Newton’s Law of gravitation states that force of attraction between two point masses is proportional to the product of the masses and inversely proportional to the square of the distance between them. I know that the force of attraction between two spheres turns out to be of the same mathematical form as a consequence of Newton’s law. But I am not able to prove how the force between any two rigid masses is only proportional to the product of their masses (as my teacher says) and the rest depends upon the spatial distribution of the mass. So $F$ is ONLY proportional to $Mmf(r)$ where $f(r)$ maybe be some function based on the specifics of the situation.
[Physics] Why is the Force of Gravitational Attraction between two “Extended” bodies proportional to the product of their masses
forcesmassnewtonian-gravitynewtonian-mechanics
Best Answer
The statement
is not true in general, or at least it is misleading. The shapes of the mass distributions and their relative positions matter when computing the gravitational force.
It is true that once you hold constant the shapes of the mass distributions and their relative positions, then the force will be proportional to the product of the total masses of the bodies.
There are certain situations where treating two extended massive bodies as point sources can be exactly correct (in the context of Newtonian gravity). For a spherically symmetric mass distribution, the gravitational potential outside of it is the same as that arising from a point source of the same mass. This is an application of Gauss' law.
In general, one can build up an increasingly good approximation of the gravitational potential arising from a given mass distribution via a multi-pole expansion. . The leading-order term, which drops off least rapidly with distance (force $\propto r^{-2}$), is that of a monopole like what arises for a point mass or outside a spherically symmetric system. But a general mass distribution will have contributions from higher-order terms (dipole, quadropole, octopole...), all of which drop off increasingly rapidly with distance. As one considers two bodies at increasing separation, reducing them both to their monopole terms becomes increasingly more accurate.
Finally, the fact that the gravitational force of attraction on an extended body due to another body can vary with position is essential when considering phenomena such as tidal forces.