Question

The acceleration due to gravity and density of the earth are related by which of the following relations ? (where G is the gravitational constant and $R_{E}$ is the radius of the earth)

• A. $\rho = \frac{4\pi GR_{E}}{3g}$
• B. $\rho = \frac{3g}{4\pi GR_{E}}$
• C. $\rho = \frac{3G}{4\pi gR_{E}}$
• D. $\rho = \frac{4\pi gR_{E}}{3G}$

Answer 1

Answer: (B)

$\rho = \frac{3g}{4\pi GR_{E}}$

Answer 2

Accelerations due to gravity on earth is

$g = \frac{GM_{E}}{R_{E}^{2}}$ ….. (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

$g = \frac{G\left( \rho\frac{4}{3}\pi R_{E}^{3} \right)}{R_{E}^{2}} = \frac{4}{3}\pi\rho GR_{E}or\rho = \frac{3g}{4\pi GR_{E}}$