Calculus_answer

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

When I printed out the enlarged picture of the plotter view panel, I found the curve to cover 9.5 cm. Because of this, I divided it into 19 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).
The program displays the plot from 0 to 1 square foot, so I converted to (0 to 144) square inches. I then divided the displayed 144 inches by the measured 9.5 cm to find a scaling factor of 15.1579. Since my measurements had been entered into a spreadsheet, multiplying by the scaling factor was very easy.
I repeated the procedure with the height and then multiplied the scaled values to find the total area under the curve. This gave a value of 740.23 lbs which represented the lift force.
I found the following functions to represent the curves (shown in Figure 2):
I finished by solving the following integral:

26.81 + 700.46 = 727.27 lbs
The true value as recorded by FoilSim is 731.6 lbs, so both answers compare favorably.

Figure 1 Figure 2

Table 1

Measured

Scaled

Rectangle

Width (cm)

Height (cm)

Width (sq. in.)

Height (lbs/sq. in.)

Area (lbs)

1

0.50

5.13

7.58

6.75

51.18

2

0.50

6.45

7.58

8.49

64.35

3

0.50

6.31

7.58

8.31

62.95

4

0.50

5.89

7.58

7.75

58.76

5

0.50

5.62

7.58

7.40

56.07

6

0.50

5.21

7.58

6.86

51.98

7

0.50

4.89

7.58

6.44

48.78

8

0.50

4.69

7.58

6.17

46.79

9

0.50

4.40

7.58

5.79

43.90

10

0.50

4.19

7.58

5.52

41.80

11

0.50

3.83

7.58

5.04

38.21

12

0.50

3.50

7.58

4.61

34.92

13

0.50

3.15

7.58

4.15

31.42

14

0.50

2.80

7.58

3.69

27.93

15

0.50

2.49

7.58

3.28

24.84

16

0.50

2.09

7.58

2.75

20.85

17

0.50

1.73

7.58

2.28

17.25

18

0.50

1.23

7.58

1.62

12.27

19

0.50

0.60

7.58

0.79

5.99

Total (in lbs)

740.23

Please send any comments to:
Web Site Related: Dale Morris (Dale.J.Morris@grc.nasa.gov), Technology Related: Tom Benson(Tom Benson@lerc.nasa.gov)

	Measured		Scaled
Rectangle	Width (cm)	Height (cm)	Width (sq. in.)	Height (lbs/sq. in.)	Area (lbs)
1	0.50	5.13	7.58	6.75	51.18
2	0.50	6.45	7.58	8.49	64.35
3	0.50	6.31	7.58	8.31	62.95
4	0.50	5.89	7.58	7.75	58.76
5	0.50	5.62	7.58	7.40	56.07
6	0.50	5.21	7.58	6.86	51.98
7	0.50	4.89	7.58	6.44	48.78
8	0.50	4.69	7.58	6.17	46.79
9	0.50	4.40	7.58	5.79	43.90
10	0.50	4.19	7.58	5.52	41.80
11	0.50	3.83	7.58	5.04	38.21
12	0.50	3.50	7.58	4.61	34.92
13	0.50	3.15	7.58	4.15	31.42
14	0.50	2.80	7.58	3.69	27.93
15	0.50	2.49	7.58	3.28	24.84
16	0.50	2.09	7.58	2.75	20.85
17	0.50	1.73	7.58	2.28	17.25
18	0.50	1.23	7.58	1.62	12.27
19	0.50	0.60	7.58	0.79	5.99

				Total (in lbs)	740.23