Answer :
To find the location of the hydrostatic force from the liquid surface on the submerged rectangular gate, we need to determine the center of pressure. The center of pressure is the point where the total hydrostatic force is considered to act.
Step-by-step Solution:
Understand the Gate Dimensions and Position:
- The gate is 1.5 meters wide and 3 meters high.
- The top edge of the gate is 2 meters below the water surface.
Calculate the Hydrostatic Force (F):
The hydrostatic force on a submerged surface is given by the formula:
[tex]F = \rho \cdot g \cdot A \cdot \bar{h}[/tex]- Where:
- [tex]\rho[/tex] is the density of water (approximately 1000 kg/m³),
- [tex]g[/tex] is the acceleration due to gravity (9.81 m/s²),
- [tex]A[/tex] is the area of the gate ([tex]1.5 \times 3 = 4.5[/tex] m²),
- [tex]\bar{h}[/tex] is the depth of the centroid of the submerged surface from the surface of the liquid (which is 2 + 1.5 = 3.5 meters).
Substitute these values into the formula:
[tex]F = 1000 \times 9.81 \times 4.5 \times 3.5[/tex]
[tex]F = 154147.5 \text{ N}[/tex]- Where:
Find the Depth to the Center of Pressure (h_cp):
The depth to the center of pressure is given by:
[tex]h_{cp} = \bar{h} + \frac{I_{G}}{A \cdot \bar{h}}[/tex]- Where [tex]I_{G}[/tex] is the second moment of area about the horizontal axis through the centroid, given by:
[tex]I_{G} = \frac{b \cdot h^3}{12}[/tex]- Here, [tex]b = 1.5[/tex] m, [tex]h = 3[/tex] m.
[tex]I_{G} = \frac{1.5 \times 3^3}{12} = 3.375 \text{ m}^4[/tex]
- Here, [tex]b = 1.5[/tex] m, [tex]h = 3[/tex] m.
- Substitute into the formula:
[tex]h_{cp} = 3.5 + \frac{3.375}{4.5 \times 3.5}[/tex]
[tex]h_{cp} = 3.5 + 0.2143[/tex]
[tex]h_{cp} = 3.7143 \text{ meters}[/tex]
- Where [tex]I_{G}[/tex] is the second moment of area about the horizontal axis through the centroid, given by:
So, the location of the hydrostatic force from the water surface is approximately 3.7143 meters.
Therefore, the force is applied at approximately 3.7143 meters from the water surface.
