# Fluid Mechanics about Hydrostatic Pressure

The rectangular gate will open authomatically, when the depth of water becomes very large. What is the minimum value of d that will cause the gate to just open? The width of the gate is 2 m.

• This looks like homework or an exam / test question. Show what you have done so far. Apr 29, 2023 at 12:09
• I used for a formula F= (specific weight x h x A). Apr 29, 2023 at 12:25

The hydrostatic pressure can be divided into constant part causing a force per width $$\rho\cdot g\cdot h\cdot d$$ plus linear part with force per width $$\rho\cdot g\cdot \frac{h^2}{2}$$, where $$h=3m$$ is the height of the gate. The constant part can be replaced by a force acting in the center of the gate, so with lever arm $$L_1 = h/2-1.35$$ above the hinge. The force from linear pressure part will act at the $$h/3$$ from the bottom, so with a lever arm $$L_2 = 1.35-h/3$$ below the hinge. When the resulting moment is zero, the moments on the hinge from both forces must be equal:

$$h\cdot d\cdot L_1 = \frac{h^2}{2}\cdot L_2$$

From that: $$d = \frac{h}{2}\frac{L_2}{L_1} = 3.5m$$

The hydrostatic pressureat any depth, as you said, is

$$p=\rho HA$$

• H the depth measured from the bottom.
• A is the width =2m

the total hydrostaic force apllied at H/3 is a triangular pressure diagram. $$F_h= \frac{H^2}{2} \rho A$$

the gate will open if the CG of hydrostatic force triangle falls above the hinge.

$$H/3> 1.2+1.35=2.55m$$ $$H\geq 3*255 \geq 7.65$$ $$d\geq (H -(1.20+3)) \geq 7.65-(1.2+3)\geq 3.45m$$

• Aren't we going to use this equation (yp= yc + (Ixc/yc*A))? Apr 29, 2023 at 15:35
• @AyçaArırt, the proof is in the pudding. why don't you plug in the d and see where the hydrostatic force acts? Apr 29, 2023 at 17:19
• Okey i gottcha. Thank u for helping. I appreciated. Apr 29, 2023 at 17:26
• @AyçaArırt, please accept my answer then, so it helps other people looking for this kind of question. Apr 29, 2023 at 17:34
• I did it. Just one more question. Can I solve the this type of questions taking moment? Apr 29, 2023 at 18:05