Question

A body of density $\rho$ floats with a volume $V_1$ of its total volume V immersed in one liquid of density ${\rho }_1$ and with the remainder of volume $V_2$ immersed in another liquid of density ${\rho }_2$ where. ${\rho }_1>{\rho }_2$ find $V_1$ and $V_2$

• A. $V_1=V\left(\frac{\rho -{\rho }_2}{{\rho }_1-{\rho }_2}\right)V_2=V\left(\frac{{\rho }_1-\rho }{{\rho }_1-{\rho }_2}\right)$
• B. $V_1=V\left(\frac{{\rho }_1-{\rho }_2}{{\rho }_1-\rho }\right),V_2=V\left(\frac{{\rho }_1-\rho }{{\rho }_1-{\rho }_2}\right)$
• C. $V_1=V\left(\frac{\rho -{\rho }_2}{{\rho }_1-\rho }\right),V_2=V\left(\frac{{\rho }_1-{\rho }_2}{\rho -{\rho }_2}\right)$
• D. $V_1=V\left(\frac{\rho -{\rho }_2}{{\rho }_1-\rho }\right),V_2=V\left(\frac{\rho -{\rho }_2}{\rho -{\rho }_1}\right)$

Answer 1

The correct option is A $V_1=V\left(\frac{\rho -{\rho }_2}{{\rho }_1-{\rho }_2}\right)V_2=V\left(\frac{{\rho }_1-\rho }{{\rho }_1-{\rho }_2}\right)$

According to question..............

$\begin{array}{c}Giventhedensityofthebody=\rho andvolume=V\\ Thereforethemassofthebody=V\rho \\ Forafloatingbodyintwoliquids,\\ Weightofthebody=Buoyantforceofliquid1+Buoyantforceofliquid2\\ Weightofthebody=V\rho g\\ \\ Given,volumeofbodyimmersedinliquidofdensity{\rho }_1=V_1\\ and,volumeofbodyimmersedinliquidofdensity{\rho }_2=V_2\\ therefore,\\ Buoyantforceofliquid1=V_1{\rho }_{1}g[accordingtothelawoffloatation.\\ Buoyantforceofliquid2=V_2{\rho }_{2}g\\ Weightofthebody=Buoyantforceofliquid1+Buoyantforceofliquid2\\ V\rho g=V_1{\rho }_{1}g+V_2{\rho }_{2}g\\ \Rightarrow V\rho =V_1{\rho }_1+V_2{\rho }_2----------\left(i\right)\\ \Rightarrow V=V_1+V_2-----------\left(ii\right)\\ now,multiplyingequ\left(ii\right)with{\rho }_2.weget,\\ \Rightarrow V{\rho }_2=V_1{\rho }_2+V_2{\rho }_2----------\left(iii\right)\\ Now,subtractingequ\left(iii\right)fromequ\left(i\right).weget,\\ \Rightarrow V\rho -V{\rho }_2=\left[V_1{\rho }_1+V_2{\rho }_2\right]-\left[V_1{\rho }_2+V_2{\rho }_2\right]\\ \Rightarrow V\left(\rho -{\rho }_2\right)=V_1{\rho }_1-V_1{\rho }_2\\ \Rightarrow V\left(\rho -{\rho }_2\right)=V_1\left({\rho }_1-{\rho }_2\right)\\ \therefore V_1=\left(\frac{\rho -{\rho }_2}{{\rho }_1-{\rho }_2}\right)V-------------\left(A\right)\\ similarly,onmultiplyingequ\left(ii\right)by{\rho }_1.weget,\\ \Rightarrow V{\rho }_1=V_1{\rho }_1+V_2{\rho }_1--------\left(4\right)\\ \\ subtractingequ\left(4\right)fromequ\left(ii\right).weget,\\ \Rightarrow V{\rho }_1-V\rho =\left[V_1{\rho }_1+V_2{\rho }_1\right]-\left[V_1{\rho }_1+V_2{\rho }_2\right]\\ \Rightarrow V\left({\rho }_1-\rho \right)=\left[V_2{\rho }_1-V_2{\rho }_2\right]\\ \Rightarrow V\left({\rho }_1-\rho \right)=V_2\left[{\rho }_1-{\rho }_2\right]\\ \therefore V_2=\left(\frac{{\rho }_1-\rho }{{\rho }_1-{\rho }_2}\right)V------------\left(B\right)\\ so,thatthecorrectoptionisA.\end{array}$