Derive expression for force of attraction between two bodies and then define gravitational constant.

“Everybody 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.” Let us consider two bodies A and B of masses m1 and m2 which are separated by a distance r. Then the force of gravitation (F) acting on the two bodies is given by

F ∝ m₁ × m₂          … (1)

and      F ∝ {1 \over 𝑟^2}    … (2)

Combining (1) and (2), we get

F ∝ {𝑚_1×𝑚_2 \over 𝑟^2}

Or      F = G × {𝑚_1𝑚_2 \over 𝑟^2}  … (3)

where G is a constant known as universal gravitational constant.

Here, if the masses m₁ and m₂ of the two bodies are of 1 kg and the distance (r) between them is 1 m, then putting m₁ = 1 kg, m₂ = 1 kg and r = 1m in the above formula, we get

G = F

Thus, the gravitational constant G is numerically equal to the force of gravitation which exists between two bodies of unit masses kept at a unit distance from each other.