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 \[{{r}_{2}}<R\]
Correct Answer: A
Solution :
\[g=\frac{4}{3}\pi \rho Gr\] \ \[g\propto r\] if \[r<R\] \[g=\frac{GM}{{{r}^{2}}}\] \ \[g\propto \frac{1}{{{r}^{2}}}\] if \[r>R\] 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\[\frac{{{F}_{1}}}{{{F}_{2}}}=\frac{{{g}_{1}}}{{{g}_{2}}}={{\left( \frac{{{r}_{2}}}{{{r}_{1}}} \right)}^{2}}\]
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