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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