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Chapter 3: Problem 53

Consider a semicylindrical trough of radius \(R\) and length \(L\) Develop general expressions for the magnitude and line of action of the hydrostatic force on one end, if the trough is partially filled with water and open to atmosphere. Plot the results (in nondimensional form) over the range of water depth \(0 \leq d / R \leq 1\).

Short Answer

Expert verified
The hydrostatic force exerted on one end of the trough is given by \( F = \gamma \cdot d \cdot \frac{1}{2} \pi R^{2} \), and it acts through the centroid of the end of the trough, which is at a depth of \( d - \frac{4R}{3\pi} \) from the water surface. The force increases linearly with the nondimensional water depth from 0 to 1.

Step by step solution

01

Define Variables

Let's denote the unit weight of water as \( \gamma \) and the depth of the water as \( d \). The semicylindrical trough has radius \( R \) and length \( L \).
02

Calculate Hydrostatic Force

The hydrostatic force exerted on the end of the trough can be calculated using the formula for the force exerted by a fluid on a surface, \( F = p \cdot A \), where \( p \) is the pressure and \( A \) is the area. Since the pressure due to a fluid column is given by \( p = \gamma \cdot d \), and the area of one end of the semicylindrical trough is \( A = \frac{1}{2} \pi R^{2} \), we find \( F = \gamma \cdot d \cdot \frac{1}{2} \pi R^{2} \).
03

Compute Line of Action

The line of action of the force is through the centroid of the area of the end of the trough, due to the symmetry of pressure distribution above and below the centroid of the semicircle. As we know, for a semicircle, the distance of the centroid from the base (line of diameter) is \( \frac{4R}{3\pi} \). The centroid is thus a depth of \( d - \frac{4R}{3\pi} \) from the water surface.
04

Dimensionless Depth and Its Variation

The nondimensional depth of water, given as \( d/R \), varies in the range 0 to 1. This means that we are considering all cases from the trough being empty (0) to the trough being completely full (1).
05

Plot the Results

Upon graphing the magnitude of the hydrostatic force against \( d/R \), we would find that the hydrostatic force linearly increases with \( d/R \), indicating that the force exerted by the water increases as the water depth increases. Note that this is a theoretical graph and real-world observations may vary due to factors like water turbulence.

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