Blog #6, Worksheet #3.1, Problem 2, Shell Theory

2. For the ensuing problems and future endeavours in physics and astronomy, you will need knowledge of a very useful property of gravity called the Shell Theorem. It is a statement about how a spherical shell of uniform density gravitationally affect objects exterior and internal to the shell: 

A spherical shell of uniform density and mass Mshell gravitationally affects objects external to the shell exactly as a point mass (Mshell) would at the shell centre. The shell exerts no net gravitational effect on objects internal to the shell. 

If instead of shell, you had a solid sphere with some spherically symmetric mass distribution, what would the Shell Theorem imply about the force of a gravity on a hypothetical object inside/outside the sphere. Reason this out from Shell Theorem qualitatively with diagrams and/or squiggle math.

We assume that there is a spherical symmetry between the object and our central mass. The larger object is hypothetically broken up into infinite shells of of a very small mass. Depending on which of these shells our smaller object resides on, there will be a different ratio of gravitational pull on the object.

We learned that any shell beyond the distance of our object does not exert a gravitational force on the object. Thus only shells at a position \(<r\) will contribute to the force felt by the small object. If the object is beyond the radius, \(R\) of the larger object, then the larger object is just considered a point mass and the simple gravitational relationship still exists: \[F = ma = \frac{GmM}{r^2}\]However when our smaller mass is within the radius of the larger mass, the gravitational pull must be modeled as a ratio of the volume enclosed vs. the total volume of the body. Only a portion of the mass of the larger object contributes to the gravity and we are left with \[F = ma = \frac{Gm (\frac {r}{R})^3M}{r^2}\]