X-Component of the Resultant Force

In order to determine the magnitude of the resultant force exerted on the curved surface AB, first find its force components F_Rx and F_Ry.

The horizontal projection of the curved surface AB is the plane area AC. The x-component of the resultant force is given by the normal force acting on this plane area. That is,

     F_Rx = ρgh_c A_AC
           = (1000 kg/m³) (9.8 m/s²) (4.5 m) (3 m) (8 m)
           = 1,058 kN

Note that h_c is the vertical distance to the centroid of plane area AC.

The y-component of the resultant force is the weight of the water directly above the curved surface (i.e., imaginary volume ABEF).

     F_Ry = ρg Vol_ABEF = ρg (Vol_ADEF + Vol_ABD)
           = (1000 kg/m³) (9.8 m/s²) [(3 m) (3 m) (8 m)
                 + (π 3²/4) m² (8 m) ]
           = 1,260 kN

Hence, the resultant force is given by

     F_R = (F_Rx² + F_Ry²)^0.5
          = [(1,058 kN)² + (1,260 kN)²]^0.5
          = 1,645 kN

And the angle θ is given by

     θ = tan ^-1 (F_Ry / F_Rx)
        = tan ^-1 (1,260 kN / 1,058 kN)
        = 50^o

Also, the resultant force F_R has to pass through point D since all the pressure forces are perpendicular to the curved surface.