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JEE Main & Advanced Physics Gravitation / गुरुत्वाकर्षण Question Bank Motion of Satellite

  • question_answer
    In the following four periods                              [AMU 2000] Time of revolution of a satellite just above the earth?s surface \[({{T}_{st}})\] Period of oscillation of mass inside the tunnel bored along the diameter of the earth \[({{T}_{ma}})\] Period of simple pendulum having a length equal to the earth?s radius in a uniform field of 9.8 N/kg \[({{T}_{sp}})\] Period of an infinite length simple pendulum in the earth?s real gravitational field \[({{T}_{is}})\]

    A)             \[{{T}_{st}}>{{T}_{ma}}\]

    B)               \[{{T}_{ma}}>{{T}_{st}}\]

    C)             \[{{T}_{sp}}<{{T}_{is}}\]

    D)               \[{{T}_{st}}={{T}_{ma}}={{T}_{sp}}={{T}_{is}}\]

    Correct Answer: C

    Solution :

                    (i)  \[{{T}_{st}}=2\pi \sqrt{\frac{{{(R+h)}^{3}}}{GM}}\]\[=2\pi \sqrt{\frac{R}{g}}\] [As h <<R and \[GM=g{{R}^{2}}]\] (ii) \[{{T}_{ma}}=2\pi \sqrt{\frac{R}{g}}\] (iii) \[{{T}_{sp}}=2\pi \sqrt{\frac{1}{g\left( \frac{1}{l}+\frac{1}{R} \right)}}=2\pi \sqrt{\frac{R}{2g}}\]            [As l = R] (iv) \[{{T}_{is}}=2\pi \sqrt{\frac{R}{g}}\]  \[[As\,\,l=\infty ]\]


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