = v 0 + g x t (the value of the initial speed can be equal to zero, then the speed will be equal to the product of the acceleration of gravity and the time value during which the body moves).

The movement of bodies under the influence of gravity: problems and methods for solving them

When solving many problems related to gravity, we recommend using the following plan:

To determine a convenient inertial reference system for yourself, it is usually customary to choose the Earth, because it meets many of the requirements for ISO.
Draw a small drawing or picture that shows the main forces acting on the body. The motion of a body under the influence of gravity involves a sketch or diagram that shows in which direction the body moves when subjected to an acceleration equal to g.
The direction for projecting the forces and the resulting accelerations must then be selected.
Write down unknown quantities and determine their direction.
Finally, using the problem solving formulas above, calculate all the unknown quantities by substituting the data into the equations to find the acceleration or distance traveled.

Ready solution to an easy task

When we are talking about such a phenomenon as the movement of a body under the influence of what is the most practical way to solve a given problem, it can be difficult. However, there are several tricks, using which you can easily solve even the most difficult task. So, let's look at live examples of how to solve this or that problem. Let's start with an easy to understand problem.

A certain body was released from a height of 20 m without an initial speed. Determine how long it will take it to reach the surface of the earth.

Solution: we know the path traveled by the body, we know that the initial speed was equal to 0. We can also determine that only the force of gravity acts on the body, it turns out that this is the movement of the body under the influence of gravity, and therefore we should use this formula: S = V 0 x t + a x t 2 /2. Since in our case a = g, then after some transformations we obtain the following equation: S = g x t 2 / 2. Now all that remains is to express time through this formula, we find that t 2 = 2S / g. Let's substitute the known values (we assume that g = 10 m/s 2) t 2 = 2 x 20 / 10 = 4. Therefore, t = 2 s.

So, our answer: the body will fall to the ground in 2 seconds.

The trick to quickly solving the problem is as follows: you can notice that the described movement of the body in the above problem occurs in one direction (vertically downward). It is very similar to uniformly accelerated motion, since no force acts on the body except gravity (we neglect the force of air resistance). Thanks to this, you can use an easy formula to find the path during uniformly accelerated motion, bypassing the images of drawings with the arrangement of forces acting on the body.

An example of solving a more complex problem

Now let's see how best to solve problems on the movement of a body under the influence of gravity, if the body does not move vertically, but has a more complex nature of movement.

For example, the following task. An object of mass m moves with unknown acceleration down an inclined plane whose coefficient of friction is equal to k. Determine the value of acceleration that occurs during the movement of a given body if the angle of inclination α is known.

Solution: You should use the plan described above. First of all, draw a drawing of an inclined plane depicting the body and all the forces acting on it. It turns out that three components act on it: gravity, friction and the support reaction force. The general equation of resultant forces looks like this: Friction F + N + mg = ma.

The main highlight of the problem is the condition of inclination at an angle α. When ox and axis oy it is necessary to take into account this condition, then we get the following expression: mg x sin α - F friction = ma (for the ox axis) and N - mg x cos α = F friction (for the oy axis).

Friction F is easy to calculate using the formula for finding the friction force, it is equal to k x mg (friction coefficient multiplied by the product of body mass and gravitational acceleration). After all the calculations, all that remains is to substitute the found values into the formula, and you will get a simplified equation for calculating the acceleration with which a body moves along an inclined plane.

According to Newton's second law, the prerequisite for the configuration of motion, in other words, the prerequisite for the acceleration of bodies, is force. Mechanics deals with forces of various physical natures. Many mechanical phenomena and processes are determined by the action of forces gravity. Law of Global Gravity was discovered by I. 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 guess, between all bodies of the Universe there are forces of attraction (gravitational forces) directed along the strip connecting centers of mass(Fig. 1.10.1). For a body in the form of a homogeneous ball, the center of gravity coincides with the center of the ball.

In the following years, Newton tried to find a physical explanation for the laws of planetary motion, discovered by the astrologer I. Kepler in the early 17th century, and give a quantitative expression for gravitational forces. Knowing how the planets move, Newton wanted to find what forces act on them. This path is called reverse mechanics problem. If the main task of mechanics is to determine the coordinates of a body of known mass and its speed at any moment in time based on known forces acting on the body and given initial conditions ( simple mechanics problem), then when solving a reverse problem, you need to find the forces acting on the body, if it is clear how it moves. The solution to this problem led Newton to the discovery of the law of global 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:

The proportionality coefficient G is similar for all bodies in nature. He is called gravitational constant

Many phenomena in nature are explained by the action of global gravitational forces. The movement of planets in the solar system, the movement of artificial satellites of the Earth, the flight lines of ballistic missiles, the movement of bodies near the surface of the Earth - all these phenomena are explained on the basis of the law of global gravitation and the laws of dynamics. One of the manifestations of the force of global gravity is gravity. This is the common name for the force of attraction of bodies towards the Earth near its surface. If M is the mass of the Earth, RЗ is 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:

Gravity is oriented 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 different points on the Earth's surface is 9.81 m/s2. Knowing the acceleration of gravity and the radius of the Earth (RЗ = 6.38·106 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 backwards in 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 spaceship as he moves away from the Earth. The force with which the astronaut is attracted to the Earth near its surface is taken to be 700 N.

An example of a system of two interacting bodies is the Earth-Moon system. The Moon is located at a distance from the Earth rЛ = 3.84·106 m. This distance is approximately 60 times greater than the Earth’s radius RЗ. As follows, the acceleration of gravity aL, 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. As follows, this acceleration is centripetal acceleration. It can be calculated using the kinematic formula for centripetal acceleration (see §1.6):

where T = 27.3 days is the period of the Moon’s orbit around the Earth. The coincidence of the results of calculations performed by different methods 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 gL 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 gА will be 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 huge jumps. 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 issue of artificial Earth satellites. Artificial satellites move outside the Earth's atmosphere, and are only affected by gravitational forces from the Earth. Depending on the initial speed, the line of motion of the galactic body can be different (see §1.24). We will consider here only the case of an artificial satellite moving radially 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 RЗ. 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 υ1. This speed is called first cosmic speed. Using the kinematic formula for centripetal acceleration (see §1.6), we obtain:

Moving at such a speed, the satellite would circle the Earth in a time. In fact, the period of the satellite's orbit in a radial orbit near the Earth's surface slightly exceeds the indicated 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 difference lies solely in the fact that the speed of the satellite is so high that the radius of curvature of its line of motion is equal to the radius of the Earth. For satellites moving along radial trajectories at a significant distance from the Earth, the Earth's gravity weakens backwards in proportion to the square of the radius r of the line of motion. The satellite speed υ is found from the condition

Thus, in large orbits the speed of satellites is less than in low-Earth orbit. The call period T of such a satellite is equal to

Here T1 is the period of the satellite's calling in low-Earth orbit. The satellite's calling period increases with increasing orbital radius. It is easy to calculate that with an orbital radius r equal to approximately 6.6RZ, the satellite calling period will be equal to 24 hours. A satellite with such a calling period, launched in the equatorial plane, will hover motionlessly over a certain point on the earth's surface. Such satellites are used in cosmic radio communication systems. An orbit with radius r = 6.6R3 is called geostationary.

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Topic 3.3. The movement of celestial bodies under the influence of gravitational forces.

The law of universal gravitation. Disturbances in the motion of solar system bodies. Mass and density of the Earth. Determination of the mass of celestial bodies. Movement of artificial Earth satellites and spacecraft to the planets.

Description of the features of the motion of solar system bodies under the influence of gravitational forces in orbits with different eccentricities. Explanation of the causes of tides on Earth and disturbances in the movement of bodies in the Solar System. Understanding the peculiarities of the movement and maneuvers of spacecraft for studying the bodies of the Solar System.

3.3.1. The law of universal gravitation.

According to the law of universal gravitation, studied in the physics course,

all bodies in the Universe 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:

Where t 1 And t 2- body masses;r - the distance between them;G - gravitational constant.

The discovery of the law of universal gravitation was greatly facilitated by the laws of planetary motion formulated by Kepler and other achievements of astronomy in the 17th century. Thus, knowledge of the distance to the Moon allowed Isaac Newton (1643-1727) to prove the identity of the force that holds the Moon as it moves around the Earth and the force that causes bodies to fall to the Earth.

After all, if the force of gravity varies in inverse proportion to the square of the distance, as follows from the law of universal gravitation, then the Moon, located from the Earth at a distance of approximately 60 of its radii, should experience an acceleration 3600 times less than the acceleration of gravity on the Earth's surface, equal to 9. 8 m/s. Therefore, the acceleration of the Moon should be 0.0027 m/s 2 .

At the same time, the Moon, like any body moving uniformly in a circle, has an acceleration

Where ω - its angular velocity,r - the radius of its orbit. If we assume that the radius of the Earth is 6400 km, then the radius of the lunar orbit will ber= 60 6 400 000 m = 3.84 10 6 m. Sidereal period of the Moon's revolution T= 27.32 days, in seconds is 2.36 10 6 With. Then the acceleration of the orbital motion of the Moon

The equality of these two acceleration values proves that the force holding the Moon in orbit is the force of gravity, weakened by 3600 times compared to that acting on the surface of the Earth.

You can also be convinced that when the planets move, in accordance with Kepler’s third law, their acceleration and the gravitational force of the Sun acting on them are inversely proportional to the square of the distance, as follows from the law of universal gravitation. Indeed, according to Kepler’s third law, the ratio of the cubes of the semimajor axes of the orbitsd and squares of circulation periods T there is a constant value:

The acceleration of the planet is

From Kepler's third law it follows

therefore the acceleration of the planet is equal

So, the force of interaction between the planets and the Sun satisfies the law of universal gravitation.

3.3.2. Disturbances in the motion of solar system bodies.

Kepler's laws are strictly satisfied if the motion of two isolated bodies (the Sun and the planet) under the influence of their mutual attraction is considered. However, there are many planets in the Solar System; they all interact not only with the Sun, but also with each other. Therefore, the motion of planets and other bodies does not exactly obey Kepler's laws. Deviations of bodies from moving along ellipses are called disturbances.

These disturbances are small, since the mass of the Sun is much greater than the mass of not only an individual planet, but also all planets as a whole. The greatest disturbances in the movement of bodies in the solar system are caused by Jupiter, whose mass is 300 times greater than the mass of the Earth. The deviations of asteroids and comets are especially noticeable when they pass near Jupiter.

Currently, disturbances are taken into account when calculating the position of the planets, their satellites and other bodies of the Solar System, as well as the trajectories of spacecraft launched to study them. But back in the 19th century. calculation of disturbances made it possible to make one of the most famous discoveries in science “at the tip of a pen” - the discovery of the planet Neptune.

Conducting another survey of the sky in search of unknown objects, William Herschel in 1781 he discovered a planet, later named Uranus. After about half a century, it became obvious that the observed motion of Uranus does not agree with the calculated one, even when taking into account disturbances from all known planets. Based on the assumption of the presence of another “subauranian” planet, calculations were made of its orbit and position in the sky. We solved this problem independentlyJohn Adams in England and Urbain Le Verrier in France. Based on Le Verrier's calculations, the German astronomer Johann Halle On September 23, 1846, he discovered a previously unknown planet in the constellation Aquarius - Neptune. This discovery became the triumph of the heliocentric system, the most important confirmation of the validity of the law of universal gravitation. Subsequently, disturbances were noticed in the movement of Uranus and Neptune, which became the basis for the assumption of the existence of another planet in the solar system. Her search was crowned with success only in 1930, when, after viewing a large number of photographs of the starry sky, the planet farthest from the Sun, Pluto, was discovered.

3.3.3. Mass and density of the Earth.

The law of universal gravitation made it possible to determine the mass of our planet. Based on the law of universal gravitation, the acceleration of gravity can be expressed as follows:

Let's substitute the known values of these quantities into the formula:

g = 9.8 m/s, G = 6.67 10 -11 N m 2 /kg 2, R = 6370 km - and we find that the mass of the Earth is M = 6 10 24 kg

Knowing the mass and volume of the globe, we can calculate its average density: 5.5 10 3 kg/m 3 . With depth, due to increasing pressure and the content of heavy elements, the density increases.

3.3.4. Determination of the mass of celestial bodies.

A more accurate formula for Kepler's third law, which was obtained by Newton, makes it possible to determine one of the most important characteristics of any celestial body - mass. Let us derive this formula, assuming (to a first approximation) the orbits of the planets to be circular.

Let two bodies, mutually attracting and revolving around a common center of mass, have massesm 1 And m 2 , are located at a distance from the center of massr 1 And r 2and revolve around it with a period T. Distance between their centersR= r 1 + r 2 . Based on the law of universal gravitation, the acceleration of each of these bodies is equal to:

The angular velocity of revolution around the center of mass is . Then the centripetal acceleration will be expressed for each body as follows:

Having equated the expressions obtained for accelerations, expressing from themr 1 And r 2 and adding them term by term, we get:

where

Since the right side of this expression contains only constant quantities, it is valid for any system of two bodies interacting according to the law of gravity and revolving around a common center of mass - the Sun and a planet, a planet and a satellite. Let's determine the mass of the Sun, for this we write the expression:

Where M- mass of the Sun;m 1 - mass of the Earth; t 2- mass of the Moon;T 1 Anda 1 - the period of revolution of the Earth around the Sun (year) and the semimajor axis of its orbit; T 2 And a 2- the period of revolution of the Moon around the Earth and the semimajor axis of the lunar orbit.

Neglecting the mass of the Earth, which is negligible compared to the mass of the Sun, and the mass of the Moon, which is 81 times less than the mass of the Earth, we obtain:

Substituting the corresponding values into the formula and taking the mass of the Earth to be 1, we get that the Sun is approximately 333,000 times larger in mass than our planet.

The masses of planets that do not have satellites are determined by the disturbances that they have on the movement of asteroids, comets or spacecraft flying in their vicinity.

3.3.5. Causes of tides on Earth

Under the influence of mutual attraction of particles, the body tends to take the shape of a ball. If these bodies rotate, they are deformed and compressed along the axis of rotation.

In addition, a change in their shape also occurs under the influence of mutual attraction, which is caused by phenomena called tides Known on Earth for a long time, they were explained only on the basis of the law of universal gravitation.

Let us consider the accelerations created by the attraction of the Moon at various points on the globe (Fig. 3.13). Since the points A, B are at different distances from the Moon, the accelerations created by its gravity will be different.

The difference in acceleration caused by the attraction of another body at a given point and at the center of the planet is called tidal acceleration.

Tidal accelerations at points A And IN directed from the center of the Earth. As a result, the Earth, and primarily its water shell, is stretched in both directions along a line connecting the centers of the Earth and the Moon. At points A And IN there is a high tide, and along a circle, the plane of which is perpendicular to this line, an ebb tide occurs on Earth. The Sun's gravity also causes tides, but due to its greater distance, they are smaller than those caused by the Moon. Tides are observed not only in the hydrosphere, but also in the atmosphere and lithosphere of the Earth and other planets.

Due to the daily rotation of the Earth, it tends to drag tidal humps along with it, while at the same time, due to the gravity of the Moon, which revolves around the Earth in a month, the tidal band should move along the earth's surface much more slowly. As a result, tidal friction occurs between the huge masses of tidal water and the ocean floor. It slows down the Earth's rotation and causes an increase in the length of the day, which in the past was much shorter (5-6 hours). At the same time, the tides caused by the Earth on the Moon have slowed down its rotation, and it now faces the Earth with one side. The same slow rotation is characteristic of many satellites of Jupiter and other planets. The strong tides caused by the Sun on Mercury and Venus appear to be the reason for their extremely slow rotation on their axis.

3.3.6. Movement of artificial Earth satellites and spacecraft to the planets.

The possibility of creating an artificial Earth satellite was theoretically substantiated by Newton. He showed that there is such a horizontally directed speed at which a body, falling to the Earth, will nevertheless not fall on it, but will move around the Earth, remaining at the same distance from it. At this speed, the body will approach the Earth due to its attraction just as much as it will move away from it due to the curvature of the surface of our planet (Fig. 3.14). This speed, which is called the first cosmic (or circular), is known to you from a physics course:

It turned out to be practically possible to launch an artificial Earth satellite only two and a half centuries after Newton’s discovery - October 4, 1957. In more than forty years since that day, which is often called the beginning of the space age of mankind, about 4,000 satellites have been launched in many countries around the world various devices and purposes. Orbital stations have been created on which crews consisting of cosmonauts from different countries work for a long time, replacing each other. American astronauts repeatedly visited the Moon; automatic interplanetary stations explored all the planets of the Solar System, with the exception of the most distant planet Pluto.

Spacecraft (SV), which are sent to the Moon and planets, experience attraction from the Sun and, according to Kepler's laws, just like the planets themselves, move in ellipses. The Earth's orbital speed is about 30 km/s. If the geometric sum of the speed of the spacecraft, which was reported to it at launch, and the speed of the Earth is greater than this value, then the spacecraft will move in an orbit that lies outside the Earth's orbit. If less, inside it. In the first case, when it flies to Mars or another outer planet, the energy costs will be minimal if the spacecraft reaches the orbit of this planet at its maximum distance from the Sun - at aphelion (Fig. 3.15). In addition, it is necessary to calculate the launch time of the spacecraft so that by this moment the planet arrives at the same point in its orbit. In other words, the initial speed and launch day of the spacecraft must be chosen in such a way that the spacecraft and the planet, each moving in its own orbit, simultaneously approach the meeting point. In the second case - for the inner planet - the meeting with the spacecraft should occur at the perihelion of its orbit (Fig. 3.16). Such flight trajectories are called semi-elliptical. The major axes of these ellipses pass through the Sun, which is at one of the foci, as expected by Kepler's first law.