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A wide vessel of uniform cross-section with a small hole in the bottom is filled with 40 \mathrm{~cm} thick layer of water and 30 \mathrm{~cm} thick layer of kerosene. The relative density of kerosene is 0.8 . The inital velocity of flow of water streaming out of the hole is \mathrm{(take \, \, g=10\, \, \mathrm{ms}^{-2} )}

Option: 1

\mathrm{\frac{2}{\sqrt{5}} \mathrm{~ms}^{-1}}


Option: 2

\mathrm{\frac{4}{\sqrt{5}} \mathrm{~ms}^{-1}}


Option: 3

\mathrm{\frac{6}{\sqrt{5}} \mathrm{~ms}^{-1}}


Option: 4

\mathrm{\frac{8}{\sqrt{5}} \mathrm{~ms}^{-1}}


Answers (1)

best_answer

Let \rho_w and \rho_k be the densities of water and kerosene. The initial weight of the liquid in the vessel \mathrm{=h_w \rho_w a g+h_k \rho_k a g} where \mathrm{h_w} and \mathrm{h_k} are the thicknesses of water and kerosene layers and a is the cross-sectional area of the vessel. Let this weight be equivalent to water layer of thickness h, then

\mathrm{ h \rho_w a g=h_w \rho_w a g+h_k \rho_k a g }
or
\mathrm{ \begin{aligned} h & =\left\{h_w+h_k\left(\frac{\rho_k}{\rho_w}\right)\right\} \\\\ & =0.4+0.3 \times 0.8=0.64 \mathrm{~m} \end{aligned} }

From Torricelli's theorem, the velocity of efflux is

\mathrm{ \begin{aligned} & v=\sqrt{2 g h} \\\\ & =\sqrt{2 \times 10 \times 0.64}=\frac{8}{\sqrt{5}} \mathrm{~ms}^{-1} \end{aligned} }

Hence the correct choice is (d).

Posted by

Divya Prakash Singh

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