Problem Statement
The figure shows a 1 cm diameter U-tube containing mercury. If 20 cc of water is poured into the right leg, determine the levels of the free liquid surfaces in the two tubes.
Given data:
- Diameter of U-tube = 1 cm
- Volume of water added = 20 cc (20 ml)
- Specific weight of mercury (γmercury) = 13.6 × 9.81 = 133.416 KN/m3
- Specific weight of water (γwater) = 9.81 KN/m3
Solution
1. Volume to Height Conversion
Calculate the height (\(h\)) of the water column from the volume added:
\(h = \frac{\text{Volume of water added}}{\text{Cross-sectional area}}\)
\(h = \frac{20 \times 10^{-6}}{\frac{\pi}{4} \times (0.01)^2} = 0.254 \text{ m}\)
2. Pressure Balance
Equate pressure at the same horizontal level in both legs of the U-tube:
\(γ_{mercury} \times 2y = γ_{water} \times h\)
\(133.416 \times 2y = 9.81 \times 0.254\)
\(y = \frac{9.81 \times 0.254}{2 \times 133.416} = 0.0093 \text{ m} = 0.93 \text{ cm}\)
3. Final Heights
Calculate the levels of free surfaces in both legs:
Height of free mercury level in the left leg:
\(18 + 0.93 = 18.93 \text{ cm}\)
Height of free water level in the right leg:
\(18 – 0.93 + 25.4 = 42.47 \text{ cm}\)
Final heights:
- Left leg mercury level: 18.93 cm
- Right leg water level: 42.47 cm
Explanation
- Volume to Height: The height of the water column is determined using the volume of water added and the cross-sectional area of the tube.
- Pressure Balance: At equilibrium, the pressure at the same horizontal level in both legs must be equal, leading to the relationship between mercury displacement and water column height.
- Final Levels: The rise in the left leg and the drop in the right leg are equal due to the incompressible nature of the mercury. The addition of water in the right leg further increases its free surface level.
Physical Meaning
This numerical demonstrates how liquid equilibrium is maintained in a U-tube. The interplay between the specific weights and the added water volume causes shifts in the mercury levels, which can be calculated using fundamental principles of fluid statics. Such analyses are essential in designing manometers and understanding fluid behavior in connected systems.
