Set up integrals to solve problems involving hydrostatic force.
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 \(F=PA.\)
Fact6.7.2.
Pascal’s principle states that the pressure on a submerged object depends only on its depth and not its orientation.
Activity6.7.3.
Suppose you submerge a trapezoidal plate laying horizontally 4 feet under freshwater. Your goal is to determine the total force of the water on the top of the trapezoidal plate.
Figure165.A trapezoidal plate.
(a)
What is the area \(A \) of the trapezoid? Be sure to give the correct units.
Answer.
\(A=\dfrac{5+3}{2}\cdot 2=8 \) ft\(^2\)
(b)
The weight density of fresh water is \(\rho= 62.4 \) pounds per cubic foot. What unit of measure is needed to convert from weight density \(\rho \) to pressure \(P \) in this context?
pounds
feet
square feet
square inch
Answer.
B
(c)
What physical quantity achieves the required unit from part (b)?
force
length
height
depth
Answer.
D
(d)
Using the results of parts (a), (b), and (c), calculate the force on the plate \(F\) using the formula \(F=PA\text{.}\)
Now consider that the trapezoidal plate from the previous activity is submerged vertically into freshwater so that the top side of the trapezoid is 4 feet under water.
Draw and label a horizontal rectangle across the middle of the plate of width \(l_i\) and height \(x_i\text{.}\) What is the area \(A_i\) of this rectangle?
Answer.
\(A_i=l_ix_i\)
(b)
Let \(F_i=P_iA_i\) represent the force on any such rectangle. Which of the following represent an approximation of the total force on the plate?
Again, consider a trapezoidal plate that is submerged vertically into freshwater so that the top side of the trapezoid is 4 feet under water.
(a)
Draw a picture of this situation, being sure to show the correct orientation and the correct side lengths.
(b)
Create a one-dimensional coordinate system with the origin at the water level and positive direction corresponding to positive depth.
Instructor Note.
Each team should have an axis that has the downward direction as positive.
(c)
As done in Activity 6.7.4, draw and label a rectangle to approximate the force on a small portion of the plate located at \(x_i\text{.}\) Use \(\Delta x_i\) to represent the height of the rectangle. According to your coordinate system, what is the depth \(d_i\) of this rectangle?
(d)
Using \(\rho=62.4\) lb/ft\(^3\text{,}\) write \(P_i\) in terms of \(x_i\text{.}\)
Answer.
\(P_i=62.4x_i\)
(e)
Recall that \(A_i=l_i\Delta x_i\text{.}\) The value of \(l_i\) should change linearly according to an equation \(l(x)\text{,}\) where \(l(4)=5\) and \(l(6)=3\text{.}\) Find the point-slope form of this linear equation. Then replace \(x\) with \(x_i\) to get \(l_i\text{.}\)
Answer.
\(l_i=9-x_i\)
(f)
Now, combine the results of parts (d) and (e) to calculate \(F_i=P_iA_i\) in terms of only \(x_i\) and \(\Delta x_i\text{.}\)
Answer.
\(F_i=(62.4x_i)(9-x_i)\Delta x_i\)
(g)
Find \(F=\displaystyle\int_a^b F(x)\,dx\) using the approximation formula \(F\approx \displaystyle\sum_i F_i\) by converting it to an integral through replacing \(x_i\) with \(x\) and \(\Delta x_i\) with \(dx\text{.}\) You will also have to choose appropriate value for \(a\) and \(b\text{.}\)
\begin{equation*}
F=\int_a^b \rho x l(x)\,dx,
\end{equation*}
where \(\rho\) is the weight density of the fluid, and \(l(x)\) is the function that gives the length of the approximating rectangle at location \(x\text{.}\)
Warning6.7.7.
When using Observation 6.7.6, it is required that the coordinate system be set up with the origin at the water level and with the positive direction pointing downward. Other setups will require a complete re-derivation of the formula (see: Activity 6.7.9).
Activity6.7.8.
Suppose a trapezoidal dam has height 40 feet, top width of 115 feet and bottom width of 70 feet. Water is pressed against the entire surface of the dam. Find an integral which computes the force exerted against this dam. Recall that the weight density of freshwater is \(\rho=62.4\) lb/ft\(^3\text{.}\)
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.
Figure167.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?
