How do you find the total force of water pressing on a curved surface that's underwater?

To find the total force of water pressing on a curved surface that is underwater, we break down the complex force distribution into its horizontal and vertical components. This method simplifies the problem by converting it into calculations involving flat projected areas and volumes of water. Each component is determined separately, and then they are combined to find the resultant total force.

The horizontal component of the force, often denoted as `F_x`, is equivalent to the hydrostatic force exerted on the vertical projection of the curved surface. The vertical projection is the area that would be created if the curved surface were projected onto a vertical plane, like a shadow cast by horizontal light. To calculate `F_x`, you find the pressure at the centroid of this vertical projected area and multiply it by the magnitude of the projected area. The centroid is the geometric center of an area. The line of action for this horizontal force, which is the point where the force effectively acts, passes through the center of pressure of this vertical projected area. The center of pressure is generally located below the centroid for submerged plane surfaces because pressure increases with depth.

The vertical component of the force, often denoted as `F_y`, is equal to the weight of the actual or imaginary column of water directly above the curved surface, extending up to the free surface of the water. The free surface is the top surface of the water open to the atmosphere. If the curved surface is oriented such that water is physically above it (e.g., the top surface of a submerged object), `F_y` is the weight of that real volume of water acting downwards. If the curved surface is oriented such that water is physically below it (e.g., the underside of a curved gate or the bottom surface of a submerged object), `F_y` is the weight of an *imaginarycolumn of water that would be above the surface if it were filled up to the free surface. This imaginary volume of water is the displaced volume of water between the curved surface and the free surface, and its weight corresponds to the buoyant force, acting upwards. In both scenarios, `F_y` is calculated using the formula `ρ g V`, where `ρ` is the density of the water, `g` is the acceleration due to gravity, and `V` is the volume of water (real or imaginary) directly above the curved surface up to the free surface. The line of action for this vertical force passes through the centroid of this volume of water.

Finally, to determine the total resultant force, `F_R`, you combine these two perpendicular components, `F_x` and `F_y`, using vector addition. The magnitude of the total resultant force is calculated using the Pythagorean theorem: `F_R = sqrt(F_x^2 + F_y^2)`. The direction of the total resultant force can be found by determining the angle `θ` it makes with the horizontal, where `tan(θ) = F_y / F_x`. The precise orientation of this resultant force depends on the individual directions of `F_x` and `F_y` relative to the curved surface.