Q. A vessel, whose bottom has round holes with diameter of $1\, mm$ is filled with water. Assuming that surface tension acts only at holes, then the maximum height to which the water can be filled in vessel without leakage is: (Given, surface tension of water is $75\times 10^{-3}N/m$ and $g=10\, m/s^{2}$)
J & K CETJ & K CET 2002
Solution:
The height $(h)$ to which water rises in a capillary is
$h=\frac{2 T \cos \theta}{r \rho g}$
Where $T$ is surface tension,
$r$ the radius of capillary,
$p$ the density and
$g$ the acceleration due to gravity.
For maximum height $\theta=0$
$\therefore \cos \theta=1$
$\Rightarrow h=\frac{2 T}{r \rho g}$
Given, $R=\frac{1}{2} mm =\frac{10^{-3}}{2} m\, d=10^{3} kg / m ^{3}$
$h =\frac{2 \times 75 \times 10^{-3} \times 2}{10^{-3} \times 10^{3} \times 10} m$ $\Rightarrow h=3 \times 10^{-2} m$
$\Rightarrow h=3\, cm$
