Determine the force due to hydrostatic pressure on the flat vertical side of a tank which has the shape in feet of the boundaries y=0, y = 8 ln x, y = 8 ln(−x) and the line y=8 ln18.
Note that water has density 62.4 lb/ft3.
Can somebody help me with this? My problem is that I have absolutely no idea where to start (and the textbook is not helping matters...)
The tank is a volume bounded by surfaces. One of the bounding surfaces is a vertical plane, and the boundaries of that surface are four curves or lines in the xy-plane.
IT IS NOT CLEAR WHICH IS THE TOP AND WHICH IS THE BOTTOM, but in the absence of that bit of information I will assume that
\(\displaystyle y_1=0\) is the bottom, and
\(\displaystyle y_2=8 \ln 18\) is the top.
Assuming also that the tank is full, we will find the total force by integrating the incremental Force
\(\displaystyle \displaystyle \int_{y_1}^{y_2} F(y)\ dy \).
How
wide is the tank at a height \(\displaystyle y\) above the bottom? It looks symmetric in the x-direction, with the two side given by
\(\displaystyle \displaystyle x = -e ^{y/8} \text{ and }x = +e ^{y/8} \)
What is the incremental Area (square feet) between \(\displaystyle y\) and \(\displaystyle y+dy\)?
Find the pressure \(\displaystyle P(y)\) as the weight of a 1-sq.ft. column of water
above \(\displaystyle y\), and
use the relationship \(\displaystyle F = P \times A\) to find the incremental force.
If you need more help,
please show us your work.
