Question

The acceleration due to gravity $g$ and density of the earth $\rho$ are related by which of the following relations.? (where G is the gravitational constant and $R_E$ is the radius of the earth)

• A. $\rho =\frac{4\pi G{R}_E}{3g}$
• B. $\rho =\frac{3g}{4\pi G{R}_E}$
• C. $\rho =\frac{3G}{4\pi g{R}_E}$
• D. $\rho =\frac{4\pi g{R}_E}{3G}$

Answer 1

The correct option is B $\rho =\frac{3g}{4\pi G{R}_E}$

Acceleration due to gravity on earth is $g=\frac{G{M}_E}{R_E^2}$ ....(i)

As

$\rho =\frac{M_E}{\frac{4}{3}\pi R_E^3}\Rightarrow M_E=\rho \frac{4}{3}\pi R_E^3$

Substituting this value in Eq. (i), we get

$g=\frac{G\left(\rho \frac{4}{3}\pi R_E^3\right)}{R_E^2}=\frac{4}{3}\pi \rho G{R}_E$

or $\rho =\frac{3g}{4\pi G{R}_E}$