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NEET NEET SOLVED PAPER 2015 (C)

  • question_answer
    Kepler's third law states that square of period of revolution (T) of a planet around the sun, is proportional to third power of average distance r between the sun and planet i.e.   \[{{T}^{2}}=K{{r}^{3}},\] here K is constant. If the masses of the sun and planet are M and m respectively, then as per Newton's law of gravitation force of attraction between them is \[F=\frac{GMm}{{{r}^{2}}}\], here G is gravitational constant. The relation between G and K is described as

    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 \[\frac{4{{\pi }^{2}}{{r}^{3}}}{GM}=K{{r}^{3}}\Rightarrow GMK=4{{\pi }^{2}}\]


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