Two large, identical water tanks, 1 and 2, kept on the top of a building of height $H$, are filled with water up to height $h$ in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as that of the hole is…

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Two large, identical water tanks, 1 and 2, kept on the top of a building of height $H$, are filled with water up to height $h$ in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as that of the hole is connected to tank 2 , and the pipe ends at the ground level. When the water flows from the tanks 1 and 2 through the holes, the times taken to empty the tanks are $t_1$ and $t_2$, respectively. If $H=\left(\frac{16}{9}\right) h$, then the ratio $t_1 / t_2$ is _______ .

Aicharya · Accepted Answer

$\begin{aligned} & \mathrm{Av}=\mathrm{av}_1 \ & \mathrm{~A}\left(-\frac{\mathrm{dy}}{\mathrm{dt}}ight)=\mathrm{a} \sqrt{2 \mathrm{gy}} ; \mathrm{dt}=\frac{\mathrm{A}}{\mathrm{a} \sqrt{2 \mathrm{~g}}} \cdot \frac{-d y}{\sqrt{y}} \ & \int_0^{t_1} \mathrm{dt}=\frac{\mathrm{A}}{\mathrm{a} \sqrt{2 \mathrm{~g}}} \int_{\mathrm{h}}^0-\frac{d y}{\sqrt{y}} \ & t_1=\frac{\mathrm{A}}{\mathrm{a} \sqrt{2 \mathrm{~g}}} 2 \sqrt{\mathrm{h}} ; \mathrm{t}_1=\frac{\mathrm{A}}{\mathrm{a}} \sqrt{\frac{2 h}{g}}\end{aligned}$ $A v^{\prime}=\mathrm{av}_2$ $\begin{aligned} & A\left(-\frac{d y}{d t}ight)=a \sqrt{2 g(H+y)} \ & d t=-\frac{A}{a \sqrt{2 g}} \frac{d y}{\sqrt{H+y}} \ & \int_0^{t_2} d t=-\frac{A}{a \sqrt{2 g}} \int_H^0 \frac{d y}{\sqrt{H+y}} \ & t_2=\frac{A}{a \sqrt{2 g}}(2)(\sqrt{H+h}-\sqrt{H}) \quad \& \quad H=\frac{16 h}{9} \ & =\frac{A}{a} \sqrt{\frac{2 h}{g}}\left(\frac{5}{3}-\frac{4}{3}ight) \ & t_2=\frac{A}{a} \sqrt{\frac{2 h}{g}}\left(\frac{1}{3}ight)\end{aligned}$ ratio $\frac{\mathrm{t}_1}{\mathrm{t}_2}=3$ ^

Two large, identical water tanks, 1 and 2, kept on the top of a building of height $H$, are filled with…

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