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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.

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

2 Answers 2

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

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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.

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

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

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