Total force on a curved surface is given by

Concept:

Let us consider a curved surface AB sub-merged in a static liquid as displayed here in the following figure.

Let us consider one small strip area dA at a depth of h from the free surface of the liquid. We have the following data from the above figure.

A = Total area of the curved surface

ρ = Density of the liquid

g = Acceleration due to gravity

Pressure intensity on small area dA = ρ g h

Hydrostatic force on a small area dA will be given by following the formula as mentioned here.

dF= ρ g h x dA

The direction of this hydrostatic force will be normal to the curved surface and will vary from point to point. Therefore, to secure the value of total hydrostatic force we will not integrate the above equation.

We will secure the value or expression for the total hydrostatic force on the curved surface by resolving the force dF in its two components or we can say that the dF force will be resolved in the X direction i.e. dx and in the Y direction i.e. fly.

dF_x = dF Sin θ = ρ g h x dA Sin θ

dF_y = dF Cos θ = ρ g h x dA Cos θ

Total force in X- direction and in Y- direction will be given as mentioned here.

F_x = ∫ dF_x = ∫ ρghdA sin θ = ρg ∫ hdA sin θ

Fy = ∫ dFy = ∫ ρghdA cos θ = ρg ∫ hdA cos θ

Let us analyze the above equation: 

FG will be dA Sin θ or vertical projection of area dA. Therefore, the expression for F_x will be the total pressure force on the projected area of the curved surface on the vertical plane.

F_x = Total pressure force on the projected area of the curved surface on the vertical plane.

EG will be dA Cos θ or horizontal projection of dA. Therefore, the expression for F_y will be the weight of the liquid contained between the curved surface extended up to the free surface of the liquid.

F_y = Weight of the liquid contained between the curved surface extended up to the free surface of the liquid.

The total force on a curved surface is given by F = \(\sqrt{F_x^2 + F_y^2}\)

where  F_x = Horizontal force on curved surface and is equal to total pressure force on the projected area of the curved surface on the vertical plane,

= ρgAh̅

and F_y = Vertical force on submerged curved surface and is equal to the weight of liquid actually or imaginary supported by the curved surface.