Calculation for the location of support of vertical circular gate

The support must be located at a point where the moments due to F_oil and F_air are in equilibrium.
$F_{oil} = p_{cg}A$

$F_{oil} = 9.81(0.8)(8) \times \pi (4^2)$

$F_{oil} = 3155.87 ~ \text{kN}$
 

$F_{air} = p_{air}A$

$F_{air} = 40 \times \pi (4^2)$

$F_{air} = 2010.62 ~ \text{kN}$
 

Since F_oil > F_air, the hinge must be located below the point where F_oil is acting. The hinge is indicated by letter H in the figure shown.
 

 

Eccentricity
$e = \dfrac{I_g}{A\bar{Y}}$

$e = \dfrac{\frac{1}{4}\pi (4^4)}{\pi (4^2) \times 8}$

$e = 0.5 ~ \text{m}$
 

For equilibrium, sum of moment at H must be zero
$\Sigma M_H = 0$

$F_{air}(4 - y) = F_{oil}(4 - e - y)$

$2010.62(4 - y) = 3155.87(4 - 0.5 - y)$

$8042.48 - 2010.62y = 11\,045.545 - 3155.87y$

$1145.25y = 3003.065$

$y = 2.62 \, \text{m}$   ←   answer