Newton's law of universal gravitation. The law and force of universal gravitation

Newton was the first to establish that the fall of a stone to the Earth, the movement of planets around the Sun, and the movement of the Moon around the Earth are caused by force or gravitational interaction.

Interaction between bodies at a distance occurs through the gravitational field they create. Thanks to a number of experimental facts, Newton was able to establish the dependence of the force of attraction of two bodies on the distance between them. Newton's law, called the law of universal attraction, states that any two bodies are attracted to each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. The law is called universal or universal, as it describes the gravitational interaction between a pair of any bodies in the Universe that have mass. These forces are very weak, but there are no barriers to them.

The law in literal expression looks like:

Gravity

The globe imparts the same acceleration g = 9.8 m/s2 to all bodies falling on the Earth, called the acceleration of gravity. This means that the Earth acts, attracts, all bodies with a force called gravity. This is a special type of universal gravitational force. The force of gravity is , depends on body mass m, measured in kilograms (kg). The value g = 9.8 m/s2 is taken as an approximate value; at different latitudes and at different longitudes its value changes slightly due to the fact that:

• the radius of the Earth changes from the pole to the equator (which leads to a decrease in the value of g at the equator by 0.18%);
• The centrifugal effect caused by rotation depends on the geographic latitude (reduces the value by 0.34%).

Weightlessness

Suppose that a body falls under the influence of gravity. Other forces do not act on it. This movement is called free fall. During that period of time when only F heavy acts on the body, the body will be in weightlessness. In free fall, a person's weight disappears.

Weight is the force with which the body stretches the suspension or acts on a horizontal support.

The state of weightlessness is experienced by a parachutist during a jump, a person during a ski jump, and an airplane passenger falling into an air pocket. We feel weightlessness only for a very short time, just a few seconds. But astronauts in a spacecraft flying in orbit with the engines turned off experience weightlessness for a long time. The spacecraft is in a state of free fall, and the bodies cease to act on the support or suspension - they are in weightlessness.

Artificial earth satellites

It is possible to overcome the gravity of the Earth if the body has a certain speed. Using the law of gravity, we can determine the speed at which a body of mass m, revolving in a circular orbit around the planet, will not fall on it and will become its satellite. Consider the motion of a body in a circle around the Earth. The body is acted upon by the force of gravity from the Earth. From Newton's second law we have:

Since a body moves in a circle with centripetal acceleration:

Where r is the radius of the circular orbit, R = 6400 km is the radius of the Earth, and h is the height above the Earth’s surface on which the satellite is moving. The force F acting on a body of mass m is equal to , where Mz = 5.98*1024 kg - the mass of the Earth.
We have: . Expressing speed it will be called The first cosmic speed is the lowest speed at which a body is transmitted, it becomes an artificial Earth satellite (AES).

It is also called circular. We take the height equal to 0 and find this speed, it is approximately equal to:
It is equal to the speed of an artificial satellite revolving around the Earth in a circular orbit in the absence of atmospheric resistance.
From the formula you can see that the speed of a satellite does not depend on its mass, which means that any body can become an artificial satellite.
If you give a body greater speed, it will overcome Earth's gravity.

The second cosmic velocity is the lowest speed that allows a body, without the influence of any additional forces, to overcome gravity and become a satellite of the Sun.

This speed was called parabolic; it corresponds to the parabolic trajectory of a body in the Earth’s gravitational field (if there is no atmospheric resistance). It can be calculated from the formula:

Here r is the distance from the center of the Earth to the launch site.
Near the surface of the Earth . There is another speed, with which a body can leave the solar system and roam the expanses of space.

The third escape velocity, the lowest speed that allows a spacecraft to overcome the Sun's gravity and leave the Solar System.

This speed

In physics, there are a huge number of laws, terms, definitions and formulas that explain all natural phenomena on earth and in the Universe. One of the main ones is the law of universal gravitation, which was discovered by the great and well-known scientist Isaac Newton. Its definition looks like this: any two bodies in the Universe are mutually attracted to each other with a certain force. The formula for universal gravitation, which calculates this force, will have the form: F = G*(m1*m2 / R*R).

History of the discovery of the law

For a very long time people have studied the sky. They wanted to know all its features, everything that reigns in inaccessible space. They made a calendar based on the sky and calculated important dates and dates of religious holidays. People believed that the center of the entire Universe was the Sun, around which all celestial objects revolved.

Truly vigorous scientific interest in space and astronomy in general appeared in the 16th century. Tycho Brahe, a great astronomer, during his research observed the movements of the planets, recorded and systematized his observations. By the time Isaac Newton discovered the law of universal gravitation, the Copernican system had already been established in the world, according to which all celestial bodies revolve around a star in certain orbits. The great scientist Kepler, based on Brahe's research, discovered the kinematic laws that characterize the motion of planets.

Based on Kepler's laws, Isaac Newton discovered his and found out, What:

• The movements of the planets indicate the presence of a central force.
• The central force causes the planets to move in their orbits.

Parsing the formula

There are five variables in the Newton's law formula:

How accurate are the calculations?

Since Isaac Newton's law is a mechanics law, calculations do not always reflect as accurately as possible the actual force with which objects interact. Moreover , this formula can only be used in two cases:

• When two bodies between which interaction occurs are homogeneous objects.
• When one of the bodies is a material point, and the other is a homogeneous ball.

Gravitational field

According to Newton's third law, we understand that the forces of interaction between two bodies are equal in value, but opposite in direction. The direction of forces occurs strictly along a straight line that connects the centers of mass of two interacting bodies. The interaction of attraction between bodies occurs due to the gravitational field.

Description of interaction and gravity

Gravity has very long-range interaction fields. In other words, its influence extends over very large, cosmic distances. Thanks to gravity, people and all other objects are attracted to the earth, and the earth and all the planets of the solar system are attracted to the Sun. Gravity is the constant influence of bodies on each other; it is a phenomenon that determines the law of universal gravitation. It is very important to understand one thing - the more massive the body, the more gravity it has. The Earth has enormous mass, so we are attracted to it, and the Sun weighs several million times more than the Earth, so our planet is attracted to the star.

Albert Einstein, one of the greatest physicists, argued that gravity between two bodies occurs due to the curvature of space-time. The scientist was sure that space, like fabric, can be pressed through, and the more massive the object, the more strongly it will press through this fabric. Einstein became the author of the theory of relativity, which states that everything in the Universe is relative, even such a quantity as time.

Calculation example

Let's try, using the already known formula of the law of universal gravitation, solve a physics problem:

• The radius of the Earth is approximately 6350 kilometers. Let's take the acceleration of free fall as 10. It is necessary to find the mass of the Earth.

Solution: The acceleration of gravity near the Earth will be equal to G*M / R^2. From this equation we can express the mass of the Earth: M = g*R^2 / G. All that remains is to substitute the values ​​into the formula: M = 10*6350000^2 / 6.7 * 10^-11. In order not to worry about degrees, let’s reduce the equation to the form:

• M = 10* (6.4*10^6)^2 / 6.7 * 10^-11.

After doing the math, we find that the mass of the Earth is approximately 6*10^24 kilograms.

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.

… Let mortals rejoice that such an adornment of the human race lived among them.

(Inscription on Isaac Newton's grave)

Every schoolchild knows the beautiful legend about how Isaac Newton discovered the law of universal gravitation: an apple fell on the great scientist’s head, and instead of getting angry, Isaac wondered why this happened? Why does the Earth attract everything, but what is thrown always falls down?

But most likely it was a beautiful legend invented later. In reality, Newton had to do difficult and painstaking work to discover his law. We want to tell you about how the great scientist discovered his famous law.

The principles of the natural scientist

Isaac Newton lived at the turn of the 17th and 18th centuries (1642-1727). Life at this time was completely different. Europe was rocked by wars, and in 1666, England, where Newton lived, was struck by a terrible epidemic called the “Black Death.” This event would later be called the “Great Plague of London.” Many of the sciences were just emerging; there were few educated people, as well as what they knew.

For example, a modern weekly newspaper contains more information than the average person at that time would learn in his entire life!

Despite all these difficulties, there were people who strived for knowledge, made discoveries and moved progress forward. One of them was the great English scientist Isaac Newton.

The principles that he called “rules of philosophizing” helped the scientist make his main discoveries.

Rule 1.“No other causes should be accepted in nature other than those that are true and sufficient to explain phenomena... nature does nothing in vain, and it would be in vain for many to do what can be done by fewer. Nature is simple and does not luxury with superfluous causes of things...”

The essence of this rule is that if we can exhaustively explain a new phenomenon by existing laws, then we should not introduce new ones. This rule in general form is called Occam's razor.

Rule 2.“In experimental physics, propositions derived from occurring phenomena using induction (that is, the method of induction), despite the possibility of assumptions contrary to them, should be revered as true, either exactly or approximately, until such phenomena are discovered by which they are further clarified or will be subject to exclusion.” This means that all laws of physics must be proven or disproved experimentally.

In his principles of philosophizing, Newton formulated the principles scientific method. Modern physics successfully explores and applies phenomena whose nature has not yet been clarified (for example, elementary particles). Since Newton, natural science has developed in the firm belief that the world can be known and that Nature is organized according to simple mathematical principles. This confidence became the philosophical basis for the tremendous progress of science and technology in human history.

Shoulders of Giants

You probably haven't heard of the Danish alchemist Quiet Brahe. However, it was he who was Kepler's teacher and the first to compile an accurate table of planetary movements based on his observations. It should be noted that these tables merely represented the coordinates of the planets in the sky. Quietly bequeathed them Johannes Kepler, to his student, who, after carefully studying these tables, realized that the movement of the planets is subject to a certain pattern. Kepler formulated them as follows:

1. All planets move around in an ellipse, with the Sun at one of the focuses.
2. The radius drawn from the Sun to the planet “sweeps” equal areas in equal periods of time.
3. The squares of the periods of two planets (T 1 and T 2) are related as the cubes of the semi-major axes of their orbits (R 1 and R 2):

What immediately strikes the eye is that the Sun plays a special role in these laws. But Kepler could not explain this role, just as he could not explain the reason for the movement of planets around the Sun.

Isaac Newton will once say that if he saw further than others, it was only because he stood on the shoulders of giants. He undertook to find the root cause of Kepler's laws.

World Law

Newton realized that in order to change the speed of a body, it is necessary to apply a force to it. Today every schoolchild knows this statement as Newton's first law: the change in the speed of a body per unit time (in other words, acceleration a) is directly proportional to the force (F), and inversely proportional to the mass of the body (m). The greater the mass of the body, the more effort we must expend to change its speed. Please note that Newton uses only one characteristic of a body - its mass, without considering its shape, what it is made of, what color it is, etc. This is an example of the use of Occam's razor. Newton believed that body mass is a necessary and sufficient “factor” to describe the interaction of bodies:

Newton imagined the planets as large bodies that move in a circle (or nearly a circle). In everyday life, he often observed a similar movement: children played with a ball to which a thread was tied, they twirled it over their heads. In this case, Newton saw the ball (planet) and that it was moving in a circle, but did not see the thread. Drawing a similar analogy and using his rules of philosophizing, Newton realized that it was necessary to look for a certain force - a “thread” that connects the planets and the Sun. Further reasoning was simplified after Newton applied his own laws of dynamics.

Newton, using his first law and Kepler's third law, obtained:

Thus, Newton determined that the Sun acts on the planets with force:

He also realized that all planets revolve around the Sun, and considered it natural that the mass of the Sun should be taken into account in the constant:

It was in this form that the law of universal gravitation corresponded to Kepler's observations and his laws of planetary motion. The value G = 6.67 x 10 (-11) H (m/kg) 2 was derived from observations of the planets. Thanks to this law, the movements of celestial bodies were described, and, moreover, we were able to predict the existence of objects invisible to us. In 1846, scientists calculated the orbit of a previously unknown planet, which by its existence influenced the movement of other planets in the solar system. It was .

Newton believed that simple principles and “mechanisms of interaction” underlie the most complex things. That is why he was able to discern a pattern in the observations of his predecessors and formulate it into the Law of Universal Gravitation.

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 Earth’s surface 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 (the launch of a manned spacecraft from the surface of the Earth, a high-speed elevator rising upward).