Repeat Prob. 11-61 for a total water height of 2 m . 11-63E...

Repeat Prob. 11-61 for a total water height of 2 m . 11-63E A semicircular 30-ft-diameter tunnel is to be built under a 150 -ft-deep, $800-\mathrm{ft}$-long lake, as shown in the figure. Determine the total hydrostatic force acting on the roof of the tunnel.
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Key Concepts

Geometric Considerations and Coordinate Systems

Solving problems involving curved submerged surfaces often requires the use of appropriate coordinate systems, such as polar or parametric coordinates, to accurately describe the geometry of the object. This aids in the formulation of the integral for calculating the hydrostatic force by providing clear expressions for the area elements and their orientation, which is essential when dealing with semi-circular or other curved geometries.

Analysis of Curved Surfaces

In scenarios where the submerged surface is curved, such as a semicircular tunnel roof, the integration process must account for the geometry of the surface. The orientation of each differential area element relative to the direction of the hydrostatic pressure is important, as it affects the component of the pressure acting in the direction of interest. This geometric analysis ensures that only the appropriate component of pressure contributes to the net hydrostatic force.

Hydrostatic Pressure Distribution

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the gravitational force acting on it. It increases linearly with depth and is commonly expressed as p = ?gh, where ? is the fluid density, g is the gravitational acceleration, and h is the depth below the fluid surface. This concept is fundamental in determining the force exerted on submerged surfaces since pressure varies over the area, necessitating integration of pressure over that area to find the total force.

Integration Over Submerged Surfaces

When the pressure on a submerged surface is not constant because of its variation with depth, integration is used to compute the total force. This involves setting up an integral that sums up small differential forces, with each differential force being the product of the local pressure and an elemental area. This concept is crucial for calculating forces on surfaces with non-uniform pressure distributions, such as the curved surface of a semicircular tunnel roof.

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