derive an expression for the gravitational force exerted by the earth on a point mass located below the surface of earth by applying shell theorem
Answers
A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its centre.
If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object's location within the shell.
A corollary is that inside a solid sphere of constant density, the gravitational force within the object varies linearly with distance from the centre, becoming zero by symmetry at the centre of mass. This can be seen as follows: take a point within such a sphere, at a distance {\displaystyle r} r from the centre of the sphere. Then you can ignore all the shells of greater radius, according to the shell theorem. So, the remaining mass {\displaystyle m} m is proportional to {\displaystyle r^{3}} r^{3} (because it is based on volume), and the gravitational force exerted on it is proportional to {\displaystyle m/r^{2}} m/r^2 (the inverse square law), so the overall gravitational effect is proportional to {\displaystyle r^{3}/r^{2}=r} r^3/r^2 =r, so is linear in {\displaystyle r} r.