A cylindrical vessel open at the top is $20$ $cm $ high and $10$ $cm$ in diameter. A circular hole whose cross-sectional area $1$ $cm^2$ is cut at the centre of the bottom of the vessel. Water flows from a tube above it into the vessel at the rate $100$ $cm^3$ $s^{^{-1}}$. The height of water in the vessel under steady state is ....... $cm$ (Take $g$ $=$ $1000 $ $cm s^{^{-2}})$
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In steady state,
Volume flow rate entering the vessel
$=$ volume flow rate leaving the vessel
$\therefore Q=a v=a \sqrt{2 g h}$ or $h=\frac{Q^{2}}{2 g a^{2}}$
$=\frac{\left(10^{2}\right)^{2}}{(2 \times 1000)(1)^{2}}=5 \mathrm{cm}$
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