Water rises up to height ' x ' in a capillary tube immersed vertically in water. When the whole arrangement is taken to a depth ' d ' in a mine, the water level rises up to height

Question

Water rises up to height ' x ' in a capillary tube immersed vertically in water. When the whole arrangement is taken to a depth ' d ' in a mine, the water level rises up to height Y . If R is the radius of the earth then the ratio Y : x is

TestHub · Accepted Answer

The height of capillary rise is given by $h = \frac{2 T cos θ}{r ρ g}$

This implies $h \propto \frac{1}{g}$

At the surface of the earth, the capillary rise is $x \propto \frac{1}{g}$

At a depth $d$ in a mine, the acceleration due to gravity is $g^{'} = g (1 - \frac{d}{R}) = g (\frac{R - d}{R})$

The capillary rise at depth $d$ is $Y \propto \frac{1}{g ^{'}}$

Taking the ratio of $Y$ to $x$ :

$\frac{Y}{x} = \frac{g}{g ^{'}} = \frac{g}{g ( \frac{R - d}{R} )} = \frac{R}{R - d}$

Thus, $Y : x = R : (R - d)$

Answer: $R : (R - d)$

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