Why 'r' in gravitational force of attraction between two bodies i.e. F = GmM/r^2 , is consideted to be the distance between COM of the two bodies ? Why it is not considered to be the distance between their COG ?
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There are relevant answers to your previous question .
The center of mass of a solid body is the same in all inertial frames. The center of gravity depends on the gravitational field existing in a given inertial frame. In physics one uses invariant quantities to transformations between inertial frames.
Mathematically assuming that all the mass is at the center of mass is a first order approximation to the body, higher order mass density distribution enters when accuracy is required in gravitational field calculation.
anna v
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In general, the $r$ in the gravitational force formula is not the distance between the centers of mass, or the distance between the centers of gravity.
However, for two spherically symmetric mass distributions (good approximations for, say, Earth and Sun), the $r$ is the distance between their centers (which in this case happen to be the centers of mass). The reason is the shell theorem. Proving it was one of Newton’s motivations for inventing calculus!
Otherwise, if the distributions aren’t spherically symmetric, then in general you have to integrate the attractions between each pair of infinitesimal masses in each object, treating these like point masses. You can think about it as every atom in one mass distribution attracts every atom in the other mass distribution.
Of course, within a single mass distribution every atom also attracts every atom. But these forces cancel and give no net force, which makes for an uninteresting calculation.
G. Smith
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