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, the 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 $\theta$, then the value of $h$ will be ($g$ is the acceleration due to gravity)

An open-ended U-tube of uniform cross-sectional area contains water (density ${10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$ ). Initially the water level stands at $0.29 \mathrm{m}$ from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density $800 \mathrm{kg} {\mathrm{m}}^{-3}$ is added to the left arm until its length is $0.1 \mathrm{m}$, as shown in the schematic figure below. The ratio $\left(\frac{h_1}{h_2}\right)$ of the heights of the liquid in the two arms is

When water is filled carefully in a glass, one can fill it to a height $h$ above the rim of the glass due to the surface tension of water. To calculate $h$ just before water starts flowing, model the shape of the water above the rim as a disc of thickness $h$ having semicircular edges, as shown schematically in the figure. When the pressure of water at the bottom of this disc exceeds what can be withstood due to the surface tension, the water surface breaks near the rim and water starts flowing from there. If the density of water, its surface tension and the acceleration due to gravity are ${10}^3 \mathrm{kg} {\mathrm{m}}^{-3},0.07 \mathrm{N} {\mathrm{m}}^{-1}$ and $10 \mathrm{m} {\mathrm{s}}^{-2}$ respectively, the value of $h$ (in $\mathrm{mm}$) is ________.