Problem Statement
For the system shown in the figure, calculate the height \( H \) of water at which the rectangular hinged gate will just begin to rotate anticlockwise. The width of the gate is 0.5m.
Solution
1. Calculate Force Due to Water (\( F_1 \))
\[ F_1 = \gamma A \bar{y} \]
\[ = 9810 \times (1.2 \times 0.5) \times (H – 0.6) \]
\[ = 5886(H – 0.6) \]
2. Compute Center of Pressure (CP) of \( F_1 \)
\[ I_G = \frac{1}{12} \times 0.5 \times 1.2^3 = 0.072 \text{ m}^4 \]
\[ y_p1 = \bar{y} + \frac{I_G}{A \bar{y}} \]
\[ = (H – 0.6) + \frac{0.072}{1.2 \times 0.5(H – 0.6)} \]
\[ = \frac{(H – 0.6)^2 + 0.12}{(H – 0.6)} \]
3. Calculate Force Due to Air Pressure (\( F_2 \))
\[ F_2 = P_A \times A \]
\[ = 40 \times 1000 \times 1.2 \times 0.5 \]
\[ = 24000 N \]
4. Taking Moment About Hinge
\[ F_1 [y_p1 – (H – 1.2)] = F_2 \times 0.6 \]
\[ 5886(H-0.6) \left[ \frac{(H-0.6)^2+0.12}{(H-0.6)} – (H-1.2) \right] = 24000 \times 0.6 \]
\[ (H-0.6)^2 + 0.12 – (H-1.2)(H-0.6) = 2.446 \]
Final Result:
- Height of water (\( H \)) = 1.6m
Explanation
The gate will rotate when the moment due to the hydrostatic force about the hinge is equal to the moment due to the applied air pressure. This calculation accounts for the center of pressure, force application points, and equilibrium condition.
Physical Meaning
This calculation is essential for designing gates in dams and reservoirs. The hinge mechanism helps control water flow, and knowing the exact height at which the gate starts rotating helps engineers design safe and efficient hydraulic structures.
