A) \[\frac{\rho -{{\rho }_{2}}}{{{\rho }_{2}}-\rho }\]
B) \[\frac{{{\rho }_{2}}-\rho }{\rho -{{\rho }_{1}}}\]
C) \[\frac{\rho +{{\rho }_{1}}}{\rho +{{\rho }_{2}}}\]
D) \[\frac{\rho +{{\rho }_{2}}}{\rho +{{\rho }_{1}}}\]
E) \[\frac{\sqrt{{{\rho }_{1}}\,{{\rho }_{2}}}}{\rho }\]
Correct Answer: A
Solution :
Let\[{{V}_{1}}\] and\[{{V}_{2}}\]be the volumes, then \[{{V}_{1}}+{{V}_{2}}=V\] As ball is floating. Weight of ball = upthrust on ball due to two liquids \[V\rho g={{V}_{1}}{{\rho }_{1}}g+{{V}_{2}}{{\rho }_{2}}g\] \[\Rightarrow \] \[V\rho ={{V}_{1}}{{\rho }_{1}}+(V-{{V}_{1}}){{\rho }_{2}}\] \[\Rightarrow \] \[{{V}_{1}}=\left( \frac{\rho -{{\rho }_{2}}}{{{\rho }_{1}}-{{\rho }_{2}}} \right)V\] Fraction in upper part\[=\frac{{{V}_{1}}}{V}=\frac{\rho -{{\rho }_{2}}}{{{\rho }_{1}}-{{\rho }_{2}}}\] Fraction in lower part\[=1-\frac{{{V}_{1}}}{V}=1-\frac{\rho -{{\rho }_{2}}}{{{\rho }_{1}}-{{\rho }_{2}}}\] \[=\frac{{{\rho }_{1}}-\rho }{{{\rho }_{1}}-{{\rho }_{2}}}\] \[\therefore \]Ratio of lower and upper parts \[=\frac{\rho -{{\rho }_{2}}}{{{\rho }_{1}}-\rho }\]
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