Mechanics

# Universal gravitation

Universal gravitation is a law, demonstrated by Isaac Newton, that relates the force of attraction between the Sun, planets, and other celestial bodies in the Solar System.

Universal gravitation is a law developed by Isaac Newton to explain the circular orbits of the planets in the Solar System and the attractive force between them. His formula was obtained based on Kepler’s laws, and the constant of universal gravitation (G) that appears in the equation is the result of the torsion balance experiment developed by Henry Cavendish.

The discovery of this law further expanded the minds of scientists, and, based on it, Newton was able to explain the shape of the Earth, the tides, the orbit of comets, among others.

## Summary on universal gravitation

• Based on the law of universal gravitation, we were able to determine the attractive gravitational force between two bodies.
• To calculate the gravitational force, it is necessary to have knowledge of the masses of bodies and the distance between them.
• The universal gravitation formula can be obtained based on Kepler’s laws .
• The universal gravitation constant is a constant of proportionality whose magnitude is 6.67408∙ 10 – 11 ∙ m² kg ².

## What is the law of universal gravitation?

The law of universal gravitation is a law that was described by the physicist Sir Isaac Newton (1643-1727), in his work Philosophiae naturalis principia mathematica , published in 1687. It describes that two bodies will mutually suffer the action of an attractive force proportional to their masses and inversely proportional to the square of the distance between them.

The statement of the law of universal gravitation says the following:

Two bodies attract each other by a force directly proportional to the product of their masses and inversely proportional to the square of the distance separating them.

To get an idea of ​​the importance of this law for Physics, in book III of the Principia , Newton applies it:

• in the discussion of the movement of the natural satellites and planets of the Solar System ;
• in the demonstration of the calculation of the masses of the planets in relation to the mass of the Earth;
• in calculating the effect of the Earth’s rotation on its flattened shape;
• in the explanation of the tides ;
• in calculating the orbit of comets, etc.

## What is the formula for universal gravitation?

In universal gravitation, we use the formula for gravitational force, namely:

FG M∙ mdtwoF=G M∙md2

• F   is the magnitude of the gravitational force of attraction, measured in Newtons [ ].NoN
• GG   is the universal gravitational constant, it is  .67   10− 11 No.mtwo/lol _two6,67 ∙ 10−11 N.m2/kg2
• MM   is the mass of body 1, measured in kilograms [ ] .lol _kg
• mm   is the mass of body 2, measured in kilograms [ ] .lol _kg
• dtwod2   is the distance between the planets, measured in meters [ ].mm

## Universal gravitation and Kepler’s laws

The law of universal gravitation has a direct connection with Kepler’s laws , mainly the law of periods ( 2nd law ) and the harmonic law ( 3rd law ), since, through them, it is possible to demonstrate the formula of the law of universal gravitation , demonstrating its confirmation.

## History of universal gravitation

At the beginning of the 17th century, Newton wanted to know how bodies kept themselves in orbit in the Solar System. From this, he studied the reason that made the Moon revolve around the Earth and, later, studied about the planetary movements described by Johannes Kepler (1571-1630), Tycho Brache (1546-1601) and Galileo Galilei (1564-1642 ). ).

Based on the principles of his predecessors, he developed the theory that all bodies that have mass are attracted to each other , and, based on Kepler’s 2nd law, he discovered that planets only describe circular orbits around the Sun if they are subject to to uniform motion with centripetal acceleration and an attractive force between them.

Adding to this Kepler’s 3rd law, Newton came to the conclusion that the force is proportional to the mass of the planet and the mass of the Sun , but also inversely proportional to the square of the distance separating them. Thus, he developed the equation of the law of universal gravitation, which, although it was developed in relation to the Sun and the planets of the Solar System, is also valid for the other celestial bodies , therefore, it is universal.

## The universal gravitational constant

The universal gravitational constant, also known as the Newtonian constant of gravitation, is a physical constant whose value is . As the attraction between two common bodies has a very small value and can be neglected, it was only through the torsion balance experiment, developed by scientist Henry Cavendish (1731-1810), between 1797 and 1798, that the value could be determined. of that constant.6,674081011Nm²/kg²6,67408∙10−11N∙m²/kg²

Note : Do not confuse the G  (capital) of the constant of universal gravitation with the g  (lowercase) of the acceleration of the Earth’s gravity , whose value is, approximately,  .9,81 m/s29,81 m/s2

## Solved exercises on universal gravitation

question 1

(PUC-SP) The intensity of the gravitational force with which the Earth attracts the Moon is F. If the mass of the Earth and the Moon were doubled and if the distance separating them were reduced by half, the new force would be:

a) 16F

b) 8F

c) 4F

d) 2F

e) F

Resolution: Alternative A

Using the law of universal gravitation, taking  to be the mass of the Earth and  the mass of the Moon:MMmm

F=G Mmd2F=G M∙md2

F=G MTmLd2F=G MT∙mLd2

Now, we’ll use the law of universal gravitation again, but with a new force  , and substitute the statement data into it:FF′

F=G MTmLd2F′=G MT∙mLd2

F=G 2MT2mL(d2)2F′=G 2MT∙2mL(d2)2

F=G 4MTmLd24F′=G 4∙MT∙mLd24

F=G 4MTmLd24F′=G 4∙MT∙mLd2∙4

F=16G MTmLd2F′=16∙G MT∙mLd2

Remembering that  , then:F=G MTmLd2F=G MT∙mLd2

F=16FF′=16∙F

question 2

(UPE) Consider the mass of the Sun , the mass of the Earth  , the Earth–Sun distance (center to center) approximately and the universal gravitation constant . The order of magnitude of the gravitational force of attraction between the Sun and the Earth is N:MS=21030kgMS=2·1030kgmT=61024mT=6·1024 dTS=11011mdTS=1·1011m G=6,671011 N.m2/kg2G=6,67∙10−11 N.m2/kg2

a) 10 23

b) 10 32

c) 10 54

d) 10 18

e) 10 21

Resolution: Alternative A

The gravitational force between the Sun and the Earth is given by the law of universal gravitation:

F=G Mmd2F=G M∙md2

F=G MSmTd2F=G MS∙mTd2

Substituting the given values ​​for the statement, we have:

F=6,671011 (21030)(61024)(11011)2F=6,67∙10−11 (2·1030)∙(6·1024)(1·1011)2

F=6,671011 (21030)(61024)11022F=6,67∙10−11 (2·1030)∙(6·1024)1·1022

F=80,041011103010241022F=80,04∙10−11∙1030∙1024∙10−22

F=80,041011+30+2422 F=80,04∙10−11+30+24−22

F=8,0041011021 F=8,004∙101∙1021

F=8,004101+21  F=8,004∙101+21

F=8,0041022 N  F=8,004∙1022 N

The order of magnitude of the force of attraction is  , since  is greater than  . 1023

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