Cross Section
Volume 7 (2014)
EFFECTS OF WING LENGTH ON THE ROTATIONAL
VELOCITY OF FALLING PAPER COPTERS
Ian Cox, Department of Physics, Weber State University
Received 2014-04-24
Abstract: The rotation of falling paper copters was investigated in a controlled manner
through the use of high speed video capture. In particular, the dependence of rotational
velocity on wingspan was determined.
INTRODUCTION
The descent of winged bodies which pivot symmetrically around a central mass has been
examined through history by many scholars and aspiring academics. The implications of such
research help society to understand the spawning potential of oak trees, build high-performance
helicopters, or explain angular momentum, torque, and air-resistance to children in an
invigorating, thought-provoking manner. In this research study I sought to examine the effects
of varying wingspan on the rotational period of falling copters.
METHODS
In order to study the rotation of moving objects, the methods vary based on what is rotating.
To determine the rotational period of the earth about its axis, you could simply use a stopwatch
and a point of reference in the sky, and obtain the rotational period with a fairly high degree of
accuracy. To determine the rotational velocity of an atom, the methods would perhaps need to be
a bit more delicate. Paper copters fall somewhere in the middle.
http://dewey.weber.edu/crossection/
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Cox
The method which I chose to obtain measurements of the rotational velocity of copters was
high-speed video (60 frames per second) and a running clock in the frame of the video (along
with a falling copter). As opposed to having multiple copters of varying wingspans, I made a
single copter with the longest wingspan that I planned to test, and cut the wings down after each
set of data were collected at a specified wingspan (see Figures 1 & 2).
Figure 1: Copter design prior to being cut out
Figure 2: Completed copter
with largest wingspan
This method alleviated the potential for variance in copter construction, thus allowing more
experimental control. The wings were trimmed using a paper slicer with a depth stop, such that
the same amount of paper was removed with every cut. The wings were folded up together at
the time of trimming so that both wings were identically cut. The remnants of what was trimmed
off were collected and placed inside the “body” of the copter, such that the total mass of the copter
remained invariant1 between wingspan changes. The “body” of the copter was constructed by
folding the lower portion of the copter in to thirds. The copter body was folded in a manner in
1
If the mass were to decrease with each trial, the copter would have less force pulling it down due to the attractive
force of gravity that pulls the copter towards Earth. Changing the force that is causing the rotation in the first place
would fundamentally alter the experiment, and the data would be meaningless.
Cross Section 2014
Rotational velocity of falling copters
3
which the leading edge of rotation was a fold and not a free end (see Figure 3). Even if the effective
drag caused by having a free end as the leading edge were negligible, this folding procedure
would give a consistent method, and thus reduce one more variable.
Figure 3: Top view of copter body with
exaggerated folds to show layering
A bobby pin was used both as an additional mass and a clip to hold the copter body
together. The extra masses from the trimmed wings were inserted in to the folds of the body,
and held securely in place by the bobby pin. A center-line was drawn down the body of the
copter, and the pin was aligned along the central axis so that the pin did not impede rotational
motion2 by adding mass further out from the center-line. The pin’s location was verified before
each drop of the copter.
The copters were dropped from the same location in a room with no air flow. The descent
was captured on video with a frame rate of 60 fps. A stopwatch was accessed through an
internet browser, which was kept in the frame of the video for timing purposes.
Each copter was dropped six times, and the video was then analyzed frame by frame to
obtain an angular velocity (rotations per second) of each descent. An average was then taken of
the six trials for each wing length (see Appendix A). A graphical representation of the results is
displayed in Figure 4.
RESULTS
Rotation of Falling Copters as Affected by Changes in
Wingspan
Angular Velocity (rev/min)
900
800
700
600
500
400
300
8
9
10
11
12
13
14
15
16
Wing Span (a.u.)
Figure 4: Rotational velocity data showing an increase in angular velocity as a result of a decrease in wing span
2
http://physics.bu.edu/~duffy/py105/Torque.html
http://dewey.weber.edu/crossection/
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Cox
A base wing length was determined, and given a value of 1. This was then doubled, tripled,
quadrupled, etc. to obtain the wingspans that were tested above. For the interested reader, 1
arbitrary wingspan unit corresponded to 1.50 cm of physical length. Above a certain wingspan
threshold, the copter was too heavy to rotate effectively, and below a certain threshold it rotated
too quickly to make accurate velocity measurements with the equipment available. These
parameters gave way to the range of wing spans tested above.
DISCUSSION
From Figure 4, you can see that as copter wing length decreases, the rotational velocity is
increased. The question is, what causes this? I feel there are a number of factors contributing to
the increase in rotational velocity with a decrease in wingspan. The first reason would be that
reducing the wingspan of the copter means that less torque is required to rotate the copter.
Although the total mass of the copter was held fixed, the location of that mass plays a critical role
in the rotation of the body. Compare this to an ice skater throwing their arms or legs toward and
away from their body to control spin rate, or the difference between holding a dumbbell at arm’s
length versus right next to your body. The other contributing factor is the force being applied.
The source of this force is the earth’s gravitational pull acting on the mass as it falls towards the
earth’s surface, thus causing the copter to interact with and push against all of the air molecules
that surround it. If this were done in a vacuum chamber, the copter wouldn’t rotate at all. This
force is directly dependent on the mass of the falling object, which was held fixed in this
experiment. The location where the force was being applied however, was altered. The area of
the wing was reduced when the wingspan was reduced. If the same force is being applied to a
smaller area, there is more pressure, causing the copter to rotate faster. I do not feel that this is
the primary contributor to the decrease in rotational velocity though.
When you look at the graph, it seems as though you could apply a linear or exponential fit,
but maybe the real (physical) result is neither of those. If we consider the limit as the wing length
goes to zero, what happens? If you were to take a simple strip of paper, and attach a paper clip
to it and allow it to fall to the ground, it would plummet straight down without even rotating. I
would imagine that with very tiny slivers of wings, the same thing would happen. What if you
dropped the copter a further distance? Would the tiny-winged copter eventually start spinning
really fast? I’m not entirely certain. I feel that there is some interesting behavior to be investigated
in limit that the wingspan of a descending copter approaches zero.
These tests successfully isolated wingspan and rotational velocity from all other controllable
variables. Without making conjectures based on inferences, it can be safely said that for the range
of wingspans tested, without approaching arbitrarily small wingspans, rotational velocity is
directly dependent on wingspan. As the wingspan of a paper copter in free fall is decreased, the
rotational period increases linearly.
Cross Section 2014
Rotational velocity of falling copters
5
APPENDIX A
Wingspan (a.u.)
Length
Width
16
16
16
16
16
16
15
15
15
15
15
15
14
14
14
14
14
14
13
13
13
13
13
13
12
12
12
12
12
12
11
11
11
11
11
11
10
10
10
10
10
10
9
9
9
9
9
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Trial
#
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
Time (s)
Start
End
10.051
28.852
38.416
49.052
57.352
5.818
30.162
38.295
46.995
56.529
5.729
23.996
33.777
43.043
2.543
13.043
30.810
39.978
4.371
12.788
21.106
29.038
37.488
45.637
16.059
24.209
31.926
39.392
46.658
54.427
24.257
34.224
42.524
52.225
59.790
7.258
10.641
19.841
27.175
35.559
50.375
6.476
40.477
52.876
4.341
11.976
19.341
27.409
11.185
30.185
39.551
50.184
58.152
6.952
31.196
39.328
48.162
57.528
6.762
24.996
34.777
43.911
3.477
13.843
31.878
40.922
5.172
13.571
22.170
29.971
38.438
46.572
16.726
25.093
32.691
40.059
47.544
55.191
24.989
34.956
43.358
52.890
60.524
7.989
11.476
20.708
27.608
36.141
51.141
6.974
41.007
53.540
4.942
12.641
20.007
28.092
Rotations
#
Angular velocity
(rpm)
7
8
7
7
5
7
7
7
8
7
7
7
8
7
7
6
8
7
7
7
9
8
8
8
6
8
7
6
8
7
8
8
9
7
8
8
10
10
5
7
9
6
7
9
8
9
9
9
370.4
360.1
370.0
371.0
375.0
370.4
406.2
406.6
411.3
420.4
406.6
420.0
480.0
483.9
449.7
450
449.4
444.9
524.3
536.4
507.5
514.5
505.3
513.4
539.7
543.0
549.0
539.7
542.1
549.7
655.7
655.7
647.5
631.6
654.0
656.6
718.6
692.0
692.8
721.0
705.0
722.9
792.5
812.6
798.7
812.0
810.8
791.2
http://dewey.weber.edu/crossection/
Averages
(a.u.)
369.5
411.8
459.7
516.9
543.9
650.2
708.7
803.0
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Cox
Cross Section 2014