## Tuesday, May 31, 2011

### MrsDrPoe: Hydrostatic Pressure Forces

Last week we talked about how pressure changes in a static body of fluid. If you recall, we noted that pressure increases as the fluid depth increases and that this pressure variation is linearly related to depth, or in mathematical terms: pressure = specific weight * depth. We call this pressure hydrostatic pressure. Today we will be briefly discussing one result of its existence.

If anything is submerged in a static body of fluid, the fluid applies a force on the object due to the hydrostatic pressure. This hydrostatic pressure force is equal to the pressure acting on the surface of the object times the surface area of the object; it always acts normally (perpendicularly) into the surface.

If we examine a swimming pool for instance, the water in the pool causes a hydrostatic pressure force on the bottom and the sides of the pool. Since the bottom of the pool is all at the same depth, the pressure acting at each point on this surface is equal. This causes a rectangular pressure distribution like the top image in the figure below. Each side of pool varies in depth, so the pressure acting at each point on the surface is not the same. This causes a triangular distribution like the one in the bottom image of the figure below.

We can calculate the resulting pressure force on these surfaces by employing the pressure prism method. This means that we will find the volume of the imaginary prism created by these pressure distributions. For the pool bottom, a rectangular prism is formed; the force is equal to the width of the pool into the page times the length of the pool times the pressure on the surface. For one side of the pool, a triangular prism is formed; the force is equal to the width of the pool into the page times 0.5 times the submerged length of the pool side times the pressure acting on the deepest point on the side of the pool (the bottom of the side).

To find the location of these resulting forces on the pool surfaces, we must find the centroids of the pressure prisms. For a rectangular prism, this would be in the very center of the surface (1/2 width, 1/2 length) and directed downward (normal to the surface). For a triangular prism, this would be at 1/2 of the width and 2/3 of the submerged length of the pool side from the top of the water.

The process for finding hydrostatic pressure forces on curved surfaces is similar but a bit more complicated, so we won't talk about that at this time.