Question

A square plate of side 3 m is submerged in water at an incline of 30 degrees with the horizontal. Calculate the fluid force on one side of the plate if the top of the plate lies at a depth of 6 m.

(* The only thing i can think to use is the equation: p * g * (S from a to b of (y * f(y) dy)) but i have no idea what to do!)

Answer 1

The hydrostatic pressure at a distance of x, measured along the plate from the top of the plate is
P(x)=ρg(6m + x sin(30°))

For a horizontal strip of width dx, and length w (=3 m), the force is
dF=pressure × area
=P(x)*w*dx
=P(x)*w*dx

Total force
= ∫dF
=∫P(x)wdx
=∫ρg(6+x sin(x))wdx
=ρgw∫(6+x sin(x))dx
where the integration limits are the length of the plate from 0 to 3 m.