I am particularly confused about finding pressure forces (specifically vertical force) on a curve surface. Although I understand the general method is to use sum of forces in the x/y direction (ie by considering the liquid weight above and so on) and find resultant force that way. I have been practising a lot of problems lately and this is one of them:
So here are all my attempts:
For the pressure diagram I did, this because pressure vary linearly.
For the horizontal force: $F_H=\:\frac{1}{2} \times \rho \times g \times h^2 \times b$ Or in this case, $h = D = 4$ and $b = B = 5$
Vertical force (my confusion):
Is the sum of forces in the y-direction equals the weight of the liquid? Or does this also include the unknown $F$? In other words, is $F_y = F + W$ or just $F_y = W$?
Also what is the equation $x = \frac{y^2}{A}$ for? I am guessing you can use that to find the self weight? ie $W = mg = \rho V g = \gamma A w$, in this case $A = A_2$
for $D$ = 4, $x = \frac{D^2}{4}=4$
$A_1 + A_2 = x \times D$
$A_1$ can be found by integration, $A_1=\:\int _0^D\:\frac{y^2}{4}dy\:\:=\:\frac{1}{12}D^3,\:\:D\:=\:4,\:A_1\:=\:\frac{1}{12}\left(4\right)^3$
So is$F_y = W$ Only?
d) For the last part, I assume you need to make moment at the hinge? But then how can I calculate this distance from the self weight of the fluid to the hinge?