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Getting a Little "Lift" out of Calculus Part
I: Answers
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- 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).
- 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.
- 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.
- I found the following functions to represent the curves (shown
in Figure
2):
f1(x) = 14.81
f2(x) = .0065 x + 13.87
- I finished by solving the following integral:
Integral (X=0 to X=144) [f1(x) - f2(x)] dx
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135.3 - 66.3 = 69.0
lbs
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- The value calculated by FoilSim is
66 lbs, so both
answers compare favorably.
Figure 1
Figure 2
Table 1
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Measured
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Scaled
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Rectangle
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Width (cm)
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Height (cm)
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Width (sq. in.)
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Height (lbs/sq. in.)
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Area (lbs)
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1
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0.50
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4.2
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11.07
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.7
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7.7
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2
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0.50
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5.0
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11.07
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.83
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9.2
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3
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0.50
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4.6
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11.07
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.766
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8.5
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4
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0.50
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4.4
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11.07
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.733
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8.1
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5
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0.50
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4.1
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11.07
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.683
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7.5
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6
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0.50
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3.6
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11.07
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.6
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6.6
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7
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0.50
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3.1
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11.07
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.516
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5.7
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8
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0.50
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2.6
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11.07
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.433
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4.8
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9
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0.50
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2.1
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11.07
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.35
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3.9
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10
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0.50
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1.7
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11.07
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.283
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3.1
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11
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0.50
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1.1
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11.07
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.183
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2.0
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12
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0.50
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.7
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11.07
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.117
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1.3
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Total (in lbs)
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68.4
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Related Pages:
Standards
Activity
Worksheet
Lesson Index
Aerodynamics Index
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