A) \[\frac{{{g}_{1}}}{{{g}_{2}}}=\frac{1}{2}\]
B) \[\frac{{{g}_{1}}}{{{g}_{3}}}=3\]
C) \[\frac{{{v}_{1}}}{{{v}_{2}}}=2\]
D) \[\frac{{{v}_{1}}}{{{v}_{3}}}=\frac{1}{3}\]
Correct Answer: C
Solution :
[c] \[g=\frac{GM}{{{R}^{2}}}=\frac{G\left( \frac{4}{3}\pi {{R}^{3}}\rho \right)}{{{R}^{2}}}\] \[\therefore \,g\propto R\] \[\therefore \,\,\,\frac{{{g}_{1}}}{{{g}_{2}}}=\frac{{{R}_{1}}}{{{R}_{2}}}=2\] \[\therefore \] (1) is not true. Further \[v=\sqrt{gR}\propto \sqrt{R(R)}\] or \[V\propto R\] \[\frac{{{v}_{1}}}{{{v}_{2}}}=\frac{{{R}_{1}}}{{{R}_{2}}}=2\] and \[\frac{{{v}_{1}}}{{{v}_{3}}}=\frac{{{R}_{1}}}{{{R}_{3}}}=3\] \[\therefore \] (4) is not trueYou need to login to perform this action.
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