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Deriviation of why mass can be thought of as to act at the centre

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The last equations where Feynman explains why the mass of an object can be taken to be acting at the centre i.e where he has a hollow spheres, in the section ‘Gravitational field of large objects’. I cant understand the following:If we call xx the distance of a certain plane section from the center, then all the mass that is in a slice dxdx is at the same distance rr from PP, and the potential energy due to this ring is −Gm′dm/r..

Also, why are the limits taken as such i.e R-a and R+a. And the part where he explains how the potential is the same inside the sphere.

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