Problem Statement
A vertical rectangular gate, 1.4 m high and 2 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:
\( A = 2 \times 1.4 = 2.8 \, \text{m}^2 \)
2. Location of Center of Gravity (CG)
The CG is located at:
\( y_{\text{CG}} = 3 + \frac{1.4}{2} = 3.7 \, \text{m} \)
3. Resultant Force on the Gate
The resultant force is:
\( F = \gamma \cdot A \cdot y_{\text{CG}} \)
\( F = 9810 \times 2.8 \times 3.7 = 101631 \, \text{N} = 101.631 \, \text{kN} \)
4. Location of Center of Pressure (CP)
Using the moment of inertia about the CG:
\( I_G = \frac{1}{12} \times 2 \times 1.4^3 = 0.457 \, \text{m}^4 \)
The location of the CP is:
\( y_p = y_{\text{CG}} + \frac{I_G}{A \cdot y_{\text{CG}}} \)
\( y_p = 3.7 + \frac{0.457}{2.8 \cdot 3.7} = 3.74 \, \text{m} \)
Results:
- Resultant Force: \( F = 101.631 \, \text{kN} \)
- Center of Pressure: \( y_p = 3.74 \, \text{m} \)
Explanation
- Area Calculation: The area of the gate is derived from its height and width, which determines the exposed surface area.
- Center of Gravity (CG): The location of the CG is crucial for determining the depth at which the resultant force acts.
- Resultant Force: The force exerted by the water on the gate is calculated using fluid pressure principles, which depend on the depth of the CG and the specific weight of water.
- Center of Pressure (CP): The CP is located slightly below the CG due to the pressure distribution increasing with depth.
Physical Meaning
This calculation demonstrates the principles of hydrostatics, particularly the distribution of pressure on submerged surfaces. Engineers use such analyses to design gates, dams, and other hydraulic structures to ensure stability and safety.
