Problem 76 - Essential Calculus

Question

Define fluid force against a submerged vertical plane region.

Step-by-Step Solution

Answer

The fluid force against a submerged vertical plane region is the force exerted by a fluid on a vertically submerged plane surface and is calculated using the formula \( F = \rho g h_{c} A \), where \( F \) is the fluid force, \( \rho \) is the fluid density, \( g \) is the acceleration due to gravity, \( h_{c} \) is the distance from the surface to the centroid of the submerged area, and \( A \) is the area of the submerged surface.

Step 1: Concept Understanding

Start with understanding the concept. Fluid pressure at any point in a static fluid is a scalar quantity. It only has magnitude and acts in all directions at a point in the fluid. The pressure varies with depth according to the equation \( P = P_0 + \rho gh \), where \( P_0 \) is the external pressure - usually atmospheric pressure, \( \rho \) is the fluid density, \( g \) is the acceleration due to gravity, and \( h \) is the height within the fluid above the point in question.

Step 2: Calculation of Fluid Force

The pressure \( P \) against a submerged plane surface is not constant and exerts a force normal to the surface. The force due to the pressure is given by the integral of the pressure over the surface area. Mathematically, it can be shown using calculus that the fluid force \( F \) exerted by a fluid on a vertically submerged plane surface is given by the equation \( F = \rho g h_{c} A \), where \( \rho \) is the fluid density, \( g \) is the acceleration due to gravity, \( h_{c} \) is the distance from the surface to the centroid of the submerged area, and \( A \) is the area of the submerged surface.

Step 3: Interpretation

Thus, the fluid force against a submerged vertical plane region is determined by the density of the fluid, the depth of the centroid to the surface of the fluid, the gravitational force, and the area of the submerged region. Hence, greater the depth and larger the surface area, greater the fluid force acting against it.