KVPY Sample Paper KVPY Stream-SX Model Paper-7

  • question_answer
    Water is filled in a tank up to height h. Bulk modulus of water is B and its density at surface is\[{{\rho }_{0}}\]. Density at depth h is

    A) \[\frac{{{p}_{0}}}{\left( 1-{{\rho }_{0}}gh/B \right)}\]    

    B)  \[\frac{{{p}_{0}}}{\left( 1-B/{{\rho }_{0}}gh \right)}\]

    C) \[{{p}_{0}}\left( 1-\frac{{{\rho }_{0}}gh}{B} \right)\]                 

    D) \[{{\rho }_{0}}\left( 1-\frac{B}{{{p}_{0}}gh} \right)\]

    Correct Answer: A

    Solution :

    Let mass m has volume \[{{V}_{0}}\]at surface and volume \[{{V}_{0}}-\Delta V\] at depth h.
    So, density at depth h is
    \[\rho =\frac{m}{{{V}_{0}}-\Delta V}\]
    \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\frac{\rho }{{{\rho }_{0}}}=\frac{{{V}_{0}}}{{{V}_{0}}-\Delta V}=\frac{1}{1-\Delta V/{{V}_{0}}}\]
    As, bulk modulus B is
    \[B=P\frac{\Delta V}{{{V}_{0}}}\Rightarrow \frac{\Delta V}{{{V}_{0}}}=\frac{P}{B}\]
                So,\[\frac{\rho }{{{\rho }_{0}}}=\frac{1}{1-\frac{P}{B}}\] or \[\rho =\frac{{{\rho }_{0}}}{\left( 1-\frac{{{\rho }_{0}}gh}{B} \right)}\]

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