Discovery

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 .

By comparing the acceleration due to gravity of the earth with the acceleration required to keep the moon in it's orbit around the earth, he was able to arrive at the basic law of gravitation.

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?

Then , I would say that force is always proportional to acceleration and if we say impact of acceleration or force , both would be same. Remember

F=ma         (a)

Where F= force on the body, exerted by the body

m= mass of the body

a= acceleration of the body i.e. how the velocity of the body is changing.

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 ,

where w is radial velocity given by

thus , substituting the values we get

(1)

Thus ,

Also ,

.              (2)

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.

Thus,

from (a) we can say

therefore,
 Gravitational Law

.

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

Or

.

Where G is called universal gravitational constant and it's value is

.

Above equation is the expression for Newton 's law of gravitation.