A) \[\frac{\rho B}{B-\left( n-1 \right){{P}_{0}}}\]
B) \[\frac{\rho B}{B+\left( n-1 \right){{P}_{0}}}\]
C) \[\frac{\rho B}{B-n{{P}_{0}}}\]
D) \[\frac{\rho B}{B+n{{P}_{0}}}\]
Correct Answer: A
Solution :
| Bulk modulus, \[B=-\frac{\Delta P}{\left( \frac{\Delta V}{{{V}_{0}}} \right)}\,\,\Rightarrow \,\,\Delta V=-{{V}_{0}}\frac{\Delta P}{B}\] |
| or \[V-{{V}_{0}}=-{{V}_{0}}\frac{\Delta P}{B}\] (Here \[{{V}_{0}}\]= volume at the surface and V = volume at the depth) |
| or \[V={{V}_{0}}-{{V}_{0}}\frac{\Delta P}{B}\,\,\,\,\Rightarrow \,\,\,\,\,V={{V}_{0}}\left( 1-\frac{\Delta P}{B} \right)\] |
| \[\therefore \] Density, \[\rho '=\frac{m}{V}=\frac{m}{{{V}_{0}}\left( 1-\frac{\Delta P}{B} \right)}\] |
| \[=\frac{m}{\frac{m}{\rho }\left( 1-\frac{n{{P}_{0}}-{{P}_{0}}}{B} \right)}\] \[(\because \,\,\,\Delta P=n{{P}_{0}}-{{P}_{0}})\] |
| \[\rho '=\frac{\rho B}{B-(n-1){{P}_{0}}}\] |
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