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 and height h=20 cm. There's a circular hole in the bottom of the glass of radius r. The maximum pressure that pushes the water back into the hole is roughly (on the order of) p=σ/r, where σ=0.072 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 such that water doesn't drip out of the glass.

Details and assumptions
The gravitational acceleration is g=−9.8 m/s2 and the glass is placed vertically.
Neglect any other effects that can influence the pressure from other external sources.

ρgh=σ/r

r = σ/ ρgh=0.0072/1000•9.8•0.2 =
= 3.7•10⁻⁵ m= 37μm

To determine the largest possible radius of the hole in micrometers (μm) such that water doesn't drip out of the glass, we can use the concept of hydrostatic pressure and the maximum pressure that can push the water back into the hole.

We start by considering the force balance acting on a small circular area at the bottom of the hole. There are two main forces involved: the hydrostatic pressure acting upwards and the surface tension force acting downwards.

1. Hydrostatic Pressure:
The hydrostatic pressure at any point within a fluid is given by the equation: P = ρgh, where ρ is the mass density of the fluid, g is the acceleration due to gravity, and h is the depth.

In our case, since the glass is placed vertically, the depth h is equal to the height of the water in the glass. Thus, the hydrostatic pressure at the bottom of the hole is: P = ρgh.

2. Surface Tension Force:
The surface tension force acts along the circumference of the hole and tends to flatten out the surface of the water. The magnitude of this force is given by the equation: F = 2πrσ, where r is the radius of the hole and σ is the surface tension coefficient of water.

For water not to drip out of the glass, the maximum pressure pushing the water back into the hole should be greater than the surface tension force. Thus, we have: ρgh > 2πrσ.

To find the largest possible radius of the hole, we rearrange this equation to solve for r:

r < (ρgh) / (2πσ)

Now we can substitute the given values and solve for r:

ρ = 1000 kg/m^3 (density of water)
g = -9.8 m/s^2 (acceleration due to gravity, negative sign because the glass is placed vertically)
h = 20 cm = 0.2 m (height of water in the glass)
σ = 0.072 N/m (surface tension of water)

r < (1000 kg/m^3 * -9.8 m/s^2 * 0.2 m) / (2π * 0.072 N/m)

Calculating this expression gives us the largest possible radius of the hole in meters. To convert this into micrometers, we multiply by 10^6:

r < ((1000 * -9.8 * 0.2) / (2π * 0.072)) * 10^6 μm

By evaluating this expression, we get the estimated largest possible radius of the hole in micrometers, ensuring water doesn't drip out of the glass.