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Gravitational Force

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Gravitational Force is an attraction between two bodies by the virtue of their mass. The force of attraction between two point masses is  F = G \frac{m_1 m_2}{r^2} , where m1 and m2 are the masses of the particles and r is the distance between them. G is a universal constant having the value 6.67 × 10-11 Nm2kg-2.

Finding Gravitational force on Extended Body

To find the gravitational force on an extended body by another such body, we have to write the force on each particle of the first body by all the particles of the second body and then we have to sum up vectorially all the forces acting on the first body. For large bodies having large number of particles, we have to add quite a large number of forces. If the bodies are assummed continuous, one has to go through the integration process for the infinite summation involved. However, the integration yields a particularly simple result for a special case, such as a spherically symmetric body. [referal source]


Historical Introduction

The motion of celestial bodies such as the moon, the Sun and the Earth, etc has been of great interest for a long time for humans. Famous Indian Astronomer and Mathematician, Aryabhat, studied these motions in great detail, most likely in 5th century A.D., and wrote his conclusions in his book Aryabhatiya. He established that the earth revolves about its own axis and moves in a circular orbit about the sun, and that the moon moves in a circular orbit about the earth. (hcv|204)

About a thousand years after Aryabhat, the brilliant combination of Tycho Brahe and Johannes Kepler studies the planetary motion in great detail. Kepler formulated his important findings in his three laws of planetary motion:

  1. All planets move in elliptical orbits with the sun at a focus
  2. The radius vector from the sun to the planet sweeps equal area in equal time.
  3. The square of the time period of a planet is proportional to the cube of the semimajor axis of ellipse.

In the year 1665, Isaac Newton focussed his attention to the motion of moon about the earth. The moon makes a revolution about the earth in T = 27.3 days. The distance of the moon from the earth is R = 3.85 × 105 km. The acceleration of the moon is given by:

 a  = \omega^2 R

 = \frac{4 \pi^2 \times (3.85 \times 10^5)}{(27.3 )^2} \frac{km}{days}
 = 0.0027 ms^{-2}

Newton was curious as to what force produced such acceleration. The acceleration is towards centre of orbit i.e. towards centre of earth. Hence, force must act towards the centre of the earth. A natural guess was that earth is attracting the moon. There's a widespread saying that Newton was sitting under an apple tree when an apple fell down from the tree on the earth. This sparked the idea that the earth attracts all bodies towards its centre. The next aim was to find the law governing this force. Newton made several daring assumptions.

The laws of nature, he declared, were same for the earth as well as the celestial bodies. This statement was not a general belief in times of Newton's existence. It was a widespread believe in western part of the world that the earthly bodies are governed by certain rules and heavenly bodies are governed by different rules. In particular, the heavenly structure was thought to be so perfect that there could not be any change in sky. Tycho Brahe saw a new star in sky and refused to believe it. Newton's declarations were revolutionary.(hcv|203)

The acceleration of a body falling near the earth's surface is about 9.8 ms-1. Thus,

 \frac{ a_{apple} }{ a_{moon} } = \frac{9.8}{0.0027} \frac{ms^{-2}}{ms^{-2}} = 3600

Also,

(distance of the moon from the earth) / (distance of the apple from the earth)
 = \frac{ d_{moon} }{ d_{apple} } = \frac{ 3.85 \times 10^5}{6400} \frac{km}{km} = 60

Thus,  \frac{ a_{apple} }{ a_{moon} } = \bigg( \frac{ d_{moon} }{ d_{apple}} \bigg)^2

Newton guessed that the acceleration of a body towards earth is inversely proportional to the square of the distance of the body from the centre of the earth. Thus,  a \propto \frac{1}{ r^2 } .

Also, the force is mass times acceleration and so it is proportional to the mass of the body. Hence,  F \propto \frac{m}{ r^2 } .

By the third law of motion, the force on a body due to the earth must be equal to the force on the earth due to the body. Therefore, this force should also be proportional to the mass of the earth. Thus, the force between the earth and a body is

 F \propto \frac{Mm}{r^2}
 F = \frac{GMm}{r^2}

Newton furhter generalised the law by saying that not only earth but all material bodies in the universe attract each other according to the above equation with same value of G. The constant G is called universal constant of gravitation and its value is found to be 6.67 × 10-11 N^2kg-2. The above equation is known as the Universal law of gravitation.

In the above argument, the distance of the apple from the earth is taken to be equal to the radius of the earth. This means we have assumed the earth to be a single particle placed at its centre. This is of course not obvious. Newton had spent several years to prove that indeed this can be done. A spherically symmetric body can be replaced by a point particle of equal mass placed at its centre for the purpose of calculating gravitational force. In the process he discovered the methods of calculus.

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