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Getting a Little "Lift" out of Calculus Part I: Answers

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  1. When I printed out the enlarged picture of the plotter view panel, I found the curve to cover 6.5 cm. Because of this, I divided it into 12 rectangles with width .5 cm each (see Figure 1). I then took the height measurements of each rectangle and recorded them in a spreadsheet (see Table 1).
  2. The program displays the plot from 0 to 1 square foot, so I converted the X-axis to (0 to 144) square inches. I then divided the displayed 144 inches by the measured 6.5 cm to find an X-scaling factor of 22.15. On the Y-axis, 1.0 psi was equivalent to 6.0 cm, which gives the Y-scaling factor of .1666. On the spread sheet, I have multiplied by the approtiate scaling factors.
  3. I then multiplied the scaled values to find the total area under the curve. This gave a value of 68.4 lbs which represented the lift force.
  4. I found the following functions to represent the curves (shown in Figure 2):

    f1(x) = 14.81

    f2(x) = .0065 x + 13.87

  5. I finished by solving the following integral:

    Integral (X=0 to X=144) [f1(x) - f2(x)] dx

    135.3 - 66.3 = 69.0 lbs

  6. The value calculated by FoilSim is 66 lbs, so both answers compare favorably.
 

Graph of pressure versus chord with small boxes superimposed 
            to determine area under the curve Figure 1

Graph of pressure versus chord with straight lines superimposed on
            the graph lines Figure 2

Table 1

Measured

Scaled

Rectangle

Width (cm)

Height (cm)

Width (sq. in.)

Height (lbs/sq. in.)

Area (lbs)

1

0.50

4.2

11.07

.7

7.7

2

0.50

5.0

11.07

.83

9.2

3

0.50

4.6

11.07

.766

8.5

4

0.50

4.4

11.07

.733

8.1

5

0.50

4.1

11.07

.683

7.5

6

0.50

3.6

11.07

.6

6.6

7

0.50

3.1

11.07

.516

5.7

8

0.50

2.6

11.07

.433

4.8

9

0.50

2.1

11.07

.35

3.9

10

0.50

1.7

11.07

.283

3.1

11

0.50

1.1

11.07

.183

2.0

12

0.50

.7

11.07

.117

1.3

Total (in lbs)

68.4


Related Pages:
Standards
Activity
Worksheet
Lesson Index
Aerodynamics Index

 

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Editor: Tom Benson
NASA Official: Tom Benson
Last Updated: Jun 12 2014

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