Pressure on Vertical Plate

Calculus 2 · Axiom Academy

EXAMPLE Pressure on Vertical Plate Set up and evaluate a hydrostatic-force integral for a submerged triangular plate. A triangular plate is submerged vertically in water. Its base of lies along the water surface and its vertex points straight down at a depth of . With water density and , find the total hydrostatic force on the plate. Triangular plate, base at the top (at the water surface) Vertex at depth below the surface Nice work — you set up and evaluated the hydrostatic force on a vertical triangular plate. Here is what carried the solution: Width as a function of depth: for a triangular plate the width varies linearly with depth; similar triangles give . Pressure formula: hydrostatic pressure at depth h is , where is fluid density and g is gravitational acceleration. Setting up the integral: the total force is , integrated over the depth range of the plate. Integration technique: expand the integrand, factor out the constants, then apply the power rule to the polynomial. Result: — about 235.2 kN pressing on the plate. This method extends to any submerged vertical surface: express the width as a function of depth, then integrate the pressure against that width.

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