A non-homogeneous sphere of radius $R$has the following density variation:

$\rho =\left\{\begin{array}{l}\begin{array}{c}{\rho }_0; r\le R/3\\ {\rho }_0/2;\left(R/3\right)<r\le \left(3R/4\right)\end{array}\\ {\rho }_0/8; \left(3R/4\right)<r\le R\end{array}\right.$

The gravitational field at a distance $2R$ from the centre of the sphere is

The gravitational field at a distance $2R$ from the centre of the sphere is

$E=\frac{GM}{{\left(2R\right)}^2}$

where $M$is the mass of the whole sphere.

Here,

$$\begin{gathered} M=\frac{4}{3}\mathrm{\pi }{\left(\frac{\mathrm{R}}{3}\right)}^3{\mathrm{\rho }}_0+\left\{\frac{4}{3}\mathrm{\pi }{\left(\frac{3}{4}\mathrm{R}\right)}^3-\frac{4}{3}\mathrm{\pi }{\left(\frac{\mathrm{R}}{3}\right)}^3\right\}\frac{{\mathrm{\rho }}_0}{2}+\left\{\frac{4}{3}{\mathrm{\pi R}}^3-\frac{4}{3}\mathrm{\pi }{\left(\frac{3}{4}\mathrm{R}\right)}^3\right\}\frac{{\mathrm{\rho }}_0}{8} \\ =0.402{\mathrm{\pi R}}^3{\mathrm{\rho }}_0 \\ \therefore \mathrm{E}=\frac{\mathrm{G}\left(0.402{\mathrm{\pi R}}^3{\mathrm{\rho }}_0\right)}{{\left(2\mathrm{R}\right)}^2}=0.1\mathrm{\pi }GR{\mathrm{\rho }}_0 \end{gathered}$$