A Review problem. With reference to Figure 14.5, show that...

A Review problem. With reference to Figure $14.5,$ show that the total torque exerted by the water behind the dam about a horizontal axis through $O$ is $\frac{1}{6} \rho g w H^{3} .$ Show that the effective line of action of the total force exerted by the water is at a distance $\frac{1}{3} H$ above $O$ .

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Key Concepts

Torque (Moment) Calculation

Torque, or moment, is a measure of the force's tendency to cause rotation about an axis. In problems involving hydrostatic forces, it is determined by integrating the product of the local pressure force and its perpendicular distance (lever arm) from the axis of rotation.

Center of Pressure

The center of pressure is the point at which the total resultant force on a submerged surface is considered to act. It is typically located below the centroid of the area due to the higher pressure at greater depths, and its determination involves equating moments to find the effective line of action of the force.

Hydrostatic Pressure Distribution

In a fluid at rest, the pressure increases linearly with the depth below the free surface, described by the equation p = ?g h. This concept is essential as it forms the basis for determining the force exerted by the fluid on submerged surfaces, such as the face of a dam.

Resultant Force on a Submerged Surface

The total force on a submerged surface is found by integrating the pressure distribution over the area of the surface. This integration accounts for the variation in pressure with depth and yields the overall force exerted by the fluid on the surface.

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