A) \[({{\rho }_{2}}-\rho )(\rho -{{\rho }_{1}})\]
B) \[({{\rho }_{2}}+\rho )/(\rho +{{\rho }_{1}})\]
C) \[{{\rho }_{1}}{{\rho }_{2}}/\rho \]
D) \[{{\rho }_{1}}/{{\rho }_{2}}\]
Correct Answer: A
Solution :
[A]The weight of a floating body is equal to the weight of the displaced fluid. If V and v represent the total volume of the piece of granite and volume of granite in water respectively, we have \[V\rho g=v{{\rho }_{1}}g+(V+v){{\rho }_{2}}g\] |
Or, \[v({{\rho }_{1}}-{{\rho }_{2}})=V(\rho -{{p}_{2}})\] |
Therefore, \[v/V=(\rho -{{p}_{2}})/({{\rho }_{1}}-{{\rho }_{2}})/({{\rho }_{1}}-{{\rho }_{2}})\] |
The ratio required in the question is \[v/(V-v)\]and is given by \[v/(V-v)=(\rho -{{p}_{2}})/[({{\rho }_{1}}-{{\rho }_{2}})/(\rho -{{\rho }_{2}})]\] |
or, \[v/(V-v)=(\rho -{{p}_{2}})/({{\rho }_{1}}-\rho )=(\rho -{{\rho }_{1}})\] |
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