Chapter 9.2, Problem 9.65P

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 has a magnitude of $P=\gamma A\bar{y}\sin \theta$, where γ is the specific weight of the liquid. The couples are M_x′= ( $\gamma {\bar{I}}_{x^{\prime }}$ sin θ)i and M_y′ = ( $\gamma {\bar{I}}_{x^{\prime }{y}^{\prime }}$ sin θ)j, where ${\bar{I}}_{x^{\prime }{y}^{\prime }}$ = ${\displaystyle \int x^{\prime }{y}^{\prime }}$ dA (see Sec. 9.3). Note that the couples are independent of the depth at which the area is submerged.

Fig. P9.65