Problem Statement
An inclined rectangular gate (1.5 m wide) contains water on one side. Determine the total resultant force acting on the gate and the location of the center of pressure.
Solution
1. Calculate Area of the Gate
The area of the gate is:
\( A = 1.5 \times 1.2 = 1.8 \, \text{m}^2 \)
2. Location of Center of Gravity (CG)
The CG is located at:
\( y_{\text{CG}} = 2.4 + \frac{1.2 \cdot \sin 30}{2} = 2.7 \, \text{m} \)
3. Resultant Force on the Gate
The resultant force is:
\( F = \gamma \cdot A \cdot y_{\text{CG}} \)
\( F = 9810 \times 1.8 \times 2.7 = 47676 \, \text{N} = 47.676 \, \text{kN} \)
4. Location of Center of Pressure (CP)
Using the moment of inertia about the CG:
\( I_G = \frac{1}{12} \times 1.5 \times 1.2^3 = 0.216 \, \text{m}^4 \)
The location of the CP is:
\( y_p = y_{\text{CG}} + \frac{I_G \cdot \sin^2 \theta}{A \cdot y_{\text{CG}}} \)
\( y_p = 2.7 + \frac{0.216 \cdot \sin^2 30}{1.8 \cdot 2.7} = 2.71 \, \text{m} \)
Results:
- Resultant Force: \( F = 47.676 \, \text{kN} \)
- Center of Pressure: \( y_p = 2.71 \, \text{m} \)
Explanation
- Area Calculation: The area of the gate is determined from its width and height, which represents the exposed surface area submerged in water.
- Center of Gravity (CG): The CG represents the average depth of the gate, calculated considering the inclined position.
- Resultant Force: The force is calculated based on the hydrostatic pressure principle, where pressure increases with depth.
- Center of Pressure (CP): The CP is slightly below the CG due to the uneven pressure distribution caused by the depth variation along the inclined gate.
Physical Meaning
This problem highlights the principles of hydrostatics for an inclined surface. The calculations are crucial for understanding pressure distributions and designing stable structures like gates and barriers in hydraulic systems.
