Directions : In the following question more than one of the answers given may be correct. Select the correct answer and mark it according to the code:
The magnitudes of gravitational field at distance\[{{r}_{1}}\]and\[{{r}_{2}}\]from the centre of a uniform solid sphere of radius R and mass M are\[{{F}_{1}}\]and\[{{F}_{2}}\]respectively. Then (1) \[\frac{{{F}_{1}}}{{{F}_{2}}}=\frac{{{r}_{1}}}{{{r}_{2}}};if\,{{r}_{1}}<R\,and\,\,{{r}_{2}}<R\] (2) \[\frac{{{F}_{1}}}{{{F}_{2}}}=\frac{r_{2}^{2}}{r_{1}^{2}};if\,{{r}_{1}}>R\,and\,\,{{r}_{2}}>R\] (3) \[\frac{{{F}_{1}}}{{{F}_{2}}}=\frac{{{r}_{1}}}{{{r}_{2}}};if\,{{r}_{1}}>R\,and\,\,{{r}_{2}}>R\] (4) \[\frac{{{F}_{1}}}{{{F}_{2}}}=\frac{r_{1}^{2}}{r_{2}^{2}};if\,{{r}_{1}}<R\,and\,\,{{r}_{2}}<R\]A) 1, 2 and 3 are correct
B) 1 and 2 are correct
C) 2 and 4 are correct
D) 1 and 3 are correct
Correct Answer: B
Solution :
If\[r>R,\]the gravitational field due to a sphere \[F=\frac{GM}{{{r}^{2}}}\] \[\therefore \] \[{{F}_{1}}=\frac{GM}{r_{1}^{2}}\]and\[{{F}_{2}}=\frac{GM}{r_{2}^{2}}\] \[\therefore \] \[\frac{{{F}_{1}}}{{{F}_{2}}}=\frac{r_{2}^{2}}{r_{1}^{2}}\] Hence (2) is a correct option If\[r<R,\]the gravitational field due to a sphere \[F=\frac{GMr}{{{R}^{3}}}\] \[\therefore \] \[{{F}_{1}}=\frac{GM{{r}_{1}}}{{{R}^{3}}}\]and\[{{F}_{2}}=\frac{GM{{r}_{2}}}{{{R}^{3}}}\] \[\therefore \] \[\frac{{{F}_{1}}}{{{F}_{2}}}=\frac{{{r}_{1}}}{{{r}_{2}}}\] Hence (1) is a correct option
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