A rectangular plate of height h feet and base b feet is...

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 $F=w k h b$.

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Key Concepts

Hydrostatic Pressure

In fluids at rest, pressure increases linearly with depth due to the weight of the overlying fluid. This pressure is given by the product of the fluid's weight density and the depth, and it forms the basis for calculating forces on submerged surfaces.

Integration of Pressure Over a Surface

The total force on a submerged surface is obtained by integrating the pressure distribution across its entire area. This integration accounts for the fact that pressure varies with depth, leading to a net force that is the sum of differential pressures acting over tiny elements of the surface.

Centroid and Average Pressure

For uniformly shaped surfaces, such as a rectangle, the average pressure acting on the surface can be represented by the pressure at its centroid. This simplifies the computation of the total force, allowing it to be expressed as the product of the area and the pressure at the midpoint of the submerged surface.

Weight Density of Fluid

The weight density of a fluid represents the weight per unit volume and is a key parameter in determining the hydrostatic pressure. It links the depth of immersion to the pressure exerted at that depth, directly affecting the calculation of fluid force on submerged objects.

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