Dependence of the force of attraction on distance. Gravitational forces

According to Newton's second law, the cause of a change in motion, that is, the cause of 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 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, between all bodies of the Universe there are forces of attraction (gravitational forces) 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 in early XVII 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 according to 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 satellites of the Earth, the flight paths of 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- weight given body, then the force of gravity is

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 spaceship 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 motion artificial satellite in a circular manner near-Earth orbit. Such satellites fly at altitudes of about 200-300 km, and we can approximately take the distance to the center of the Earth 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 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 on the 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. For bodies free form and its size cannot be used. Calculation of gravitational force in 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 a gravitational force, 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:

As you know, weight is the force with which a body presses on a support due to gravity towards the Earth.

According to the second law of mechanics, the weight of any body is related to the acceleration of gravity and the mass of this body by the relation

The weight of a body is due to the resultant of all the forces of attraction between each particle of the body and the Earth. Therefore, the weight of any body must be proportional to the mass of this body, as it is in reality. If we neglect the influence daily rotation Earth, then according to Newton’s law of gravity the weight is determined by the formula

where is the gravitational constant, the mass of the Earth, the distance of the body from the center of the Earth. Formula (3) shows that the weight of a body decreases with distance from the earth's surface. Average

The radius of the Earth is equal; therefore, when raised by weight, it decreases by 0.00032 of its value.

Since the earth’s crust is heterogeneous in density, in areas under which dense rocks lie in the depths of the earth’s crust, the force of gravity is somewhat greater than in areas (at the same latitude) whose bed is composed of less dense rocks. Massifs of mountains cause a deviation of the plumb line towards the mountains.

Comparing equations (2) and (3), we obtain an expression for the acceleration of gravity without taking into account the influence of the Earth’s rotation:

Each body lying calmly on the surface of the Earth, participating in the daily rotation of the Earth, obviously has a common centripetal acceleration with the given area, lying in a plane parallel to the equator and directed towards the axis of rotation (Fig. 48). The force with which the Earth attracts any body lying calmly on its surface is partly manifested statically in the pressure that the body exerts on the support (this component is called “weight”; another geometric component of the force manifests itself dynamically, imparting centripetal acceleration to the body, involving it in daily rotation of the Earth. For the equator this acceleration is greatest; for the poles it is equal to zero. Therefore, if any body is transferred from the pole to the equator, it will “lose” some weight.

Rice. 48. Due to the rotation of the Earth, the force of attraction to the Earth has static (weight) and dynamic components.

If the Earth were exactly spherical, then the weight loss at the equator would be:

where is the peripheral speed at the equator. Let denote the number of seconds in a day, then

Hence, taking into account that we find the relative weight loss:

Therefore, if the Earth were exactly spherical, then each kilogram of mass transferred from the Earth's pole to the equator would lose approximately in weight (this could be detected by weighing on a spring balance). The actual weight loss is even greater (about ) since the Earth has a somewhat flattened shape and its poles are located closer to the center of the Earth than areas lying on the equator.

The centripetal acceleration of daily rotation lies in a plane parallel to the equator (Fig. 48); it is directed at an angle to the radius drawn from the given area to the center of the Earth (latitude of the area). We consider centripetal force as one component of the gravitational force; weight as another geometric component of the same force. Therefore, the direction of the plumb line for all areas except the equator and poles does not coincide with the direction of the straight line drawn to the center of the Earth. However, the angle between them is small because the centripetal component of the gravitational force is small compared to the weight. The compression of the Earth that occurs as a result of the daily rotation is precisely such that a plumb line (and not a straight line drawn to the center of the Earth) is everywhere perpendicular to the surface of the Earth. The shape of the Earth is a triaxial ellipsoid.

The most accurate dimensions of the earth's ellipsoid, calculated under the guidance of prof. F.N. Krasovsky, are as follows:

To calculate the acceleration of gravity depending on geographical latitude terrain and, consequently, to determine the weight of bodies at sea level altitude, the International Geodetic Congress adopted the formula in 1930

We present the values ​​of gravitational acceleration for different latitudes (at sea level):

At latitude 45° (“normal acceleration”)

Let's consider how the force of gravity changes as we move deeper into the Earth. Let be the average radius of the earth's spheroid. Let us consider the gravitational force at point K, located at a distance from the center of the Earth.

The attraction at this point is determined by the total action of the outer spherical layer of thickness and the inner sphere of radius. An exact mathematical calculation shows that the spherical layer does not have any effect on the material points located inside it, since the attractive forces caused by its individual parts are mutually balanced. Thus, all that remains is the action of the inner spheroid of radius and, therefore, less mass than the mass globe.

If the globe were uniform in density, then the mass inside the sphere would be determined by the expression

where is the average density of the Earth. In this case, the acceleration of gravity, numerically equal to the force acting on a unit mass in the gravitational field, will be equal to

and therefore will decrease linearly as it approaches the center of the Earth. The acceleration of gravity has its maximum value on the surface of the Earth.

However, due to the fact that the Earth's core consists of heavy metals (iron, nickel, cobalt) and has an average density of more than the average density of the Earth's crust, then near the Earth's surface initially even increases slightly with depth and reaches its maximum value at a depth of about i.e. . at the boundary of the upper layers of the earth's crust and the ore shell of the Earth. Further, the force of gravity begins to decrease as it approaches the center of the Earth, but somewhat slower than required by the linear dependence.

Is significant interest the history of one of the instruments designed to measure the acceleration of gravity. In 1940, at an international conference of gravimetrists, the device of the German engineer Haalck was examined. During the debate, it became clear that this device is fundamentally no different from the so-called “universal barometer” designed by Lomonosov and described in detail in his work “On the relationship between the amount of matter and weight,” published in 1757. Lomonosov’s device was designed as follows (Fig. 49).

This makes it possible to take into account very minor changes in the acceleration of gravity.

Between any bodies in nature there is a force of mutual attraction called force of universal gravity(or gravitational forces).

was discovered by Isaac Newton in 1682. When he was still 23 years old, he suggested that the forces that keep the Moon in its orbit are of the same nature as the forces that make an apple fall to Earth. (Gravity mg ) is directed vertically strictly to the center of the earth g; Depending on the distance to the surface of the globe, the acceleration of gravity is different. At the Earth's surface in mid-latitudes its value is about 9.8 m/s 2 . as you move away from the Earth's surface

decreases. – Body weight (weight strength)is the force with which a body acts on horizontal support or stretches the suspension. It is assumed that the body motionless relative to the support or suspension. Let the body lie on a horizontal table motionless relative to the Earth. Denoted by the letter.

R Body weight and gravity differ in nature:

The weight of a body is a manifestation of the action of intermolecular forces, and the force of gravity is of gravitational nature. If acceleration a = 0 , then the weight is equal to the force with which the body is attracted to the Earth, namely ..

[P] = N

• If the condition is different, then the weight changes: if acceleration A 0 not equal , then the weight P = mg - ma (down) or P = mg + ma
• (up); if the body falls freely or moves with free fall acceleration, i.e.g(Fig. 2), then the body weight is equal 0 (P=0 ). The state of the body in which its weight equal to zero, called weightlessness.

IN weightlessness There are also astronauts. IN weightlessness For a moment, you too find yourself when you jump while playing basketball or dancing.

Home experiment: Plastic bottle with a hole at the bottom and fills with water. We release it from our hands from a certain height. While the bottle falls, water does not flow out of the hole.

Weight of a body moving with acceleration (in an elevator) A body in an elevator experiences overloads

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 gravity turns out to be different in different places on the Earth and depends on the geographic latitude of the place where the reference system associated with the Earth is located, relative to which the acceleration of gravity is determined.

Measurements carried out at different latitudes showed that the numerical values ​​of the acceleration due to gravity 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 a 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 free fall is less than the weight of a body at rest, and the weight of a body whose acceleration is opposite to the direction of the acceleration of free fall is greater than the weight of a 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., freely fall), they are in a state weightlessness. A characteristic feature This state is the absence of deformations in freely falling bodies and internal stresses, 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).