question_answer

A satellite is rotating around a planet in  orbit of radius r with time period\[T\]if gravitational force changes according to \[{{r}^{5/2}},\] then \[{{\text{T}}^{\text{2}}}\]will be                           

A)  \[\propto {{r}^{3}}\]                     

B)  \[\propto {{r}^{7/2}}\]

C)  \[\propto {{r}^{9/2}}\]                 

D)  \[\propto {{r}^{3/2}}\]

Correct Answer: B

Solution :

                 For a satellite rotating around a planet in circular motion                 Gravitational force = centripetal force                 \[\frac{GM}{{{r}^{5/2}}}=\frac{m{{v}^{2}}}{r}\] \[\Rightarrow \]               \[{{v}^{2}}=\frac{GM}{{{r}^{3/2}}}\] \[\Rightarrow \]               \[{{T}^{2}}={{\left( \frac{2\pi r}{v} \right)}^{2}}\]                                 \[=\frac{4{{\pi }^{2}}{{r}^{2}}}{GM/{{r}^{3/2}}}=\frac{4{{\pi }^{2}}{{r}^{7/2}}}{GM}\] \[\Rightarrow \]               \[{{T}^{2}}\propto \,{{r}^{7/2}}\]