A solid sphere of mass $m$ and radius $r$ is placed inside a hollow thin spherical shell of mass $M$

Question

A solid sphere of mass $m$ and radius $r$ is placed inside a hollow thin spherical shell of mass $M$ and radius $R$. A particle of mass $m'$ is placed on the line joining the two centres at a distance $x$ from the point of contact of the sphere and the shell. Find the magnitude of the resultant gravitational force on this particle due to the sphere and the shell if $2r < x < 2R$.

Professor PC Thomas Classes · Answer

For $2r < x < 2R$, the particle is located inside the hollow shell but outside the solid sphere. By the shell theorem, the gravitational force due to the outer shell is zero. The force is solely due to the solid sphere, acting from its centre (at a distance $r$ from the contact point). The effective distance to the particle is $(x-r)$, so $F = \frac{Gmm'}{(x-r)^2}$.