KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    Two smooth spherical non conducting shells each of radius R having uniformly distributed charge \[Q\And -\,Q\] on their surfaces are released on a smooth non-conducting surface when the distance between their centres is 5 R. The mass of A is m and that of B is 2 m. The speed of A just before A and B collide is:
    [Neglect gravitational interaction]
    \[(take\,\,K=\frac{1}{4\pi {{\in }_{0}}})\]

    A) \[\sqrt{\frac{2\,k{{Q}^{2}}}{5\,mR}}\]

    B) \[\sqrt{\frac{4\,k{{Q}^{2}}}{5\,mR}}\]

    C) \[\sqrt{\frac{8\,k{{Q}^{2}}}{5\,mR}}\]

    D) \[\sqrt{\frac{16\,k{{Q}^{2}}}{5\,mR}}\]

    Correct Answer: A

    Solution :

    From conservation of momentum, if speed of sphere A is v, then speed of sphere B is \[\frac{v}{2}.\]
    From conservation of energy\[\frac{1}{2}m{{v}^{2}}+\frac{1}{2}(2m)\,\,{{\left( \frac{v}{2} \right)}^{2}}=\frac{-\,k{{Q}^{2}}}{5R}+\frac{k{{Q}^{2}}}{2R}\]or            \[\frac{3}{4}m{{v}^{2}}=\frac{3}{10}\frac{k{{Q}^{2}}}{R}\]
    or         \[v=\sqrt{\frac{2}{5}\frac{k{{Q}^{2}}}{mR}}\]

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