spring, a brass weight, and an ultrasonic position sensor, as shown in Fig. 1. Pulling the brass weight down a little bit, and then releasing it, will start the oscillation. Due to very small frictional forces, the amplitude of the observed SHM will slowly decrease in time, but we can neglect this effect when we analyze just a few cycles.

We have used springs specially made for the study of Hooke’s law. One good supplier of such springs is Home Sci-ence Tools ($1.95, http://www.hometrainingtools.com). We have also used a brass weight holder with a base diameter of 4.4 cm and brass slotted weights. The computerized data col-lection was done with the help of an HC-SR04 ultrasonic dis-tance sensor ($5.18 from Amazon, http://www.amazon.com), an Arduino Uno microcontroller board ($29.95 from Jameco, http://www.jameco.com), a USB cable, and four male-to-female breadboard jumper wires ($0.69 each from Vetco, http://www.vetco.net). We have also tested a Parallax PING))) ultrasonic distance sensor ($29.95 from Jameco) with similar results. We have repeated the SHM experiment with a variety of springs and masses with excellent results. The data report-ed in this article were obtained with the spring from Home Science Tools using a total mass of 250 grams and using the HC-SR04 ultrasonic distance sensor.

The HC-SR04 ultrasonic distance sensor, shown in Fig. 2, uses the principle of echolocation. The sensor has four pins: 5 V (power supply), 0 V (ground), trigger pulse input, and echo pulse output. A trigger signal of 5 V, lasting for at least 10 ms, will cause the sensor to emit a short ultrasonic signal. After the transmission of this short ultrasonic signal, con-sisting of eight pulses at 40 kHz, the sensor will listen for the echo. During this time the echo pin is kept at 5 V. After the receiver detects the echo signal, or after a maximum wait time of 38 ms, the echo pin is reset to 0 V. The HC-SR04 ultrasonic distance sensor has an operating range of 2 cm to 5 m, with a resolution of 0.3 cm. A datasheet is posted at http://www.elec-troschematics.com/8902/hc-sr04-datasheet/.

The Arduino Uno is a mi-crocontroller board based on the Atmega328 microcontroller integrated circuit. It draws cur-rent from a USB cable, or from an ac-to-dc adapter. It has six analog input pins and 14 digital input/output pins. Detailed hardware documentation, the free Arduino software (the in-tegrated development environ-ment), and plenty of program-

An Arduino Investigation of Simple Harmonic MotionCalin Galeriu, Becker College, Worcester, MAScott Edwards and Geoffrey Esper, Bay Path Regional Vocational Technical High School, Charlton, MA

We cannot hope for a new generation of scientists and engineers if we don’t let our young students take ownership of their scientific and engineering

explorations, if we don’t let them enjoy the hands-on cycle of design and production, and if we don’t let them implant their creativity into a technologically friendly environment. With this educational philosophy in mind, Massimo Banzi1 and his team have developed and popularized the open source Arduino microcontroller board. The Arduino board has helped countless people in their science, electronics, robot-ics, or engineering projects, allowing them to build things that we have not even dreamed of. Physics instructors have also realized the advantages of using Arduino boards for lab experiments.2-4 The schools are saving money because the homemade experimental equipment is much cheaper than the commercial alternatives. The students are thankful for an educational experience that is more interesting, more loaded with STEM content, and more fun. As further proof of this new trend in physics education, Vernier5 is now documenting the use of their probes with Arduino boards. This is why we have developed an Arduino-based physics investigation of the simple harmonic motion (SHM) of a mass on a spring. The experimental data are collected with the help of an ultrasonic distance sensor and an Arduino Uno board. The data are then graphed and analyzed using Origin 9. This rich cross-curricular STEM activity integrates electronics, computer programming, physics, and mathematics in a way that is both experimentally exciting and intellectually rewarding.

The experimental investigation of SHMThe ex-

perimental setup for our investigation of SHM consists of a stand, a

Fig. 1. The experimental setup for our investi-gation of SHM. Our students Colin Wolfe and Nathan Youngs have helped us with the data collection and analysis.

Fig. 2. The HC-SR04 ultrasonic distance sensor.

DOI: 10.1119/1.4865518 The Physics Teacher ◆ Vol. 52, March 2014 157

158 The Physics Teacher ◆ Vol. 52, March 2014

example code for how to use the PING))) sensor. This code is called from the Examples|Sensors menu. A tutorial about the PING))) sensor is also posted on the official Arduino site at http://www.arduino.cc/en/Tutorial/Ping. The Arduino code that we have used is a variation of this PING))) example code.

The Arduino board is measuring the echo time with the help of the pulseIn() function, which in this case measures the total time (in ms) during which the HC-SR04 sensor keeps the Echo pin at 5 V. By multiplying this time with the speed of sound in air at room temperature (343 m/s), we obtain the to-tal distance d traveled by the ultrasonic pulse. When the HC-SR04 sensor is centered very precisely under the brass weight, this total distance is the sum of two sides in an isosceles tri-angle. The height in this triangle is the distance to the brass weight, the position x that we are looking for. The distance between the transmitter and the receiver of the HC-SR04 sen-sor is 2.6 cm, and we can use the Pythagorean theorem to cal-culate the distance x. This correction becomes very important when the distance between the sensor and the brass weight is small, just a few cm. With this correction accurate results can be obtained even with shorter stands. Our code improves the PING))) example code in three ways. Our code works for both 3 and 4 pin sensors, we implement a Pythagorean cor-rection in the calculation of the distance, and we use floating point arithmetic in the calculation and reporting of the data.

Data analysisData analysis was done using an evaluation copy of Ori-

gin 9 downloaded from http://www.originlab.com. Origin is a powerful software application specifically tailored for scientific calculations. We have used the Nonlinear Curve Fit option, with a User Defined fitting function of the form A*sin(2*pi*x/T+B)+M. For our example the amplitude A is 7.3460.026 cm, the period T is 0.767540.00018 s, the ini-tial phase B is 3.5730.014 rad, and the midline (equilibrium position) M is 43.5460.015 cm. The R2 value is very close to one, a further indication that the sinusoidal model describes very well the experimental data. The value of the period can

ming examples are provided on the official Arduino website at http://arduino.cc.

The HC-SR04 sensor is connected to the Arduino Uno board with the help of four male-to-female breadboard jumper wires. The Arduino Uno board supplies the voltage to the HC-SR04 sensor. The Vcc pin of the sensor is connected to the 5 V pin of the Arduino board, and the GND pin of the sensor is connected to a GND pin of the Arduino board. The Trig pin of the sensor is connected to the Arduino digital pin 7, and the Echo pin of the sensor is connected to the Arduino digital pin 8.

A short piece of Arduino programming code, shown in Fig. 3, is needed to collect the experimental data. The com-puter first compiles this program and then sends it to the Ar-duino board through the USB cable. The Arduino board will execute the code and will repeatedly send back to the com-puter the two experimental measurements, the internal clock time, and the distance to the brass weight. The measurements are repeated every 10 ms. The list of experimental measure-ments is copied from the Arduino Serial Monitor window and pasted into a Microsoft Excel worksheet.

If the PING))) sensor is used instead, then the trigPin and echoPin constants have to be set to the same number, 7, since this sensor uses only one pin for both the trigger and the echo signals. In this case the SIG pin of the PING))) sensor is con-nected to the Arduino digital pin 7.

The Parallax PING))) sensor has been on the market for about 10 years. During this time this sensor has become very popular, and fortunately the Arduino software comes with an

Fig. 3. The Arduino code used for data collec-tion.

Fig. 4. SHM data analysis done using Origin 9.

The Physics Teacher ◆ Vol. 52, March 2014 159

5. “A guide to using Vernier sensors with Arduino,” The Caliper 30, 1-2 (Fall 2013).

Calin Galeriu is teaching physics at Becker College and math at Bay Path Regional Vocational Technical High School. He earned a BS degree in physics from the University of Bucharest, MA degree from Clark University, and a PhD degree from Worcester Polytechnic Institute.Becker College, 61 Sever St., Worcester, MA 01609; Calin.Galeriu@becker.edu

Scott Edwards is an electronics instructor and vocational lead teacher at Bay Path Reg. Voc. Tec. High School. He has a BS in engineering from UConn.Bay Path Reg. Voc. Tec. High School, 57 Old Muggett Hill Rd., Charlton, MA 01507; SEdwards@baypath.net

Geoffrey Esper is an electronics instructor at Bay Path Reg. Voc. Tec. High School. He has a BS and MA degree in physics from Clark University.Bay Path Reg. Voc. Tec. High School, 57 Old Muggett Hill Rd., Charlton, MA 01507; GEsper@baypath.net

be used to calculate the elastic constant of the spring: k = m(2.p/T)2 < 16.75 N/m.

The same experimental setup can be used to investigate Hooke’s law. This will produce an independent measurement of the elastic constant of the spring. We have used total mass-es from 100 g to 450 g. The graph of the applied force mg as a function of position x displays the anticipated linear behavior, and the slope of the line of best fit gives the elastic constant. The value obtained this way, k <16.77 N/m, is very close to the value determined from the period of the SHM. Another important observation is that the spring has a residual force, and it won’t stretch if the mass of the brass weight is less than 100 g.

ConclusionsWe have developed an Arduino-based exploration of the

simple harmonic motion of a mass on a spring. By using an Arduino Uno board and an HC-SR04 ultrasonic distance sensor, this very important physics experiment can be per-formed on a budget, with an excellent match between the experimental data and the theoretical model. This integrated STEM activity has also uncovered some fundamental connec-tions between physics, electronics, computer programming, and mathematics. This Arduino-based investigation of SHM has helped the students to improve both their experimental and theoretical skills. We would like to thank the anonymous referee for very valuable comments.

References1. Massimo Banzi, Getting Started with Arduino, 2nd ed. (O’Reilly

Media, 2011).2. K. Zachariadou, K. Yiasemides, and N. Trougkakos, “A low-

cost computer-controlled Arduino-based educational labora-tory system for teaching the fundamentals of photovoltaic cells,” Eur. J. Phys. 33, 1599–1610 (Nov. 2012).

3. Eric Ayars, “Applications of Arduino Microcontrollers in Un-dergraduate Laboratories,” invited talk presented at the AAPT national meeting in New Orleans, LA, Jan. 2013.

4. Calin Galeriu, “An Arduino-controlled photogate,” Phys. Teach. 51, 156–158 (March 2013).

Fig. 5. Experimental investigation of Hooke’s law.

2nd Editionby Clarence Bakken

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