The attraction of bodies to each other. All bodies are attracted to each other. But why? Description of the law of gravity
Questions.
1. What was called universal gravitation?
Universal gravitation was called the mutual attraction of all bodies in the universe.
2. What is another name for the forces of universal gravitation?
The forces of universal gravitation are otherwise called gravitational (from the Latin gravitas - "gravity").
3. Who and in what century discovered the law of universal gravitation?
The law of universal gravitation was discovered by Isaac Newton in the 17th century.
4. How is the law of universal gravitation read?
Any two bodies are attracted to each other with a force that is directly proportional to the product of their masses, and inversely proportional to the square of the distance between them.
5. Write down the formula expressing the law of universal gravitation.
6. In what cases should this formula be used to calculate gravitational forces?
The formula can be used to calculate gravitational forces if the bodies can be taken as material points: 1) if the dimensions of the bodies are much smaller than the distances between them; 2) if two bodies are spherical and homogeneous; 3) if one body, spherical in shape, is many times larger in mass and size than the second.
7. Is the Earth attracted to an apple hanging on a branch?
In accordance with the law of universal gravitation, an apple attracts the Earth with the same force as the Earth - an apple, only in the opposite direction.
Exercises.
1. Give examples of the manifestation of gravity.
The fall of bodies to the ground under the action of gravity, the attraction of celestial bodies (Earth, Moon, sun, planets, comets, meteorites) to each other.
2. The space station flies from the Earth to the Moon. How does the modulus of the vector of the force of its attraction to the Earth change in this case? to the moon? Is the station attracted to the Earth and the Moon with the same or different in modulus forces when it is in the middle between them? Justify all three answers. (It is known that the mass of the Earth is about 81 times the mass of the Moon).
.jpg)
3. It is known that the mass of the Sun is 330,000 times the mass of the Earth. Is it true that the Sun pulls the Earth 330,000 times more strongly than the Earth pulls the Sun? Explain the answer.
No, the bodies attract each other with the same forces, because the force of attraction is proportional to the product of their masses.
4. The ball thrown by the boy moved up for some time. At the same time, its speed decreased all the time until it became equal to zero. Then the ball began to fall down, with increasing speed. Explain: a) whether the force of attraction to the Earth acted on the ball during its upward movement; down; b) what caused the decrease in the speed of the ball when it moves up; increasing its speed when moving down; c) why, when the ball moves up, its speed decreases, and when it moves down, it increases.
a) yes, the force of attraction acted all the way; b) the universal force of gravity (gravity of the Earth); c) when moving up, the speed and acceleration of the body are in different directions, and when moving down, they are co-directed.
5. Is a person standing on Earth attracted to the Moon? If yes, then to what is it attracted more: to the Moon or to the Earth? Is the moon attracted to this person? Justify answers.
Yes, all bodies are attracted to each other, but the force of attraction of a person to the Moon is much less than to the Earth, because. The moon is much further away.
DEFINITION
The law of universal gravitation was discovered by I. Newton:
Two bodies are attracted to each other with , which is directly proportional to their product and inversely proportional to the square of the distance between them:
Description of the law of gravity
The coefficient is the gravitational constant. In the SI system, the gravitational constant has the value:
This constant, as can be seen, is very small, so the gravitational forces between bodies with small masses are also small and practically not felt. However, the motion of cosmic bodies is completely determined by gravity. The presence of universal gravitation or, in other words, gravitational interaction explains what the Earth and planets “hold” on, and why they move around the Sun along certain trajectories, and do not fly away from it. The law of universal gravitation allows us to determine many characteristics of celestial bodies - the masses of planets, stars, galaxies and even black holes. This law allows us to calculate the orbits of the planets with great accuracy and create a mathematical model of the Universe.
With the help of the law of universal gravitation, it is also possible to calculate cosmic velocities. For example, the minimum speed at which a body moving horizontally above the Earth's surface will not fall on it, but will move in a circular orbit is 7.9 km / s (the first space velocity). In order to leave the Earth, i.e. to overcome its gravitational attraction, the body must have a speed of 11.2 km / s, (the second cosmic velocity).
Gravity is one of the most amazing natural phenomena. In the absence of gravitational forces, the existence of the Universe would be impossible, the Universe could not even arise. Gravity is responsible for many processes in the Universe - its birth, the existence of order instead of chaos. The nature of gravity is still not fully understood. To date, no one has been able to develop a worthy mechanism and model of gravitational interaction.
The force of gravity
A special case of the manifestation of gravitational forces is gravity.
Gravity is always directed vertically downward (towards the center of the Earth).
If the force of gravity acts on the body, then the body performs. The type of movement depends on the direction and module of the initial speed.
We deal with the force of gravity every day. , after a while it is on the ground. The book, released from the hands, falls down. Having jumped, a person does not fly away into outer space, but falls down to the ground.
Considering the free fall of a body near the Earth's surface as a result of the gravitational interaction of this body with the Earth, we can write:
whence the free fall acceleration:
The free fall acceleration does not depend on the mass of the body, but depends on the height of the body above the Earth. The globe is slightly flattened at the poles, so bodies near the poles are slightly closer to the center of the earth. In this regard, the acceleration of free fall depends on the latitude of the area: at the pole it is slightly greater than at the equator and other latitudes (at the equator m / s, at the North Pole equator m / s.
The same formula allows you to find the free fall acceleration on the surface of any planet with mass and radius .
Examples of problem solving
EXAMPLE 1 (the problem of "weighing" the Earth)
| The task | The radius of the Earth is km, the acceleration of free fall on the surface of the planet is m/s. Using these data, estimate the approximate mass of the Earth. |
| Solution | Acceleration of free fall at the surface of the Earth: whence the mass of the Earth: In the C system, the radius of the Earth Substituting the numerical values of physical quantities into the formula, we estimate the mass of the Earth:
|
| Answer | Mass of the Earth kg. |
m.
EXAMPLE 2
| The task | An Earth satellite moves in a circular orbit at an altitude of 1000 km from the Earth's surface. How fast is the satellite moving? How long does it take for a satellite to make one complete revolution around the earth? |
| Solution | According to , the force acting on the satellite from the side of the Earth is equal to the product of the mass of the satellite and the acceleration with which it moves:
From the side of the earth, the force of gravitational attraction acts on the satellite, which, according to the law of universal gravitation, is equal to: where and are the masses of the satellite and the Earth, respectively. Since the satellite is at a certain height above the surface of the Earth, the distance from it to the center of the Earth: where is the radius of the earth. |
where G=6.67×10 -11 N×m 2 /kg 2 is the universal gravitational constant.
This law is called the law of universal gravitation.
The force with which bodies are attracted to the Earth is called gravity. The main feature of gravity is the experimental fact that this force all bodies, regardless of their mass, reports the same acceleration directed to the center of the Earth.
It follows that the ancient Greek philosopher Aristotle was wrong when he claimed that heavy bodies fall to Earth faster than light ones. He did not take into account that in addition to the force of gravity, the body is affected by the force of resistance against the air, which depends on the shape of the body.
A musket bullet and a heavy cannonball, thrown by the Italian physicist Galileo Galilei from the famous 54.5 m high tower located in the city of Pisa, reached the Earth's surface almost simultaneously, i.e. fell with the same acceleration (Fig. 4.27).
Calculations carried out by G. Galileo showed that the acceleration acquired by bodies under the influence of the Earth's gravity is 9.8 m/s 2 .
Further more accurate experiments were carried out by I. Newton. He took a long glass tube, into which he placed a lead ball, cork and feather (Fig. 4.28).
This tube is now called "Newton's tube". Turning the tube over, he saw that the ball fell first, then the cork, and only then the feather. If, however, the air is first evacuated from the tube using a pump, then after turning the tube over all the bodies will fall to the bottom of the tube at the same time. And this means that in the second case, all bodies increased their speed in the same way, i.e. get the same acceleration. And this acceleration was imparted to them by a single force - the force of attraction of bodies to the Earth, i.e. the force of gravity. The calculations made by Newton confirmed the correctness of G. Galileo's calculations, since he also obtained the value of the acceleration acquired by freely falling bodies in the "Newton's tube", equal to 9.8 m / s 2. This constant acceleration is called free fall acceleration on Earth and is denoted by the letter g(from the Latin word "gravitas" - heaviness), i.e. g \u003d 9.8 m / s 2.
Free fall is understood as the movement of a body that occurs under the influence of a single force - gravity (drag forces against air are not taken into account).
On other planets or stars, the value of this acceleration is different, as it depends on the masses and radii of the planets and stars.
Here are the values of the free fall acceleration on some planets of the solar system and on the moon:
1. Sun g = 274 N/kg
2. Venus g \u003d 8.69 N / kg
3. Mars g = 3.86 N/kg
4. Jupiter g = 23 N/kg
5. Saturn g = 9.44 N/kg
6. Moon (Earth satellite) g = 1.623 N/kg
How can one explain the fact that the acceleration of all bodies freely falling to the Earth is the same? After all, the greater the mass of the body, the greater the force of gravity acting on it. You and I know that 1 N is a force that imparts an acceleration equal to 1 m / s 2 to a body with a mass of 1 kg. At the same time, the experiments of G. Galileo and I. Newton showed that gravity changes the speed of any body 9.8 times more. Consequently, a force of 9.8 N acts on a body with a mass of 1 kg, and a force of gravity equal to 19.6 N will act on a body with a mass of 2 kg, etc. That is, the greater the mass of the body, the greater the force of gravity will act on it, and the proportionality coefficient will be a value equal to 9.8 N / kg. Then the formula for calculating the force of gravity will look like or in general:
Accurate measurements have shown that the acceleration of free fall decreases with height and changes slightly with changes in latitude due to the fact that the Earth is not a strictly spherical body (it is slightly flattened at the poles). In addition, it may depend on the geographical location on the planet, since the density of the rocks that make up the surface layer of the Earth is different. The latter fact makes it possible to detect mineral deposits.
Here are some values of the free fall acceleration on Earth:
1. At the North Pole g = 9.832 N/kg
2. At the equator g = 9.780 N/kg
3. At latitude 45 about g \u003d 9.806 N / kg
4. At sea level g = 9.8066 N/kg
5. At the Khan-Tengri peak, 7 km high, g = 9.78 N/kg
6. At a depth of 12 km g = 9.82 N/kg
7. At a depth of 3000 km g = 10.20 N/kg
8. At a depth of 4500 km g = 6.9 N/kg
9. At the center of the Earth g = 0 N/kg
The attraction of the Moon leads to the formation of ebbs and flows in the seas and oceans on Earth. The tide in the open ocean is about 1 m, and off the coast of the Bay of Fundy in the Atlantic Ocean it reaches 18 meters.
The distance from the Earth to the Moon is enormous: about 384,000 km. But the gravitational force between the Earth and the Moon is large and amounts to 2 × 10 20 N. This is due to the fact that the masses of the Earth and the Moon are large.
When solving problems, if there are no special reservations, the value of 9.8 N/kg can be rounded up to 10 N/kg.
The lag of clock pendulums synchronized on the first floor of a high-rise building is associated with a change in the value g. Since the value g decreases as the height increases, then the clock on the top floor will begin to lag behind.
Example. Determine the force with which a steel bucket weighing 500 g, with a volume of 12 liters, completely filled with water, presses on the support.
The force of gravity is equal to the sum of the force of gravity of the bucket itself, equal to F heavy1 = m 1 g, and the gravity of the water poured into the bucket, equal to F heavy1 = m 2 g= ρ2 V 2 g, i.e.
F strand = m 1 g + p2 V 2 g
Substituting numerical values, we get:
F strand \u003d 0.5 kg 10N / kg + 10 3 kg / m 3 12 10 -3 m 3 10N / kg = \u003d 125 N.
Answer: F strand = 125 N
Questions for self-control:
1. What force is called gravitational? What is the reason for this power?
2. What does the law of universal gravitation say?
3. What force is called gravity? What is its main feature?
4. Does gravity exist on other planets? Justify the answer.
5. For what purpose did G. Galileo conduct experiments on the Leaning Tower of Pisa?
6. What do the experiments that Newton conducted with the "Newton's tube" prove to us?
7. What acceleration is called free fall acceleration?
8. You have two identical sheets of paper. Why does a crumpled sheet fall to the ground faster, despite the fact that each sheet has the same force of gravity?
9. What is the fundamental difference in the explanation of free fall by Aristotle and Newton?
10. Give a presentation on how Aristotle, Galileo and Newton studied free fall.
Sir Isaac Newton, having been hit on the head with an apple, deduced the law of universal gravitation, which reads:
Any two bodies are attracted to each other with a force directly proportional to the product of the masses of the body and inversely proportional to the square of the distance between them:
F = (Gm 1 m 2)/R 2 , where
m1, m2- masses of bodies
R- distance between the centers of bodies
G \u003d 6.67 10 -11 Nm 2 / kg- constant
Let us determine the acceleration of free fall on the surface of the Earth:
F g = m body g = (Gm body m Earth)/R 2
R (radius of the Earth) = 6.38 10 6 m
m Earth = 5.97 10 24 kg
m body g = (Gm body m Earth)/R 2 or g \u003d (Gm Earth) / R 2
Note that the acceleration due to gravity does not depend on the mass of the body!
g \u003d 6.67 10 -11 5.97 10 24 / (6.38 10 6) \u003d 398.2 / 40.7 \u003d 9.8 m / s 2
We said earlier that the force of gravity (gravitational attraction) is called weighing.
On the surface of the Earth, weight and mass of a body have the same meaning. But as you move away from the Earth, the weight of the body will decrease (since the distance between the center of the Earth and the body will increase), and the mass will remain constant (since mass is an expression of the inertia of the body). Mass is measured in kilograms, weight - in newtons.
Thanks to the force of gravity, celestial bodies rotate relative to each other: the Moon around the Earth; Earth around the Sun; The Sun around the center of our Galaxy, etc. In this case, the bodies are held by centrifugal force, which is provided by the force of gravity.
The same applies to artificial bodies (satellites) revolving around the Earth. The circle along which the satellite revolves is called the orbit of rotation.
In this case, the centrifugal force acts on the satellite:
F c \u003d (m satellite V 2) / R
Gravity force:
F g \u003d (Gm satellite m of the Earth) / R 2
F c \u003d F g \u003d (m satellite V 2) / R \u003d (Gm satellite m Earth) / R 2
V2 = (Gm Earth)/R; V = √(Gm Earth)/R
Using this formula, you can calculate the speed of any body rotating in an orbit with a radius R around the Earth.
The natural satellite of the Earth is the Moon. Let us determine its linear velocity in orbit:
Mass of the Earth = 5.97 10 24 kg
R is the distance between the center of the earth and the center of the moon. To determine this distance, we need to add three quantities: the radius of the Earth; the radius of the moon; distance from the earth to the moon.
R moon = 1738 km = 1.74 10 6 m
R earth \u003d 6371 km \u003d 6.37 10 6 m
R zl \u003d 384400 km \u003d 384.4 10 6 m
The total distance between the centers of the planets: R = 392.5 10 6 m
Linear speed of the moon:
V \u003d √ (Gm of the Earth) / R \u003d √6.67 10 -11 5.98 10 24 / 392.5 10 6 \u003d 1000 m / s \u003d 3600 km / h
The moon moves in a circular orbit around the earth with a linear velocity of 3600 km/h!
Let us now determine the period of revolution of the Moon around the Earth. During the period of revolution, the Moon overcomes a distance equal to the length of the orbit - 2πR. Orbital speed of the moon: V = 2πR/T; on the other hand: V = √(Gm Earth)/R:
2πR/T = √(Gm Earth)/R hence T = 2π√R 3 /Gm Earth
T \u003d 6.28 √ (60.7 10 24) / 6.67 10 -11 5.98 10 24 \u003d 3.9 10 5 s
The period of revolution of the Moon around the Earth is 2,449,200 seconds, or 40,820 minutes, or 680 hours, or 28.3 days.
1. Vertical rotation
Earlier in circuses there was a very popular trick in which a cyclist (motorcyclist) made a full turn inside a circle located vertically.
What is the minimum speed the trickster must have in order not to fall down at the top point?
To pass the top point without falling, the body must have a speed that creates such a centrifugal force that would compensate for the force of gravity.
Centrifugal force: F c \u003d mV 2 / R
The force of gravity: F g = mg
F c \u003d F g; mV 2 /R = mg; V = √Rg
And again, note that there is no body mass in the calculations! It should be noted that this is the speed that the body should have at the top!
Let's say that a circle with a radius of 10 meters is set in the circus arena. Let's calculate the safe speed for the trick:
V = √Rg = √10 9.8 = 10 m/s = 36 km/h
This law, called the law of universal gravitation, is written in mathematical form as follows:
where m 1 and m 2 are the masses of the bodies, R is the distance between them (see Fig. 11a), and G is the gravitational constant equal to 6.67.10-11 N.m 2 /kg2.
The law of universal gravitation was first formulated by I. Newton when he tried to explain one of I. Kepler's laws, which states that for all planets the ratio of the cube of their distance R to the Sun to the square of the period T of revolution around it is the same, i.e.
Let us derive the law of universal gravitation as Newton did, assuming that the planets move in circles. Then, according to Newton's second law, a planet with a mass mPl moving along a circle of radius R with a speed v and a centripetal acceleration v2/R must be acted upon by a force F directed towards the Sun (see Fig. 11b) and equal to:
The speed v of the planet can be expressed in terms of the radius R of the orbit and the period of revolution T:
Substituting (11.4) into (11.3) we obtain the following expression for F:
It follows from Kepler's law (11.2) that T2 = const.R3 . Therefore, (11.5) can be transformed into:
Thus, the Sun attracts the planet with a force directly proportional to the mass of the planet and inversely proportional to the square of the distance between them. Formula (11.6) is very similar to (11.1), only the mass of the Sun is missing in the numerator of the fraction on the right. However, if the force of attraction between the Sun and the planet depends on the mass of the planet, then this force must also depend on the mass of the Sun, which means that the constant on the right side of (11.6) contains the mass of the Sun as one of the factors. Therefore, Newton put forward his famous assumption that the gravitational force should depend on the product of the masses of the bodies and the law became the way we wrote it down in (11.1).
The law of universal gravitation and Newton's third law do not contradict each other. According to formula (11.1), the force with which body 1 attracts body 2 is equal to the force with which body 2 attracts body 1.
For bodies of ordinary size, gravitational forces are very small. So, two adjacent cars are attracted to each other with a force equal to the weight of a raindrop. Since G. Cavendish in 1798 determined the value of the gravitational constant, formula (11.1) has helped to make a lot of discoveries in the "world of huge masses and distances." For example, knowing the magnitude of the free fall acceleration (g=9.8 m/s2) and the radius of the Earth (R=6.4.106 m), we can calculate its mass m3 as follows. Each body with mass m1 near the surface of the Earth (i.e. at a distance R from its center) is affected by the gravitational force of its attraction equal to m1g, substitution of which in (11.1) instead of F gives:
whence we obtain that m З = 6.1024 kg.
Review questions:
· Formulate the law of universal gravitation?
· What is the gravitational constant?
Rice. 11. (a) - to the formulation of the law of universal gravitation; (b) - to the derivation of the law of universal gravitation from Kepler's law.
§ 12. GRAVITY FORCE. WEIGHT. WEIGHTLESSNESS. FIRST SPACE VELOCITY.