Question

Fluid Force on a Rectangular Plate A rectangular plate of height \(h\) feet and base \(b\) feet is submerged vertically in a tank of fluid that weighs \(w\) pounds per cubic foot. The center is \(k\) feet below the surface of the fluid, where \(h \leq k / 2\). Show that the fluid force on the surface of the plate is \(\boldsymbol{F}=w k h b\)

Short answer

The fluid force on the surface of the plate \(F = w * k * h * b\) pounds.

Step-by-step solution

Identify the pressure

Firstly, identify the pressure at the center of the plate. The pressure \(p\) at depth \(k\) in a fluid is given by the equation \(p = w * k\) where \(w\) is the weight per cubic foot of the fluid and \(k\) is the depth from the surface.

Calculate the force

Since the pressure is constant over the height of the plate due to its center being more than half the height from the surface, the force \(F\) is given by the product of pressure \(p\) and the area of the plate \(A = h * b\). Hence, the fluid force on the plate is \(F = p * A = w * k * h * b\).