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 $g=\frac{GM}{R^2}$ , where $M$ is the mass of the sphere]

• A. $\frac{mg}{2}$
• B. $\frac{3mg}{8}$
• C. $\frac{mg}{16}$
• D. None of these

Answer 1

Answer: (B)

Gravitation field at mass $m$ due to a full solid sphere
${\overrightarrow{E}}_1=-\frac{g_0\overrightarrow{r}}{R}=-\frac{g_0\hat{r}}{2}$
Gravitational field due to the cavity at mass $m$
$M^{\prime }=\frac{{\left(R/2\right)}^3}{R^3}M$ $\Rightarrow M^{\prime }=\frac{M}{8}$
${\left(\overrightarrow{E}\right)}_2=-\left(\frac{-{\left(\text{GM}\right)}^{{}^{\prime }}\hat{\text{r}}}{{\left(\text{R}\right)}^2}\right)\text{,}$
$=\frac{GM\hat{\text{r}}}{8{\text{R}}^2}=\frac{{\text{g}}_0\hat{r}}{8}$
${\left(\overrightarrow{E}\right)}_{net}=\left(\frac{-{\left(\text{g}\right)}_0}{2}+\frac{{\left(\text{g}\right)}_0}{8}\right)\hat{\text{r}}$
$=\frac{-3{\text{g}}_0\hat{r}}{8}$
$\Rightarrow {\overrightarrow{F}}_{\text{net}}=-\frac{3{\text{mg}}_0\hat{r}}{8}$