A vertical plane with a width of 2 ft and a height of 6ft is submerged in a tank of water and is subjected to gage air pressure of 13 psi, as illustrated in Figure EP 2.27. (a) Determine the magnitude of the resultant hydrostatic force acting on the vertical plane. (b) Determine the location of the resultant hydrostatic force acting on the vertical plane.

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Accepted Answer

(a) The magnitude of the hydrostatic resultant force acting on the vertical plane is determined by applying Equation 2.156 [latex]F= \int_{0}^{L}{pdA} =\int_{0}^{L}{(\gamma h+p_{g} )(wdh)} [/latex] as follows:

[latex]W:= 2 ft [/latex] [latex]L: = 6 ft [/latex] [latex]A:= w. L = 12 ft^{2} [/latex] [latex]\gamma :=62.417 \frac{Ib}{ft^{3} } [/latex]

[latex]P_{g} :=13 \frac{Ib}{in^{3} } \frac{(12in)^{2} }{(1in)^{2} } =1.872 imes 10^{3} \frac{Ib}{ft^{2} } [/latex]
[latex]F:= \int_{0}^{L}{(\gamma .h+ P_{g})w.dh} =2.471 imes 10^{4}Ib [/latex]

Alternatively, the magnitude of the hydrostatic resultant force acting on the vertical plane is determined by applying Equation 2.157 [latex]F=(p_{ca} + p_{g})A=(\gamma h_{ca} + p_{g})A=(\gamma L/2+ p_{g})(wL)[/latex] as follows:

[latex]h_{ca} := \frac{L}{2} =3ft [/latex] [latex]F:= (\gamma .h_{ca}+ P_{g} ).A=2.471 imes 10^{4}Ib [/latex]

(b) The location of the resultant hydrostatic force acting on the vertical plane is determined by applying Equation 2.159 [latex]h_{F} =\frac{M}{F} [/latex] as follows:

[latex]h_{F}:= \frac{\int_{0}^{L}{h.(\gamma .h+P_{g}).wdh} }{\int_{0}^{L}{(\gamma .h+ P_{g}). wdh} } =3.091 ft [/latex]

Alternatively, the location of the resultant hydrostatic force acting on the vertical plane is determined by applying Equation 2.162 [latex]h_{F} =\frac{\gamma (I_{x-ca}+A x^{2}_{ca} )+p_{g}h_{ca}A }{(\gamma h_{ca}+ p_{g})A}= \frac{\gamma (wL^{3}/12+(wL)(L/2)^{2} )+p_{g}(L/2)(wL)}{(\gamma L/2+p_{g})wL} [/latex] as follows:

[latex]I_{xca} :=\frac{w. L^{3} }{12} [/latex] [latex] h_{F}:= \frac{\gamma .(I_{xca}+A. h^{2}_{ca} )+ P_{g}. h_{ca}.A}{(\gamma .h_{ca}+ P_{g}).A} =3.091 ft [/latex]

Question 2.27: A vertical plane with a width of 2 ft and a height of 6ft is...

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