The magnitudes of the gravitational force at distances $r_{1}$ and $r_{2}$ from the centre of a uniform sphere of radius R and mass M are $F_{1}$ and $F_{2}$ respectively. Then

Moderate

A

$\frac{F_{1}}{F_{2}} = \frac{r_{1}}{r_{2}} if r_{1} < R and r_{2} < R$
B

$\frac{F_{1}}{F_{2}} = \frac{r_{1}^{2}}{r_{2}^{2}} if r_{1} > R and r_{2} > R$
C

$\frac{F_{1}}{F_{2}} = \frac{r_{1}}{r_{2}} if r_{1} > R and r_{2} > R$
D

$\frac{F_{1}}{F_{2}} = \frac{r_{2}^{2}}{r_{1}^{2}} if r_{1} < R and r_{2} < R$

Solution

$\begin{array}{l} g = \frac{4}{3} πρGr ∴ g \propto r if r < R \\ g = \frac{GM}{r^{2}} ∴ g \propto \frac{1}{r^{2}} if r > R \end{array}$
$If r_{1} < R and r_{2} < R then \frac{F_{1}}{F_{2}} = \frac{g_{1}}{g_{2}} = \frac{r_{1}}{r_{2}}$
$If r_{1} > R and r_{2} > R then \frac{F_{1}}{F_{2}} = \frac{g_{1}}{g_{2}} = {(\frac{r_{2}}{r_{1}})}^{2}$

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