Physics - Gravitation Question with Solution | TestHub

Gravitation field at mass $m$ due to a full solid sphere

${\overrightarrow{\text{E}}}_1=-\frac{{\text{g}}_0\overrightarrow{\text{r}}}{\text{R}}=-\frac{{\text{g}}_0\widehat{\text{r}}}{2}$

Gravitational field due to the cavity at mass $m$

$M\text{'}=\frac{{\left({\displaystyle \raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^3}{R^3}M$ $\Rightarrow M\text{'}=\frac{M}{8}$

${\overrightarrow{\text{E}}}_2=-\left(\frac{-{\text{GM}}^{\text{'}}\widehat{\text{ r}}}{{\text{R}}^2}\right)\text{,}$

$=\frac{GM\widehat{\text{r}}}{8{\text{R}}^2}=\frac{{\text{g}}_0\widehat{r}}{8}$

${\overrightarrow{\text{E}}}_{net}=\left(\frac{-{\text{g}}_0}{2}+\frac{{\text{g}}_0}{8}\right)\widehat{\text{r}}$

        $=\frac{-3{\text{g}}_0\widehat{r}}{8}$

$\Rightarrow {\overrightarrow{\text{F}}}_{\text{net}}=-\frac{3{\text{mg}}_0\widehat{r}}{8}$