Shell Theorem - Inside A Shell

Inside A Shell

For a point inside the shell the difference is that for θ equal to zero φ takes the value π radians and s the value R - r. When then θ increases from 0 to π radians φ decreases from the initial value π radians to zero and s increases from the initial value R - r to the value R + r.

This can all be seen in the following figure

Inserting these bounds in the primitive function

one gets that in this case

saying that the net gravitational forces acting on the point mass from the mass elements of the shell, outside the measurement point, cancel out.

Generalization: If the resultant force inside the shell is:

The above results into being identically zero if and only if

Outside the shell (i.e r>R or r<-R) :