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SIR,CAN U PLEASE HELP ME TO DERIVE EQUATIONS RELATED TO GRAVITATION: 1)NEWTONS UNIVERSAL LAW OF GRAVITATION(F=GMm/d square) 2)FREE FALL(g=G M/R square) 3)WEIGHT OF AN OBJECT ON MOON

1) The Universal law of Gravitation was given by Newton. So it is also known as Newton's Law of  gravitation.

According to Universal Law of Gravitation : Every body in the Universe attracts every other body with a force which is directly proportional to the product of their masses and inversely  proportional to the square of the distance between them.

Consider  two bodies A and B of masses m₁ and m₂ are lying at a distance r from each other. Let the force of attraction between these two bodies be F.  Now, according to the Universal law of Gravitation.

(i). The force between two bodies is directly proportional to the product of  their masses.That is ,
                F ? m₁m₂   ___________________ (1)
(ii) The force between two bodies is inversely proportional to the square of the distance between them.That is,

                F ?   1/ r² ___________________ (2)
Combining (1) and (2), we get
                F ? m₁m₂ /  r²
Gravitational force, F = Gm₁m₂/ r²

where, G is a constant known as "Universal Gravitational Constant".

2)Expression for the acceleration due to gravity on the surface of the earth

Let m be the mass of a body, M the mass of the earth, R the radius of the earth and g the acceleration due to gravity.Let F be the force acting on the body.

By the Newton's second law of motion,

The force acting on the body'

               F = ma

                 = mg      -------------------------(i)

By the Newton's Law of Gravitation,

          F = G Mm /R²   ---------------------(ii)

From equations (i) and (ii),we get

       mg = GMm/R²

                         g = GM/R²

^{3)Derivation for the weight of an object on the moon}

                          Let m be the mass of an object.Then its weight on the earth (W_e) is given by 

W_e = F_e = G x M_e x m / R_e²                   ----------------------(1)

                  (By Universal Law of Gravitation)

where M_e and R_e are the mass and radius of the earth, respectively

The weight of the same object on the moon(W_m) is given by ,

W_m = F_m = G x M_m x m / R_m²                 --------------------(2)

                     (By Universal Law of Gravitation)

where M_m and R_m are the mass and radius of the moon,respectively

Dividing equation (2) by (1) , we get,

W_m /W_e = M_mR_e² / M_eR_m² = M_m/M_e = (R_e / R_m)²

The mass of earth is about 100 times that of moon and the radius of earth is four times that of the moon.Thus,we have 

       M_e / M_m = 100

and R_e / R_m = 4

Then,

W_m / W_e = 1/100 x (4)²

                 = 16 / 100

                 ?    1/6

i.e.,Weight on the moon / Weight on the earth = 1/6

or, W_m = W_e / 6

Thus, the weight of an object on moon is nearly 1/6th that on the surface of the earth