Problem Statement
Gate AB in the figure is 1 m long and 0.7 m wide. Calculate the force \( F \) on the gate and the position \( X \) of the center of pressure (CP).
Solution
1. Specific Weight of Oil
The specific weight of oil is:
\( \gamma = 0.81 \times 9810 = 7946 \, \text{N/m}^3 \)
2. Calculate Area of the Gate
The area of the gate is:
\( A = 0.7 \times 1 = 0.7 \, \text{m}^2 \)
3. Location of Center of Gravity (CG)
The CG is located at:
\( y_{\text{CG}} = 3 + 1 \cdot \sin 50 + \frac{1 \cdot \sin 50}{2} = 4.15 \, \text{m} \)
4. Resultant Force on the Gate
The resultant force is:
\( F = \gamma \cdot A \cdot y_{\text{CG}} \)
\( F = 7946 \times 0.7 \times 4.15 = 23083 \, \text{N} = 23.08 \, \text{kN} \)
5. Location of Center of Pressure (CP)
Using the moment of inertia about the CG:
\( I_G = \frac{1}{12} \times 0.7 \times 1^3 = 0.058 \, \text{m}^4 \)
The vertical distance of CP from the free surface is:
\( y_p = y_{\text{CG}} + \frac{I_G \cdot \sin^2 \theta}{A \cdot y_{\text{CG}}} \)
\( y_p = 4.15 + \frac{0.058 \cdot \sin^2 50}{0.7 \cdot 4.15} = 4.161 \, \text{m} \)
The vertical distance between the CP and CG is:
\( \Delta y = 4.161 – (3 + 1 \cdot \sin 50) = 0.395 \, \text{m} \)
The horizontal distance \( X \) is:
\( X = \frac{\Delta y}{\sin 50} = \frac{0.395}{\sin 50} = 0.515 \, \text{m} \)
Results:
- Resultant Force: \( F = 23.08 \, \text{kN} \)
- Horizontal Position of CP: \( X = 0.515 \, \text{m} \)
Explanation
- Specific Weight: The specific weight of oil is calculated based on its relative density and the standard weight of water.
- Area Calculation: The gate’s area is determined using its given dimensions.
- CG Location: The CG’s depth is calculated by considering the inclined surface of the gate.
- Resultant Force: The hydrostatic force is determined based on the depth of CG and specific weight.
- CP Location: The CP is slightly below the CG due to pressure distribution, and its horizontal position \( X \) is calculated using trigonometry.
Physical Meaning
This problem demonstrates the calculation of hydrostatic forces on an inclined rectangular gate. These principles are crucial in designing structures that can withstand fluid forces while ensuring stability and integrity.
