A) \[{{v}_{1}}={{v}_{2}}\]
B) \[{{v}_{1}}>{{v}_{2}}\]
C) \[{{v}_{1}}<{{v}_{2}}\]
D) \[\frac{{{v}_{1}}}{{{r}_{1}}}=\frac{{{v}_{2}}}{{{r}_{2}}}\]
Correct Answer: C
Solution :
Key Idea: Gravitational force provides the required centripetal force. The gravitational force exerted on the planet is \[{{F}_{1}}=\frac{GMm}{r_{1}^{2}}=\frac{g{{R}^{2}}m}{r_{1}^{2}}\] where \[GM=g{{R}^{2}}\] Similarly \[{{F}_{2}}=\frac{GMm}{r_{2}^{2}}=\frac{g{{R}^{2}}m}{r_{2}^{2}}\] This gravitational force provides the necessary centripetal force, hence we have \[\frac{g{{R}^{2}}m}{r_{1}^{2}}=\frac{mv_{1}^{2}}{{{r}_{1}}}\] \[\Rightarrow \] \[v_{1}^{2}=\frac{g{{R}^{2}}}{{{r}_{1}}}\] \[\Rightarrow \] \[{{v}_{1}}=\sqrt{\frac{g{{R}^{2}}}{{{r}_{2}}}}\] Similarly \[{{v}_{2}}=\sqrt{\frac{g{{R}^{2}}}{{{r}_{2}}}}\] \[\therefore \] \[\frac{{{v}_{1}}}{{{v}_{2}}}=\sqrt{\frac{{{r}_{2}}}{{{r}_{1}}}}\] Given, \[{{r}_{1}}>{{r}_{2}}\] \[{{v}_{2}}>{{v}_{1}}\] Note: Greater the distance of planet above earth's surface, smaller is the speed of planet. Because speed of planet does not depend upon the mass of planet.You need to login to perform this action.
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