Question

The ratio of surface tensions of mercury and water is given to be $7.5$ , while the ratio of their densities is $13.6$ . Their contact angles, with glass, are close to $135^\circ $ and $0^\circ $ , respectively. If it is observed that mercury gets depressed by an amount $h$ in a capillary tube of radius $r_{1}$ , while water rises by the same amount $h$ in a capillary tube of radius $r_{2}$ , then the ratio $\frac{r_{1}}{r_{2}}$ is close to

• A. $\frac{3}{5}$
• B. $\frac{2}{3}$
• C. $\frac{4}{5}$
• D. $\frac{2}{5}$

Answer 1

Answer: (D)

$\left|\right.h_{Hg}\left|\right.=\left|\right.h_{water}\left|\right.$
$\frac{2 S_{1} c o s \theta _{1}}{\rho _{1} r_{1} g}=\frac{2 S_{2} c o s \theta _{2}}{\rho _{2} r_{2} g}$
Therefore, the ratio of the radii of the respective tubes is,
$\frac{r_{1}}{r_{2}}=\frac{\rho _{2}}{\rho _{1}}\times \frac{S_{1} cos \theta _{1}}{S_{2} cos \theta _{2}}$
$\frac{r_{1}}{r_{2}}=\frac{1}{13.6}\times 7.5\times \frac{1}{\sqrt{2}}$
$\frac{r_{1}}{r_{2}}\approx\frac{2}{5}$