A gate in an irrigation canal is constructed in the form of a trapezoid $4 \mathrm { ft }$ wide at the bottom, $8 \mathrm { ft }$ wide at the top, and $2 \mathrm { ft }$ high. It is placed vertically in the canal, with the water extending to its top. Find the hydrostatic force on one side of the gate.
A) $166 \mathrm { lb }$
B) $21 \mathrm { lb }$
C) $67 \mathrm { lb }$
D) $666 \mathrm { lb }$
E) None of these

Question

Quizplus · Accepted Answer

The hydrostatic force on the gate can be calculated using the formula for the force on a submerged surface: $ F = \rho \cdot g \cdot A \cdot \bar{h} $, where $\rho$ is the density of water, $g$ is the acceleration due to gravity, $A$ is the area of the surface, and $\bar{h}$ is the depth of the centroid of the surface. The area of the trapezoid is $\frac{1}{2} \times (4 + 8) \times 2 = 12 \, \text{ft}^2$. The centroid is at a depth of 1 ft. Using $\rho = 62.4 \, \text{lb/ft}^3$ and $g = 32.2 \, \text{ft/s}^2$, the force is approximately $666 \, \text{lb}$.

A Gate in an Irrigation Canal Is Constructed in the Form