A sloping plane with a width of 8ft and a height of 10 ft is submerged at a depth of 6 ft in a tank of water and is subjected to a gage pressure of 14 psi, making an 40^◦ angle with the horizontal, as illustrated in Figure EP 2.29. (a) Determine the magnitude of the resultant hydrostatic force

Question

A sloping plane with a width of 8ft and a height of 10 ft is submerged at a depth of 6 ft in a tank of water and is subjected to a gage pressure of 14 psi, making an [latex]40^{\circ } [/latex] angle with the horizontal, as illustrated in Figure EP 2.29. (a) Determine the magnitude of the resultant hydrostatic force acting on the sloping plane. (b) Determine the location of the resultant hydrostatic force acting on the sloping plane.

Accepted Answer

(a) The magnitude of the hydrostatic resultant force acting on the sloping plane is determined by applying Equation 2.171 [latex] F= \int{pdA} =\int_{0}^{L}{(\gamma h(y)+ p_{g})(wdy)} =\int_{0}^{L}{(\gamma (y_{o}+ y)\sin \alpha +p_{g} )(wdy)} [/latex], integrating over the length of the sloping planar gate using the top of the plane as a reference point for integration as follows:

[latex]W:= 8 ft [/latex] [latex]L: 10 ft [/latex] [latex]A:=w.L = 80 ft^{2} [/latex] [latex]h_{o} := 6 ft [/latex]

[latex] \alpha := 40 deg [/latex] [latex] y_{o}: = \frac{ h_{o}}{\sin (\alpha )} =9.334 ft [/latex] [latex] h(y):= (y_{o}+y) \sin (\alpha ) [/latex]

[latex]p_{g} :=14 \frac{Ib}{in^{2} } \frac{(12in)^{2} }{(1in)^{2}} =2.016 imes 10^{3} \frac{Ib}{ft^{2} } [/latex] [latex] \gamma :=62.417 \frac{Ib}{ft^{3} }[/latex]
[latex]F:= \int_{0}^{L}{(\gamma .h(y)+ p_{g}).wdy} =207288 Ib [/latex]

Alternatively, the magnitude of the hydrostatic force acting on the sloping plane may be determined by applying Equation 2.172 [latex]F= (p_{ca}+p_{g})A=(\gamma h_{ca}+p_{g})A=(\gamma ( y_{o} + y_{ca})\sin \alpha + p_{g})A= (\gamma ( y_{o} + L/2)\sin \alpha + p_{g})(wL) [/latex] as follows:

[latex]y_{ca}:= \frac{L}{2} =5 ft [/latex] [latex]h(ca):= y_{o}+ y _{ca} \sin (\alpha )= 9.214 ft [/latex]
[latex]F:= (\gamma .h(ca)+ p_{g} ).A=2.073 imes 10^{5} Ib [/latex]

(b) The location of the resultant hydrostatic force acting on the sloping plane is determined by applying Equation 2.174 [latex]y_{f} = \frac{M}{F} [/latex] as follows:

[latex]y_{F}:= \frac{\int_{0}^{L}{y.(\gamma .h(y)+ p_{g}).wdy} }{\int_{0}^{L}{(\gamma .h(y) + p_{g} ).wdy} } =5.129 ft [/latex] [latex]h(F):= y_{o}+ y_{F} \sin (\alpha ) =9.297 ft [/latex]

Alternatively, the location of the resultant force acting determined by applying Equation 2.177 [latex]z_{F} = \frac{\gamma (I_{x-ca}+ A z^{2}_{ca} )\sin \alpha + p_{g} z_{ca}A}{(\gamma ( z_{ca} \sin \alpha)+ p_{g})A}= \frac{\gamma (wL^{3}/12 +wl (y_{o}+L/2 )^{2})\sin \alpha + p_{g}(y_{o}+L/2 )(wL)}{(\gamma (y_{o}+L/2 )\sin \alpha+ p_{g} )(wL)} [/latex] as follows:

[latex]I_{xca} := \frac{w. L^{3} }{12} =666.667 ft^{4} [/latex] [latex] z(ca):= ( y_{o}+ y_{ca} )=14.334 ft [/latex]
[latex] z(F):= \frac{\gamma .(I_{xca} +A. z^{2}_{ca} ) \sin (\alpha ) + p_{g}. z(ca).A}{[\gamma (z(ca) \sin (\alpha )) + p_{g}].A} = 14.463 ft [/latex]
[latex] y(F):= z(F)- y_{o}=5.129 ft [/latex] [latex] h(F):= z(F) \sin (\alpha )= 9.297 ft [/latex]

Question 2.29: A sloping plane with a width of 8ft and a height of 10 ft is...

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