In the year 1665, the university of Cambridge had to be closed because of the black plague which swept in England . Newton , a scholar of Trinity college, University of Cambridge , went home. It was during his stay at home that one day he saw an apple falling down from a tree in his orchard. Newton who was only 24 years old at that time, began to think that force which attracted apple towards the earth might be the same as the force attracting the moon towards the earth. At that time , it was discovered that moon revolves around the earth due to some attractive force .
Now you may have doubt that in first paragraph I have talked about force and in second paragraph I am talking about acceleration, how it can be possible?
Where F= force on the body, exerted by the body
m= mass of the body
Now we will talk about how Newton discovered the expression of law of gravitation.
The acceleration of a body falling near the earth's surface is about 9.8m/s 2 (we will see it in detail later). Also, moon makes a revolution about the earth in T=27.3 days. The distance of the moon from the earth is R=3.85*10 5 km. The acceleration of the moon is therefore ,
thus , substituting the values we get
Note- distanceof any thing from earth or any planet means distance from centre of earth to that thing. since apple was not at significant distance from earth thus we have assumed d apple as radius of earth.
Comparing (1) and (2), it is concluded that
Newton guessed that the acceleration of the body towards the earth is inversely proportional to the square of the distance of the body from the centre of the earth as evident from above expression.
from (a) we can say
Also by third law of motion , the force on a body due to the earth must be equal to the force on the earth due to body. Therefore, this force should be proportional to the mass of the earth (M). Thus, the force between the earth and a body is given by
Where G is called universal gravitational constant and it's value is
Above equation is the expression for Newton 's law of gravitation.