The contractor ran out of gunite mixture and finished the...

The contractor ran out of gunite mixture and finished the deep corner of a 5 -m-wide swimming pool with a quarter circle piece of PVC pipe, labeled $A B$ in Fig. P2.97. Compute the horizontal and vertical water forces on the curved panel $A B$

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest, which increases linearly with depth. It is calculated as the product of the fluid’s density, gravitational acceleration, and the vertical depth below the free surface. This concept is fundamental when determining the forces exerted by water on submerged surfaces, as the pressure at each point on the surface varies with its depth.

Integration of Pressure Over a Surface

When a submerged surface is subject to hydrostatic pressure, the total force on that surface is obtained by integrating the pressure over the entire area. This involves calculating the differential force on small elements of the surface (taking into account that each element experiences a different pressure due to its depth) and summing these contributions. This technique is essential for analyzing water forces on both flat and curved panels.

Force Decomposition into Horizontal and Vertical Components

For curved surfaces, the force due to hydrostatic pressure is generally not aligned with the coordinate axes. Therefore, it is necessary to decompose the resultant force into horizontal and vertical components. This is done by resolving the differential force vectors into their horizontal and vertical parts before integrating over the entire surface. This decomposition is crucial in many engineering applications, such as determining the stability and structural demands on submerged panels.

Projection Methods for Curved Surfaces

A common technique in fluid mechanics is to reduce the integration complexity for curved surfaces by projecting them onto a plane. Notably, the horizontal component of the hydrostatic force on a curved surface is equal to the force on its vertical projection. This simplification is leveraged in design and analysis, allowing for easier calculation of the force magnitudes while accounting for the non-uniform pressure distribution on the actual curved surface.

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