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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 21

  • question_answer
    Three planets of same density and with radii \[{{R}_{1}},{{R}_{2}}\] and \[{{R}_{3}}\] such that \[{{R}_{1}}=2{{R}_{2}}=3{{R}_{3}}\] have gravitation fields on the surface \[{{g}_{1}},{{g}_{2}}\] and \[{{g}_{3}}\] and escape velocities \[{{v}_{1}},{{v}_{2}}\] and \[{{v}_{3}}\] respectively. Then

    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 true  


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