PROBLEM 9.65 Show that the system of hydrostatic forces acting on a submerged plane area Acan be reduced to a force Pat the centroid Cof the area and two couples. The force Pis perpendicular to the area and is of magnitude sinPAyγθ=, where is the specific weight of the liquid, and the couples are ( )sinxxI′′=Miand ( )sinyxyI′=Mj, where xyIxy dA=∫(see Sec. 9.8). Note that the couples are independent of the depth at which the area is submerged. SOLUTION The pressure pat an arbitrary depth ( )sinyis ( )sinpy=so that the hydrostatic force dFexerted on an infinitesimal area dAis ( )sindFydAγθ=Equivalence of the force P and the system of infinitesimal forces dFrequires :sinsinFP dFy d
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