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ykricky
A square plate with sides a is submerged vertically in water as shown. Express the hydrostatic force against one side of the plate as an integral and evaluate it. (Use delta for δ, the weight density constant.)
F = ∫∫p(x,y) dA p(x,y) = ρgy, where y is the depth of the water, the limits on the integral would be y from x - √[2]a to -x and x would go from 0 to ½√[2]a This is for the right side of the plate, to get the total force just multiply by 2 integrating w.r.t. y: ½ρgy² from x - √[2]a to -x, gives ½ρg[( x - √[2]a )² - (-x)² ) ½ρg[x² + 2 - 2√[2]x - x² ) ½ρg[2a² - 2a√[2]x) integrating w.r.t. x: ½ρg[2a²x - √[2]ax²], from 0 to ½√[2]a ½ρg[(2a²(½√[2]a) - √[2]a(½√[2]a)²) - (2a²(0) - √[2]a(0)²)], ½ρg[a³√[2] - a³] ½ρga³[√[2] - 1] multiply by two to get the total force: F = ρga³[√[2] - 1]