The Conical Pendulum: The Tethered Airplane

Anthony P. Mazza, student, <amazza2@gmail.com>;
William E. Metcalf, student, <wildbillmetcalf@gmail.com>;
Anthony D. Cinson, student, <tony.cinson@gmail.com>;
John J. Lynch, Assistant Professor, <jlynch@wju.edu>;
Physics Department, Wheeling Jesuit University, Wheeling, West Virginia, USA.

PACS Codes: 01.50.Lc, 01.50.Pa, 01.50.Wg.

Keywords: Centripetal, conical, pendulum, circular, video.

Abstract

	The introductory physics lab curriculum usually has one experiment on uniform circular motion. Physics departments typically have several variable-speed rotators in storage that, if they work, no longer work well. Replacing these rotators with new ones is costly, especially when they are only used once a year. This work describes how an inexpensive (.$10) tethered airplane powered by a small electric motor can be used to study uniform circular motion. The airplane is easy to see and entertaining to watch.

	For a given string length and airspeed, a tethered airplane quickly finds a stable, horizontal, circular orbit. Using a Digital Video (DV) camcorder, VideoPoint Capture, QuickTime Player, meter sticks, and a stopwatch, data on the airplane’s motion was obtained. The length of the string was varied from 120cm to 340cm while the airspeed ranged from 200cm/s to 480cm/s. For each string length and airspeed, the period of the orbit and the diameter of the path were carefully measured. Theoretical values of path radii were also calculated using Newton’s 2nd Law. The agreement between experiment and theory was usually better than 2%.

The Conical Pendulum

	In theory, the conical pendulum1-3 consists of a bob of mass m at the end of a string of length L moving with constant speed v in a horizontal circle of radius r. See Fig.1. The bob’s acceleration results from two forces: the tension FT from the string which points upward toward the pivot and the weight mg which points downward. Along the radial and vertical directions Newton’s 2nd law yields
	
respectively, where θ is the angle between the string and the vertical. Using 
    	(1)
where T is the period of the orbit and

it can be shown that the radius r of the orbit relates to the string length and the period as

	(2)

	In practice, a small tangential force is required to overcome air drag. In this experiment, that force is provided by a propeller driven by a small DC motor.

The Tethered Airplane

	The idea of using a tethered airplane to study uniform circular motion is an obvious one.4-5 This work describes how we have used it. The airplane used in this experiment is pictured in Fig.2. It is a 1/40th scale toy Cessna 210 made in China by Novatoy. Its wingspan is 9in, its length is 8½in, and its weight is 4.8ozs (with two AA batteries). It attaches to a nylon string that attaches to a pivot that is mounted on the ceiling.

	The quantity L mentioned above is in fact a little longer than the length of the string. It is the distance between the pivot and the center of mass (CM) of the airplane. The CM can be determined by hanging the airplane from various points using a string, as in Fig.3, and visually projecting a line along this string. Faint lines that intersect at the CM can be etched into the body of the airplane.

	The period of the orbit T was obtained by timing 10 periods with a stopwatch. See Fig.4. Several such measurements of 10T were made and an average was taken provided there was agreement to within three digits. 

	The most difficult measurement to make was the path diameter 2r because the airplane is moving fast--200cm/s to 480cm/s. The path diameter, as well as the string length, were usually greater than two meters–-the length of our longest meter sticks. This made it necessary to tape meter sticks together. For 2r, the meter stick was positioned horizontally along a line just below the center of the circular orbit. The location of this center was clearly indicated with a bob hanging from the pivot above it as in Fig.5. Using a DV camcorder the airplane was filmed flying just over the meter stick at either end. See Fig.6. Several passes were filmed and the video was recorded to the hard drive of a computer in the QuickTime (.MOV) format using VideoPoint Capture set up for 60 frames per second. Afterwards, the movies were analyzed frame-by-frame using QuickTime Player to determine the two positions along the meter stick over which the CM passed. The orbits were quite stable with the airplane flying over the same two positions repeatedly. Assessing the radial positions of the CM required careful observation of the two movies and was usually made by at least two people. The diameter 2r was the difference between these two positions.

	The reader might wonder why QuickTime Player was used instead of video analysis programs such as VideoPoint or VideoGraph. In this experiment we wanted to have large orbits that took up much of the room. In fact our longest string length exceeded the height of the room. These two programs, however, limit the size of the orbit. They require the camcorder to point directly upward so the line-of-sight is perpendicular to the plane of the motion. To work well, the entire orbit needs to be video captured. Even with our camcorder’s widest angle the orbital diameter was limited to 120cm or less–-this with the camcorder on the floor, 320cm below the pivot. In principle, video capturing just a quarter of the orbit could yield r and T from fits to x=x(t) and y=y(t) data. Our first attempts at doing this with a large orbit, however, were discouraging.

	In order to obtain many data points, we also needed to vary the airspeed. The airplane was not designed for this so we did a number of things. (1) We alternated between using one and using two batteries. In this model of airplane the batteries are in parallel. The batteries were located very near the CM so the effect of adding or removing one of them was ignored. (2) We tried different combinations of weaker and stronger batteries. We rated all of our batteries by how much current they drew from a 10-Ω resistor. (3) We attached post-its to the main wing to increase drag. (4) We removed the main wing to reduce drag as in Fig.7. This wing provides no lift and its contribution to the CM was negligible. Using various combinations of the above we were able to vary the airspeed between 200cm/s and 480cm/s. 

Results

	The 22 data points are listed in Table I. Measurements were taken for five string length’s varying from 120cm to 340cm. For each string length, measurements of T and 2r were taken for a range of airspeeds. Airspeeds were calculated using Eqn.(1) and theoretical radii were calculated using Eqn.(2). We used a value of g=980.2m/s2 for the acceleration due to gravity corresponding to our latitude of 40oN. The data are also presented graphically in Fig.8 alongside the corresponding theoretical curves. The experimental discrepancies were generally less than 2%.

Application

	The procedure described above for measuring r is accurate but it is too involved for a two-hour introductory lab. The following modifications are suggested. 

(1) Adjust the string length so that the airplane is flying just over the heads of two students at opposite ends of the orbit. The students could be standing on the floor or standing on a stool. They could also be lying on lab benches or lying on the floor. Once positioned, they can hold a meter stick just above their heads and watch the airplane fly directly over. Viewed from so close, it will not be much more than a blur. Students can still record the point over which the CM is passing. It is recommended that the main wing be removed because it causes the blur to be very wide. It helps that the airplane has a small light on its belly. In fact, it is easy to see the path of this light with the ceiling lights dimmed. A third student can check that the meter stick is lined up radially––passing under the central bob––as described earlier. This student can also operate the stopwatch.

(2) Sargent-Welch sells a tethered airplane for $50. Though more expensive, it comes rigged with a spring scale which reads the string tension FT while the airplane is moving.6 Students can easily compare it with

The spring scale circumvents the need to measure r which can be complicated and time-consuming.

(3) The Student Force Sensor from Vernier Software has also been used with the conical pendulum.7 With the Force Sensor students can obtain the radial component of the string tension FT(r) and compare it with

(4) Film the motion from below and capture to the hard drive as mentioned earlier. The camcorder needs to be pointing upward and a meter stick or some object of known length must be in the plane of the motion. The video can then be analyzed using a program such as VideoPoint or VideoGraph. Unfortunately, the size of the orbit is limited by the camera angle and the height of the ceiling.

	It is also recommended not to bother with varying the airspeed because it is too time-consuming. A wide enough range of orbits is possible by simply varying L. Consider also having the students calculate g using 

instead calculating r. By the time they are studying circular motion they are familiar with g and can make a meaningful comparison with the known value.

Conclusions
	The tethered airplane described above offers an inexpensive and entertaining way to study uniform circular motion in the introductory lab. The orbits can be large and the experimental discrepancies small.

References

1.	Karen Cummings, Priscilla W. Laws, Edward F. Redish, and Patrick J. Cooney, Understanding Physics (John Wiley & Sons, Inc., 2004), p.125.

2.	Douglas C. Giancoli, Physics for Scientists and Engineers, 3rd ed. (Prentice-Hall, 2000), p.116.

3.	Vernon Barger and Martin Olsson, Classical Mechanics: A Modern Perspective (McGraw-Hill, Inc., 1995), p.81.

4.	Frank Butcher, “Circular Motion Studies with a Toy Airplane,” Phys. Teach. 25, 572-573 (Dec.1987).  

5.	Bill Franklin, "More on Toy Airplanes," <letter to the editor> Phys. Teach. 26, 264 (May 1988).

6.	Lee Larson and Roderick Grant, “The Airplane Experiment,” Phys. Teach. 29, 564-565 (Dec.1991).

7.	Thomas Moses and Natalie L. Adolphi, “A New Twist for the Conical Pendulum,” Phys. Teach. 36,372-373 (Sept.1998).

Fig.1. The conical pendulum.

Fig.2. The Tethered Cessna 210 Action Model airplane by Novatoy. 

Fig.3. Determining the location of the center of mass (CM). 

Fig.4. Stopping the stopwatch. 

Fig.5. Bob marking the center of the orbit.

Fig.6. Airplane lined up with the meter stick.

Fig.7. Airplane (with main wing removed) lined up with the meter stick.

Table I. <See below>

Fig.8. Comparison of experimental and theoretical orbital radii for L=119.2cm,173.7cm,227.7cm,281.8cm,341.1cm.

-----------------------------------------------------------------------------
Table I. Measurements of T and 2r were taken for a range of string length's L and airspeeds v. Experimental and Theoretical r’s are compared.
-----------------------------------------------------------------------------
                                      Experimental   Theoretical Experimental
Length   Period  Diameter Airspeed        Radius        Radius    Discrepancy
  L(cm)     T(s)    2r(cm)   v(cm/s)         r(cm)         r(cm)	
-----------------------------------------------------------------------------
 119.2    1.984    134.2    212.6          67.1          68.3         -1.8%
 119.2    1.857    164.0    277.4          82.0          82.9         -1.1%
 119.2    1.721    183.8    335.5          91.9          93.8         -2.0%
 119.2    1.624    197.2    381.5          98.6          99.6         -1.1%
-----------------------------------------------------------------------------
 173.7    2.498    158.5    199.3          79.3          78.5          0.9%
 173.7    2.476    165.5    210.0          82.8          83.7         -1.1%
 173.7    2.345    212.6    284.8         106.3         107.4         -1.0%
 173.7    2.215    241.7    342.8         120.9         123.8         -2.4%
 173.7    1.943    288.2    466.0         144.1         146.2         -1.5%
-----------------------------------------------------------------------------
 227.7    2.873    203.6    222.6         101.8          99.2          2.6%
 227.7    2.799    235.7    264.5         117.9         118.4         -0.4%
 227.7    2.727    265.4    305.8         132.7         133.3         -0.5%
 227.7    2.608    299.6    360.9         149.8         152.7         -1.9%
 227.7    2.334    359.2    483.5         179.6         183.2         -2.0%
-----------------------------------------------------------------------------
 281.8    3.245    215.7    208.9         107.9         105.3          2.4%
 281.8    3.173    262.1    259.5         131.1         130.1          0.7%
 281.8    3.085    309.4    315.1         154.7         153.5          0.8%
 281.8    2.917    369.3    397.7         184.7         186.5         -1.0%
 281.8    2.789    407.6    459.1         203.8         205.2         -0.7%
-----------------------------------------------------------------------------
 341.1    3.517    292.8    261.5         146.4         148.4         -1.4%
 341.1    3.487    328.2    295.7         164.1         158.9          3.3%
 341.1    3.210    445.1    435.6         222.6         225.6         -1.4%
-----------------------------------------------------------------------------

Radius vs. Period

r(cm)

T(s)

L=119.2cm

L=173.7cm

L=227.7cm

L=281.8cm

L=341.1cm

Anthony P. Mazza
232 Center Grange Road
Aliquippa, PA 15001
<amazza2@gmail.com>
<amazza2@ignatius.wju.edu>

William E. Metcalf
4979 Ridgebury Boulevard
Lyndhurst, OH 44124 
<wildbillmetcalf@gmail.com>

Anthony D. Cinson
13843 Lincoln Street
Minerva, OH 44657
<tony.cinson@gmail.com>
<acinson2@ignatius.wju.edu>