73_Problem CHAPTER 9

# 73_Problem CHAPTER 9 - PROBLEM 9.65 Show that the system of...

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PROBLEM 9.65 Show that the system of hydrostatic forces acting on a submerged plane area A can be reduced to a force P at the centroid C of the area and two couples. The force P is perpendicular to the area and is of magnitude sin PA y γ θ = , where is the specific weight of the liquid, and the couples are ( ) sin xx I ′′ = Mi and ( ) sin yx y I = Mj , where xy I xy dA = (see Sec. 9.8). Note that the couples are independent of the depth at which the area is submerged. SOLUTION The pressure p at an arbitrary depth ( ) sin y is ( ) sin py = so that the hydrostatic force dF exerted on an infinitesimal area dA is ( ) sin dF y dA γθ = Equivalence of the force P and the system of infinitesimal forces dF requires :s i n s i n FP d F y d
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