Problem Statement
A manometer is attached to a tank containing three different fluids as shown in the figure. What will be the difference in elevation of the mercury column in the manometer (\( y \))?
Solution
Given:
- Specific weight of water (\( \gamma \)) = \( 9.81 \, \text{kN/m}^3 \)
- Specific weight of mercury (\( \gamma_{\text{m}} \)) = \( 13.6 \times 9.81 = 133.416 \, \text{kN/m}^3 \)
- Specific weight of oil (\( \gamma_{\text{oil}} \)) = \( 0.82 \times 9.81 = 8.0442 \, \text{kN/m}^3 \)
Pressure Balance Equation:
Using the pressure balance between points A and B:
\( P_A = P_B \)
Substitute the terms for the fluids:
\( 35 + \gamma_{\text{oil}} h_{\text{oil}} + \gamma h = \gamma_{\text{m}} y \)
Substitute the known values:
\( 35 + 8.0442 \times 3 + 9.81 \times 3 = 133.416 y \)
Simplify:
\( 35 + 24.1326 + 29.43 = 133.416 y \)
\( 88.5626 = 133.416 y \)
Solve for \( y \):
\( y = \frac{88.5626}{133.416} \)
\( y = 0.66 \, \text{m} \)
Explanation
This problem determines the difference in elevation of the mercury column in a manometer using hydrostatic principles:
- The pressure at point A is balanced by the pressure at point B, accounting for the contributions of the oil, water, and mercury columns.
- The specific weights of the fluids and their respective heights are used to compute the pressure difference and find the height difference in the mercury column.
Physical Meaning
- Manometer Height Difference (\( y \)): The calculated height represents the equilibrium condition for the pressure balance between the fluids.
- Specific Weight Contribution: The heavier mercury contributes significantly to the height difference compared to water and oil.
- Hydrostatic Balance: The pressure balance ensures that the forces exerted by the fluids remain in equilibrium.
