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Chapter 3: Problem 40

Compare the height due to capillary action of water exposed to air in a circular tube of diameter \(D=0.5 \mathrm{mm}\) and between two infinite vertical parallel plates of gap \\[ a=0.5 \mathrm{mm} \\].

Short Answer

Expert verified
The capillary rise height in both scenarios is the same, as the radius of the tube or half the distance between the plates, which is the determining factor in the equation for capillary rise, is the same in both cases.

Step by step solution

01

Calculate capillary rise in the tube

We know the diameter \(D\) of the tube is 0.5 mm, so the radius \(r\) is 0.25 mm. Given that \(T = 72.8 \times 10^{-3} N/m\), \(\theta \approx 0\) for water and air, \(\rho = 1000 kg/m^3\), and \(g = 9.81 m/s^2\), we can substitute these values into the formula \(h = \frac{2T \cos(\theta)}{r \rho g}\) to get the capillary rise.
02

Calculate capillary rise between plates

In the second scenario, we have a gap of width \(a = 0.5 mm\) between two parallel plates, so \(r\), which is half the gap, is 0.25 mm. Using the same other values for \(T, \theta, \rho, g\) as in the previous scenario, and substituting these again in the formula \(h\), we can get the capillary rise.
03

Compare the capillary rise in both scenarios

Observe the capillary rise height in both scenarios. Since the formula for capillary rise height has \(r\) in the denominator, and \(r\) is the same in both scenarios, we expect the capillary rise to be the same.

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