Chapter 9.2, Problem 9.66P

Show that the resultant of the hydrostatic forces acting on a submerged plane area A is a force P perpendicular to the area and of magnitude $P=\gamma A\bar{y}$ sin θ = $\bar{p}$A, where γ is the specific weight of the liquid and $\bar{p}$ is the pressure at the centroid C of the area. Show that P is applied at a point C_P, called the center of pressure, whose coordinates are x_p = I_xy/A $\bar{y}$ and y_p = I_x/A $\bar{y}$, where I_xy = ${\displaystyle \int xy}$ dA (see Sec. 9.3). Show also that the difference of ordinates y_p − $\bar{y}$ is equal to ${\bar{k}}_{x^{\prime }}^2/\bar{y}$ and thus depends upon the depth at which the area is submerged.

Fig. P9.66