Forces On a Curved Surface

 Finding the force on a curved surface

Resolving the pressure forces on a curved surface directly is difficult, so instead we draw horizontal and vertical planes around the curved surface, as shown in the first diagram, and consider the forces acting on those.

FH, the horizontal component of the overall force, is equal to Fx and can be found using the method of a flat vertical surface, finding the centroid and average pressure, then multiplying by area. Most geometries you will see will have a rectangular cross section, so this is found by finding the pressure half way down the imaginary surface, and multiplying by the height and width of the surface.

FV, the vertical component of the overall force, is equal to Fy (the force given by the pressure at the height of the horizontal imaginary surface) minus the weight of the water above it. This requires knowledge of the geometry of the curved surface to find the weight of the water.

Alternatively, FV is equal to the upthrust on the surface, which is equal to the weight of the water displaced. Again this required knowledge of geometry, but does not require knowledge of the pressure at this depth. Image 2 shows a diagram of the displacement. The green shows the displaced water.

So overall: