Consider a completely submerged curve surface under water. Consider an elementary area da situated at a depth ‘h’ below free water surface. As shown

Total pressure acting on elementary area (dA)

dp  =  γ. h. dA

Total pressure,   ∫ dp  = ∫ γhd A

P  =   ∫ γhd A

But in case of curved surface the eh direction f the total pressure on the elementary area is not in the same direction, but varies from point to point. Thus the integration of Equation for curved surface is impossible. Sothat this problem will be solved by resolving the force P into horizontal and vertical components P_H and P_V.

Then total force on the curved surface is,

P =  √(P_H² + P_v²)

Direction of resultant force,  θ = tan^-1  (P_V / P_H)

Horizontal components,         P_H = ∫ γ.h. dA. sin θ

γ ∫ h dA sin θ  = The total pressure force ont eh projected area of the curved surface on the vertical plane.

P_H = the total pressure force on the projected area of the curved surface of the

Vertical components,

P_V  = ∫ γ.h. dA. cos θ

But da sin θ = vertical projected area of the curved surface as shown in.

Now, da cos θ = horizontal projected area of the curve surface as shown in.

∫ .h. dA. cos θ  = total volume of liquid contained between curved surface and free surface.

γ. ∫ hdA cos θ. = total weight of liquid contained between curved surface and free surface.

P_V  = total weight of liquid contained between curved surface and free surface.

For more help in Hydrostatic Forces on Cured Surface click the button below to submit your homework assignment