The magnitudes of the gravitational force on a point mass m 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 the correct statement is/are
Statement A: $\frac{F_{1}}{F_{2}} = \frac{r_{1}}{r_{2}}$ if $r_{1} < R$ and $r_{2} < R$
Statement B: $\frac{F_{1}}{F_{2}} = \frac{r_{1}^{2}}{r_{2}^{2}}$ if $r_{1} > R$ and $r_{2} > R$
Statement C: $\frac{F_{1}}{F_{2}} = \frac{r_{1}}{r_{2}}$ if $r_{1} > R$ and $r_{2} > R$
Statement D: $\frac{F_{1}}{F_{2}} = \frac{r_{2}^{2}}{r_{1}^{2}}$ if $r_{1} < R$ and $r_{2} < R$

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both C and D

both A and B

only A

only B

answer is A.

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Detailed Solution

$g = \frac{4}{3} π ρ G r ∴ g α r i f r < R g = \frac{G M}{r^{2}} ∴ g α \frac{1}{r^{2}} i f r > R I f r_{1} < R a n d r_{2} < R t h e n \frac{F_{1}}{F_{2}} = \frac{g_{1}}{g_{2}} = \frac{r_{1}}{r_{2}} I f r_{1} > R a n d r_{2} > R t h e n \frac{F_{1}}{F_{2}} = \frac{g_{1}}{g_{2}} = {(\frac{r_{2}}{r_{1}})}^{2}$