A magic glass

A typical science question young people have is "Why can a glass can contain water?" and a typical answer is "Because the distance between molecules that make up the glass is smaller than the size of each water molecule." This isn't quite right though. Imagine making a small hole in the bottom of a bottle that is full of water. If the hole is small enough, the water will not come out unless you squeeze the bottle a bit. So, answering that question with molecular distances and sizes is science overkill -- a glass can contain water even if there are holes in it. However, there's a limit on how big the holes can be.

Consider a glass with full of water of mass density ρ=1,000 kg/m3\rho=1,000~\mbox{kg/m}^3 and height h=20 cmh=20~\mbox{cm}. There's a circular hole in the bottom of the glass of radius rr. The maximum pressure that pushes the water back into the hole is roughly (on the order of) p=σ/rp=\sigma/r, where σ=0.072 N/m\sigma=0.072 ~\mbox{N/m} is the water's surface tension. This extra pressure comes from the curvature of the water surface, and it tends to flatten out the surface.

Estimate the largest possible radius of the hole in μm\mu \mbox{m} such that water doesn't drip out of the glass.

Details and assumptions

  • The gravitational acceleration is g=9.8 m/s2g=-9.8~\mbox{m/s}^2 and the glass is placed vertically.
  • Neglect any other effects that can influence the pressure from other external sources.

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