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)

$|h_{Hg}|=|h_{water}|$
$\frac{2S_{1}cos{\theta }_1}{{\rho }_1{r}_{1}g}=\frac{2S_{2}cos{\theta }_2}{{\rho }_2{r}_{2}g}$
Therefore, the ratio of the radii of the respective tubes is,
$\frac{r_1}{r_2}=\frac{{\rho }_2}{{\rho }_1}\times \frac{S_{1}cos{\theta }_1}{S_{2}cos{\theta }_2}$
$\frac{r_1}{r_2}=\frac{1}{13.6}\times 7.5\times \frac{1}{\sqrt{2}}$
$\frac{r_1}{r_2}\approx \frac{2}{5}$