Set up integrals to solve problems involving force and/or pressure.
Subsection6.7.1Activities
Fact6.7.1.
Recall that pressure is measured as force over area:
\begin{equation*}
P=F/A.
\end{equation*}
Rewriting this, we have that \(F=PA.\)
Activity6.7.2.
Consider a trapezoid-shaped dam that is 60 feet wide at its base and 90 feet wide at its top. Assume the dam is 20 feet tall with water that rises to its top. Water weighs 62.4 pounds per cubic foot and exerts \(P=62.4d\) lbs/ft\(^2\) of pressure at depth \(d\) ft. Consider a rectangular slice of this dam at height \(h_i\) feet and width \(b_i\text{.}\)
A slice at height \(h_i\) of width \(\Delta h\text{,}\) with base \(b_i\) of a damn with base 60 ft, top 90 ft, 20 ft tall.
Figure165.A slice at height \(h_i\) of width \(\Delta h\text{.}\)
(a)
At a height of \(h_i\) feet, what is the base of the rectangle \(b_i\text{?}\)
(b)
What is the area of a rectangle with base \(b_i\) feet and height \(\Delta h\) feet?
(c)
Using a depth of \(20-h_i\) feet, how much pressure is exerted on this rectangle?
(d)
Using the pressure found in (c), the area in (b), and Fact 6.7.1, how much force is exerted on this rectangle?
