question_answer

Equal volume of two immiscible liquids of densities p and 2 p are filled in a vessel as shown in figure. Two small holes are punched at depths \[\frac{h}{2}\] and\[\frac{3h}{2}\] from the surface of lighter liquid. If \[{{\operatorname{v}}_{1}}\]and\[{{\operatorname{v}}_{2}}\]are the velocities of efflux at these two holes.

A) \[\frac{1}{2\sqrt{2}}\]               

B) \[\frac{1}{2}\]

C) \[\frac{1}{4}\]                          

D) \[\frac{1}{\sqrt{2}}\]

Correct Answer: D

Solution :

\[{{\operatorname{v}}_{1}}=\sqrt{2g\times \frac{h}{2}}\] Since density is different, apply Bernoulli theorem for \[{{\operatorname{v}}_{2}}\] Since holes are open to atmosphere pressure \[={{P}_{0}}\] (atmosphere) \[={{P}_{0}}+\rho gh+2\rho \frac{h}{2}={{P}_{0}}+\frac{1}{2}2\rho {{\operatorname{v}}^{2}}_{2}\] \[{{\operatorname{v}}_{2}}=\sqrt{2gh}\] \[\frac{{{\operatorname{v}}_{1}}}{{{\operatorname{v}}_{2}}}=\frac{1}{\sqrt{2}}\]