Question

A cavity of the radius $R/2$ is made inside a solid sphere of radius $R$ . The centre of the cavity is located at a distance $R/2$ from the centre of the sphere. The gravitational force on a particle of mass $m$ at a distance $R/2$ from the centre of the sphere on the line joining both the centres of sphere and cavity is (opposite to the centre of the cavity).
[Here $g=\frac{G M}{R^{2}}$ , where $M$ is the mass of the sphere]

• A. $\frac{m g}{2}$
• B. $\frac{3 m g}{8}$
• C. $\frac{m g}{1 6}$
• D. None of these

Answer 1

Answer: (B)

Gravitation field at mass $m$ due to a full solid sphere
$\overset{ \rightarrow }{E}_{1}=-\frac{g_{0} \overset{ \rightarrow }{r}}{R}=-\frac{g_{0} \hat{r}}{2}$
Gravitational field due to the cavity at mass $m$
$M'=\frac{\left(R/2\right)^{3}}{R^{3}}M$ $\Rightarrow M'=\frac{M}{8}$
$\left(\overset{ \rightarrow }{\textit{E}}\right)_{2}=-\left(\frac{- \left(\textit{GM}\right)^{'} \hat{ \, \textit{r}}}{\left(\textit{R}\right)^{2}}\right)\text{,}$
$=\frac{GM \hat{\textit{r}}}{8 \textit{R}^{2}}=\frac{\textit{g}_{0} \hat{r}}{8}$
$\left(\overset{ \rightarrow }{\textit{E}}\right)_{net}=\left(\frac{- \left(\textit{g}\right)_{0}}{2} + \frac{\left(\textit{g}\right)_{0}}{8}\right)\hat{\textit{r}}$
$=\frac{- 3 \textit{g}_{0} \hat{r}}{8}$
$\Rightarrow \overset{ \rightarrow }{\textit{F}}_{\text{net}}=-\frac{3 \textit{mg}_{0} \hat{r}}{8}$