# UNIVERSAL GRAVITATIONAL FORCE

All the credit of this outstanding work goes to sir Isaac Newton, it is believed that Newton found the concept of Gravitation , when, once he had sat under a apple tree , and suddenly a apple fall from tree on him.

Gravitation is the part of the classical mechanics and was formulated in Newton’s work *philosophae naturalis principia mathematica *(simply *principia mathematica*) , and first published on 5 July 1687.

Now takes a look on this famous laws;

Newton laws of UNIVERSAL gravitation usually states that :

Every particles attracts each other particles in this universe with a force which is directly proportional to the product of their respective masses and inversely proportional to the square of shortest distance between their centers. This is a general law derived from the empirical observations , and Newton called this observation inductive reasoning.

And in today’s language this laws is stated as , every point mass attracts every other point mass with a force acting along with the adjoining line , which join their centers.

This force is directly proportional to the product of their respective masses and inversely proportional to the square of shortest distance between their centers.

The equation of gravitational force thus takes the form:

F = Gm1m2/r^2

Where F is the force acting between them, m1 and m2 are masses and r is the distance between their centers, and G is proportionality constant called universal gravitational constant.

Newton only combined this formula by his observations.

So here we are going to give a mathematical process of deriving this formula , which is as below:

## DERIVATION FOR UNIVERSAL GRAVITATIONAL FORCE

If planets revolve in circular orbit then their centripetal force is given as;

F = mv^2/r. Where v^2/r is centripetal acceleration.

Then. v = ωr. And also ω = 2π/T

Then putting values in centripetal acceleration we get;

a = ω^2.r and then, a = 4π^2.r/T^2

And according to the Kepler third laws planetary motion, we know that,

T^2 is proportional to r^3

Then. T^2 =C.r^3. Where C is constant Then putting above acceleration formula in

Below formula;

F= ma. = mr4π^2/T^2 now put C.r^3 in place of T^2

F= 4π^2.m/C.r^2

Now here

4π^2/C = GM

k = 4π^2/C = GM. , [T^2 = 4π^2.r^3/GM]

And by this 4π^2/GM = C , ie. Constant.

Then putting the value of C = 4π^2/GM in the formula of force. We get;

F = GMm/r^2

M is the mass of bigger object , on around planets is revolving.

This is the required UNIVERSAL GRAVITATIONAL FORCE.