A tank is filled with water of density $1\,g$ per $cm^3$ and oil of density $0.9\,g$ per $cm^3$ . The height of water layer is $100\,cm$ and of the oil layer is $400\,cm.$ If $g = 980\,cm/sec^2,$ then the velocity of efflux from an opening in the bottom of the tank is
Diffcult
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Let $\mathrm{d}_{\mathrm{w}}$ and $\mathrm{d}_{\mathrm{o}}$ be the densities of water and oil; then the pressure at the bottom of the tank
$=\mathrm{h}_{\mathrm{w}} \mathrm{d}_{\mathrm{w}} \mathrm{g}+\mathrm{h}_{\mathrm{o}} \mathrm{d}_{\mathrm{o}} \mathrm{g}$
Let this pressure be equivalent to pressure due to water of height $h.$ Then,
$\mathrm{hd}_{\mathrm{w}} \mathrm{g}=\mathrm{h}_{\mathrm{w}} \mathrm{d}_{\mathrm{w}} \mathrm{g}+\mathrm{h}_{\mathrm{o}} \mathrm{d}_{\mathrm{o}} \mathrm{g}$
$\therefore \mathrm{h}=\mathrm{h}_{\mathrm{w}}+\frac{\mathrm{h}_{\mathrm{o}} \mathrm{d}_{\mathrm{o}}}{\mathrm{d}_{\mathrm{w}}}=100+\frac{400 \times 0.9}{1}$
$=100+360=460$
According to Toricelli's theorem,
$\mathrm{v}=\sqrt{2 \mathrm{gh}}=\sqrt{2 \times 980 \times 460} \mathrm{cm} / \mathrm{sec}$
$=\sqrt{920 \times 980} \mathrm{cm} / \mathrm{sec}$
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