Capillarity

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius 0.1 mm is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume $V_0$ to $\dfrac{100}{101}V_0$ with the movable piston. Considering air as an ideal gas, the height $(h)$ of the liquid column in the capillary above the liquid level in cm is _____.

[Given: Surface tension of the liquid is $0.075\ \mathrm{N\,m^{-1}}$, atmospheric pressure is $10^5\ \mathrm{N\,m^{-2}}$, acceleration due to gravity $(g)$ is $10\ \mathrm{m\,s^{-2}}$, density of the liquid is $10^3\ \mathrm{kg\,m^{-3}}$ and contact angle of capillary surface with the liquid is zero]

A cylindrical capillary tube of 0.2 mm radius is made by joining two capillaries T1 and T2 of different materials having water contact angles of 0° and 60°, respectively. The capillary tube is dipped vertically in water in two different configurations, case I and II as shown in figure. Which of the following option(s) is(are) correct?

[Surface tension of water = 0.075 N/m, density of water = 1000 kg/m³, take $g = 10\ \mathrm{m/s^2}$]

A uniform capillary tube of inner radius $r$ is dipped vertically into a beaker filled with water. The water rises to a height $h$ in the capillary tube above the water surface in the beaker. The surface tension of water is $\sigma$. The angle of contact between water and the wall of the capillary tube is $\theta$. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

The following observations were taken for determining surface tension T of water by capillary method :

diameter of capillary, $D=1.25\times10^{-2}$ m

rise of water, $h=1.45\times10^{-2}$ m.

Using $g=9.80$ m/s$^2$ and the simplified relation $T=\dfrac{rhg}2\times10^3$ N/m, the possible error in surface tension is closest to :

A glass capillary tube is of the shape of a truncated cone with an apex angle $\alpha$ so that its two ends have cross sections of different radii. When dipped in water vertically, water rises in it to a height $h$, where the radius of its cross section is $b$. If the surface tension of water is $S$, its density is $\rho$, and its contact angle with glass is $\theta$, the value of $h$ will be ($g$ is the acceleration due to gravity)

A capillary tube is immersed vertically in water and the height of the water column is $x$. When this arrangement is taken into a mine of depth $d$, the height of the water column is $y$. If $R$ is the radius of earth, the ratio $\dfrac xy$ is :