Problem Statement
Find the minimum size of a glass tube that can be used to measure a water level if the capillary rise in the tube is to be restricted to 2 mm. Consider the surface tension of water in contact with air as 0.073575 N/m.
Given Data
- Maximum Capillary Rise, \(h = 2.0 \, \text{mm}\)
- Surface Tension, \(\sigma = 0.073575 \, \text{N/m}\)
- Fluid: Water in contact with air
- Density of water, \(\rho = 1000 \, \text{kg/m}^3\)
- Angle of contact for water-glass, \(\theta = 0^\circ\)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)
Solution
1. Convert Units to SI
The capillary rise is given in millimeters, so we convert it to meters for consistency with other SI units.
2. Apply the Capillary Rise Formula
The height \(h\) of capillary rise is given by the formula, assuming an angle of contact \(\theta = 0^\circ\) for water and glass (\(\cos(0^\circ) = 1\)):
To find the minimum diameter \(d\), we rearrange the formula:
3. Substitute Values and Calculate
Now, we substitute the known values into the rearranged formula to find the diameter \(d\).
4. Convert Diameter to Centimeters
It is often more convenient to express the result in centimeters.
The minimum diameter of the glass tube should be \( d = 1.5 \, \text{cm} \).
Explanation of Capillary Action
Capillary Action describes the movement of a liquid up a narrow tube against the force of gravity. This phenomenon is driven by the adhesive forces between the liquid (water) and the surface of the tube (glass), combined with the liquid's own surface tension. The liquid climbs the tube until the upward force from surface tension is balanced by the downward force from the weight of the liquid column.
Physical Meaning
The calculation shows that to limit the capillary rise to 2 mm, the tube must have a diameter of at least 1.5 cm. This demonstrates the inverse relationship between the tube's diameter and the height of the capillary rise (\(h \propto 1/d\)).
- If a tube with a diameter smaller than 1.5 cm were used, the capillary rise would exceed 2 mm, leading to an inaccurate measurement of the water level.
- If a tube with a diameter larger than 1.5 cm were used, the capillary rise would be less than 2 mm.
Therefore, 1.5 cm is the minimum required size to ensure the error from capillary action is kept within the specified limit.
