Question

Why is the denominator in Newton's Law second powered? (M1M2/d^2) Is this not a surface area function? What does surface area have to do with abstract distance between M1M2?

Answer 1

The field of gravity is spherically symmetric, right? Look at the shape of the moon, it screams "gravity works spherically". So imagine the size of a spherical surface equidistant from the center of the moon. Now go another foot "up" away from the center, you are on a new spherical surface where force of gravity is the same everywhere, but less than before. What is different? The size of the sphere, the magnitude of the force ... they are proportional.

Answer 2

ok, one way of looking at any force is its ability to invoke a "sphere of influence". This is applicable to both the newtonian and Einsteinian way of looking at gravity( remember the tarpaulin and ball explanation of general relativity). The simple logic predicts that the force is inversely proportionally to the distance( so it appears in the denominator). consider a point on that sphere of influence. Since any point on that surface can be assumed to be infinitesimally on a 2 D surface( remember the curved surface area of a sphere is obtained by integrating a unit area over the radius!), the denomination should predict such a mathematics, and hence, its a powered 2.