Why do we integrate along the whole length while finding the gravitational force between a object of

fetsBedscurce4why1 2022-05-10 Answered
Why do we integrate along the whole length while finding the gravitational force between a object of mass (m) and a rod of length L and mass M? Can't we simply use F = G m M r 2 ??
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Answers (1)

empatteMattmkezo
Answered 2022-05-11 Author has 22 answers
We have to consider the contribution of each little piece of mass d M = ρ d V and how far it is from the other mass, m resulting in a small force d F. If you have uniform distribution of mass, ρ = M / ( L A ), where A is the cross sectional area of the rod, and we probably don't integrate over that because it's a small diameter rod. So d V = A d x. Then we have a 1 / r 2 force behavior where r is the distance from each d M
d F = G m r 2 d M = G m r 2 M L d x
To find r, you have to give a specific geometry, and let x be the variable of integration that runs along the length and integrate from 0 to L. r will be a function of x. The geometry dictates what that function is. Because of the nature of 1 / r 2 , the force does not change linearly, but the distance from rod to mass m might (again, the geometry is important).
In your comment, you mentioned the sphere situation. That means that for a sphere of mass M and radius R 0
d M = ρ d V = M V s p R 2 sin θ d R d θ d ϕ
0 < R < R 0
where R is a spherical variable of integration, along with θ and ϕ. The distance r will be a function of R, θ, and ϕ. Without going into the details, an examination of the form of d M shows that the mass is not linearly distributed, so there's a chance, at least, that a 1 / r 2 force will reduce to something simple.
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