Question $11$: Compare the gravitational force inside a solid sphere, at the surface of a solid sphere, and outside of a solid sphere at various distances using Newton's Universal Law of Gravitation. $[$Hint: The answer is an algebraic expression for all three cases, but one case is calculated to an exact number.$]$

Constants: $G=6\mathrm{.}67{10}^{-11}\frac{N{m}^2}{k{g}^2},$ Radius of the Earth $r_E=6\mathrm{.}378{10}^{6}m$

A$.$ Inside: ${\displaystyle \underset{?}{\overset{?}{}}}F=\frac{GmM}{r^2},$ Surface: ${\displaystyle \underset{?}{\overset{?}{}}}F=0,$ Outside: ${\displaystyle \underset{?}{\overset{?}{}}}F=\frac{GmM}{r^2}=\frac{GmM}{(r_E+h{)}^2}$

B$.$ Inside: ${\displaystyle \underset{?}{\overset{?}{}}}F=0,$ Surface: ${\displaystyle \underset{?}{\overset{?}{}}}F=\frac{GmM}{r_E^2},$ Outside: ${\displaystyle \underset{?}{\overset{?}{}}}F=\frac{GmM}{r^2}=\frac{GmM}{(r_B+h{)}^2}$

C$.$ Inside: ${\displaystyle \underset{?}{\overset{?}{}}}F=\frac{GmM}{r^2}=\frac{GmM}{h^2},$ Surface: ${\displaystyle \underset{?}{\overset{?}{}}}F=\frac{GmM}{r_E^2},$ Outside: ${\displaystyle \underset{?}{\overset{?}{}}}F=\frac{GmM}{r^2}=\frac{GmM}{(r_E+h{)}^2}$

D$.$ Inside: ${\displaystyle \underset{?}{\overset{?}{}}}F=\frac{GmM}{r^2}=\frac{GmM}{(r_E-h{)}^2}$ Surface: ${\displaystyle \underset{?}{\overset{?}{}}}F=\frac{GmM}{r_E^2},$ Outside: ${\displaystyle \underset{?}{\overset{?}{}}}F=\frac{GmM}{r^2}=\frac{GmM}{(r_E+h{)}^2}$