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 earth then the ratio X Y ​ is

TestHub · Accepted Answer

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

At the surface of the earth, the height is $X = \frac{2 T cos θ}{ρ g r}$ .

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

The height of capillary rise at this depth is $Y = \frac{2 T cos θ}{ρ g ^{'} r}$ .

Taking the ratio of $Y$ to $X$ , we get:

$\frac{Y}{X} = \frac{g}{g ^{'}}$

Substituting the value of $g^{'}$ , we get:

$\frac{Y}{X} = \frac{g}{g ( 1 - \frac{d}{R} )} = (1 - \frac{d}{R})^{- 1}$

Answer: $(1 - \frac{d}{R})^{- 1}$

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