Imagine a hole drilled along the radius of the Earth. A uniform rod of length equal to the radius (R) of the Earth is inserted into this hole. Choose the CORRECT statement(s) (here  ${\text{g}}_0$ is value of acceleration due to gravity at surface of Earth).

$x_{cg}=\frac{{\displaystyle \int xdw}}{{\displaystyle \int dw}}$

Where  $dw=$ Weight of an infinitesimal element whose x co-ordinate is x.
Let  $\lambda =$ mass per unit length of the rod. Consider an element at a distance x from the centre
$dw=\left(\lambda dx\right)\left(g\frac{x}{R}\right)$

[Acceleration due to gravity at a distance x from the centre of the Earth is $=g\frac{x}{R}$]

$\therefore x_{cg}=\frac{\frac{\lambda g}{R}{\displaystyle {\int }_0^R{x}^{2}dx}}{\frac{\lambda g}{R}{\displaystyle {\int }_0^{R}xdx}}=\frac{\frac{R^3}{3}}{\frac{R^2}{2}}=\frac{2R}{3}$