Chapter 6, Problem 31RE

To determine

To Find: The hydrostatic force on the side of the gate.

Answer to Problem 31RE

The net force is $458\text{lb}$ .

Explanation of Solution

Given:

The gate in the irrigation canal is constructed in the form of the trapezoid 3 ft wide and the bottom is 5 ft wide at the top and 2 ft high.

Calculation:

The given diagram is shown in Figure 1

  

Figure 1

Consider the vertical $x-$ axis at the centre of the gate such that the origin is the surface of the water and consider that $ABCD$ be the gate.

Then,

  $\begin{array}{c}\left|EF\right|=\left|BC\right|\\ =3\text{ft}\end{array}$

Then,

  $\begin{array}{c}\left|AD\right|=\left|5\text{ft}\right|\\ =\left|EF\right|+\left|AE\right|+\left|FD\right|\\ =\left|EF\right|+2\left|AE\right|\\ =3+2\left|AE\right|\end{array}$

Then,

  $\left|AE\right|=1\text{ft}$

Consider the distance of the height of strip from the origin is,

  $x_i^{*}$

Then, by the triangle property,

  $\begin{array}{l}\frac{a}{2-x_i^{*}}=\frac{1}{2}\\ a=\frac{1}{2}\left(2-x_i^{*}\right)\end{array}$

Then, the width of the strip is,

  $\begin{array}{c}3+2a=3+\left(2-x_i^{*}\right)\\ =5-x_i^{*}\end{array}$

Then, the area of the strip is,

  $A_o=\left(5-x_i^{*}\right)\Delta x$

Then, the pressure on the strip is,

  $\begin{array}{c}P=\delta d_i\\ =62.5x_i^{*}\end{array}$

Then, the force on the strip is,

  $F_i=62.5x_i^{*}\left(5-x_i^{*}\right)\Delta x$

Then, the total force on the side length is,

  $\begin{array}{c}F=\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}62.5x_i^{*}\left(5-x_i^{*}\right)\Delta x}\\ ={\displaystyle {\int }_0^{2}62.5x\left(5-x\right)dx}\\ =62.5\left(\frac{22}{3}\right)\text{lb}\\ =458\text{lb}\end{array}$