What is the best way to approach this problem?
Does the problem change since the gate is only partially submerged?
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$\begingroup$There are at least a couple ways to set this up, but a convenient one is to start by calling the water level $ \ y = 0 \ $, measuring $ \ y \ $ "downward" from there as the "depth" variable. The center of the circle, of which the semicircle is part, is then located at $ \ x = 0 , y = 1 \ , $ making the equation for the circle is then $x^2 + (y-1)^2 \ = \ 2^2$.
The hydrostatic pressure at depth $y$ is given by $P(y) = \rho g y \ $. The infinitesimal amount of force applied to the gate at that depth is then $ \ dF = P(y) \ dA = P(y) \cdot w(y) \ dy \ , $ where $w(y)$ is the "width function" for the gate. We find this function by solving the equation of the circle for $x$, and doubling that distance to the circular edge of the gate from its symmetry axis. Thus,
$$x^2 \ = \ 4 \ - \ (y-1)^2 \ \Rightarrow \ w(y) \ = \ 2 \cdot \sqrt{4 - (y-1)^2} $$
$$dF \ = \ \rho g y \ \cdot \ 2 \cdot \sqrt{4 - (y-1)^2} \ dy \ ,$$
which we would then integrate "downward" in depth from $y = 0$ to $y = 1$ .
The set-up does change depending upon the water level, but we would still measure from the surface of the water as $y = 0$ in this approach.
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