pISSN 0374-4914 eISSN 2289-0041

Research Paper

New Phys.: Sae Mulli 2021; 71: 1067-1075

Published online December 31, 2021 https://doi.org/10.3938/NPSM.71.1067

Learning the Resistance Coefficients of Pendulum Motion Using Balls of Various Sizes

Kyung-Ryul LEE, Young-Gu JU*

Department of Physics Education, Kyungpook National University, Daegu 41566, Korea

Correspondence to:ygju@knu.ac.kr

Received: May 7, 2021; Revised: November 11, 2021; Accepted: November 15, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

An optical system with a photogate was developed to measure the speed of a pendulum at the lowest point of motion to obtain the damping and the resistance coefficients of the pendulum. The photogate consisted of a photoresistor, a laser, a mechanical body, and a pendulum ball. A 3D printer was used to produce the mechanical body and pendulum balls of various sizes. Furthermore, an Arduino was used to automate the measurement of the speed at the lowest point of motion and increase the precision. It was found that the resistance coefficient was found to be proportional to the size of the balls, which is regardless of ball mass, consistent with the drag equation for a small Reynolds number. The developed instrument and program were applied to the experiment class, and positive results in education about air resistance was obtained.

Keywords: Damping coefficient, Resistance coefficient, Simple pendulum, Fluid dynamics, Arduino, 3D printer

I. INTRODUCTION

A pendulum motion is one of the most common topics in physics textbooks, especially for explaining Newtonian mechanics, including the concepts of periodic motion, gravity, oscillation, and friction [13]. For small-angle oscillations, the pendulum motion reduces to a harmonic oscillator problem. Many theoretical and experimental studies related to simple pendulums have been conducted to investigate their nonlinear effects and improve the precision of measurements [46].

The pendulum motion can be described by a sinusoidal function of time if there is no air resistance. The frictional force associated with air resistance causes damping of the pendulum motion, which can be described by the same equation used for a damped oscillator. Since the air resistance is generally not sufficiently large to cause overdamping or critical damping, a damped pendulum can be considered as an underdamped oscillator, which has an amplitude that decays exponentially in time with a damping coefficient [1,2]. The damping coefficient is equal to the resistance coefficient of the friction term in the equation of motion, divided by two times the mass. In most textbooks, this friction term is assumed to be the product of the resistance coefficient and the velocity of the bob or ball of the pendulum. Since air friction is a type of drag force, it should depend on the size of the pendulum ball. It is interesting to investigate the dependence of the resistance coefficient on the size of the pendulum ball.

In this study, we measured the damping and resistance coefficients for pendulum balls of various sizes. We examined the dependence of the resistance coefficient on the ball size and examined the compatibility of the results with the drag equation, which is frequently used to describe the air resistance acting on the motion of spherical objects [7,8].

Another salient feature of this research is the achievement of high accuracy in and simplification of the pendulum experiment for educational purposes. We employed a microcontroller-based laboratory method [9] and used an Arduino [10], which is one of the most popular opensource electronics platform that offers easy-to-use hardware and software. This electronics platform has considerable advantages in terms of cost, availability, and ease of learning [11,12], which are especially helpful for teaching students basic experiments such as an experiment for understanding pendulum motion. An Arduino can be used to automate data acquisition and increase measurement accuracy by sampling more data with ease.

In addition to Arduino, we employed a 3D printer to fabricate the mechanical structure of a photogate and balls of various sizes. 3D printers have become less costly and are available in some secondary schools and many colleges. The combination of an Arduino and a 3D printer can help upper-level physics instructional laboratory students perform many basic experiments without having to purchase expensive instruments and provide students with an opportunity to learn coding.

According to fluid dynamics, the frictional force (or drag) for a spherical object in motion depends on the properties of the fluid and on the size, shape, and speed of the object. This relationship is described by the drag equation, which is expressed as Eq. (1), where Fd,CD, ρ,A, and v are the drag force, drag coefficient, density of the fluid, cross-sectional area, and velocity of the object relative to the fluid, respectively [7,8].

$Fd=12CDρAυ2$

In addition, CD depends on the Reynolds number, which can be derived using Eq. (2), where L is a characteristic diameter or linear dimension of the object such as the diameter of a sphere and µ is the dynamic viscosity of the fluid.

$Re=ρLυμ$

If the Reynolds number Re is about 1200 or less, the drag coefficient CD is roughly inversely proportional to it [7,13]. This means that the drag force is proportional to the velocity as in Eq. (3), where k is the proportional coefficient between CD and 1/Re and c is the resistance coefficient.:

$Fd=12kμρLυρAυ2=12kμALυ=cυ$

Therefore, for a pendulum motion moving with a small Reynolds number, the frictional force Fd due to air resistance is proportional to the velocity of the object as shown in Eq. (3) where the resistance coefficient c is independent of the velocity but is proportional to the size of a sphere. In our experiment, Re can be estimated using the size and velocity of the pendulum ball and by assuming that the density and viscosity of air are approximately 1.225 kg/m3 and 1.81 ×10-5 kg/(m·s), respectively. The estimated Re value of 296 – 927 was obtained, which was less than 1200; thus, the drag was considered to be proportional to the velocity, as shown in Eq. (3).

Using the drag equation, the following differential equation for pendulum motion can be obtained:

where m, c, g, l, and θ are the mass, resistance coefficient, standard gravity, pendulum length, and angular displacement (θ = 0 at the bottom position), respectively. [1,2] For a small-angle oscillation, this expression can be approximated as

$mθ¨+cθ˙+mg/lθ=0$

The solution of this differential equation has the form of Eq. (6), where θm, γ, and ω are the maximum angle, damping coefficient, and angular frequency, respectively. Furthermore, the angular frequency can be expressed in terms of the standard gravity and pendulum length, as shown in Eq. (7). The damping coefficient is related to the resistance coefficient according to Eq. (8):

$θt=θme−γtcosωt$
$ω=gl$
$γ=c2m$

The expression for the angular velocity of the pendulum ball [Eq. (9)] can be obtained by differentiating Eq. (6) with respect to time:

At the lowest point of motion, cos(ωt) becomes zero and sin(ωt) becomes ±1, and the following expression can be obtained:

$θ˙tlow=±θmωe−γtlow$

By multiplying both sides of Eq. (10) by l, we can obtain the following expressions for the linear velocity and pendulum speed at the lowest point:

$υtlow=±υme−γtlow$
$υtlow=υme−γtlow$
$υm=θmlω$

The speed at the lowest point is also the maximum speed of the pendulum in any given period of motion. The speed of the pendulum decreases exponentially with time, and the decay constant equals the damping constant of the motion. Taking the natural logarithm of both sides of Eq. (12) gives

When ln |v(tlow)| is plotted as a function of tlow, the slope of the line corresponds to the damping coefficient with the opposite sign. Thus, the damping coefficients of pendulum motion can be obtained from the measured speed of the ball at the lowest point.

Notably, there was a study [14] that measured air friction and its dependency on the size of balls from the pendulum experiment. This preceding research defined and measured the quantity $ai*$ :

$ai*≡υi−12−υi+122υiTi$

where vi is the velocity measured at the bottommost position and i-th period and Ti is the i-th period. The fit parameters were defined as $ai*=c1υi+c2υi2$ where a = c1m and $b=3πc2m8$ if the force of air friction can be modeled as f = av + bv2. Although the definition and derivation of Eq. (15) is complicated and sophisticated, $ai*$ of Eq. (15) is approximately equal to 2viγ = vic/m in our study. Although our study assumed that friction force was proportional only to speed, the resistance coefficient c may not always be constant. At high velocities, it may instead be dependent on speed. However, the maximum angle of swing was relatively small and the maximum speed was not high for our experiment, so resistance coefficient c stayed nearly constant. Therefore, the study performed in this paper shows an alternative approach to measuring the air friction of simple pendulums and their size dependencies, which may be easier for undergraduate students to understand and perform.

III. EXPERIMENTS

To measure the speed of a pendulum ball at its lowest point, we fabricated a photogate using an Arduino and a 3D printer in the laboratory. The photogate consisted of a diode laser, a photoresistor, a mechanical structure for holding the two devices, and an Arduino circuit, as shown in Fig. 1. We used Cubify Invent as the computeraided design (CAD) software to design the photogate body and pendulum balls. The photogate body was Ushaped with two holes at the ends [Fig. 1(a)] for holding the diode laser and photoresistor. The pendulum ball swung between these two devices blocking the laser beam and decreasing the laser light arriving at the photoresistor.

Figure 1. (Color online) Photogate fabricated using the 3D printer and an Arduino: (a) a schematic of an assembly view of the photogate body and photographs of (b) the photogate comprising the laser, photoresistor, and Arduino circuit and (c) the detailed wiring of the divider with the photoresistor and a resistor connected to the Arduino board. The dimension of the photogate was 80(length) × 67 (height) × 24 (width) mm3.

The photoresistor formed the upper part of a voltage divider with a fixed resistor on a breadboard. The upper and lower ends of the voltage divider were connected to the 5 V pin and 0 V (GND) pin of an Arduino board, respectively, as shown in Fig. 1(b) and (c). The center of the voltage divider was tied to an analog input labeled “A2” on the Arduino board. When the laser beam was not blocked, the light reached the photoresistor and decreased its resistance, thereby reducing the voltage drop across the photoresistor; this caused the voltage at the middle point of the voltage divider to remain high. As the ball began to block the laser beam, the process was reversed: the voltage at the middle point and the A2 pin of the Arduino board decreased. In short, a lower light intensity at the sensor resulted in a lower voltage at the Arduino input and vice versa. Thus, the voltage of the photosensor indicated whether the ball was blocking the laser beam.

Since the Arduino board could check the sensor very frequently and measure time with an accuracy of a tenth of a millisecond, we could accurately measure the time span between the instant when the ball began blocking the laser beam and when the blocking ended. The speed of the ball was the diameter of the ball divided by the blockage time span.

The experimental setup is shown in Fig. 2. The ball was suspended by two strings instead of a single string to prevent the rotation of the plane of the pendulum motion during the measurements.

Figure 2. (Color online) Photograph of the setup for the simple pendulum experiment.

Pendulum balls of various sizes were designed and fabricated using CAD and the 3D printer, and they are shown in Fig. 3. Basically, a single spherical metal ball, shown in Fig. 3(c), was used throughout the experiment with strings of a fixed length. The ball was enclosed by plastic shells of different sizes to vary the bob size. This approach eliminated possible changes in the experimental conditions such as the length of strings. The plastic shell consisted of two halves that could be attached to each other with two pins. A half-shell had two holes on the flat side for holding the pins, as shown in Fig. 3(a). The plastic shell also had space at its top for two strings to pass through. The outer diameters of the spherical shells were 18.7, 24.1, 30.0, and 36.0 mm. Balls with five different sizes, including the metal ball (which had a diameter of 11.5 mm), were used in the pendulum experiment.

Figure 3. (Color online) Balls used in the experiment. (a) the 3D design of a plastic ball and photographs of (b) plastic balls of various sizes produced by the 3D printer and (c) the metal ball with a string strung through a passage at its top.

The measurement of pendulum motion required the programming of the Arduino board to acquire time data at the exact instants when the ball began to block the laser path and when the laser path was unblocked. The program used in the experiment is shown in the Appendix. It enabled the board to check whether the voltage at the A2 pin was high or low. If the voltage was greater than 2 V, then the program recognized the state as HIGH. Otherwise, the state was identified as LOW. The HIGH and LOW states corresponded to high and low light intensities at the photoresistor, respectively. Since the voltage input at the A2 pin was digitized into an integer value from 0 to 1023, the threshold value in the code was set to 400. The measured timing data were stored in the memory until the number of data reached 100, whereupon the board sent the stored data in a single batch through serial communication to the main computer to prevent unnecessary and irregular latency in the communication between data acquisition intervals.

The clock frequency of the Arduino board was 16 MHz, and the number of coding lines in the execution loop was less than 10. Therefore, theoretically, the time taken to check the photosensor voltage should be less than a microsecond. However, the time to execute a loop was measured to be about 130 µs. It emerged that reading the voltage from an analog input was relatively slower than the execution of the other instructions and consumed most of the execution time. Since the laser path was blocked for a time span of a few tens of milliseconds, the relative error in the time measurement was on the order of 1/100 or less. The relative error in the measurement of the ball diameter was on the order of 1/1000 or less since the diameter was about a few tens of millimeters and the precision of the vernier calipers was 0.01 mm. Therefore, the precision of the time span was critical to the measurement precision for the speed of the ball.

The experiment was performed as follows. First, the ball was pulled to a distance of 10 cm. After the activation of the Arduino board, the ball was released and allowed to oscillate. After the pendulum passed the optical sensor 100 times, Arduino sent the data to a notebook that displayed the information on its serial monitor. The data from the serial monitor was transferred to a spreadsheet program for additional analysis. The lengths of the strings were 50 cm. The experiment was conducted five times under the same conditions for every ball. To reduce the error, we took adequate care to ensure that there was no initial speed at the specified width of swing when releasing the pendulum ball. We used a ruler to stop a ball before the commencement of its motion. Since the ruler was fastened by a screw, loosening the screw allowed the ball to move. This method helped initiate the motion under the same conditions in terms of the width of the swing and initial speed.

IV. RESULTS AND DISCUSSION

Using the fabricated photogate, we measured the speeds of the balls of various sizes during pendulum motion. In the experiment, the change in the speed of the oscillating bob with time was measured. From the rate of reduction of the speed, the damping and resistance coefficients could be determined. A change in the ball diameter resulted in changes in the resistance and damping coefficients, which were obtained from the speed of the pendulum at its lowest point. From plots of ln |v(tlow)| as a function of time, the slope and damping coefficient were obtained using Eq. (14). Plots of ln |v(tlow)| vs. time for the various ball sizes are shown in Fig. 4. The plot for each ball size represents one result out of five measurements taken, to serve as an example.

Figure 4. (Color online) Plot of ln |v(tlow)| as a function of time for the simple pendulum motion of balls of various sizes: 11.5, 18.7, 24.1, 30.0, and 36.0 mm. Each plot represents the result of one measurement out of the five measurements taken for that ball size. |v(tlow)| was measured in m/s.

Notice from Fig. 4 that the data points are well fitted by a straight trend line. This implies that the damping coefficient and the resistance coefficient are constant, and not dependent on speed. If the term proportional to the square of the speed affected air friction, the slope in Fig. 4 and the corresponding damping coefficient should have changed with time due to a dependency on the speed which decreased with time. Therefore, our assumption that the damping coefficient is constant is valid within the range of ball speeds measured in the experiment.

The resistance coefficient was obtained from 2 by using Eq. (8). The damping and resistance coefficients measured for the various ball sizes are presented in Table 1. As the ball size increased, the resistance coefficient increased. However, the damping coefficient decreased with an increase in the ball size. This implies that when the size of the ball increased, the increase in c was less than that in the mass of the ball.

Measured damping and resistance coefficients for various ball sizes. The numbers in parenthesis represent the standard deviations of five time measurements.

Ball radius (mm)Mass (g)Damping coefficient γ (1/s)Resistance coefficient c (kg/s)
5.756.360.00960 (0.00015)1.22 × 10-4(2 × 10-6))
9.358.310.00861 (0.00005)1.43 × 10-4(2 × 10-6))
12.0512.350.00697 (0.00036)1.72 × 10-4(9 × 10-6))
15.0016.290.00674 (0.00004)2.19 × 10-4(2 × 10-6))
18.0023.480.00564 (0.00014)2.65 × 10-4(7 × 10-6))

A plot of the experimentally observed resistance coefficient as a function of the ball radius is shown in Fig. 5. A linear equation fits the experimental data, except for the smallest ball radius. The photo of the metal ball in Fig. 3(c) shows the attached structure on top of the ball. The structure not only allows a string to pass through but also deviates the pendulum bob from its ideal spherical shape. This deviation from a spherical shape may have caused additional air resistance when compared with the ball surrounded by plastic shells. Therefore, the data from the metal ball without a plastic shell should be excluded when fitting the data with a trend line in Fig. 5. Furthermore, the y-intercept of the fitted curve in Fig. 5 is close to zero, which also indicates the experimental data conforms with the drag equation for a low Reynolds number regime.

Figure 5. (Color online) Plot of the resistance coefficient c vs. the ball radius R. The error bar indicates the standard deviation. The data points without noticeable error bars have standard deviations too low to see. The data from the metal ball without a shell was excluded from the trend line, due to its nonspherical shape.

We compared our measured resistance coefficient with the coefficient a as shown in the reference [14]. According to the previous literature [14], the diameter of 40 mm had the resistance coefficient a of 2.7×10-4 kg/s, which agrees well with our value c of 2.9 × 10-4 kg/s as estimated from the trend line at the radius of 20 mm in Fig. 5.

To examine the effect of the developed instrument on experimental education, a survey and an interview were conducted after experimenting with five second-year high school students pursuing general physics. The survey included five multiple-choice questions and one subjective question. Moreover, an additional interview was conducted. The contents of the questionnaire and its results are shown in the Appendix. First, in response to the survey question about the adequacy of the content level of the learning instrument, four out of five students answered: “appropriate.” In terms of the difficulty of the experiment, three students answered as “appropriate, ” and two students answered as “easy.”. In the interview, the students answered that the experiment was appropriate for checking the previously learned contents. They also responded that the experiment was appropriate or easy because the Arduino program automated data acquisition. The experimental data could be easily checked using a spreadsheet program that shows a plot of data and extracts the damping coefficients compared to the existing pendulum experiment. Besides, they said that the experiment was more convenient than the conventional pendulum experiment using a 3D printer and an Arduino. It was found that making an experimental instrument and using an Arduino program had a positive effect on arousing the students’ interest.

The experiment resulted in significant advancements in the understanding of air resistance in a pendulum motion. Before the experiment, in the questionnaire, three out of five participants predicted that the damping coefficient increased as the size of the pendulum ball increased. In both the damping and resistance coefficient predictions, a student provided incorrect answers. Before the experiment, only one student provided the correct answer with correct reasoning. However, after the experiment, the increasing order of damping coefficients was correctly answered by all the students with a proper explanation in the questionnaire. It appears to have had a positive effect on improving the learning about air resistance by confirming previously known knowledge through experiments.

V. CONCLUSIONS

We fabricated a photogate using Arduino and a 3D printer to measure the resistance coefficient of a simple pendulum. Pendulum balls of various sizes were fabricated by inserting a metal ball into 3D-printed plastic shells of various sizes. These balls were used to determine the relationship between the resistance coefficient and ball size. The laboratory-made photogate and Arduino board could measure the time span of the blocking of the laser path using a ball with sufficient accuracy. From the measured time spans, the speed of the balls at their lowest point, which was expected to decrease with the damping coefficient because of air friction, was determined.

The diameters of the balls used in the experiment herein were 11.5, 18.7, 24.1, 30.0, and 36.0 mm. The experimentally measured damping coefficients of these balls were 0.00960, 0.00861, 0.00697, 0.00674, and 0.00564 1/s, respectively. The resistance coefficients of the balls were obtained by multiplying the damping coefficient by twice the mass of the balls, and the values obtained were 1.22×10-4, 1.43×10-4, 1.72×10-4, 2.19× 10-4, and 2.65 × 10-4 kg/s, respectively.

From a plot of the resistance coefficient as a function of the ball radius, a linear relationship between these parameters was obtained and it was observed to agree with the drag equation for small Reynolds numbers. The smallest diameter was an outlier to the fitted curve because the attached structure on top of the ball could sufficiently deviate the pendulum bob from its ideal spherical shape and cause an additional air resistance. Thus, measuring the damping and resistance coefficients for pendulum motion was possible using a 3D printer and Arduino.

Furthermore, we applied the developed instrument and program to five second-year high school students pursuing general physics. It turns out that the instrument and the program were educationally effective. The survey after the experiment showed positive responses from the students in regard to the instrument and the method based on Arduino and 3D printer. Further, the survey showed that the knowledge about the damping coefficient improved after the experiment.

This study shows the possibility of designing physics experiments based on Arduino and a 3D printer for educating students in multidisciplinary approaches, such as those pertaining to mechanics, optics, electronics, and software coding. Furthermore, such experiments cover many aspects of a discipline, such as oscillation, damping, and fluid mechanics. Our results show how multidisciplinary education can be provided to students and educators in science, technology, engineering, and mathematics.

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