Question

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.)

Answer 1

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]