Gravity: formula, definition. Gravitational forces. The law of universal gravitation. Body weight
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 ; 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 g
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.; 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 a = 0 ((Fig. 2), then the body weight is equal ). P=0 The state of a body in which its weight is zero is called.
weightlessness IN weightlessness IN There are also astronauts. IN
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
Physics - 10th grade"
Why does the Moon move around the Earth?
What happens if the moon stops?
Why do the planets revolve around the Sun? Chapter 1 discussed in detail that Earth imparts to all bodies near the surface of the Earth the same acceleration - the acceleration of gravity. But if the globe imparts acceleration to a body, then, according to Newton’s second law, it acts on the body with some force. The force with which the Earth acts on a body is called. First we will find this force, and then we will consider the force of universal gravity.
Acceleration in absolute value is determined from Newton's second law:
weightlessness general case it depends on the force acting on the body and its mass. Since the acceleration of gravity does not depend on mass, it is clear that the force of gravity must be proportional to mass:
The physical quantity is the acceleration of gravity, it is constant for all bodies.
Based on the formula F = mg, you can specify a simple and practically convenient method for measuring the mass of bodies by comparing the mass of a given body with a standard unit of mass. The ratio of the masses of two bodies is equal to the ratio of the forces of gravity acting on the bodies:
This means that the masses of bodies are the same if the forces of gravity acting on them are the same.
This is the basis for determining masses by weighing on spring or lever scales. By ensuring that the force of pressure of a body on a pan of scales, equal to the force of gravity applied to the body, is balanced by the force of pressure of weights on another pan of scales, equal to the force of gravity applied to the weights, we thereby determine the mass of the body.
The force of gravity acting on a given body near the Earth can be considered constant only at a certain latitude near the Earth's surface. If the body is lifted or moved to a place with a different latitude, then the acceleration of gravity, and therefore the force of gravity, will change.
The force of universal gravity.
Newton was the first to strictly prove that the cause of a stone falling to the Earth, the movement of the Moon around the Earth and the planets around the Sun are the same. This force of universal gravity, acting between any bodies in the Universe.
Newton came to the conclusion that if it were not for air resistance, then the trajectory of a stone thrown from a high mountain (Fig. 3.1) at a certain speed could become such that it would never reach the surface of the Earth at all, but would move around it like the way the planets describe their orbits in celestial space.
Newton found this reason and was able to accurately express it in the form of one formula - the law of universal gravitation.
Since the force of universal gravitation imparts the same acceleration to all bodies regardless of their mass, it must be proportional to the mass of the body on which it acts:
“Gravity exists for all bodies in general and is proportional to the mass of each of them... all planets gravitate towards each other...” I. Newton
But since, for example, the Earth acts on the Moon with a force proportional to the mass of the Moon, then the Moon, according to Newton’s third law, must act on the Earth with the same force. Moreover, this force must be proportional to the mass of the Earth. If the force of gravity is truly universal, then from the side of a given body a force must act on any other body proportional to the mass of this other body. Consequently, the force of universal gravity must be proportional to the product of the masses of interacting bodies. From this follows the formulation of the law of universal gravitation.
Law of universal gravitation:
The force of mutual attraction between two bodies is directly proportional to the product of the masses of these bodies and inversely proportional to the square of the distance between them:
The proportionality factor G is called gravitational constant.
The gravitational constant is numerically equal to the force of attraction between two material points weighing 1 kg each, if the distance between them is 1 m. Indeed, with masses m 1 = m 2 = 1 kg and a distance r = 1 m, we obtain G = F (numerically).
It must be borne in mind that the law of universal gravitation (3.4) as universal law valid for material points. In this case, the forces of gravitational interaction are directed along the line connecting these points (Fig. 3.2, a).
It can be shown that homogeneous bodies shaped like a ball (even if they cannot be considered material points, Fig. 3.2, b) also interact with the force determined by formula (3.4). In this case, r is the distance between the centers of the balls. The forces of mutual attraction lie on a straight line passing through the centers of the balls. Such forces are called central. The bodies that we usually consider falling to Earth have dimensions much smaller than the Earth's radius (R ≈ 6400 km).
Such bodies can, regardless of their shape, be considered as material points and determine the force of their attraction to the Earth using the law (3.4), keeping in mind that r is the distance from a given body to the center of the Earth.
A stone thrown to the Earth will deviate under the influence of gravity from a straight path and, having described a curved trajectory, will finally fall to the Earth. If you throw it at a faster speed, it will fall further." I. Newton
Determination of the gravitational constant.
Now let's find out how to find the gravitational constant. First of all, note that G has a specific name. This is due to the fact that the units (and, accordingly, the names) of all quantities included in the law of universal gravitation have already been established earlier. The law of gravitation gives a new connection between known quantities with certain names of units. That is why the coefficient turns out to be a named quantity. Using the formula of the law of universal gravitation, it is easy to find the name of the unit of gravitational constant in SI: N m 2 / kg 2 = m 3 / (kg s 2).
To quantify G, it is necessary to independently determine all the quantities included in the law of universal gravitation: both masses, force and distance between bodies.
The difficulty is that the gravitational forces between bodies of small masses are extremely small. It is for this reason that we do not notice the attraction of our body to surrounding objects and the mutual attraction of objects to each other, although gravitational forces are the most universal of all forces in nature. Two people with masses of 60 kg at a distance of 1 m from each other are attracted with a force of only about 10 -9 N. Therefore, to measure the gravitational constant, fairly subtle experiments are needed.
The gravitational constant was first measured by the English physicist G. Cavendish in 1798 using an instrument called a torsion balance. The diagram of the torsion balance is shown in Figure 3.3. A light rocker with two identical weights at the ends is suspended from a thin elastic thread. Two heavy balls are fixed nearby. Gravitational forces act between the weights and the stationary balls. Under the influence of these forces, the rocker turns and twists the thread until the resulting elastic force becomes equal to the gravitational force. By the angle of twist you can determine the force of attraction. To do this, you only need to know the elastic properties of the thread. The masses of the bodies are known, and the distance between the centers of interacting bodies can be directly measured.
From these experiments the following value for the gravitational constant was obtained:
G = 6.67 10 -11 N m 2 / kg 2.
Only in the case when bodies of enormous mass interact (or at least the mass of one of the bodies is very large) does the gravitational force reach of great importance. For example, the Earth and the Moon are attracted to each other with a force F ≈ 2 10 20 N.
Dependence of the acceleration of free fall of bodies on geographic latitude.
One of the reasons for the increase in the acceleration of gravity when the point where the body is located moves from the equator to the poles is that the globe is somewhat flattened at the poles and the distance from the center of the Earth to its surface at the poles is less than at the equator. Another reason is the rotation of the Earth.
Equality of inertial and gravitational masses.
The most striking property of gravitational forces is that they impart the same acceleration to all bodies, regardless of their masses. What would you say about a football player whose kick would be equally accelerated by an ordinary leather ball and a two-pound weight? Everyone will say that this is impossible. But the Earth is just such an “extraordinary football player” with the only difference that its effect on bodies is not of the nature of a short-term blow, but continues continuously for billions of years.
In Newton's theory, mass is the source of the gravitational field. We are in the Earth's gravitational field. At the same time, we are also sources of the gravitational field, but due to the fact that our mass is significantly less than the mass of the Earth, our field is much weaker and surrounding objects do not react to it.
The extraordinary property of gravitational forces, as we have already said, is explained by the fact that these forces are proportional to the masses of both interacting bodies. The mass of a body, which is included in Newton’s second law, determines the inertial properties of the body, i.e. its ability to acquire a certain acceleration under the influence of a given force. This inert mass m and.
It would seem, what relation can it have to the ability of bodies to attract each other? The mass that determines the ability of bodies to attract each other is the gravitational mass m r.
It does not at all follow from Newtonian mechanics that the inertial and gravitational masses are the same, i.e. that
m and = m r . (3.5)
Equality (3.5) is a direct consequence of experiment. It means that we can simply talk about the mass of a body as a quantitative measure of both its inertial and gravitational properties.
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. Consider what happens when some weight is placed on 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 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).
Man has long known the force that makes all bodies fall to the Earth. But until the 17th century. It was believed that only the Earth has special property attract bodies located near its surface. In 1667, Newton suggested that in general forces of mutual attraction act between all bodies. He called these forces the forces of universal gravitation.
Newton discovered the laws of motion of bodies. According to these laws, motion with acceleration is possible only under the influence of force. Since falling bodies move with acceleration, they must be acted upon by a force directed downward toward the Earth.
Why don’t we notice the mutual attraction between the bodies around us? Maybe this is explained by the fact that the attractive forces between them are too small?
Newton was able to show that the force of attraction between bodies depends on the masses of both bodies and, as it turned out, reaches a noticeable value only when the interacting bodies (or at least one of them) have a sufficiently large mass.
The acceleration of gravity is distinguished by the curious feature that it is the same in a given place for all bodies, for bodies of any mass. At first glance, this is a very strange property. After all, from the formula expressing Newton’s second law,
it follows that the acceleration of a body should be greater, the smaller its mass. Bodies with low mass must fall with greater acceleration than bodies with large mass. Experience has shown (see § 20) that the accelerations of freely falling bodies do not depend on their masses. The only explanation that can be found for this amazing
fact, is that the very force with which the Earth attracts a body is proportional to its mass i.e.
Indeed, in this case, for example, doubling the mass will also double the force, but the acceleration, which is equal to the ratio, will remain unchanged. Newton made this only correct conclusion: the force of universal gravity is proportional to the mass of the body on which it acts. But bodies attract each other. And according to Newton’s third law, forces of equal absolute value act on both attracting bodies. This means that the force of mutual attraction must be proportional to the masses of each of the attracting bodies. Then both bodies will receive accelerations that do not depend on their masses.
If the force is proportional to the masses of each of the interacting bodies, then this means that it is proportional to the product of the masses of both bodies.
What else does the force of mutual attraction between two bodies depend on? Newton suggested that it should depend on the distance between the bodies. It is well known from experience that near the Earth the acceleration of free fall is equal and it is the same for bodies falling from a height of 1, 10 or 100 m. But from this we cannot yet conclude that the acceleration does not depend on the distance to the Earth. Newton believed that distances should be counted not from the surface of the Earth, but from its center. But the radius of the Earth is 6400 km. It is clear, therefore, that several tens or hundreds of meters above the Earth’s surface cannot noticeably change the acceleration of gravity.
To find out how the distance between bodies affects the force of their mutual attraction, you need to know with what acceleration bodies move at great distances from the Earth’s surface.
It is clear that it is difficult to measure the vertical acceleration of free fall of bodies located at an altitude of several thousand kilometers above the Earth's surface. It is more convenient to measure the centripetal acceleration of a body moving around the Earth in a circle under the influence of the force of gravity towards the Earth. Let us remember that we used the same technique when studying the elastic force. We measured the centripetal acceleration of a cylinder moving in a circle under the influence of this force.
In studying the force of universal gravity, nature itself came to the aid of physicists and made it possible to determine the acceleration of a body moving in a circle around the Earth. Such a body is the Earth's natural satellite - the Moon. After all, if Newton’s assumption is correct, then we must assume that the centripetal acceleration of the Moon as it moves in a circle around the Earth is imparted by the force of its attraction to the Earth. If the force of gravity between the Moon and the Earth did not depend on the distance between them, then the centripetal acceleration of the Moon would be the same as the acceleration
free fall of bodies near the Earth's surface. In fact, the centripetal acceleration with which the Moon moves in its orbit is equal, as we already know (see Exercise 16, Problem 9), . And this is approximately 3600 times less than the acceleration of falling bodies near the Earth. At the same time, it is known that the distance from the center of the Earth to the center of the Moon is 384,000 km. This is 60 times the radius of the Earth, i.e. the distance from the center of the Earth to its surface. Thus, an increase in the distance between attracting bodies by 60 times leads to a decrease in acceleration by 602 times. From this we can conclude that the acceleration imparted to bodies by the force of universal gravity, and therefore this force itself, is inversely proportional to the square of the distance between the interacting bodies.
Newton came to this conclusion.
Therefore, we can write that two mass bodies are attracted to each other with a force, the absolute value of which is expressed by the formula
where is the distance between bodies, y is the coefficient of proportionality, the same for all bodies in nature. This coefficient of universal gravitation is called the gravitational constant.
The above formula expresses the law of universal gravitation discovered by Newton:
All bodies 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.
Under the influence of universal gravity, both planets move around the Sun and artificial satellites around the Earth.
But what should be understood by the distance between interacting bodies? Let's take two bodies of arbitrary shape (Fig. 109). The question immediately arises: what distance should be substituted into the formula for the law of universal gravitation? Distance between
the farthest points of the surface of both bodies or, conversely, the distance between the nearest points? Or maybe the distance between some other points of the body?
It turns out that formula (1), expressing the law of universal gravitation, is valid when the distance between bodies is so large compared to their sizes that the bodies can be considered material points. When calculating the gravitational force between them, the Earth and the Moon, the planets and the Sun can be considered material points.
If the bodies have the shape of balls, then even if their sizes are comparable to the distance between them, they attract each other as material points located at the centers of the balls (Fig. 110). In this case, this is the distance between the centers of the balls.
Formula (1) can also be used when calculating the force of attraction between a ball of large radius and a body of arbitrary shape small sizes located close to the surface of the ball (Fig. 111). Then the dimensions of the body can be neglected in comparison with the radius of the ball. This is exactly what we do when we consider the attraction of various bodies to the globe.
The force of gravity is another example of a force that depends on the position (coordinates) of the body on which this force acts, relative to the body that has the effect. After all, the force of gravity depends on the distance between bodies.
In this section we will talk about Newton's amazing guess, which led to the discovery of the law of universal gravitation.
Why does a stone released from your hands fall to Earth? Because he is attracted by the Earth, each of you will say. In fact, the stone falls to the Earth with the acceleration of gravity. Consequently, a force directed towards the Earth acts on the stone from the Earth. According to Newton's third law, the stone acts on the Earth with the same magnitude force directed towards the stone. In other words, forces of mutual attraction act between the Earth and the stone.
Newton's conjecture
Newton was the first to first guess and then strictly prove that the reason that causes a stone to fall to the Earth, the movement of the Moon around the Earth and the planets around the Sun is the same. This is the force of gravity acting between any bodies in the Universe. Here is the course of his reasoning, given in Newton’s main work, “The Mathematical Principles of Natural Philosophy”: “A stone thrown horizontally will deflect
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1
/ /
U
Rice. 3.2
under the influence of gravity from a straight path and, having described a curved trajectory, will finally fall to the Earth. If you throw it at a higher speed, ! then it will fall further” (Fig. 3.2). Continuing these reasonings, Newton came to the conclusion that if it were not for air resistance, then the trajectory of a stone thrown from a high mountain at a certain speed could become such that it would never reach the surface of the Earth at all, but would move around it “just as the planets describe their orbits in celestial space.”
Now we have become so familiar with the movement of satellites around the Earth that there is no need to explain Newton’s thought in more detail.
So, according to Newton, the movement of the Moon around the Earth or the planets around the Sun is also a free fall, but only a fall that lasts, without stopping, for billions of years. The reason for such a “fall” (whether we are really talking about the fall of an ordinary stone to the Earth or the movement of planets in their orbits) is the force of universal gravity. What does this force depend on?
Dependence of gravitational force on the mass of bodies
§ 1.23 talked about the free fall of bodies. Galileo's experiments were mentioned, which proved that the Earth imparts the same acceleration to all bodies in a given place, regardless of their mass. This is possible only if the force of gravity towards the Earth is directly proportional to the mass of the body. It is in this case that the acceleration of gravity, equal to the ratio of the force of gravity to the mass of the body, is a constant value.
Indeed, in this case, increasing the mass m, for example, by doubling will lead to an increase in the modulus of force F, also doubling, and accelerating
F
ratio, which is equal to the ratio -, will remain unchanged.
Generalizing this conclusion for gravitational forces between any bodies, we conclude that the force of universal gravity is directly proportional to the mass of the body on which this force acts. But at least two bodies are involved in mutual attraction. Each of them, according to Newton’s third law, is acted upon by gravitational forces of equal magnitude. Therefore, each of these forces must be proportional to both the mass of one body and the mass of the other body.
Therefore, the force of universal gravity between two bodies is directly proportional to the product of their masses:
F - here2. (3.2.1)
What else does the gravitational force acting on a given body from another body depend on?
Dependence of gravitational force on the distance between bodies
It can be assumed that the force of gravity should depend on the distance between the bodies. To check the correctness of this assumption and find the dependence of the gravitational force on the distance between bodies, Newton turned to the movement of the Earth's satellite, the Moon. Its movement was studied much more accurately in those days than the movement of the planets.
The rotation of the Moon around the Earth occurs under the influence of the gravitational force between them. Approximately, the orbit of the Moon can be considered a circle. Consequently, the Earth imparts centripetal acceleration to the Moon. It is calculated by the formula
l 2
a = - Tg
where B is the radius of the lunar orbit, equal to approximately 60 radii of the Earth, T = 27 days 7 hours 43 minutes = 2.4 106 s is the period of revolution of the Moon around the Earth. Considering that the radius of the Earth R3 = 6.4 106 m, we obtain that the centripetal acceleration of the Moon is equal to:
2 6 4k 60 ¦ 6.4 ¦ 10
M „ „„„. , O
a = 2 ~ 0.0027 m/s*.
(2.4 ¦ 106 s)
The found acceleration value is less than the acceleration of free fall of bodies at the Earth's surface (9.8 m/s2) by approximately 3600 = 602 times.
Thus, an increase in the distance between the body and the Earth by 60 times led to a decrease in the acceleration imparted by gravity, and, consequently, the force of gravity itself by 602 times.
An important conclusion follows from this: the acceleration imparted to bodies by the force of gravity towards the Earth decreases in inverse proportion to the square of the distance to the center of the Earth:
ci
a = -k, (3.2.2)
R
where Cj is a constant coefficient, the same for all bodies.
Kepler's laws
A study of the movement of planets showed that this movement is caused by the force of gravity towards the Sun. Using careful long-term observations by Danish astronomer Tycho Brahe, the German scientist Johann Kepler in early XVII V. established the kinematic laws of planetary motion - the so-called Kepler's laws.
Kepler's first law
All planets move in ellipses, with the Sun at one of the focuses.
An ellipse (Fig. 3.3) is a flat closed curve, the sum of the distances from any point of which to two fixed points, called foci, is constant. This sum of distances is equal to the length of the major axis AB of the ellipse, i.e.
FgP + F2P = 2b,
where Fl and F2 are the foci of the ellipse, and b = ^^ is its semimajor axis; O is the center of the ellipse. The point of the orbit closest to the Sun is called perihelion, and the point farthest from it is called p
IN
Rice. 3.4
"2
B A A aphelion. If the Sun is at focus Fr (see Fig. 3.3), then point A is perihelion, and point B is aphelion.
Kepler's second law
The radius vector of the planet describes equal areas in equal periods of time. So, if the shaded sectors (Fig. 3.4) have the same areas, then the paths si> s2> s3 will be traversed by the planet in equal periods of time. It is clear from the figure that Sj > s2. Hence, linear speed The motion of the planet at different points of its orbit is not the same. At perihelion the planet's speed is greatest, at aphelion it is least.
Kepler's third law
The squares of the periods of revolution of the planets around the Sun are related to the cubes of the semimajor axes of their orbits. Having designated the semimajor axis of the orbit and the period of revolution of one of the planets by bx and Tv and the other by b2 and T2, Kepler’s third law can be written as follows:
From this formula it is clear that the further a planet is from the Sun, the longer its period of revolution around the Sun.
Based on Kepler's laws, certain conclusions can be drawn about the accelerations imparted to the planets by the Sun. For simplicity, we will consider the orbits not elliptical, but circular. For planets solar system this replacement is not too rough an approximation.
Then the force of attraction from the Sun in this approximation should be directed for all planets towards the center of the Sun.
If we denote by T the periods of revolution of the planets, and by R the radii of their orbits, then, according to Kepler’s third law, for two planets we can write
t\ L? T2 R2
Normal acceleration when moving in a circle is a = co2R. Therefore, the ratio of the accelerations of the planets
Q-i GD.
7G=-2~- (3-2-5)
2 t:r0
Using equation (3.2.4), we obtain
T2
Since Kepler's third law is valid for all planets, the acceleration of each planet is inversely proportional to the square of its distance from the Sun:
Oh oh
a = -|. (3.2.6)
VT
The constant C2 is the same for all planets, but does not coincide with the constant C2 in the formula for the acceleration imparted to bodies by the globe.
Expressions (3.2.2) and (3.2.6) show that the force of gravity in both cases (attraction to the Earth and attraction to the Sun) imparts to all bodies an acceleration that does not depend on their mass and decreases in inverse proportion to the square of the distance between them:
F~a~-2. (3.2.7)
R
Law of Gravity
The existence of dependencies (3.2.1) and (3.2.7) means that the force of universal gravity is 12
TP.L Sh
F~
R2? TTT-i TPP
F = G
In 1667, Newton finally formulated the law of universal gravitation:
(3.2.8) R
The force of mutual attraction between two bodies is directly proportional to the product of the masses of these bodies and inversely proportional to the square of the distance between them. The proportionality coefficient G is called the gravitational constant.
Interaction of point and extended bodies
The law of universal gravitation (3.2.8) is valid only for bodies whose dimensions are negligible compared to the distance between them. In other words, it is valid only for material points. In this case, the forces of gravitational interaction are directed along the line connecting these points (Fig. 3.5). This kind of force is called central.
To find the gravitational force acting on a given body from another, in the case when the sizes of the bodies cannot be neglected, proceed as follows. Both bodies are mentally divided into elements so small that each of them can be considered a point. By adding up the gravitational forces acting on each element of a given body from all elements of another body, we obtain the force acting on this element (Fig. 3.6). Having performed such an operation for each element of a given body and adding up the resulting forces, the total gravitational force acting on this body is found. This task is difficult.
There is, however, one practically important case when formula (3.2.8) is applicable to extended bodies. Can you prove
m^
Fi Fig. 3.5 Fig. 3.6
It should be noted that spherical bodies, the density of which depends only on the distances to their centers, when the distances between them are greater than the sum of their radii, are attracted with forces whose moduli are determined by formula (3.2.8). In this case, R is the distance between the centers of the balls.
And finally, since the sizes of bodies falling to the Earth are many smaller sizes Earth, then these bodies can be considered as point bodies. Then R in formula (3.2.8) should be understood as the distance from the given body to the center of the Earth.
Between all bodies there are forces of mutual attraction, depending on the bodies themselves (their masses) and on the distance between them.
? 1. The distance from Mars to the Sun is 52% greater than the distance from Earth to the Sun. How long is a year on Mars? 2. How will the force of attraction between the balls change if the aluminum balls (Fig. 3.7) are replaced with steel balls of the same mass? "same volume?
