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.
36.73
wrong,its 54.5
its wrong tell the right answer please
To estimate the largest possible radius of the hole in micrometers (μm) such that water doesn't drip out, we need to consider the balance between the pressure due to the water column and the maximum pressure provided by the water's surface tension.
We can start by calculating the pressure due to the water column. The pressure at any point in a fluid is given by the formula:
P = ρgh,
where P is the pressure, ρ is the density of the fluid (1,000 kg/m3 in this case), g is the acceleration due to gravity (-9.8 m/s2), and h is the height of the water column (20 cm or 0.2 m in this case).
Plugging in the values, we have:
P = (1,000 kg/m3) * (-9.8 m/s2) * (0.2 m)
P = -1960 N/m2.
Now, we can consider the maximum pressure provided by the water's surface tension, which is given as p = σ/r, where σ is the water's surface tension (0.072 N/m) and r is the radius of the hole.
To prevent water from dripping out, the pressure due to the water column should be higher than the maximum pressure from surface tension. So we equate the two values:
-1960 N/m2 = (0.072 N/m) / r.
Solving for r, we get:
r = (0.072 N/m) / (-1960 N/m2)
r ≈ -3.67 x 10^-5 m.
Since we're looking for the radius in micrometers, we need to convert the result to μm by multiplying by 10^6:
r ≈ -3.67 x 10^-5 m * 10^6
r ≈ -36.7 μm.
However, negative values for radius do not make physical sense in this context. We can disregard the negative sign and take the absolute value:
r ≈ 36.7 μm.
Therefore, the largest possible radius of the hole in micrometers (μm) such that water doesn't drip out of the glass is approximately 36.7 μm.