Question

A fixed container of height ${}^{\prime }{H}^{\prime }$ with large cross-sectional area ${}^{\prime }{A}^{\prime }$ is completely filled with water. orifice of cross-sectional area ${}^{\prime }{a}^{\prime }$ are made, one at the bottom and the other on the vertical wall container at a distance $H/2$ from the top of the container. Find the time taken by the water level of a height of $H/2$ to get reduced to ground level .

Answer 1

$t=\frac{A}{a}(\sqrt{H_1}-\sqrt{H_2})\sqrt{\frac{2}{g}}$

$t_1$-time taken to reduce water level to $\frac{H}{2}$ when only upper orifice is open

$t_1=\frac{A}{a}\left(\sqrt{H}\right)\sqrt{\frac{2}{g}}{t}_1=\frac{A}{a}\sqrt{\frac{2H}{g}}$

Similarly,

$t_2=\frac{A}{a}(\sqrt{H}-\sqrt{\frac{H}{2}})\sqrt{\frac{2}{g}}=\frac{A}{a}\left(\frac{\sqrt{2}-1}{\sqrt{2}}\right)\sqrt{\frac{2H}{g}}$

$T$-time taken when both are open

$\frac{1}{T}=\frac{1}{t_1}+\frac{1}{t_2}T=\frac{\frac{\sqrt{2}-1}{\sqrt{2}}\sqrt{\frac{2H}{g}}}{1+\frac{\sqrt{2}-1}{\sqrt{2}}}T=\frac{\sqrt{2}-1}{2\sqrt{2}-1}\sqrt{\frac{2H}{g}}$