Forces of universal gravitation. Law of Gravity
You are already familiar with the phenomenon of universal gravitation from the ninth grade course. We also talked about the fact that all bodies with mass attract each other. From Newton's second law it follows that any resultant force is equal to the product of the mass of the body and the acceleration imparted by this force: .
So, for example, the force of gravity is equal to the product of the mass of the body and the acceleration of gravity:
But, as we have already found out, the acceleration of gravity does not depend on the mass of the body, from which we can conclude that gravitational forces impart acceleration that does not depend on the mass of the body!
This amazing property can only be explained by the fact that gravitational forces are proportional to the mass of the body on which they act.
And now, let's remember Newton's third law:
bodies act on each other with forces equal in magnitude, directed along one straight line in opposite directions
: .
In particular, if the Earth acts on the Moon with a certain force, then the Moon must also act on the Earth with this force. This means that the gravitational force arising between two bodies is proportional to the masses of both bodies
.
Let us now consider the acceleration with which the Moon moves. Let us recall that any curvilinear motion is accelerated. The movement of the Moon around the Earth is well known to people: the period of revolution of the Moon around the Earth is approximately 27.3 Earth days, and the average radius of the Moon’s orbit is 384 thousand kilometers. Based on this, we can calculate the centripetal acceleration of the Moon:
If we now compare the resulting value with the acceleration of free fall on Earth, we will see that the acceleration of the Moon is approximately 3600 times less than the acceleration of free fall on Earth:
Now let’s compare the radius of the Earth with the distance between the Earth and the Moon:
It turns out that the radius of the Earth is about 60 times smaller than the distance between the Earth and the Moon. Note that 602 is 3600. From this we can conclude that the gravitational force between two bodies decreases in proportion to the square of the distance between these bodies
.
Based on all of the above, the formulation of the law of universal gravitation is as follows: the force of mutual attraction between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between them
.
It should be noted that if the bodies are not material points, then the distance between them is taken to be the distance between the centers of gravity of these bodies. In the formula we see a coefficient of proportionality, which is called the gravitational constant
.
The gravitational constant is numerically equal to the force of attraction between two material points of mass 1 kg, if the distance between them is 1 meter.
The gravitational constant is a very important constant, since it was with its help that people were able to calculate the mass of the Earth, Moon, Sun, and so on. But how to calculate the gravitational constant itself?
The gravitational constant was first measured by Henry Cavendish in 1798. Using torsion balances, he was able to determine the value of the gravitational constant quite accurately (it almost coincides with the value accepted today).
The torsion balance consists of the following setup: a light rocker arm with two balls at the ends is suspended on a thin elastic thread. Two much heavier balls are fixed nearby (in Cavendish’s experiment, the light balls had a mass of 775 g, and the heavy ones – 49.5 kg). As a result of gravitational interaction, the rocker turned and twisted the thread. Knowing the elastic properties of the thread, Cavendish was able to measure the force of attraction. Since the masses of the balls were known to him, as well as the distance between them, Cavendish was able to calculate the gravitational constant.
It should be noted that the law of universal gravitation gives an accurate result in three cases:
1)
If both bodies have the shape of a ball and are homogeneous.
2)
If the sizes of the bodies are negligible compared to the distance between them.
3)
If one of the bodies has the shape of a ball and its dimensions are many times larger than the dimensions of the second body of any shape.
Examples of problem solving.
Task 1.
Jupiter has a mass and radius of 69,911 km. Determine the acceleration of gravity on Jupiter.
Since the law of universal gravitation is one of the universal laws of nature, it is equally true for all bodies with mass. Let's consider an arbitrary body (the mass and dimensions of which are negligible compared to the mass and dimensions of Jupiter).
Task 2.
Calculate at what height above the Earth's surface the geostationary satellite should be located.
At first glance, it may seem that the problem has no initial data. But if we look closely at the situation, we will be convinced that we have some data. First of all, this is the period of revolution of the satellite: of course, it must be equal to the period of revolution of the Earth around its axis. And most importantly: the satellite must move at a constant speed, since the Earth also moves at a constant speed.
It should be noted that in the problems we took the orbits of the planets as circles to simplify the calculations. In reality, the planets of the Solar System move in elliptical orbits, so when we talk about the distance from a planet to the Sun, we mean the average radius of its orbit.
Lesson "The Law of Universal Gravitation"
Lesson "The Law of Universal Gravitation"
Target:
contribute to the formation of ideas about the law of universal gravitation as a fundamental law of nature.
Tasks:
Educational:
form the concept of gravitational forces;
show the universal nature of the law of universal gravitation
limits of applicability of the law
introduce the experimental determination of the gravitational constant;
Developmental:
develop speech and thinking;
improve mental activity: conduct analysis, synthesis; put forward a hypothesis, observe, highlight significant features, compare, draw conclusions, check results;
Educational:
form a system of views on the world;
cultivate interest in creative and research work.
Equipment:
projection equipment, presentation “The Law of Universal Gravitation”, handouts.
Lesson type:
lesson on learning new material
During the classes
1. Organizational stage.
Greetings
2.
Repetition. Checking homework.
Let's start with what we already know. Let's remember and answer the following questions:
What is free fall of a body?
What is the acceleration of gravity?
Why does a piece of cotton wool fall in the air with less acceleration than an iron ball?
Who was the first to come to the conclusion that free fall is a uniformly accelerated motion?
Does gravity act on a body thrown upward during its ascent?
With what acceleration does a body thrown upward move in the absence of air resistance?
Newton's first law.
Newton's second law.
The stage of preparing students for active conscious assimilation of knowledge.
Statement of educational problem.
Teacher's word: In front of you on the slide are photographs depicting various objects: a falling apple, the movement of planets around the Sun, the Earth and the Moon, a man jumping.
Let's find out what can unite the depicted objects into a single whole. (approximate answer: bodies are attracted to each other, gravitational forces act)
Teacher's word: Now let's answer the question, what unites all these subjects into a single whole? (approximate answer: law of universal gravitation)
Formulation together with students of questions (goals and objectives) for studying new material.
What is the purpose of our lesson?
-Familiarize yourself with the law of universal gravitation;
-Identify the area of application of the law of universal gravitation and show its universality.
The stage of acquiring new knowledge.
Today in our lesson we will study the law of universal gravitation and show its practical significance. Let's expand on the concept of interaction of bodies using this law as an example and get acquainted with the range of action of gravitational forces.
Today we will talk about the great force of nature - the force of universal gravity. For thousands of years people have complained about this power. It did not allow the construction of multi-kilometer towers (the upper floors with their weight pressed on the lower ones - the structure was destroyed), bridges over wide rivers (the engineers miscalculated a little - and they collapsed with a roar). Meanwhile, the man had no idea how much he owed to this power.
Open lesson on the topic “The Law of Universal Gravitation” 9th grade.
Now we open our notebooks and write down the date and topic of the lesson (The story is accompanied by a presentation).The law could not have been discovered without the work of other scientists:
The Danish astronomer Tycho Brahe (1546-1601), who observed the movements of the planets for many years, accumulated a huge amount of interesting data, but was unable to process it.
Nicolaus Copernicus (1473-1543) discovered the Heliocentric system of the world
Johannes Kepler (1571-1630), using Copernicus's idea of the heliocentric system and the results of Tycho Brahe's observations, established the laws of planetary motion around the Sun, but could not explain the dynamics of this motion.
After the discovery of these laws, the search began for the patterns that govern the movement of planets around the Sun.
Isaac Newton discovered this law at the age of 23, but did not publish it for 9 years, since the incorrect data available at that time about the distance between the Earth and the Moon did not confirm his idea. Only in 1667, after this distance was clarified, the law of universal gravitation was finally published.
Newton's hypothesis: “The cause that causes a rock to fall to the Earth, the movement of the Moon around the Earth and the planets around the Sun is the same.”
Newton suggested that a number of phenomena that seemed to have nothing in common (the fall of bodies to the Earth, the revolution of the planets around the Sun, the movement of the Moon around the Earth, the ebb and flow of the tides) were caused by one reason.
Taking a single mental look at the “earthly” and “heavenly”, Newton suggested that there is a single law of universal gravitation, to which all bodies in the Universe are subject - from apples to planets!
Now research by historians shows that such a guess was expressed by scientists even before Newton. However, it was he who made a particular but very important conclusion from this hypothesis: there must be a connection between the centripetal acceleration of the Moon and the acceleration of gravity on Earth. This relationship had to be established numerically and verified.
And this was done in 1687. Newton established one of the fundamental laws of mechanics
“Any two bodies are attracted to each other with a force whose modulus is directly proportional to the product of their masses and inversely proportional to the square of the distance between them:
F= G m1m2 / r2
where m1 and m2 are the masses of interacting bodies, r is the distance between the bodies, G is the coefficient of proportionality, the same for all bodies in nature and called the universal gravitation constant, or gravitational constant.”
Gravitational interaction is an interaction characteristic of all bodies of the Universe and manifests itself in their mutual attraction to each other.
Is force a vector or scalar quantity?
Now let's find out where the force of gravitational interaction is directed?
It is directed along a straight line connecting the centers of gravity of the bodies. And according to Newton's Third Law, bodies interact with forces equal in magnitude and opposite in direction. F12=-F21
How do bodies that are far from each other attract?
The answer is simple through the gravitational field.
Gravity field—
a special type of matter that carries out gravitational interaction.
What else can we call this law?
Now let’s find out the limits of applicability of this law
- For material points
- For two spheres in which the density is uniformly distributed
- For a large ball and a material point
It is approximately fulfilled for any bodies if the distance between them is significantly greater than their sizes.
When Newton discovered the law of universal gravitation, he did not know a single numerical value for the masses of celestial bodies, including the Earth. He also did not know the value of the constant G.
Meanwhile, the gravitational constant G has the same value for all bodies in the Universe and is one of the fundamental physical constants. How can one find its meaning?
Shows Cavendish's experience.
Thus G=6.67·10-11 m/kg·s2
This law is of great importance
-The law of universal gravitation underlies celestial mechanics - the science of planetary motion.
-With the help of this law, the positions of celestial bodies in the firmament for many decades in advance are determined with great accuracy and their trajectories are calculated.
-The law of universal gravitation is also used in calculating the motion of artificial Earth satellites and interplanetary automatic vehicles.
Versatility
The universality lies in the fact that absolutely all particles of the Universe participate in gravity, and it manifests itself at any distance. Unlike other fundamental interactions, in which either not all particles participate, or it manifests itself only at ultra-short distances.
“The law of gravity is universal. It extends over vast distances. And Newton, who was interested in the Solar System, could well have predicted what would come out of Cavendish’s experiment, for Cavendish’s scales, two attracting balls, are a small model of the Solar System. If we magnify it ten million million times, we get the solar system. Let's increase it another ten million million times - and here you have galaxies that attract each other according to the same law. When embroidering her pattern, Nature uses only the longest threads, and any, even the smallest, sample of it can open our eyes to the structure of the whole” (Richard Phillips Feynman).
12/15/2015 (9th grade) 12/15/2015 (9th grade) The Law of Gravity. - presentation
(9th grade) (9th grade) The Law of Universal Gravitation
Plan for the study of physical laws: 1. The history of the discovery of the law 2. The range of phenomena described by this law 3. The formulation and mathematical expression of the law; 4. Experiments confirming the validity of the law; 5. Examples of accounting and application in practice; 6. Conditions (limits) of applicability of the law;
Danish astronomer Tycho Brahe (), who observed the movements of the planets for many years, accumulated a huge amount of interesting data, but was unable to process it. 1. How the law of universal gravitation was discovered
Johannes Kepler (), using Copernicus's idea of the heliocentric system and the results of observations by Tycho Brahe, established the laws of planetary motion around the Sun, but he also could not explain the dynamics of this motion.. How the law of universal gravitation was discovered
Isaac Newton discovered this law at the age of 23, in 1658, but did not publish it for 9 years, since the incorrect data available at that time about the distance between the Earth and the Moon did not confirm his idea. Only in 1667, after this distance was clarified, the law of universal gravitation was finally published. How the law of universal gravitation was discovered
The force of universal gravitation Newton's hypothesis: “The cause 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.”
Taking a single mental look at the “earthly” and “heavenly”, Newton suggested that there is a single law of universal gravitation, to which all bodies in the Universe, from apples to planets, are subject! How the law of universal gravitation was discovered, Newton suggested that a number of phenomena that seemingly have nothing in common (the fall of bodies to the Earth, the revolution of planets around the Sun, the movement of the Moon around the Earth, ebbs and flows, etc.) are caused by one reason.
Isaac Newton is an English physicist and mathematician, creator of the theoretical foundations of mechanics and astronomy. He discovered the law of universal gravitation, developed differential and integral calculus, invented the reflecting telescope and was the author of the most important experimental works in optics. Newton is rightly considered the creator of classical physics. In 1667, Newton suggested that forces of mutual attraction act between all bodies, which he called the forces of universal gravitation.
In 1687, Newton discovered one of the fundamental laws of mechanics, called the Law of Universal Gravitation: “Any two bodies attract each other with forces whose moduli are directly proportional to the product of their masses and inversely proportional to the square of the distance between them.” where m 1 and m 2 are the masses of interacting bodies, r is the distance between the bodies, G is the coefficient of proportionality, the same for all bodies in nature and called the universal gravitation constant, or gravitational constant.”
1. Formulation of the Law of Universal Gravitation: “Any two bodies are attracted to each other with forces whose magnitudes are directly proportional to the product of their masses and inversely proportional to the square of the distance between them.” where m 1 and m 2 are the masses of interacting bodies, r is the distance between the bodies, G is the coefficient of proportionality, the same for all bodies in nature and called the universal gravitation constant, or gravitational constant. 2. Formula expressing the law:
But force is a vector quantity! ? where m 1 and m 2 are the masses of interacting bodies, r is the distance between the bodies, r unit is the unit vector in the direction from the first body to the second G is the gravitational constant
Gravitational interaction is an interaction characteristic of all bodies of the Universe and manifests itself in their mutual attraction to each other. A gravitational field is a special type of matter that carries out gravitational interaction. Remember that...
Each body of mass M creates a field around itself, which is called gravitational. If a test body of mass m is placed at a certain point in this field, then the gravitational field acts on this body with a force F, depending on the properties of the field at this point and on the magnitude of the mass of the test body. Mechanism of gravitational interaction
The speed of propagation of gravitational interaction is finite. It is equal to the speed of light and amounts to km/s. Mechanism of gravitational interaction This means that body m will not immediately “feel” a change in the direction of the gravitational force from body M when it “suddenly” moves...
4.Experiments confirming the validity of the law; In 1798, Henry Cavendish designed a torsion balance and used it to measure the force of attraction between two spheres, confirming the law of universal gravitation; determined the mass and average density of the Earth.
4. Henry Cavendish's experiment to determine the gravitational constant. English physicist Henry Cavendish in 1798 determined how strong the force of attraction between two objects is. As a result, the gravitational constant was determined quite accurately, which allowed Cavendish to determine the mass of the Earth for the first time.
Scheme of G. Cavendish's experiments
Cavendish experience
G is the gravitational constant, it is numerically equal to the force of gravitational attraction of two bodies weighing 1 kg each, located at a distance of 1 m from one another. G=6, N m 2 /kg 2 The force of mutual attraction between bodies is always directed along the straight line connecting these bodies. 5. Examples of accounting and application of the law in practice;
5. Examples of accounting and application of the law in practice: 5.1. The effect of gravity on all bodies on Earth is a manifestation of the law of universal gravitation. Calculations of orbits, speeds of planets, artificial Earth satellites (AES) (Communications, Internet, GPS, reconnaissance) 5.3. Taking into account gravity during the approach of artificial massive bodies (for example, the ISS orbital station and space trucks)
5. Examples of accounting and application of the law in practice: 5.4. The phenomenon of ebb and flow is a manifestation of the law of gravity.
6. Conditions (limits) of applicability of the law; The law of universal gravitation is applicable only to bodies that can be considered material points, or to bodies that have a spherical (spherical) shape. For bodies that cannot be considered material points, special procedures are required.
6. Limits of applicability of the law The law of universal gravitation has certain limits of applicability; it is applicable for: 1) material points; 2) bodies shaped like a ball; 3) a ball of large radius interacting with bodies whose dimensions are much smaller than the dimensions of the ball.
: 1. Why doesn't the Moon fall to Earth? 2. Why do we notice the force of attraction of all bodies towards the Earth, but do not notice the mutual attraction between these bodies themselves? 3. How would the planets move if the Sun's gravitational force suddenly disappeared? 4. How would the Moon move if it stopped in orbit? 5. Does the Earth attract a person standing on its surface? Flying plane? An astronaut on an orbital station? Think and answer
6. Some bodies (balloons, smoke, airplanes, birds) rise upward, despite gravity. Why do you think? Is there a violation of the law of universal gravitation here? 7. What needs to be done to increase the gravitational force between two bodies? 8. What force causes ebbs and flows in the seas and oceans of the Earth? 9. Why don’t we notice the gravitational attraction between the bodies around us? Think and answer
1. What force makes the Earth and other planets move around the Sun? Choose the correct statement. A. Inertial force. B. Centripetal force. C. Gravitational force. 2. What force causes ebbs and flows in the seas and oceans of the Earth? Choose the correct statement. A. The force of water pressure on the bottom of seas and oceans. B. Gravitational force. C.Atmospheric pressure force. 3. What needs to be done to increase the gravitational force between two bodies? Choose the correct statement. A. Move both bodies away from each other. B. Bring both bodies closer. Mini test
Calculation problems 1. A spacecraft weighing 8 tons approached an orbital space station weighing 20 tons at a distance of 500 m. Find the force of their mutual attraction. 2. At what distance will the force of attraction between two bodies, each weighing 1000 kg, be equal to 6.N? 3. Two identical balls are at a distance of 1 m from each other and attract with a force of 6. N. What is the mass of each ball?
Question-answer Make up questions and then give answers to fragments 1-4 in the picture
Reflection Filling out the conceptual table Last name, first name What did you know? What did you learn? What do you disagree with? What's not clear? Exchange of opinions, quotes from tables with reflection. Summing up the lesson.
Homework: § 15, exercise 15 (1,2,3) (Peryshkin A.V., Gutnik E.M. Physics. 9th grade - M.: Bustard, 2007) Homework: § 15, exercise 15 (1,2,3) (Peryshkin A.V., Gutnik E.M. Physics. 9th grade - M.: Bustard, 2007) Thank you for your attention. Thank you for the lesson
Definition of the law of universal gravitation
According to this law, all material bodies attract each other, and the force of attraction does not depend on the physical or chemical properties of the bodies. It depends, if everything is simplified as much as possible, only on the weight of the bodies and the distance between them. You also need to additionally take into account the fact that all bodies on Earth are affected by the gravitational force of our planet itself, which is called gravity (from Latin the word “gravitas” is translated as heaviness).
Let us now try to formulate and write down the law of universal gravitation as briefly as possible: the force of attraction between two bodies with masses m1 and m2 and separated by a distance R is directly proportional to both masses and inversely proportional to the square of the distance between them.
The law of universal gravitation and weightlessness of bodies
The law of universal gravitation discovered by Newton, as well as the accompanying mathematical apparatus, later formed the basis of celestial mechanics and astronomy, because with its help it is possible to explain the nature of the movement of celestial bodies, as well as the phenomenon of weightlessness. Being in outer space at a considerable distance from the force of attraction and gravity of such a large body as a planet, any material object (for example, a spaceship with astronauts on board) will find itself in a state of weightlessness, since the force of the Earth’s gravitational influence (G in the formula for the law of gravity) or some other planet will no longer influence him.
