Show that the system of hydrostatic forces acting on a

Chapter 9, Problem 9.65

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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 \(P=gA\overline{y}\sin u\), where g is the specific weight of the liquid, and the couples are \(\mathbf{M}_{x^{\prime}}=\left(\mathrm{g} \bar{I}_{x^{\prime}} \sin \mathrm{u}\right) \mathbf{i}\) and \(\mathbf{M}_{y^{\prime}}=\left(g \bar{I}_{x^{\prime} y^{\prime}} \sin \mathrm{u}\right) \mathbf{j}\), where \(\bar{I}_{x^{\prime} y^{\prime}}=\int x^{\prime} y^{\prime} d A\) (see Sec. 9.8). Note that the couples are independent of the depth at which the area is submerged.