A) \[GK=4{{\pi }^{2}}\]
B) \[GMK=4{{\pi }^{2}}\]
C) \[K=G\]
D) \[K=\frac{l}{G}\]
Correct Answer: B
Solution :
The gravitational force of attraction between the planet and sun provide the centripetal force i.e. \[\frac{GMm}{{{r}^{2}}}=\frac{m{{v}^{2}}}{r}\Rightarrow v=\sqrt{\frac{GM}{r}}\] The time period of planet will be \[T=\frac{2\pi r}{v}\] \[\Rightarrow \,\,\,\,{{T}^{2}}=\frac{4{{\pi }^{2}}{{r}^{2}}}{\frac{Gm}{r}}=\frac{4{{\pi }^{2}}{{r}^{3}}}{GM}\] ?(i) Also from Kepler's third law \[{{T}^{2}}=K{{r}^{3}}\] ?(ii) From Eqs. (i) and (ii), we get \[\frac{4{{\pi }^{2}}{{r}^{3}}}{GM}=K{{r}^{3}}\Rightarrow GMK=4{{\pi }^{2}}\]You need to login to perform this action.
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