MManually driven harmonic oscillator
M N S Silva and J T Carvalho-Neto
Departamento de Ciˆencias da Natureza, Matem´atica e Educa¸c˜ao, UniversidadeFederal de S˜ao Carlos, Caixa Postal 153, 13600-970, Araras, SP, BrasilE-mail: [email protected]
November 2019
Abstract.
Oscillations and resonance are essential topics in physics that can be exploredtheoretically and experimentally in the classroom or teaching laboratory environments.However, one of the main challenges concerning the experimental study of resonancephenomena via forced oscillations is the control of the oscillation frequency, whichdemands an electronic circuit or a fine tuned coupled mechanical system. In thiswork, we demonstrate that, in what concerns the physics teaching, such demandingaccessories are not necessary. The forced oscillations can be implemented by theteacher’s hand guided by an oscillating circle displayed in a web application loadedin a smartphone. The oscillations are applied to an ordinary spiral toy. Qualitative,as well quantitative, proposals are explored in this work with excellent results. a r X i v : . [ phy s i c s . e d - ph ] N ov anually driven harmonic oscillator
1. Introduction
Oscillations and waves are essential contentsfor understanding various natural phenomenaranging from basic physics to applied sciences.Specifically, the harmonic oscillators providegreat explanatory power as they have analyti-cal solutions in various damping and excitationregimes [1].In addition to the theoretical descriptionof these systems, didactic and illustrative ex-periments can be performed in the classroomenvironment, providing a fairly complete de-scription of oscillatory phenomena [2]. Re-garding the damped and driven harmonic os-cillator, the electromagnetic and electrome-chanical systems are the most versatile towork with and have many technological ap-plications. Consequently, they are greatly ex-plored for the teaching of forced oscillationsand resonance [2, 3, 4]. Nowadays, technolo-gies such as smartphone accelerometers havebeen widely used to study mechanical oscilla-tors [5, 6, 7, 8]. Optical sensors and countersare also employed, some of which use reason-ably complex electronic circuits [9, 10, 11].There are few purely mechanical imple-mentations for the teaching of the driven os-cillator. As an example, in [12] the authoruses the oscillations of a heavy pendulum todrive oscillations on a lighter one. Usually,electrical motors are coupled to spring-massor pendulum systems in order to drive the os-cillations [13, 14]. These devices, while inge-nious and some easy to construct, require time-consuming experimental preparations that caninhibit widespread use in the classroom.In this work, our purpose is to demon-strate that the excitation and reading of apurely mechanical driven harmonic oscillatoraimed at teaching oscillations and resonancecan be fairly well performed with a simple spi- ral toy [15, 16] driven by the own teacher’shand. Moreover, with just few more add-ons,it is possible to quantitatively explore the os-cillator amplitude response as function of theexternal force frequency in different dampingconditions. The only necessary technologicalresource is a smartphone loaded with our opensource oscillator web application, which servesas a reference in swinging the spiral toy. Amultimedia projector is useful just in case ofperforming quantitative analysis for the wholeclassroom.
2. Materials
The materials used in this work consistedof a spiral toy and a web application called harmetronome [17] that we developed specifi-cally for this work (the name makes referenceto the metronome that helps musicians to keepthe pace at a chosen beat rate). The web appli-cation consists of a circle that executes simpleharmonic oscillation. The oscillation frequencyor period can be set to a specific value or canbe swiped in a predefined range.The spiral toy consisted of a plastic helicalspring with 7.6 cm in external diameter andweighting 0.84 g per loop. Optionally, metalnuts and plastic discs can be attached to theend of the spiral toy in order to change its massand damping factor, respectively.
3. Applications
This is the simplest propose. The mode ofoperation is illustrated in figure 1 [18]. Theteacher holds the smartphone (loaded with theharmetronome app) with his hand and one endof the spiral toy with the other hand. Both anually driven harmonic oscillator
Figure 1.
Snapshots of the forced oscillations on thespiral toy (a) below resonance at ω = 2 . ω = 4 . ω = 6 . ( a ) ( b ) ( c ) moment that the spring bottom end was at itslowest position for each oscillation frequency.The red line was drew to indicate the springequilibrium position. That is, the spiral toybottom end oscillates symmetrically below andabove the red line. An unloaded spiral toywith 26 and one fourth loops was used in thisdemonstration.As can be seen in figure 1, the expectedbehavior is very well reproduced, despitethe inherently inaccurate movement performedby the teacher’s hand. Bellow and abovethe natural frequency, the spring oscillationamplitudes are smaller than the amplitude atthe natural frequency. Additionally, at ω =2 . ω = 6 . anually driven harmonic oscillator ◦ phase difference betweenthem. On resonance, at ω = 4 . ◦ phase difference.It is important to note that since the dampingfactor is significantly lower than the naturalfrequency, the resonance frequency and thenatural frequency are very close to each other. It is possible to quantitatively study theoscillator response to an external force byrunning the harmetronome web application ona computer and projecting it on the wall bymeans of a multimedia projector. Besides theoscillating circle, there is the option to showa vertical scale against which the shadow of ahanging mass is project and its position can beread by the students. Therefore, as the teacherswings the top end of the spiral toy in pacewith the harmetronome circle, the studentstake note of the minimum and maximumposition reached by the hanging mass for eachfrequency of oscillation choose by the teacher.At the end, the amplitude of oscillationfor each frequency of the external force iscalculated by taking the difference between themaximum and minimum positions.In figure 2 [19] is shown a picture of theteacher executing the experimental demonstra-tion where the harmetronome application isprojected on the blackboard. In order to makethe experiment realization more comfortablefor the teacher, the spiral toy is attached to thetop of a toy car, which in turn is moved by theteacher against the blackboard and a straightpiece of wood. The piece of wood helps to con-strain the spring movement along a straightline. With the purpose to demonstrate theeffectiveness of this propose, we performed
Figure 2.
Quantitative demonstration of the drivenharmonic oscillator. With the left hand, the teacherholds a white piece of wood to vertically constrain themovement executed by its right hand. The shadow ofthe hanging washer can be seen against the projectedscale and the oscillating circle to the left of the scale. this demonstration to 34 students of a firstyear class of a Brazilian undergraduate physicsteaching course. A piece of six and a halfloops of the spiral toy was used. Additionally,a mass of 16.5 g and a 78.5 cm plasticdisc were attached to the spring end in orderto lower the resonance frequency and themaximum oscillation amplitude, respectively.This procedure was important to avoid thespring reaching its minimum elongation. Theplastic disc was cut from the cover of apolypropylene folder.Figure 3(a) shows the graphic containingthe resonance curves recorded by all students.Figure 3(b) shows the graphic of the resonancecurve obtained by the average of all curves anually driven harmonic oscillator Figure 3. (a) Experimental measurements of thespiral toy resonance demonstration shown in figure 2.Each continuous curve was measured by one of the34 students. (b) The average resonance curve takenfrom the data of the individual students. The dotscorrespond to the average values and the error bars tothe standard deviations. ( a )( b ) equation (1) shown in figure 3(a). The amplitude ofoscillation A of the harmonic oscillator asfunction of the external force frequency ω isgiven by [1]: A ( ω ) = A (cid:113) ( ω − ω ) + 4 ω β , (1)where A = F /m , β = b/ m , ω = (cid:113) k/m , F is the external force amplitude, m is theharmonic oscillator mass, and b is the dragcoefficient. The continuous curve in figure 3(b)corresponds to the fitting of equation (1) to thestudents measurements.As result, the following parameters were obtained: ω = (5 . ± .
03) rad/s, β =(0 . ± .
03) rad/s, and A = (11 . ± .
4) m/s .The correlation coefficient of the model to theexperimental data was r = 0 . Our proposal can also be applied to thephysics teaching laboratory, where groups oftwo or three students can realize the resonanceexperiment separately. In such case, the spiraltoy can oscillate around a rigid tube (e.g. PVCwater tube) to which a cloth tape measure isattached. The top end of the spiral toy isfixed to the tube. While one student oscillatesthe tube in pace with the harmetronome, theother students take note of the minimum andmaximum amplitudes reached by the springas read on the cloth tape. A picture ofsuch configuration is shown in figure 4. Thepurpose of the rigid tube is to constrain thecoil movement along a straight line and to holdthe scale given by the cloth tape.The damping forces and masses can bechanged by attaching different plastic ringsand metal washers or nuts to the bottom endof the spiral toy. To illustrate the use of thissetup, we performed the resonance experimentfor the six configurations showed in table 1.The same piece of six and a half loops spiraltoy of section 3.2 was used. The resultingresonance curves are shown in figure 5(a) to5(f). The experimental data is more noisythan the data of figure 3(b) because we didjust one measurement for each configuration.However, it could be greatly improved byperforming more repetitions and taking theiraverage values.We also measured the spring elasticconstant, k = 0 . ± .
01 N/m, whichwas obtained using the same apparatus byreading each new spring equilibrium position anually driven harmonic oscillator Figure 4.
Experiment to be performed in groupsof students. The spiral toy is fixed to the top of theguiding white tube. The oscillations are read againstthe cloth tape in red. The pink plastic ring andthe metal washers attached to it change the oscillatordamping factor and mass, respectively. as function of the attached masses.In table 1 we compare the undampednatural frequencies ω obtained by the relation w = (cid:113) k/m eff with this same parameterobtained from the fitting of equation (1) tothe experimental data in figures 5(a) to 5(f).Since the attached masses, m , are in theorder of the spiral toy mass ( m s = 5 . . m s , such that the effectivemass was calculated as m eff = m + 0 . m s [20, 21]. As can be seen in table 1, the naturalfrequencies obtained from the two approacheswere reasonably close to each other, presentingdeviations from 2% to 7%. Another correlationthat can be observed is between the dragcoefficient b and the plastic ring area r A .
4. Conclusions
We have shown that a manually drivenharmonic oscillator can be explored in variouslevels of physics teaching.In the qualitative approach, the amplitudeof oscillation and phase difference behavior asone approaches the system’s natural frequencywas clearly observed.A quantitative proposal involving theconstruction of a resonance curve was appliedto a real class of 34 students and theexperimental data thus obtained was fittedby the theoretical model with a correlationcoefficient of 0.991. One of the main difficultiesin this approach is the realization of manydifferent oscillations by a single person (e.g.the teacher) whose arm can get tired. In thiscase, the teacher may take turns with otherstudents.At last, an implementation using a rigidtube as a guide to the spiral toy wasproposed to be applied to a physics teachinglaboratory, where each group of students can anually driven harmonic oscillator Figure 5.
Experimental data obtained with the setup of figure 4 for different values of effective mass m eff and ring area r A . The continuous curves correspond to the fitting using equation (1). The fitted parameters arelisted at table 1. ( a )( b )( c ) ( d )( e )( f ) m e ff = 13.55 g r A = 79 cm2 m e ff = 11.05 g r A = 119 cm2 m e ff = 20.05 g r A = 79 cm2 m e ff = 9.25 g r A = 79 cm2 m e ff = 15.35 g r A = 119 cm2 m e ff = 21.85 g r A = 119 cm2 make its own quantitative analysis probing theforced damped harmonic oscillator in differentconditions. References [1] Thornton S T and Marion J B 2003
ClassicalDynamics of Particles and Systems (Cengage Learning) ISBN 0534408966[2] MacLeod A 1969
Physics Education Physics Education American Journal of Physics The Physics Teacher American anually driven harmonic oscillator m eff (g) ring area(cm ) ω = (cid:113) k/m eff (rad/s) From fitting of equation (1) ω (rad/s) b (g/s)(a) 13.55 79 6.50 6.34 9.7(b) 20.05 79 5.34 5.67 13(c) 9.25 79 7.86 7.38 9.5(d) 11.05 119 7.19 6.67 20(e) 15.35 119 6.10 6.01 15(f) 21.85 119 5.12 5.42 18 Table 1.
Driven harmonic oscillator parameters used in the experimental configuration depicted in figure 4.
Journal of Physics Physics Education The Physics Teacher Physics Education American Journal ofPhysics Physics Education Physics Education et al. Physics Education PhysicsEducation PhysicsEducation https://doi.org/10.1088/0031-9120/40/6/003 [16] Pendrill A M and Eager D 2014 Physics Education https://harmetronome.glitch.me/ [18] Silva M N S and Carvalho-Neto J T2019 Qualitative manually driven har-monic oscillator demonstration https://youtu.be/Q9A-8cXByHk [19] Silva M N S and Carvalho-Neto J T2019 Quantitative manually driven har-monic oscillator demonstration https://youtu.be/74YtezSDFRo [20] Fox J and Mahanty J 1970 American Journal ofPhysics ThePhysics Teacher45