Fluid Pressure over Flat Plate of Variable Width

The resultant force is calculated by integrating the force acting on each differential area strip across the entire plate. Since pressure increases with depth, the resultant force equals the plate's total area multiplied by the pressure at its centroid depth. This resultant is equivalent to the volume under the pressure distribution curve.

The centroid represents the geometric center of the plate, while the center of pressure is where the resultant force actually acts. For plates with variable width, pressure distribution is non-uniform, causing the line of action of the resultant force to pass through the centroid of the pressure volume rather than the plate's geometric centroid.

The pressure distribution volume represents the three-dimensional shape formed by pressure values across the plate's surface. The resultant hydrostatic force equals this volume's magnitude, and its line of action passes through the centroid of this volume, intersecting the plate at the center of pressure.

Pressure increases linearly with depth in a fluid. For a differential area strip at depth, the force magnitude equals pressure times the strip's area. Since deeper portions experience greater pressure, they contribute more to the total resultant force, which is why the resultant force location depends on the pressure distribution pattern.

The resultant force magnitude equals the plate's total area multiplied by the pressure at the plate's centroid depth. This relationship simplifies calculations by converting the complex pressure distribution into a single equivalent force, making it easier to analyze the overall hydrostatic effect on the plate.

Variable width changes how pressure distributes across the plate surface. The integration process accounts for changing widths at each depth, requiring careful calculation of differential area elements. The resulting pressure distribution volume has an irregular shape, making the center of pressure location distinct from the geometric centroid.

With constant width, pressure distribution forms a uniform rectangular volume, simplifying calculations and aligning the center of pressure more predictably. Variable width plates create irregular pressure volumes where the center of pressure deviates significantly from the geometric centroid, requiring more detailed integration analysis.