A) \[\frac{R}{4}\]
B) \[\frac{R}{2}\]
C) \[\frac{R}{3}\]
D) \[\frac{R}{8}\]
Correct Answer: B
Solution :
We know that \[g=\frac{GM}{{{R}^{2}}}=\frac{G}{{{R}^{2}}}\,\left[ \frac{4}{3}\pi {{R}^{3}}d \right]\] Where, \[d=\] mean density of earth For planet, \[g'=\frac{G}{{{(R')}^{2}}}\times \left[ \frac{4}{3}\pi {{R}^{'3}}d \right]\] Given that, \[g=g'\] Therefore, \[=\frac{G}{{{R}^{2}}}\times \,\left[ \frac{4}{3}\pi {{R}^{3}}d \right]=\frac{G}{{{({{R}^{'}})}^{2}}}\times \frac{4}{3}\pi {{R}^{3}}(2d)\] Solving we get, \[R'=\left( \frac{R}{2} \right)\]
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