Answer:
Explanation:
Volume of the earth is,
[tex]V=\frac{4}{3}\pi R_E^3[/tex]
Volume of the sphere is,
[tex]V=\frac{4}{3}\pi r^3[/tex]
The density of the earth is,
[tex]\rho =\frac{M_E}{V}\\\\=\frac{M_E}{\frac{4}{3}\pi R_E^3}[/tex]
The mass of the sphere is,
[tex]M=V\rho\\\\=\frac{4}{3}\pi r^3\frac{M_E}{\frac{4}{3}\pi R_E^3}\\\\=\frac{M_Er^3}{R_E^3}[/tex]
Consider a mass [tex]m[/tex] at r<[tex]R_E[/tex]
Expression for the force is,
[tex]F=\frac{GMm}{r^2}\\\\=\frac{G(\frac{M_Er^3}{R_E^3})m}{r^2}\\\\=\frac{GM_Erm}{R_E^3}[/tex]
From newtons second law,
[tex]F=mg\\\\\frac{GM_Erm}{R_E^3}=mg\\\\g=\frac{GM_Er}{R_E^3}[/tex]
