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Universal Law of Gravitation

This discovery of the law of universal gravitation by Isaac Newton is one of the greatest landmarks in the history of science. Before Newton's time, scientist believed that the tendency of all objects to fall to the earth and the motion of planets were entirely unrelated phenomena. Newton showed that these two phenomena involved gravitation and that they could be analysed using his equations of motion.
The motion of heavenly bodies, particularly the planets, the moon and the sun, was the subject of active interest among students, including Newton, at Cambridge in 1664. In 1665, the plague broke out and the college was closed. Newton, then just 23 years of age, went home to Woolsthorpe, where he continued to think about the motion of the planets and the moon.

Acceleration due to gravity on earth and centripetal acceleration of moon

Newton compared the acceleration due to gravity of an apple with the centripetal acceleration of the moon which led him to law of gravitation. We now present his calculations and arguments.
Assuming the moon's orbit to be circular (which is a good approximation) and knowing the distance of moon from the earth and its period of revolution, Newton calculated the centripetal acceleration as follows
Radius of moon's orbit (R)=3.84 108m
Period of moon's revolution (T)=27.3 days = 27.3 24 3600 s
The speed of the moon in orbit (υ)== 1.02 103 ms-1

Centripetal acceleration
= 2.72 10-3 ms-2
The acceleration of a falling apple is g = 9.8 m s-2. Newton assumed that both the moon and the apple are accelerated towards the center of the earth. The difference in their motions arises because the moon has a tangential velocity whereas the apple does not.
He assumed that both g and ac were due to the gravitational force of the earth acting respectively on the apple and the moon.
Newton found that the value of ac(=2.72 10-3 m s-2) was about 1/3600 of the value of g(=9.8ms-2). He know that the ratio RE/R was about 1/60. Thus be observed that

where R = distance of the moon from the centre of the earth
and RE = radius of the earth.
In other words, the acceleration of a body and hence the force is inversely proportional to the square of its distance from the centre of the earth. This led Newton to postulate that the gravitational force might vary inversely as the square of the distance, i.e. the gravitational force obeys the inverse square law.

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