Question

Consider a water tank shown in the figure. It has one wall at \(x = L\) and can be taken to be very wide in the z direction. When filled with a liquid of surface tension \(S\) and density \( \rho \), the liquid surface makes angle \( \theta_0 \) (\( \theta_0 < < 1 \)) with the x-axis at \(x = L\). If \(y(x)\) is the height of the surface then the equation for \(y(x)\) is: (take \(g\) as the acceleration due to gravity) 

• A. \( \frac{d^2 y}{dx^2} = \frac{\rho g}{S} y \)
• B. \( \frac{d^2 y}{dx^2} = \sqrt{\frac{\rho g}{S}} y \)
• C. \( \frac{d^2 y}{dx^2} = \sqrt{\frac{S}{\rho g}} y \)
• D. \( \frac{d^2 y}{dx^2} = \frac{S}{\rho g} y \)

Hint:

The shape of the liquid meniscus near a wall is determined by the balance between surface tension forces (related to the curvature) and gravitational forces (related to hydrostatic pressure). The Young-Laplace equation provides the fundamental relationship, which can be approximated for small slopes.