4.5.2: Forces on Curved Surfaces - Engineering LibreTexts

The pressure is acting on surfaces perpendicular to the direction of the surface (no shear forces assumption). The element force is ( 4.5.2.1 ) 𝑑 ⁢ 𝐹 = − 𝑃 ⁢ 𝑛 ^ ⁢ 𝑑 ⁢ 𝐴 Here, the conventional notation is used which is to denote the area, 𝑑 ⁢ 𝐴 , outward as positive. The total force on the area will be the integral of the unit force ( 4.5.2.2 ) 𝐹 = − ∫ 𝐴 𝑃 ⁢ 𝑛 ^ 𝑑 𝐴 The result of the integral is a vector. So, if the 𝑦 component of the force is needed, only a dot product is needed as ( 4.5.2.3 ) 𝑑 ⁢ 𝐹 𝑦 = 𝑑 ⁢ 𝐹 ⋅ 𝑗 ^ From this analysis (eqn. 180) it can be observed that the force in the direction of 𝑦 , for example, is simply the integral of the area perpendicular to 𝑦 as ( 4.5.2.4 ) 𝐹 𝑦 = ∫ 𝐴 𝑃 ⁢ 𝑑 ⁢ 𝐴 𝑦 The same can be said for the 𝑥 direction. The force in the 𝑧 direction is ( 4.5.2.5 ) 𝐹 𝑧 = ∫ 𝐴 ℎ ⁡ 𝑔 ⁡ 𝜌 ⁢ 𝑑 ⁢ 𝐴 𝑧 Fig. 4.27. Schematic of net force on floating body. The force which acting on the 𝑧 direction is the weight of the li

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