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AMU Medical AMU Solved Paper-2013

  • question_answer
    The figure shows a spherical hollow inside a lead sphere of radius J ?; the surface of the hollow passes through the center of the sphere and touches the right side of the sphere. The mass of the sphere before hollowing was M. With what gravitatioinal force does the hollowed-out lead sphere attract a small sphere of mass m that lies at a distance d from the center of the lead sphere, on the straight line connecting the centers of the spheres and of the hollow?

    A)  \[\frac{GMm}{{{d}^{2}}}\left[ 1-\frac{1}{8{{\left( 1-\frac{R}{2d} \right)}^{2}}} \right]\]  

    B)  \[\frac{GMm}{{{d}^{2}}}\left[ 1+\frac{1}{8{{\left( 1-\frac{R}{2d} \right)}^{2}}} \right]\]

    C)  \[\frac{GMm}{{{d}^{2}}}\left[ 1-\frac{1}{4{{\left( 1+\frac{R}{2d} \right)}^{2}}} \right]\]

    D)  \[\frac{GMm}{{{d}^{2}}}\left[ 1-\frac{1}{8{{\left( 1+\frac{R}{2d} \right)}^{2}}} \right]\]   

    Correct Answer: A

    Solution :

                    From the given figure For solid \[{{F}_{1}}=\frac{GMm}{{{d}^{2}}}\] For cavity \[{{F}_{2}}=\frac{\frac{Gm}{8}(m)}{{{\left( d-\frac{R}{2} \right)}^{2}}}\]                 \[=\frac{GMm}{8{{d}^{2}}{{\left( 1-\frac{R}{2d} \right)}^{2}}}\] So, resultant force                 \[F={{F}_{1}}-{{F}_{2}}\]                 \[=\frac{GMm}{{{d}^{2}}}-\frac{GMm}{8\,{{d}^{2}}{{\left( 1-\frac{R}{2d} \right)}^{2}}}\]                 \[=\frac{GMm}{{{d}^{2}}}\left[ 1-\frac{1}{8{{\left( 1-\frac{R}{2d} \right)}^{2}}} \right]\]


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