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Chapter 6: Problem 6

Why is integration used to find the total force on the face of a dam?

Short Answer

Expert verified
Answer: Integration is used to find the total force on the face of a dam because it helps sum up the infinite number of infinitesimally small elements, each having varying force due to pressure at different depths in the fluid. Integration accounts for the varying pressure in the fluid with depth, allowing us to calculate the total force acting on the dam face.

Step by step solution

01

Understanding pressure in a fluid

Pressure in a fluid increases with depth due to the weight of the fluid above any given point. Therefore, the pressure at any given depth in the fluid can be calculated by the equation: P = 蟻gh, where P is the pressure, 蟻 is the density of the fluid, g is the acceleration due to gravity, and h is the depth of the fluid.
02

Understand the force on a small area of the dam face

The force acting on an infinitesimally small area, 螖A, of the dam face is given by the product of pressure at that depth, P, and the area 螖A. Mathematically, this can be represented as: 螖F = P螖A = 蟻gh螖A.
03

Integrate to find the total force on the dam face

To find the total force on the dam face, we need to sum up the forces acting on each infinitesimally small area, 螖A, of the dam face due to the pressure at different depths. Since the force varies with depth, integration is used as a method to sum up the forces acting on the entire dam face. The total force F is given by: F = \int_{0}^{H} 蟻ghdA, where H is the total depth of the fluid and dA is an infinitesimally small area element.
04

Interpretation

Integration helps sum up the infinite number of infinitesimally small elements, each having varying force due to the pressure at different depths in the fluid. Therefore, it is used to find the total force on the face of a dam, accounting for the varying pressure in the fluid with depth.

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