Suppose the centre of mass of 2 bodies coincide ,

Jordin Olsen

Jordin Olsen

Answered question

2022-05-02

Suppose the centre of mass of 2 bodies coincide , then how will we calculate the gravitational force between the 2 bodies ?

Answer & Explanation

gunithd5

gunithd5

Beginner2022-05-03Added 8 answers

Suppose you have two densities of mass ρ 1 and ρ 2 . The two centers of mass coincide if
R C M 1 = R 3 r ρ 1 ( r ) d 3 r R 3 ρ 1 ( r ) d 3 r R C M 2 = R 3 r ρ 2 ( r ) d 3 r R 3 ρ 2 ( r ) d 3 r
are equal, but the force is not calculated as
F = G M 1 M 2 | R C M 1 R C M 2 | 3 ( R C M 1 R C M 2 )
Instead you must sum the contribution of all the infinitesimal elements
F = G R 3 R 3 ρ 1 ( r 1 ) ρ 2 ( r 2 ) | r 1 r 2 | 3 ( r 1 r 2 ) d 3 r 1 d 3 r 2
So you are not dividing by zero like in the first formula, the integral is well defined always. It happens that in a spherical distribution of mass, the integral formula reduce to the first one, but in that case the centers of mass never coincides.

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