2. A liquid with density $\rho$ and surface tension $\sigma$ has been spilled on a horizontal plate so that it forms a very large
puddle whose depth is $h$. Consider the region near the edge of the puddle, which can be viewed to good
approximation as two-dimensional. If the contact angle is $\alpha$, derive an expression for the shape of the liquid
surface $y(x)$. Assume for simplicity that $\alpha$ is small, so that the radius of curvature of the surface is large
compared with $h$ and can be approximated by $R = \left(\frac{d^2y}{dx^2}\right)^{-1}$.
