The Physics Teacher

(The answer to this month’s “Figuring Physics” will be printed in the May issue of The Physics Teacher and is available at TPT Online, tpt.aapt.org. The answer to March’s question appears on p. 315 of this issue.) DOI: 10.1119/10.0009990 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 243

letters to the editor Resting a disk on a rough incline otherwise, the net torque w.r.t. O will not be zero and thus the disk will rotate about O. As shown in Fig. 1, there are at In a recent paper,1 De Luca works out how a thin disk can be most two such positions CM can situate (labeled as A and rested on a rough incline by adhering an additional mass at its B), agreeing with the two distinct solutions found in Ref. 1. rim. De Luca’s analysis is correct and rigorous, but I think it However, if the vertical line at O does not intersect the circle may be difficult for most high school students, and the physics of radius r (r too small or θ too large), the disk cannot be set at of the solution(s) need to be more elaborated. Here is an alter- equilibrium in any way. native, and comparatively simpler approach. Denote the center, radius, and mass of the disk as C, R, Referring to the symbols in Fig. 2, the following results are and M, respectively, the additional mass as m, and the sloping straightforward. In COD, angle of the incline as θ. The center of mass (CM) of the com- 1 = π/2 − θ, (2) posite disk-and-m object is at a mdiRst/a(nMce+rmfr)o.2mHCenocne,the line and CAD, cos 2 = CD/r = (R/r) sin θ, connecting C and m, where r = CD = R sin θ. In CBD or (3) or R/r = 1+ ρ, (1) cos 2 = (1+ ρ) sin θ. where ρ = M/m. It is obvious that for achieving a static equi- Thus, the problem is solved. librium, CM must be positioned exactly above O, the contact The angles of A and B, measured from CO, are = 1 + 2 point of the disk and the incline. Since the normal force N, and the friction fr act on the disk at O, the third force, W= and = 1 − 2, respectively. With the identity cos = (M + m)g, must point in a direction passing through O as well; cos ( 1 ± 2) = cos 1 cos 2 sin 1 sin 2, the exact Eq. (15) of Ref. 1 is derived if sin θ and cos θ are expressed in terms of t = tan θ. The equilibriums A and B exist only when the angle 2 exists, i.e., the vertical line at O cuts, or at least touches, the circle of radius r, implying the satisfaction of the condition sin θ ≤ 1/ (1+ ρ), (4) Fig. 1. Only when the mass m is placed such that corresponding to the right-half of Eq. (13) of Ref. 1. Of course, CM, the center of mass of the disk-and-m object, another critical equilibrium condition is tan θ ≤ μs, where μs is is directly above O, an equilibrium can be achieved. the static friction coefficient, ensuring the incline friction can balance the component of gravitational force along the incline, Fig. 2. By finding 1 and 2, the angular positions of the same as a block resting on a rough incline. A and B are determined. One can check the stability of the two equilibriums. Sup- pose the system is initially at A, as that shown in Fig. 1. Now the disk is made to rotate slightly clockwise (counterclock- wise), m rotates and then CM will go to the RHS (LHS) of the vertical line at O, hence the then torque due to W about O will make a further clockwise (counterclockwise) rotation, mean- ing CM will go further away from A. In other words, the equi- librium at A is unstable. By the same method, the equilibrium at B is found to be stable. Besides, those who wish to under- stand more about the stability properties of the system could refer back to Ref. 1 and the related paper by De Luca.3 Finally, I ought to say that the analytical approach in Ref. 1 is worthy nevertheless, since one can learn and appreciate the reasoning in every step of the process of solving the problem by employing simple concepts in trigonometry and geometry. Indeed, an alternative “simpler approach” of a physics prob- lem is very often unavailable. 1. Roberto De Luca, “Will the wheel stand still uphill?” Phys. Teach. 59, 430–431 (Sept. 2021). 2. See, for example, http://hyperphysics.phy-astr.gsu.edu/hbase/ cm.html. 3. R. De Luca, “Oscillation of a balanced hollow cylinder on an inclined plane,” Am. J. Phys. 89, 677–682 (July 2021). Chiu-king Ng retired, C C C Yenching College, Hong Kong 244 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

Deducing the law of reflection from Fermat’s letters principle Connecting A recent article1 proposes a student activity to demonstrate Physics equality of the angles of incidence and reflection for a light Teachers ray that is incident through a point A and reflects off a flat with the mirror so that it passes through a point B along a path that Finest Jobs requires the least travel time. Since the speed of light is con- stant along the entire path, such a path necessarily has the Find your future at shortest distance from A to B via some point on the mirror and the issue is to find the required point on the mirror. The aapt.org/careers activity consists in choosing various points on the mirror and physically measuring the sum of the distances from each of those points to A and B using a ruler. It is helpful to supplement that activity with the following one.2 Consider any two points A and C in a homogeneous me- dium. Draw any plane (perpendicular to the page) separating them, shown in edge view as the red line in Fig. 1. The path of shortest distance between points A and C is the straight light ray shown in blue, which intersects the red plane at point O. Draw the normal to the plane that passes through point O as the dashed line in Fig. 1. By construction, the angle of incidence i is equal to the angle of transmission t. If this fig- ure is drawn on a trans- lucent page, then after folding it across the red line, line segment OC is seen to coincide with line segment OB, with angle t equal to angle r. Howev- Fig. 1. er, this folding operation implies that the red plane has become a mirror, and one there- by discovers that the angle of incidence i must be equal to the angle of reflection r for the path of least time from points A to B via reflection from the mirror. 1. S. R. Pathare, B. G. Latad, R. D. Lahane, and S. S. Huli, “Why is the angle of incidence equal to the angle of reflection? An activ- ity,” Phys. Teach. 59, 650–651 (Nov. 2021). 2. P. A. Tipler and G. Mosca, Physics for Scientists and Engineers, 6th ed. (Freeman, New York, 2008), p. 1078. Carl E. Mungan U.S. Naval Academy, Annapolis MD THE PHYSICS TEACHER ◆ Vol. 60, April 2022 245

Robust Triboelectric Charging of Identical Balloons of Different Radii Francisco Vera, Rodrigo Rivera, and Manuel Ortiz, Pontificia Universidad Catolica de Valparaiso Facultad de Ciencias, Valparaiso, Chile Francisco Antonio Horta-Rangel, University of Guanajuato, Guanajuato, Mexico Electrification by rubbing different materials is a well- known phenomenon with a history that began more than five centuries B.C. ago. However, simple exper- iments can lead to contradictory or inconsistent results, and the history of this phenomena is plagued with non-intuitive results. For example, triboelectric charging by rubbing iden- tical materials is possible. In this work we want to highlight some historical aspects of triboelectricity that could enrich the discussion of electrostatics in an undergraduate physics course. We will focus on the effect of strain on the triboelec- Fig. 1. Two balloons of different sizes. Left: both balloons charged tric properties of a sample, which we think is not well known negative when initially uncharged and rubbed against hair. Right: to physics teachers. We will show that it is possible to obtain balloons charged with different polarities when rubbed against robust polarities by rubbing identical rubber balloons of dif- each other. ferent radii and we will also show that this charging method can be very useful in introductory physics courses. One of our designs included identical rollers that charged the In practical terms, one important idea was established in dome with random polarities. While discussing the physics 1757 by J. C. Whilke, who proposed to arrange materials in behind this setup, one of our teaching assistants rubbed two a triboelectric series. When two materials in the series are balloons of the same material and was surprised of the result- rubbed, the relative order of the materials in the series de- ing different polarities in them. Further experimentation led termines which one becomes positively charged and which us to discover that rubbing two identical balloons inflated to one becomes negatively charged. Thus, a given material can different sizes produced a robust polarity sign that was cor- acquire any sign of charge depending on which material is related with the relative size of the balloons. The main result is rubbed against it. The phenomenon of triboelectric charging that the larger (more stressed) balloon always gets negatively is so complex that a basic understanding of the physics behind charged. it is not clear nowadays, even in matters so basic as to whether electrons, ions, or bulk material are the particles transferred We have made experiments using several brands of rubber that are responsible of the charging process. Besides, multi- balloons, and we also used small water balloons. In every ple factors have incidence on the result of the triboelectric case we inflated identical balloons to different radii and the charging process, among them: the chemical nature of the predicted polarities where obtained. In a typical experiment surface, its morphology, the interaction with the environment, at the laboratory, we used balloons of diameters 10 cm, 20 cm, and even the past history of the samples. A simple review of and 25 cm, and we measured the polarity of the balloons using the history of static electricity and the triboelectric series can some of the polarity detectors for physics demonstrations that be read in the review written in 2019 by Lacks and Shinbrot1 we reported previously.9,10 The magnitude of the charge in and in the introduction written by Pionteck for the the balloons depended to some extent on the kind of balloons of Antistatics.2 Recently Zou et al.,3 working under Handbook used, but in all cases the polarity of the charges was consistent well-con- as described above. In another setup we began with three bal- trolled experimental conditions, built a beautiful quantitative loons initially uncharged, then they were rubbed against hair, triboelectric series. However, we must warn the reader that a and after all balloons become negatively charged we rubbed simple physical change in a sample can considerably alter its any pair of balloons. For any pair of balloons in this experi- triboelectric properties. The effects of strain on the triboelec- ment, the larger balloon gets negatively charged and the initial tric properties of materials were discussed by Jamieson in polarities change accordingly. As shown in Fig. 1, charging 19104 and, contrary to the common belief, the triboelectric balloons in this way offers a well-controlled mechanism to charging of identical materials is possible, as discussed by produce a specific sign of charges in the classroom and opens Shaw in 1926.5 More up-to-date references on this subject can the possibility to enrich the discussion of the triboelectric be obtained in the works of Shinbrot, Komatsu and Zhao,6 series, including the triboelectric charging of identical mate- Sow et al.,7 and Sow, Lacks, and Sankaran.8 At the end of rials. section III of the last reference, the authors concluded that a rigorous ordering of materials in a triboelectric series appears A usual experiment used to introduce the basic concepts to be unrealistic. of electrostatics involves the adherence to a table and poste- In the past few years, we have been building Van de Graaff rior separation of pieces of Scotch tape.11-13 However, in this generators using unconventional materials for the rollers. experiment the charging of the tape is not robust and the po- larities obtained depend on the brand of tape used and/or the table or surface to which it adheres.13,14 Thus, if this point is not explicitly discussed with students, confusion might arise 246 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 DOI: 10.1119/5.0038084

when students determine the charge of their tape by using the 9. Francisco Vera, Manuel Ortiz, Diego Romero-Maltrana, and typical plastic rod and fur experiment and different students Francisco Antonio Horta-Rangel, “Using capacitors to measure might obtain different results. Our setup has the advantage charge in electrostatic experiments,” Phys. Teach. 56, 525–527 that the larger balloon is always negatively charged and no (Nov. 2018). confusion can arise. Besides, the balloons are much easier to manipulate than the tape, since the stickiness of the later and 10. Francisco Vera, Jaime Villanueva, and Manuel Ortiz, “Revelan- its tendency to adhere to the hands and to itself is avoided. do el signo de las cargas eléctricas usando un detector de polari- The balloons also provide a nice visualization, as in Fig. 1, that dad entretenido,” Revista Eureka sobre Enseñanza y Divulgación is easier to see from all parts of a classroom if they are used as de las Ciencias 15 (2), 2401 (2018). a part of a lecture demonstration. 11. Lillian C. McDermott, Peter S. Shaffer, and the Physics Educa- The robust charging of two initially discharged balloons tion Group, Tutorials in Introductory Physics (Pearson, Wash- of different radii also offers a simple alternative to replace the ington, 2002). fleece and wool sweater in the simple experiment proposed by Rueckner in 200715 to demonstrate charge conservation. 12. Erik Mazur, Principles and Practice of Physics (Pearson, Boston, Rueckner’s improved version of the Faraday pail experi- 2015). ment shows the equal in magnitude and opposite sign of the charges acquired by two objects initially contained in a tall 13. Rachel H. Chabay and Bruce A. Sherwood, Matter & Interac- aluminum cooking pot. Our setup offers the possibility of tions, 4th ed, Vol. 2 (Wiley, New Jersey, 2015), pp. 572–575. explicitly producing a well-defined sign of charge depending on which balloon is left inside the initial cooking pot. Also, 14. Randall Harrington, “Getting a charge out of a transparent we are presently using homemade electrophori that include tape,” Phys. Teach. 38, 23–25 (Jan. 2001). balloons of different sizes to produce the high voltages in the electrodes for the experiment of electric field patterns of seeds 15. Wolfgang Rueckner, “An improved demonstration of charge in oil. To the best of our knowledge, the robust charging of conservation,” Am. J. Phys. 75, 861–863 (Sept. 2007). two rubbed balloons has not been noted before and it could be used in schools and undergraduate labs to provide a simple Concerned about mechanism to obtain specific signs of charges for a variety of experiments. the safety of your students? Acknowledgment SeBlleesrt! We would like to acknowledge financial support from Fondo Nacional de Desarrollo Científico y Tecnológico, FONDE- Promote safety awareness and encourage safe CYT Project 1181782. habits with this essential manual. Appropriate for elementary to advanced undergraduate labo- References ratories. 1. Daniel J. Lacks and Troy Shinbrot, “Long-standing and un- Members $9.50 • Nonmembers $11.50 resolved issues in triboelectric charging,” Nat. Rev. Chem. 3, order online: www.aapt.org/store or call: 301-209-3333 465–476 (2019). 2. George Wypych and Jurgen Pionteck, Handbook of Antistatics (Chem Tec Publishing, Canada, 2016). 3. H. Zou et al., “Quantifying the triboelectric series,” Nat. Com- mun. 10, 1427 (2019). 4. W. Jamieson, “The electrification of insulated materials,” Nature 2111 (83), 189 (1910). 5. P. E. Shaw, “The electrical charges from like solids,” Nature 2975 (118), 659–660 (1926). 6. T. Shinbrot, T. S. Komatsu, and Q. Zhao, “Spontaneous tribo- charging of similar materials,” Europhys. Lett. 83, 24004 (2008). 7. M. Sow, R. Widenor, A. Kumar, S. W. Lee, D. J. Lacks, and R. M. Sankaran, “Strain-induced reversal of charge transfer in contact electrification,” Angew. Chem. Int. Ed. 51, 2695–2697 (2012). 8. Mamadou Sow, Daniel J. Lacks, and R. Mohan Sankaran, “De- pendence of contact electrification on the magnitude of strain in polymeric materials,” J. Appl. Phys. 112, 084909 (2012). THE PHYSICS TEACHER ◆ Vol. 60, April 2022 247

Sterile Neutrinos: Are They Real? B. Eberly, University of Southern Maine, Portland, ME Fig. 1. Right-handed objects are defined when the angular momentum vector is parallel to the direction of motion, while D. Lincoln, Fermi National Accelerator Laboratory, Batavia, IL left-handed objects have an angular momentum vector that is antiparallel to the direction of motion. Alternatively, when Neutrinos are perhaps the least understood of the viewed so that the object is moving directly toward the observer, known denizens of the subatomic world. They have right-handed objects rotate in a counterclockwise fashion, while nearly no mass, interact only via the weak nuclear force left-handed objects rotate in a clockwise manner. and gravity, and, perhaps most surprising, the three known species of neutrinos can transform from one variant into anoth- Fig. 2. Parity conservation violation implies that certain spin er. This transformation, called neutrino oscillation, has been configurations are not observed. For instance, right-handed demonstrated only relatively recently and has led to speculation antineutrinos and left-handed neutrinos are observed. However, that there might be another, even more mysterious, neutrino if one interchanges right and left, the direction of motion will variant, called the sterile neutrino. While the sterile neutrino change, although the spin direction doesn’t change. The results— remains a hypothetical particle, it is an interesting one and left-handed antineutrinos and right-handed neutrinos—are not searches for it are a key research focus of the world’s neutrino observed in nature. scientist community. were able to prove that there exist precisely three distinct fla- Before talking about sterile neutrinos, it is perhaps useful to vors of low-mass neutrinos.6 briefly review the behaviors of ordinary neutrinos and the his- tory that revolves around them. Neutrinos were hypothesized In a series of experiments that began in 1964, Ray Davis in 1930 by Austrian physicist Wolfgang Pauli. He proposed1 measured the neutrino flux from the Sun and found that his them to solve some mysteries involving beta decay. At the time, measurements were below what were expected from calcu- beta decay was thought to be the transformation of a neutron lations.1,7 This discrepancy was known as the solar neutrino into a proton and an electron. However, careful studies of the problem. A parallel set of experiments studying neutrinos process made scientists realize that energy and angular momen- generated in Earth’s atmosphere from cosmic rays in the tum did not seem to be conserved in this process. Pauli’s con- 1980s seemed to indicate a deficit of muon neutrinos.1,8 This jecture was that a third particle, which we now call the neutrino, discrepancy is called the atmospheric neutrino anomaly. was created in the decay and carried away the missing energy These observations and others led scientists to speculate that and angular momentum. perhaps the different types of neutrinos could transform into each other via a process called neutrino oscillation. The basic Neutrinos do not interact via either electromagnetism or the process was first proposed by Bruno Pontecorvo in 1957,1,9 strong nuclear force and therefore are very difficult to detect. It and the phenomenon of neutrino oscillation was definitively wasn’t until 1956 that neutrinos emitted from nuclear reactors confirmed through a series of experiments performed at the were observed.1,2 (Technically, antineutrinos were observed in end of the 20th century.1,10 These measurements were per- this early experiment.) The following year, violation of parity formed using the SuperKamiokande detector (atmospheric conservation was observed in weak nuclear interactions.1,3 neutrinos) and the Sudbury Neutrino Observatory (SNO, Parity is conserved if one can replace (x, y, z) (–x, –y, –z) and the equations describing the interaction are unchanged. If one makes this substitution and the change is evident, then parity is not conserved under that interaction. Because parity conservation is violated in weak interactions, and because neutrinos interact only via the weak force, this implies all detectable neutrinos are left-handed, while detect- able antimatter neutrinos are all right-handed. Right-handed means that the particle’s intrinsic angular momentum (its spin) is parallel to the direction of motion, while left-handed means that it is antiparallel. Handedness is illustrated in Fig. 1, while the implications of parity conservation violation are illustrated in Fig. 2. In 1962, a collaboration led by Leon Lederman, Melvin Schwartz, and Jack Steinberger discovered that neutrinos come in more than one distinct “flavor”; specifically they found that there is an electron-type and a muon-type neutrino.1,4 Each of these is associated with the named charged lepton. With the discovery of the tau lepton in 1975,1 researchers suspected that there was a third tau-type neutrino, a conjecture that was con- firmed in 2000.5 In the early 1990s, researchers using data from the LEP accelerator, located at the CERN laboratory in Europe, 248 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 DOI: 10.1119/10.0009991

number is conserved in both example processes, in the second process after the decay there is a muon-type lepton number of –1 and an electron-type lepton number of +1. Because this type of decay is not observed in nature, we say that it is forbid- den. Neutrino oscillation is the only known phenomenon that does not respect flavor-specific lepton number conservation. If an electron-type neutrino transforms into a muon-type or tau-type, the flavor-specific lepton number changes during the transformation. There is another aspect of neutrino oscillation that can be confusing. Unlike most subatomic particles, the observed neutrinos do not have a unique mass. In contrast, any electron will have a mass of 0.511 MeV/c2 and every muon has a mass of 106 MeV/c2. The oscillation of neutrinos is essentially a quantum mechanical phenomenon that occurs because the observed neutrinos are a quantum superposition of different mass states. This can be illustrated most easily in a simplified world that has only two neutrino flavors—say the electron— n( aet)easnydstmemuoinn-wtyhpaet ( ). These form an orthogonal coordi- one might call the “neutrino plane.” In addition, the neutrino plane can be spanned by another or- thogonal coordinate system based on the mass states. Fig. 3. This figure is an image of the Sun taken by the For instance, assume that there are two neutrinos 1 and SuperKamiokande detector using only neutrinos that penetrate 2, which have mass m1 and m2, respectively. A coordinate Earth and thus illustrating the capabilities of neutrinos to effort- system based on mass can also exist in the neutrino plane, ori- lessly penetrate matter. (Figure courtesy the SuperKamiokande ented at an angle with respect to the flavor coordinate system. collaboration) This is essentially identical to how a Cartesian plane can be represented by orthogonal (x,y) and (u,v) coordinate systems, solar neutrinos). An accessible review of this phenomenon is which have an angle between them. This idea is illustrated in given in Ref. 11. Figure 3 shows an image of the Sun taken by Fig. 4. the SuperKamiokande experiment, imaged through Earth and using only neutrinos. Neutrino oscillations Fig. 4. A two-dimensional Cartesian plane can be spanned by a Neutrino oscillations solved both the atmospheric and so- number of orthogonal coordinate systems, which are traditionally labelled (x,y) and (u,v), with an angle between the x- and u-axes. lar neutrino problems, but they are probably the most surpris- Both are equally good coordinate systems. The u- and v-axes are ing and counterintuitive of all particle physics phenomena. each a mixture of x- and y-components that can be determined Unlike all other fundamental subatomic particles, neutrinos by projecting them onto the x- and y-axes. The situation is similar can, over time, change their identity. This process can only for neutrinos, for which a two-component neutrino plane can be proceed if conservation laws that were previously thought to represented by either a flavor coordinate system ( e, ) or a mass be inviolable are, in fact, violated. one ( 1, 2). Lepton number is a quantity that is conserved in essentially The fact that the mass and flavor systems are not the same the same manner as energy, momentum, and electrical charge. for neutrinos implies that if one were to identify an electron If a photon (which has a lepton number of zero, i.e., L = 0) neutrino and then subsequently attempt to measure its mass, transforms into a matter/antimatter pair and one half of the one would obtain either m1 or m2 , with the probability de- pair is an electron (L = +1), the other particle must be a posi- pending on the angle between the two coordinate systems. tron (L = –1). Similarly, lepton number is conserved if a posi- tatihnvedesleaynecxhealaemrcgpterldoensW, ncoebunotssreoirnnvoa(Lt(iLo=n=0o)+fd1ce)h,ciaa.reyg.s,eiWnimt+opalipceiot+sl+yitrrυeoes.nt(r(IiLcnt=sbot–ht1he) Neutrinos oscillate because they are created in flavor states. possible decay products.) As neutrinos propagate, the Schrödinger equation predicts that each component mass state evolves in time different- For most interactions, lepton number conservation is more ly, allowing the neutrino to be detected later in a different restrictive than that. Lepton number is conserved within lep- ton families, meaning there is an electron-type lepton num- ber, muon-type, and tau-type, and that each of these is conserved in particle decays. For example, the positive W boson decay mentioned above is observed, while the decay W+ μ++ υe is not observed. This is because while lepton THE PHYSICS TEACHER ◆ Vol. 60, April 2022 249

are correlated by Eq. (1) means that the published result is a complicated graph that allows many possible solutions. In the situation where an experiment observes no oscillation signal, then only an exclusion region can be determined. One such graph is shown in Fig. 5. For the three-neutrino flavor situation, there are more os- cilmla12t3i,oannpdaramm22e3t(eδrCs,Psipseacpifaicraalmlyete1r2,,for1b3,id2d3e,nδCinP,thems12i2m, ple two-neutrino picture, which allows neutrinos to oscillate dif- ferently than antineutrinos). In spite of this added complexity, many experimental results fit quite well to the two-neutrino oscillation described by Eq. (1). Measurement of the oscilla- tion parameters is an active research industry, and a review of the current numerical values and their uncertainties can be found in Ref. 12. Disagreement in data Prior to the definitive proof of the phenomenon of neutri- no oscillation in 1998–2001, scientists were fairly convinced Fig. 5. Final measurement of the LSND experiment16 analyzed that the phenomenon was the explanation for the myriad using the two-neutrino oscillation approximation. LSND esti- unexplained observations in neutrino data. Consequently, mates that the true values for sin2 2 and m2 are within the accelerator-based experiments were proposed to search for blue region with 90% probability and within the blue and yellow the signature of neutrino oscillation. One such experiment region with 99% probability. Also included are measurements was proposed in 1990.14 This experiment was to be performed from four other neutrino oscillation experiments at the time. at the Los Alamos National Laboratory and it was named the For each experiment, the region to the right of the red curve is Liquid Scintillator Neutrino Detector, or LSND. excluded at a 90% confidence level, while the region to the left LSND generated both muon neutrinos and antineutrinos is allowed. References for the other experiments can be found and then used their detector to search for the appearance of in Ref. 16. We see that the Karmen experiment excludes nearly their electron counterparts. Their method was to use a pos- all of the parameter space favored by LSND. itively charged pion beam from the LAMPF (Los Alamos Meson Production Facility) beamline. These pions were then fmlaevcohrasntiactse1.1Iftomc1al≠cumla2t,ewtehecapnroubsaebfialiirtlyytshiamt ptwleoqduiasntitnucmt slowed by passing through matter and most came nearly to flavor neutrinos can transform into one another. In the sim- rest. Positive pions decay into positive muons and muon neu- ple case where there are only two flavors of neutrinos, if one ttrriinnToohsse,. aPsnoedasirtmcihvuesomtnrauatonengtiysntwehuaetsnribdnaeoscsead(yi.uienp.,toπon+psoesaμirt+rcohνniμns,gefleoe+rcνtarμno_νneμxnνceee)us.s- starts with a beam of (for example) pure electron neutrinos, aonf teilneecutrtornin-otysp(eν_ea)nwtienreeuptrriondousc. eTdhienrtehaesodnecisaythcahtanino.eIlfetchtreoyn with an energy E, expressed in GeV, and a detector located L were observed, tihnetonaonneelhecytprootnhaensitsinisetuhtaritntohe(ν_mμuonν_ae)n.tAindedui-- kilometers from where the neutrino beam is produced, the trino oscillated rneesut tarnindotsh(eνree)foarree only produced by the mu- probability that any electron-type neutrino will be detected as tionally, electron should have relatively little a muon-type neutrino is on that decays at energy. Detection of higher energy electron neutrinos could be taken as evidence of oscillations from muon neutrinos cre- (1) ated by pions that decayed before they could come to rest. LSND published a series of papers, beginning in 1995.15 E The first papers reported searches for electron antineutrino appearance, while subsequent papers included electron neutri- The mathematics describing the oscillation probabilities no appearance searches. The final LSND paper was published between three distinct flavor and mass states is considerably in 2001 and it included results gleaned from both searches.16 more complicated, and the interested reader can find more Furthermore, this paper was published late enough that it information in Ref. 12. Equation (1) is given in an SI form in included information from the SuperKamiokande and SNO Ref. 13. measurements, which established that neutrino oscillations were real.10 Figure 5 is taken from this reference. One can dissect Eq. (1). Parameters L and E are determined And that’s where the situation becomes interesting. When by the experimental setup. The angle between the flavor and several experiments attempt to measure the same quantity, mass states in the neutrino plane is and it determines the one can overlay the various experiments’ results and get a maximum probability of oscillation. The mass difference m212 = m12 – m22, expressed in eV2, contributes to the fre- quency at which neutrinos oscillate. The parameters and m212 are constants of nature and can be extracted from the data. However, it is often true that any specific experiment cannot determine a unique numerical value for both of them. It is far more common that a range of parameters are allowed and, furthermore, the fact that the two parameters 250 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

more accurate estimate of the parameters being measured by A four-neutrino configuration in which the neutrino can free- combining measurements. ly oscillate between all four flavors can be made to fit much of However, it is also occasionally true that measurements the existing data. might disagree. Figure 5 shows such a situation. This plot However, there are issues. First, experiments at LEP have shows the values of sin2 2 and m2 that are allowed accord- demonstrated that there exist exactly three low-mass neutri- ing to the LSND data (yellow + blue at 99% confidence), as nos that interact via the weak force.6 Furthermore, in order well as exclusion curves for four other experiments. In all for the four-neutrino theoretical paradigm to explain any cases, those experiments exclude the region to the right of discrepancies caused by LSND data, the fourth neutrino must the respective red curve with a 90% confidence and allow the have a mass of order 1 eV or larger. This runs afoul of cos- parameter space to the left. All experiments exclude at least mological estimates of the sum of all active neutrino species, a portion of the LSND preferred parameter space, with the which conclude that ∑mν < 0.13 eV.17 Karmen and Bugey experiments excluding nearly all of it. The This has led scientists to propose18 the existence of what Karmen experiment was qualitatively similar to LSND, and are called “sterile neutrinos.” Sterile neutrinos are currently the differences are detailed in Ref. 16. hypothetical and are defined to be a class of low mass particles It is possible in the current day to perform a global fit to all that participate in neutrino oscillation, but do not participate oscillation data, excluding LSND data. That analysis finds that in the weak force. two of the mass states (×m110a–n5 deVm22]), are nearly the same While it is often said that neutrinos are only left-handed [ m122 = (7.53 ± 0.18) while the third mass state particles, it is more correct to say that the weak nuclear force (m3) is quite different [m2mis223se=v(e2r.a4l5o3rd±e0r.s03o3f )m×a1gn0–it3uedVe2l]a.r1g2er only couples to left-handed particles and right-handed anti- The LSND result for particles. Since neutrinos interact only via the weak force, it than the values determined from other, more recent measure- is therefore possible that right-handed neutrinos exist, but ments. If we assume that the measurements are accurate, and these particles do not participate in the known interactions of that the LSND result is caused by neutrino oscillations, then the Standard Model. It is not, strictly speaking, necessary that clearly the standard three-neutrino picture is not complete. sterile neutrinos are right-handed neutrinos, but most sterile neutrino models propose that they are. Sterile neutrinos The existence of right-handed neutrinos is not so implausi- In order to resolve the discrepancies between LSND and ble, given that all other known fundamental subatomic parti- cles come in both a right- and left-handed spin state. Howev- the other neutrino oscillation experiments, scientists have er, the existence of right-handed neutrinos would require an considered the possibility that there might be four neutrinos. extension of the Standard Model. Fig. 6. A photomultiplier tube used by the MicroBooNE Fig. 7. The MicroBooNE detector in the test facility. 251 experiment to trigger on scintillator light that is gener- Because MicroBooNE uses liquid argon as a detec- ated when a neutrino interacts in the detector. (Figure tion medium, the cryostat is shrouded in insulative courtesy Fermilab/R. Hahn) material. (Figure courtesy Fermilab/R. Hahn) THE PHYSICS TEACHER ◆ Vol. 60, April 2022

Fig. 8. The results of a joint search for sterile neutrinos another experiment was performed at Fermilab to study the conducted by the MINOS, MINOS+, Daya Bay, and Bugey-3 MiniBooNE results in greater detail.22 This experiment is experiments.26 They see no evidence of sterile neutrinos and called MicroBooNE, and it uses a different technology than exclude at a 90% confidence level all oscillation parameters to LSND and MiniBooNE. MicroBooNE uses liquid argon as the right of the red line. Also shown are exclusion curves from the detector medium, allowing much better reconstruction the NOMAD and KARMEN experiments. Taken together, the of interactions in the detector. Initial results from the Micro- LSND and MiniBooNE region seems to be entirely excluded. BooNE experiment23 show no indication of electron neutrino Figure adapted from Ref. 26, which also contains the relevant appearance, which is particularly puzzling because Micro- references. BooNE and MiniBooNE share the same neutrino beam. The experiment continues to analyze data as part of Fermilab’s Alternative searches for sterile neutrinos focus on neutrino physics and its final analysis is not yet avail- able. Figure 6 shows a photomultiplier tube used by Given that the existence of sterile neutrinos is outside the MicroBooNE to trigger when a neutrino interacts with the existing Standard Model paradigm, it is worth revisiting the argon in the detector. This interaction creates electrically data to consider the possibility that the data are in error. There charged particles, which then move and cause scintillation are other experiments that see discrepancies consistent with light in the liquid argon. Figure 7 shows the MicroBooNE 1-eV2-scale neutrino oscillations, including nuclear reactor detector prior to final installation, when it was being tested. experiments and radioactive source experiments that observe Because the detector uses liquid argon as a detection medium, fewer neutrinos than expected. However, the dominant evi- it is wrapped in insulative material to optimize detector per- dence for sterile neutrinos is the LSND measurement, thus it formance and minimize cooling costs. is imperative that the experiment be confirmed. There are other experiments that have searched for sig- In 1997, an experiment called MiniBooNE was proposed natures that can confirm or falsify the LSND result. These at Fermilab to check the LSND result.19 Compared to LSND, are neutrino disappearance experiments that complement the MiniBooNE detector observed neutrinos that were about the LSND appearance observations—after all, if one species 10 times more energetic, while using dramatically different of neutrino appears during oscillation, then another must detection technology. Many of the LSND collaborators were disappear. In one experiment at a nuclear reactor complex also on MiniBooNE, although the experiment mandated at its in France, scientists detect a 6% deficit in electron neutrino outset that no more than 50% of the MiniBooNE author list flux.24 This is consistent with a fourth neutrino species with could be scientists who were also on LSND. This was to pro- mass of about 1.2 eV/c2. In addition, an experiment using the tect against the possibility that the LSND measurement could Daya Bay nuclear reactor in China initially noted a deficit in have arisen from some sort of algorithmic error that might be electron neutrino flux, but a recent paper suggests that this transported from LSND and appear in MiniBooNE analyses. deficit can be traced to incorrect modeling.25 Furthermore, Daya Bay performed a joint search for sterile neutrinos with MiniBooNE’s first results were announced in 2007 and the the Bugey-3 reactor experiment in France and the MINOS experiment concluded that they did not see evidence for the and MINOS+ experiments at Fermilab.26 Figure 8 shows their kind of neutrino oscillation reported by LSND.20 However, result, in which they saw no evidence of neutrino oscillations. a subsequent follow-on analysis reported in 201821 that they This creates yet another puzzle to resolve, since these experi- saw a significant excess of electron-like events in the data and ments completely exclude the region of oscillation parameters that the derived parameters were consistent with LSND and most favored by the LSND and MiniBooNE data. furthermore that the data were consistent with oscillation behavior requiring four neutrino flavors, i.e., sterile neutrinos There are other experiments searching for sterile neutri- were needed. nos, including one27 from the IceCube detector, which uses a cubic kilometer of ice in Antarctica as the detection medium. This could be strong evidence for sterile neutrinos, and IceCube reports no evidence for the existence of sterile neu- trinos. Conclusion If sterile neutrinos were to exist, it would require a sub- stantial rewrite of our understanding of the subatomic world. There are a few experiments that have reported evidence for their existence; however, there have been many others (with higher sensitivity) that have ruled out the parameter space favored by LSND. At this point, the case for the existence of sterile neutrinos is by no means a strong one. The conflicting results might be because they exist, but are hard to find; or the handful of positive results may simply be anomalies indicative of the difficulty associated with studying neutrinos—ghosts of the subatomic world. 252 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

Fermilab, America’s flagship particle physics laboratory, wυN_μiCthLChν_iogel,h”leLsceAtnio-s1ni1tSi8vt4oit2rye(/i1n_9Pt9hu0eb)l,aihpc/tp2tep1as/r:0/a/9ni2nc/ie2s.1cih0aea9an2.0no0er4lgs./pυcdoμlfl.ecνtieoann/d continues to expand their already world-class neutrino re- 15. G. Athanassopoulos et al., “Candidates in a search for muon an- search program and relevant experiments are underway to tineutrino to electron antineutrino oscillations,” Phys. Rev. Lett. perform precise tests of the LSND and MiniBooNE results.28 75, 2650-2653 (1995); G. Athanassopoulos et al., “Evidence for We can’t say yet whether these efforts will definitively answer neutrino oscillations from muon decay at rest,” Phys. Rev. C 54, the question of the existence of the sterile neutrino, or intro- 2685–2708 (1996); G Athanassopoulos et al., “Evidence for duce even more puzzles to resolve, but it is safe to say that the Cν5e o8s, c2i4ll8a9ti–o2n5s1f1ro(1m99p8io).n decay in flight neutrinos,” results will add another fascinating chapter to the saga of the υμ Phys. enigmatic sterile neutrino. Rev. References 16. Ao11b.2sAe0gr0vu7ail(tai2or0ne0t1oa)f.lν._,μ“Eapvipdeeanrcaencfoerinneauν_treibneoaoms,c”iPllahtyiso.nRsefvr.oDm6t4h,e 1. D. Lincoln, Understanding the Universe: From Quarks to the 17. See Ref. 12, “Neutrinos in cosmology” section. Cosmos (World Scientific, Singapore, 2012); R. Crease and C. Mann, The Second Creation: Makers of the Revolution in Twen- tieth-Century Physics (Rutgers University Press, Rutgers, 1996); 18. D. O. Caldwell and R. N. Mohapatra, “Neutrino mass explana- F. Close, Neutrino (Oxford University Press, Oxford, 2012); L. tions of solar and atmospheric neutrino deficits and hot dark matter,” Phys. Rev. D 48, 3259 (1993); J. Peltoniemi and J. W. F. Lederman and D. Teresi, The God Particle: If the Universe Is Valle, “Reconciling dark matter, solar and atmospheric neutri- the Answer, What Is the Question? (Mariner Books, New York, 2016). nos,” Nucl. Phys. B 406, 409 (1993); S. M. Bilenky, C. Giunti, and 2. C. L. Cowan et al., “Detection of the free neutrino: A confirma- W. Grimus, “Phenomenology of neutrino oscillations,” Prog. Part. Nucl. Phys. 43, 1 (1999); V. Barger et al., “Fate of the sterile tion,” Science 124, 3212 (1956). neutrino,” Phys. Lett. B 489, 345 (2000); T. Goldman, G. J. Ste- 3. C. S. Wu et al., “Experimental test of parity conservation in beta phenson Jr., and B. H. J. McKellar, “Implications of quark-lep- decay,” Phys. Rev. 105, 1413 (1957). ton symmetry for neutrino masses and oscillations,” Mod. Phys. 4. G. Danby et al., “Observation of high-energy neutrino reactions Lett. A 15, 439 (2000). and the existence of two kinds of neutrinos,” Phys. Rev. Lett. 9 19. E. Church et al., “A letter of intent for an experiment to mea- (1), 36 (1962). sure νμ νe oscillations and νμ disappearance at the Fermilab 5. K. Kodama et al. (DONUT Collaboration), “Observation of tau Booster: BooNE,” arXiv:nucl-ex/9706011 (1997). neutrino interactions,” Phys. Lett. B 504, 218–224 (2001). 6. M. Tanabashi et al. (Particle Data Group), “Number of light 20. A. A. Aguilar-Arevalo et al., “Search for electron neutrino ap- neutrino types from collider experiments,” Phys. Rev. D 98, pearance at the m2 ~ 1 eV2 scale,” Phys. Rev. Lett. 98, 231801 (2007). 030001 (2018) and 2019 update (section 61). 21. A. A. Aguilar-Arevalo et al., “Observation of a significant excess 7. J. Bahcall, “Solar neutrinos. I. Theoretical,” Phys. Rev. Lett. 12, 300 (1964); R. Davis, “Solar neutrinos. II. Experimental,” Phys. of electron-like events in the MiniBooNE short-baseline neutri- no experiment,” Phys. Rev. Lett. 121 (22), 221801 (2018). Rev. Lett. 12, 303 (1964). 22. R. Acciarri et al., “Design and construction of the MicroBooNE 8. K. S. Hirata et al. (The Kamiokande Collaboration), “Experi- mental study of the atmospheric neutrino flux,” Phys. Lett. B detector,” JINST 12, P02017 (2017). 205, 416 (1988); D. Casper et al., “Measurement of the atmo- 23. P. Abratenko et al., “Search for an excess of electron neutrino interactions in MicroBooNE using multiple final state topolo- spheric neutrino composition with the IMB-3 detector,” Phys. gies,” arXiv:2110.14054v2 (2021). Rev. Lett. 66, 2561 (1991). 9. B. Pontecorvo, “Inverse beta processes and nonconservation of 24. G. Mention et al., “Reactor antineutrino anomaly,” Phys. Rev. D lepton charge,” Soviet Phys. JETP 7, 172 (1958). 83, 073006 (2011). 25. F. P. An et al., “Evolution of the reactor antineutrino flux and 10. Y. Fukuda et al. (Super-Kamiokande Collaboration), “Evidence spectrum at Daya Bay,” Phys. Rev. Lett. 118, 251801 (2017). for oscillation of atmospheric neutrinos,” Phys. Rev. Lett. 81, 1562 (1998); Q. R. Ahmad et al. (SNO Collaboration), “Mea- 26. P. Adamson et al., “Improved constraints on sterile neutrino mixing from disappearance searches in the MINOS, MINOS+, surement of the rate ve + d p + p + e- interactions produced Daya Bay, and Bugey-3 experiments,” Phys. Rev. Lett. 125, by 8B solar neutrinos at the Sudbury Neutrino Observatory,” 071801 (2020). Phys. Rev. Lett. 87, 071301 (2001); Q. R. Ahmad et al. (SNO Col- laboration), “Direct evidence for neutrino flavor transformation 27. M. G. Aartsen et al., “An eV-scale sterile neutrino search us- ing eight years of atmospheric muon neutrino data from the from neutral-current interactions in the Sudbury Neutrino Ob- IceCube Neutrino Observatory,” Phys. Rev. Lett. 125, 141801 servatory,” Phys. Rev. Lett. 89, 011301 (2002). (2020). 11. D. Lincoln and T. Miceli, “The enigmatic neutrino,” Phys. Teach. 53, 331 (Sept. 2015). 28. R. Acciarri et al., “A proposal for a three detector short-baseline neutrino oscillation program in the Fermilab booster neutrino 12. P. A. Zyla et al. (Particle Data Group), “Neutrino masses, mix- beam,” arXiv:1503.01520 (2015). ing, and oscillations,” Prog. Theor. Exp. Phys. 2020, 083C01 (2020) (section 14). 13. While the form of Eq. (1) is what is used by particle physicists, it [email protected]; [email protected] can also be written as where L is in meters, m2 is in kilograms, and E is in joules. 14. X. Q. Lu et al., “A proposal to search for neutrino oscillations THE PHYSICS TEACHER ◆ Vol. 60, April 2022 253

Impetus-Force-Like Drawings May Be Less Common Than You Think Amy D. Robertson, (she/her) Seattle Pacific University, Seattle, WA Lisa M Goodhew (she/her) and Paula R. L. Heron, (she/her) University of Washington, Seattle, WA Rachel E. Scherr, (she/her) University of Washington – Bothell Campus, Bothell, WA Perhaps the most commonly cited student idea about structional results, and (b) varied across samples. forces in the literature is the notion of an impetus This matters because what instructors think about student force,1-25 defined as the “belief that there is a force in- side a moving object that keeps it going and causes it to have thinking shapes their instructional decision-making.28,29 For some speed,” that can then “fade away as the object moves example, an instructor who assumes that the impetus force along.”15 According to the literature, even after physics in- idea is common and persistent may plan instruction to ad- struction students use impetus force reasoning to argue that dress it, be likely to hear this idea in what students say and do, forces are necessary to sustain motion5-9,22 or that motion and pay careful attention to whether or not students continue implies force.7,9-12,14,22 For example, many students drew an to use this idea in homework and on exams. An instructor upward arrow to indicate a force on a coin that was moving who believes that this idea is uncommon and/or not particu- upward after being tossed. The coin was halfway between the larly persistent may foreground other considerations. In other point of its release and its turnaround point. Interviews with words, the expected ubiquity and persistence of the impetus students in the course indicate that the arrow was meant to force idea becomes another lens—beyond an awareness of the indicate “the ‘force of the throw,’ the ‘upward original force,’” idea itself —through which instructors plan and interpret. and so on. Clement interprets these results to mean that stu- This paper has implications for that lens. dents “believe that continuing motion implies the presence of a continuing force in the same direction, as a necessary cause Methods of the motion.” A detailed methods section can be found in Appendix A.30 Here we offer an abbreviated overview of the questions we A number of other studies have been designed to elicit used, our sample, and how we analyzed student responses. impetus-force-like ideas (e.g., Refs. 6, 21, and 24). The im- This study grew out of a larger effort to identify universi- pression these articles give is that impetus-force-like ideas ty student resources for understanding physics—ideas that are persistent, common, and, in Clement’s words, concerning. can be framed as “beginnings” of sophisticated scientific This is communicated both by the language that the articles understandings.31-35 As part of that broader effort, we gave use and by the percentages they report. For example, authors students a number of conceptual questions that had been say that the impetus force idea is “particularly strong”3; “espe- used in previous studies, to see whether a resources-oriented cially common,”3 used by “many students”6,21; “shows up in a analysis would yield a different set of categories of student wider diversity of problem situations than one would expect,”3 thinking. Our analysis for this paper emerged from an initial including “a wide variety of simple situations”21; “appears to noticing that the frequency of impetus-like drawings seemed still be present in many students after they have completed a to be much lower among students in our samples than the course in mechanics” (which is then named as a “rather dis- frequencies reported in the original studies. We pursued this turbing result”3); and “appears to be a major stumbling block noticing by analyzing student responses to three questions: in the physics curriculum.”3 (This language is consistent with the modified coin toss question, the curved tubes question, and prevailing notions of misconceptions in the ’80s and ’90s, as the pendulum question, all featured in Fig. 1. stable conceptions resistant to change and that act as barriers The modified coin toss question, curved tubes question, to student learning, e.g., Refs. 4, 26, and 27.) The commonal- and pendulum question are (slightly) modified versions of ity of these ideas, as inferred from students’ drawings, ranges questions used in previous studies documenting student use from 33%21 to 75%,3 again suggesting that students use these of impetus-like drawings. The modified coin toss question was ideas frequently. adapted from a question used in a study done by Clement (mentioned in the introduction)3 who found that students The prevalence and persistence of the impetus force idea often drew upward-pointing arrows on the tossed object at has prompted large-scale curriculum development and in- point B, indicating an impetus-like “toss” or “hand” force structional planning to address it. In this paper, we report on sustaining the object’s motion. The curved tubes question is a set of preliminary results that challenges the universality of a slightly modified version of a question used by McCloskey, the assumption that many students have impetus-like ideas Caramazza, and Green.21 These authors found that students that persist through instruction. In our study, we asked ques- often drew curved paths for balls exiting curved tubes; they tions from a number of the studies we cite above, and we used interpreted these curved paths as impetus-like, since they similar methods as described in those papers to code student indicate that the ball “will continue in curved motion even drawings for impetus-like ideas. We found that the frequency when no external forces aacntdoKneist.s”,6Twhhe opeonbdsuerluvmedqtuheasttsitoun- was of impetus-like drawings (a) was consistently less than report- adapted from Sadanand ed in previous studies, including those that report post-in- 254 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 DOI: 10.1119/5.0027858

in the next section if either coder deemed it impetus-like. Even with this generous approach, we mostly agreed on which drawings were impetus-like; our percentage agreement was 98.5%. Results and discussion Figure 2 compares the per- centages of impetus-like drawings reported by the three studies discussed above (blue bars) to the percentages of impetus-like draw- ings among the students in our samples (yellow bars for individual samples, brown bars for overall). (For the curved tubes question, the darker bars are for Problem 1 and the lighter bars for Problem 2.) What Fig. 2 illustrates to us Fig. 1. Questions used in our study. Images of curved tubes from M. McCloskey, A. Caramazza, and B. Green, “Curvilinear Motion in the Absence of External Forces: Naïve Beliefs about the is that the frequency of impe- Motion of Objects,” Science 210 (4), 1139-1141 (1980). Reprinted with permission from AAAS. tus-like drawings in our study both (a) is consistently less than dents often drew a horizontal arrow in the direction of motion reported in previous studies and (b) varies across samples. for ball B, indicating that “many students invoke forces in the This is true for all three questions in our study. This calls into direction of motion even when there seems to be nothing that question, for us, an interpretation of impetus-like thinking can generate that force.” as universally persistent and common. For some samples, this We analyzed student responses to these three questions, kind of reasoning, as evidenced in drawings, is in fact quite from students enrolled in introductory physics courses at uncommon. six different U.S. universities, Universities A through F. Six What we cannot tell yet is why our results are so different hundred forty-four students answered the modified coin toss question, 214 the curved tubes question, and 429 the pen- than those from previous studies. The results Clement reports dmuolutimonq. uAesstwioendaefstcerribreelienvadnettainilsitnruAcptipoennadbixouAt,3fo0 rocuers and (reproduced in Fig. 2) are post-instructional; those that Car- study amazza et al. report are largely post-instructional (32 of 47 of likely oversamples from Asian and wealthy populations and their research subjects had previously taken a high school or college physics course); and Sadanand and Kess do not tell us undersamples from Latinx and Black populations, limiting when they gave their questions, only that the students in their the generalizability of our results. However, our primary aim study were “college-bound seniors…enrolled in an elective has not been to produce a generalizable result; it is to call into course in noncalculus-based physics.” It’s not enough (and question the impression that impetus-like responses are uni- in fact is not accurate) to say that our results are different versally common and persistent. because of when the questions were asked. The variation be- tween samples in our study supports this further, since all of Because our goal was to compare the frequencies of im- these samples were given the questions post-instruction. petus-like responses in the original studies to those in our sample, we sought to use the same methods for coding student One possibility for why our results are different than the diagrams as the original authors to the extent possible. That original studies is that the impetus force idea was never all means that we counted as impetus-like: all upward arrows at that common and/or persistent, and the papers we have cited point B in the modified coin toss question; all curved trajecto- have been overgeneralized. This seems somewhat unlikely ries for the ball as it exited the tubes in the curved trajectories given the resonance of this research with so many instructors, question; and all horizontal arrows pointing in the direction but it’s not outside the range of possible explanations. Though of motion in the pendulum question. More details about how it would not have affected our comparison, since we modeled our methods compare to those of the original authors can be our methods after those used in earlier studies, we did explore found in Appendix A.30 the possibility that earlier researchers may have overattributed Two independent coders—authors ADR and LMG —cod- impetus-like thinking to students’ drawings by looking at how ed student responses. Because we are seeking to challenge the students labeled their impetus-like arrows in our study. That literature’s read on the prevalence and persistence of impe- is, might students who drew these arrows be thinking of them tus-like drawings, we took the approach that offers the most as velocity or acceleration vectors, or as frictional forces, such generous interpretation of a drawing as impetus-like. In par- that the interpretation of all upward (or horizontal) arrows as ticular, a drawing was included in the percentages reported impetus-like thinking is an over-attribution? Appendix B30 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 255

Yet another possibility is that the impetus force idea is ef- fectively (though differentially) addressed by PER-informed instruction. Some of the original studies (e.g., Refs. 3, 21, and 24) hypothesize that physics instruction that “takes into account students’ misconceptions about motion”24 would ad- dress the prevalence of impetus-like responses. Since the early 1990s when these studies were published, the physics instruc- tional community has certainly heeded this call, and all of the courses in our study were PER-informed. One may wonder whether the modifications we made to the original questions contributed to the results in Fig. 2. We think not. It is very difficult for us to imagine the changes to the Aco3i0n),towshs iacnhdwceursveeedatsulbaregseplryotbwleemakss(tdoefsocrrmibaetdainndAcplapreitny-, dix effecting this degree of change. More substantive changes were made to the pendulum question—in particular, choos- ing a single direction of motion—but even when we include all horizontal arrows (in the direction of motion and oppo- site), the percentages come to 36.8% (University A, Course 2), 20.5% (University E, Course 3), 4.5% (University F), and 14.5% (overall). These are still consistently lower than Sad- anand and Kess’ reported percentage (63.0%) of impetus-like drawings, preserving our original claim. Perhaps even more importantly, if clarifying the questions in the ways we did were the source of the reduction in frequencies in Fig. 2—i.e., if that’s all it took—it would support, rather than challenge, our overarching message that impetus-like thinking is not univer- sally common and persistent. Fig. 2. Comparison of impetus-like drawings in original stud- Implications ies (blue) and our study (yellow, brown). “Univ.” stands for Literature on common student ideas about forces can leave “university,” “C” for “course.” instructors with the impression that many students have im- presents the results of this exploration. In short, most students petus-force-like ideas that persist through physics instruction. (about 75%) who drew impetus-like arrows labeled them in Our results challenge the universality of this interpretation, ways consistent with impetus-like thinking, suggesting that showing that the frequency of impetus-like drawings is of- in many cases these arrows do in fact indicate impetus-like ten less than in previous studies, including those that report reasoning. post-instructional data. This finding nuances the interpreta- tion of results that were produced decades ago and prompts A second possibility for why our results are different is a series of questions about why we got such different results that the impetus force idea is less common and/or persistent in our study than researchers did then. Together with our among introductory physics students in the present day. For results, answers to these questions may shift or sharpen the example, perhaps we are seeing less frequent use of this idea lens through which teachers plan for instruction and interpret post-instruction in our study because fewer students think student thinking. this way in the first place, pre-instruction. Acknowledgments The authors are grateful for feedback on portions of this draft from McKensie Mack and Raphael Mondesir, and for con- structive conversations with members of our Advisory Board, including Andrew Elby, Fred Goldberg, Steve Kanim, and Sam McKagan. This work was supported in part by NSF Grant Numbers 1608510, 1608221, 1256082, 1914603, and 1914572. References 1. M. S. Steinberg, D. E. Brown, and J. Clement, “Genius is not immune to persistent misconceptions: Conceptual difficulties impeding Isaac Newton and contemporary physics students,” Int. J. Sci. Educ. 12, 265 (1990). 256 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

2. I. Galili and V. Bar, “Motion implies force: Where to expect ves- 21. M. McCloskey, A. Caramazza, and B. Green, “Curvilinear mo- tiges of the misconception?” Int. J. Sci. Educ. 14, 63 (1992). tion in the absence of external forces: Naïve beliefs about the motion of objects,” Science 210, 1139 (1980). 3. J. Clement, “Students' preconceptions in introductory mechan- ics,” Am. J. Phys. 50, 66 (Jan. 1982). 22. J. Minstrell, FACETS, http://www.facetinnovations.com/dai- sy-public-website/fihome/home.html. 4. M. McCloskey, “Intuitive physics,” Sci. Am. 248, 124 (1983). 5. L. C. McDermott, “Research on conceptual understanding in 23. M. McCloskey and D. Kohl, “Naive physics: The curvilinear im- petus principle and its role in interactions with moving objects,” mechanics,” Phys. Today 37, 24 (1984). J. Exp. Psychol. 9, 146 (1983). 6. N. Sadanand and J. Kess, “Concepts in force and motion,” Phys. 24. A. Caramazza, M. McCloskey, and B. Green, “Naive beliefs in Teach. 28, 530 (Nov. 1990). ‘sophisticated’ subjects: Misconceptions about trajectories of 7. R. Gunstone and M. Watts, “Force and Motion,” in Children's objects,” Cognition 9, 117 (1981). Ideas in Science, edited by R. Driver (Open University Press, 25. D. M. Watts and A. Zylbersztajn, “A survey of some children’s 1985), pp. 85–104. ideas about force,” Phys. Educ. 16, 360 (1981). 8. R. J. Whitaker, “Aristotle is not dead: Student understanding of trajectory motion,” Am. J. Phys. 51, 352 (April 1983). 26. L. McDermott, “A View from Physics,” in Toward a Scientific 9. A. C. Alonzo and J. T. Steedle, “Developing and assessing a force Practice of Science Education, edited by M. Gardner, J. G. and motion learning progression,” Sci. Educ. 93, 389 (2009). Greeno, F. Reif, A. H. Schoenfeld, A. diSessa, and E. Stage (Law- 10. R. K. Thornton and D. R. Sokoloff, “Assessing student learning rence Erlbaum Associates, Hillsdale, NJ, 1990), pp. 3–30. of Newton's laws: The Force and Motion Conceptual Evaluation and the Evaluation of Active Learning Laboratory and Lecture 27. G. J. Posner, K. A. Strike, P. W. Hewson, and W. A. Gertzog, Curricula,” Am. J. Phys. 66, 338 (April 1998). “Accommodation of a scientific conception: Toward a theory of 11. D. M. Watts, “A study of schoolchildren's alternative frame- conceptual change,” Sci. Educ. 66, 211 (1982). works of the concept of force,” Eur. J. Sci. Educ. 5, 217 (1983). 12. I. Aviani, E. Natasa, and V. Mesic, “Drawing and using free body 28. L. S. Shulman, “Knowledge and teaching: Foundations of the diagrams: Why it may be better not to decompose forces,” Phys. new reform,” Harvard Educ. Rev. 57, 1 (1987). Rev. ST Phys. Educ. Res. 11, 20114 (2015). 13. R. F. Gunstone, “Student understanding in mechanics: A large 29. D. Hammer, “Misconceptions or P-prims: How may alternative population survey,” Am. J. Phys. 55, 691 (Aug. 1987). perspectives of cognitive structure influence instructional per- 14. I. A. Halloun and D. Hestenes, “Common sense concepts about ceptions and intentions?” J. Learn. Sci. 5, 97 (1996). motion,” Am. J. Phys. 53, 1056 (Nov. 1985). 15. J. Clement, “Students' Alternative Conceptions in Mechanics: 30. Readers can view the appendices at TPT Online, http://dx.doi. A Coherent System of Preconceptions?” in Proceedings of the org/10.1119/5.0027858, under the Supplemental tab. International Seminar on Misconceptions in Science and Mathe- matics, edited by H. Helm and J. D. Novak (Cornell University 31. D. Hammer, “Student resources for learning introductory phys- Press, New York, 1983), pp. 310–315. ics,” Am. J. Phys. 68, S52 (July 2000). 16. J. K. Gilbert, D. M. Watts, and J. Osborne, “Students’ concep- tions of ideas in mechanics,” Phys. Educ. 17, 62 (1982). 32. D. Hammer, A. Elby, R. E. Scherr, and E. F. Redish, “Resources, 17. D. Hestenes, M. Wells, and G. Swackhamer, “Force Concept In- Framing, and Transfer,” in Transfer of Learning from a Modern ventory,” Phys. Teach. 30, 141 (March 1992). Multidisciplinary Perspective, edited by J. P. Mestre (Informa- 18. P. J. J. M. Dekkers and G. D. Thijs, “Making productive use of tion Age Publishing, Inc., 2005), pp. 89–119. students’ initial conceptions in developing the concept of force,” Sci. Educ. 82, 31 (1998). 33. J. P. Smith III, A. A. diSessa, and J. Roschelle, “Misconceptions 19. L. Leboutet-Barrell, “Concepts of mechanics among young peo- reconceived: A constructivist analysis of knowledge in transi- ple,” Phys. Educ. 462 (1976). tion,” J. Learn. Sci. 3, 115 (1993). 20. L. Viennot, “Spontaneous reasoning in elementary dynamics,” Eur. J. Sci. Educ. 1, 205 (1979). 34. A. A. diSessa, “Toward an epistemology of physics,” Cogn. Instr. 10, 105 (1993). 35. D. Hammer and E. van Zee, Seeing the Science in Children’s Thinking: Case Studies of Student Inquiry in Physical Science (Heinemann, Portsmouth, NH, 2006). Seattle Pacific University, Physics Department, 3307 Third Ave. W, Suite 307, Seattle, WA 98119-1997; [email protected] THE PHYSICS TEACHER ◆ Vol. 60, April 2022 257

Co-deriving the Formulas for Centripetal Acceleration and Mass-Spring Period Mark Eichenlaub, Lewisburg, PA There is a close connection between simple harmonic motion and uniform circular motion. This connection is widely taught and included in standard textbooks.1-3 Here, we exploit this connection to simultaneously derive v two results from introductory mechanics: the period of a mass- spring system and the centripetal acceleration formula. Fig. 1. Two identical mass-spring systems with different initial Previously published derivations of the centripetal accelera- conditions. The left particle begins at rest and oscillates. The tion formula often rely on calculus,4-6 either via explicit com- right is given a push and moves in uniform circular motion. putation or arguing geometrically about limits. One approach relies entirely on the kinematics of vectors, without invoking v limits.7 Our approach likewise does not use limits, but is root- ed in dynamics. A similar argument to ours for the period of a spring’s oscillations appears in Baird.8 Baird takes the centripetal acceleration formula as an assumption; our work extends the argument to derive the centripetal acceleration formula at the same time. We assume instead the formulas for kinetic energy and elastic potential energy in an ideal spring. Students work- ing through this argument will combine many fundamental tools of basic mechanics, including properties of vectors, energy conservation, Newton’s second law, and formulas for energy. To set up our derivation, suppose we have two identical copies of a mass-spring system. In each, a point particle of vx mass m is connected to an ideal, zero-length spring9 of spring constant k. The other side of the spring is attached to a fric- tionless horizontal table a distance r to the left of the particle. In the left-hand system, we release the particle with zero initial velocity. The spring pulls on the particle, and it un- dergoes one-dimensional simple harmonic motion. In the right-hand system, we release the particle with an appropriate initial velocity in the y-direction so that it undergoes uniform circular motion. Fig. 2. The two motions superimposed. The x-positions and x- We examine the x-component of motion in each case. The components of the velocity are the same. force on the particle, in each case, is F = −kr, and taking the When x = 0, this energy is entirely kinetic, so x-component of ftrhoismeqthueatsiaomn,ewxedhisapvleacFexm=e−nktxa.nTdhweiptharztei-ro cles are released . (2) initial x velocity. Because both the initial x conditions and the equation of motion are the same in the x-direction for each ptiacrleti.cIlne,oitthfoerllwowortdhsa,ttxh(et)onane-ddvimx(te)nasrieonthael same for each par- Using Eqs. (1) and (2) to eliminate E and solve for v, we find harmonic motion (3) is the projection of the circular motion onto the x-axis. The circular motion has some constant speed v. When x = 0, the particle in circular motion is at the top of its orbit, and its velocity is entirely in the x-direction. Because both parti- We can also find the period of rotation. This is the distance cles have the same x velocity at all times, v is also the speed of traveled by the circular motion in one rotation divided by the the simple harmonic motion at x = 0. We can find this speed speed, or via energy conservation. The energy initially stored in the (4) spring is . (1) 258 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 DOI: 10.1119/5.0041135

Using Eq. (3) to eliminate v from Eq. (4), we find (5) 5. Ernest Zebrowski Jr., “On the derivation of the centripetal ac- celeration formula,” Phys. Teach. 10, 527 (Dec. 1972). This is also the period of the simple harmonic motion, com- 6. Carl E. Mungan, “Orbital speed of a satellite due to gravity,” pleting half the co-derivation. Phys. Teach. 56, 339 (Sept. 2018). Next, we determine the centripetal acceleration. From Newton’s second law, its magnitude is 7. Bill Wedemeyer, “Centripetal acceleration – A simpler deriva- tion,” Phys. Teach. 31, 238 (April 1993). (6) 8. L. C. Baird, “Novel derivation of the formula for the period of simple harmonic motion,” Am. J. Phys. 32, 233 (March 1964). 9. This is a spring that obeys the force law F = −kr. Its rest length is zero. Lewisburg, PA 17837; [email protected] Going back to Eq. (3) and using it to eliminate k, we find (7) The direction of centripetal acceleration is toward the origin because this is the direction the spring pulls. This completes the derivation of two common results. Educators could adapt this argument into classroom or homework exercises, adjusting the scaffolding to their stu- dents. It would provide an opportunity for students to reflect on the work that vector properties are doing in the derivation. For example, any sort of nonlinear spring, or a spring with finite rest length, will not work for this argument because the equation for force in the x-direction will include y, and the one-dimensional motion will not be a projection of the circu- lar motion. Additionally, to construct their own version of this argu- ment, students will have to combine energy conservation and Newton’s second law into the same analysis. The argument that the initial conditions and the equation of motion in the x direction are the same for each motion may be new to many introductory students, and this derivation provides a familiar context to introduce this type of reasoning. Combined, these properties mean that educators using this derivation in an activity can create opportunities for synthesis across several topics in the mechanics curriculum. Acknowledgment The author thanks Jiajia Dong for feedback on this manu- script. References 1. Halliday, Resnick, Krane, Physics, 5th ed. (Wiley, New York, 2002), p. 384 2. Douglas C. Giancoli, Physics: Principles with Applications (Pear- son, Boston, 2016), p. 292. 3. Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics, Vol. 1, Sec. 21-3. 4. Russell D. Patera, “Deriving the centripetal acceleration formu- la: a = v2/r,” Phys. Teach. 13, 547 (Dec. 1975). THE PHYSICS TEACHER ◆ Vol. 60, April 2022 259

Our Shifting Understandings of Culturally Relevant Pedagogy in Physics Clausell Mathis and Sherry Southerland, Florida State University, Tallahassee, FL In this paper, we describe the work of a teacher (Sarah) as she important to note that physics is often viewed as acultural,9 so attempted to use culturally relevant pedagogy in her physics the notion of culture impacting learning can be taboo to many classroom. Culturally relevant pedagogy (CRP) is an ap- in the physics education community.10 Some scholars, howev- proach to teaching developed by Gloria Ladson-Billings,1 with er, take the view that culture plays a role in the representation the goal of encouraging learning through drawing on students' of students in physics.10,11 Not only in physics, but science as cultural capital as a centerpiece of their instruction. CRP meets a whole has been slow in merging culture into its pedagogy the academic and cultural needs of diverse students through because of the belief that knowledge is neutral, and not in- recognizing and building upon their cultural backgrounds, fluenced by customs or other aspects of society.12 There is an experiences, and interests in instruction.1,2 We describe how abundance of literature that describes how culture, whether our experience working with Sarah broadened and honed our acknowledged or not, plays an important role in teaching and original understanding of CRP as we documented Sarah's in- learning.13 Unfortunately, teachers, and particularly physics structional successes, as well as the challenges, she encountered teachers, have not recognized the cultural capital students in her physics teaching. Based on our work with Sarah, we pro- bring into the classroom.14 vide recommendations to inform other physics teachers as they attempt to employ CRP. Given the paucity of students of color in physics, where on- ly 2% of all physics degrees were awarded to Black students,15 What is culturally relevant pedagogy? and 4% for Hispanic,16 if the field is to diversify its ranks, What is meant by the term “culturally relevant pedagogy” clearly more non-traditional approaches to instruction than those currently employed are called for. (CRP)? While this pedagogy can involve many different instruc- tional techniques,3-5 its components can be organized along We argue instructional approaches that are more aware of three domains. The first is academic excellence, which em- culture that students bring with them can be beneficial to in- phasizes teachers holding high expectations for students while creasing representation in physics. This study was conducted affirming their culture. The second domain is cultural compe- to describe and understand physics teachers’ attempts toward tence, which centers around teachers’ use of students’ cultural enacting CRP in a physics classroom. We entered this work resources as a reference to build their instruction. The third do- expecting to uncover how a successful teacher modified her main is sociopolitical consciousness, which encourages teachers curriculum to capitalize on students’ interests and resources, to incorporate important social and cultural issues dealing with but, as will be presented, while this was a component of her equity and injustice into the classroom. Ladson-Billings’ CRP work, what we discovered was that CRP was as much about suggests that teachers should merge all three domains into their her relationships with students than any sort of curricular practice. modifications. Tapping into a range of cultural knowledge, contributions, General overview experiences, and perspectives of students is critical for CRP- As an attempt to understand how CRP could be actualized based instruction.6 An obstacle many teachers have when implementing CRP is that salient aspects of the culture of in the physics classroom, we explored physics teachers' views underrepresented students are not typically represented in a of CRP.8 Most of the teachers in the survey were enthusiastic classroom. Indeed, the culture represented in most curricula about attempting CRP but were unsure of how to enact such only resonates with students from the dominant culture. Given instruction. After interviewing several physics teachers, we this, one may ask: How can you create classroom environments found one Black woman, Sarah (a pseudonym), who taught at where students’ cultural aspects are embraced and leveraged as a Title 1 high school in the Southeastern United States, to par- essential tools in instruction and learning? ticipate in our study. Working in a school that served largely Black students in poverty, Sarah stood out as a person for this In particular, the physics classroom poses a challenge of study given her success in attracting students to the discipline actualizing CRP, as there are not many tangible examples of (increasing the number of physics sections from one to five, CRP-informed lessons or activities that can be used as reference. resulting in a school offering more sections of physics than To address this, Johnson provided university physics faculty anywhere else in the school district at the time of this study). guidance on how to enact CRP in their classrooms.7 While Sarah was an award-winning teacher who had deep familiar- Johnson’s article unpacked Ladson-Billings’ major domains and ity with the context in which the school was located, thus we gave descriptions of how each should be done in the physics thought Sarah could teach us much about a teacher’s efforts classroom, unfortunately there were not any specific examples on engaging in CRP. of physics teachers’ attempts to use CRP. Given this paucity of tangible examples, there is confusion among physics teachers on For this study, we worked with Sarah to create lessons that how to interpret the CRP framework and actualize it.8 It is also built from students’ interests and concerns, lessons she would subsequently teach and then reflect with us on the outcomes. 260 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 DOI: 10.1119/5.0027583

We observed Sarah’s classes, taking field notes of class discus- about students’ hobbies, interests, and concerns. Sarah was sions, lectures, and group activities of problem solving and able to develop good relationships with her students, which projects. We also observed Sarah's interactions with students ultimately led to a positive classroom culture. Students were both during her planning periods and between classes. We at- comfortable in expressing their challenges and helped each tended Sarah’s classes daily up to six months and met with her other out in learning difficult concepts. online every other week (via Zoom) to discuss lessons both previous and upcoming. When asked how she can identify aspects of students’ cul- Findings ture and relate them to physics concepts, Sarah stated: To understand the various ways Sarah worked towards By first getting to know my students … I have CRP, we organized her actions (both in and out of the class- to get to know them first … I have to determine room) according to the three major pillars of CRP (academic what matters to them the most … what are their excellence, cultural competence, sociopolitical conscious- behaviors … extracurricular activities, home life, ness). what they like to do in their extracurricular time … that’s all in regard to culture and their behav- Academic excellence iors, what matters to them, and once I begin to We saw Sarah regularly encourage academic excellence understand my students, I have to get to know them, then I can build on making that connection from her students. First, she was responsive in her teaching with them. by showing value in the knowledge students brought to the class. Sarah commonly followed up to student responses to Sarah connected with her students, and for a few, their her questions, indicating her interest in their ideas (“Help me relationships extended outside the classroom. Sarah also understand your thinking?”). These teaching moves spoke to showed cultural competence through art by developing activ- the value she placed on students’ incoming ideas. ities where students had to learn key physics concepts while showing creativity. There were two unique art projects Sarah Sarah also gave students the opportunity to “do science” assigned her students. One was a rocket launching project, and embrace a science identity through providing a learning where students built rockets out of old two-liter bottles. The space where they did hands-on activities and performed tasks rockets were launched in the air, where students measured that required investigations. During an interview with Sarah, the range along with calculating the launching and landing she was asked how she can establish rigor, while affirming stu- angle. The second art-based project was where students had dents’ culture, and Sarah stated: to build roller coasters and determine the total energy of the cart riding at different points. Students had to make brochures I think it’s going back to knowing how to manage promoting the rollercoaster ride explaining the calculations your knowledge. If you can determine why they ex- and determining the speed on the rollercoaster at various hibit certain behaviors … then I think you can dive locations. These projects created an avenue for student ar- into helping them manage the knowledge in teach- tistic expression that would not have been obtained through ing them how to think critically to solve problems traditional teaching methods. Sarah recognized the interest … but you have to first let them learn how to man- this blending of art and physics inspired in her students and age knowledge first … that’s one of the key things I understood the reliance on art in her classroom to be one of think in them being successful … is how to manage the reasons why students flocked to her physics courses. what they know … how to relate to prior knowledge … to build on the new knowledge that you are go- Sociopolitical consciousness ing to receive. As described by Ladson-Billings and others, a major In addition, Sarah commonly made herself available component of CRP involves the teachers’ efforts to create during her planning periods and after school to assist those an awareness of their students about sociopolitical concerns students who needed additional help. Lastly, Sarah showed drawn from their lives, especially those that affect their par- academic excellence by motivating her students towards ticular cultural group, and to understand how the discipline their academic and career goals. Sarah gave students advice can be brought to address these concerns. In this pillar of CRP, on what to anticipate when going to college, indicating her teachers are encouraged to inspire their students to strive for expectation that they would pursue higher education. Taken change in their communities. For example, issues of under- together, Sarah’s approach towards academic excellence was representation in physics would invoke classroom discussions based on understanding students’ strengths and attempting to that have a critical analysis on factors contributing to the un- shape their efforts toward both education in general, and in derrepresentation issue and strive for solutions. Or if there is physics specifically. a sociopolitical issue that can be connected to physics in some form, teachers should ask themselves how they can connect Cultural competence issues within the physics context that students are interested Sarah demonstrated her cultural competence by having an in exploring. awareness of cultural norms. She was interested in learning Sarah’s use of sociopolitical consciousness as a tool to THE PHYSICS TEACHER ◆ Vol. 60, April 2022 261

engage students in physics was through creating lessons that Sarah experienced difficulty encouraging her students to es- capitalized on their cultural resources. Sarah would listen to tablish agency in their learning. To meet this challenge, Sarah key casual conversations students had to gather information attempted to build interesting but cognitively demanding as a reference for future instruction. For example, Sarah de- tasks. However, when she saw her students struggling with veloped a lesson on the issue of gun control (focusing on the this task, like many teachers she often lessened the demand of conservation of momentum), based on a classroom conver- the task.20 Sarah stepped in and lessened the demand because sation on the use of firearms. In this activity, students had to she was concerned about students becoming disengaged in use the conservation of momentum to create an argument for the face of struggle. Sarah did not want students to become or against the control of specific kinds of firearms. Students discouraged about learning physics if they always believed gathered information on the firearms and used physics-based it was too difficult. Unfortunately, in her efforts to maintain explanations to argue for legislation. student engagement, she failed to develop students’ agency in learning. The tension of supporting student agency while Aside from physics, on several occasions we also witnessed accounting for resistance to change is something that many Sarah encouraging students’ discussions on how race and teachers experience.19 Physics teachers who are new to CRP social class played a role in others’ perceptions of them. Stu- will have to seek support from colleagues and peers in order dents talked about how they were often looked down on by to stay focused on changing instruction. other students in the school district, because of their school’s status as a predominantly “Black” school. Students felt there Creating cultural connections was a perception that their school was viewed as “less than” Teachers making cultural connections is a staple of CRP in- or “easier” than other schools in the district. This general ex- perience then broadens to other instances of discrimination struction. Given that culture is a multifaceted concept, many that they faced in schools and in the community. Sarah also times teachers have difficulty understanding what can be de- shared her own experiences as a woman of color in STEM, scribed as a “cultural norm” or a “cultural aspect” of students. how expectations of her were lower than those held for her This creates challenges in implementing CRP for teachers, white colleagues, and she navigated such discriminatory acts. particularly teachers who have no experience or knowledge These unscripted and unplanned discussions, which hap- in the cultural dynamics of their students and communities. pened at least three times in our observations, were helpful in For teachers who are new to CRP, there will be a struggle in allowing students to learn from someone who has experience attempting to create relationships, develop lessons, reflect with these issues. Sarah’s students were able to relate to her on lessons, and seek effective advice, given the vagueness of experiences and express their past moments of handling dis- understanding and defining one’s culture. CRP is an evolving crimination. pedagogy based on how the cultures of students evolve.21,22 For physics teachers there will need to be support among Sarah was aware of the negative stereotypes that society peers, administrators, community members, and colleagues had for her students of color, particularly in physics. Having to understand what are the challenges and pitfalls that can be discussions on sociopolitical topics offers challenges and done to effectively use CRP in the classroom. opportunities for physics teachers. The challenging issue is learning how to transform possible bias and stereotypes of At times Sarah expressed difficulty in creating cultural others into an opportunity for learning and critical analysis of connections to physics ideas. She was successful in connecting issues. concepts such as momentum and energy to students’ cultural lives but had challenges with areas such as thermodynamics Challenges and electricity. One particular moment was when Sarah tried It is important to note that even though Sarah was a to connect students’ interests in Disney to learning about en- ergy efficiency. Students had to argue what was the best route woman of color with past success teaching physics, even she to drive to Disney World. Students were given a list of cars to experienced a number of challenges as she attempted to enact choose from and each car’s weight. From there, students had CRP. The three major challenges Sarah encountered were: to use the kinetic energy formula to calculate the energy used supporting student agency, creating cultural connections, and on the car at several points on the route. Using the collected resistance on certain sociopolitical concerns. information, students can identify the most energy efficient way to get to Disney World. The Disney World project had Supporting student agency challenges because students were confused on which speed to Student agency is a feature of learning where students are use. This caused miscommunication between some students and Sarah because students did not know which information active in the learning process and take ownership of learn- to use. Eventually Sarah lessened the demand and students ing.16 In many traditional classrooms, the teacher lectures were able to finish the class. about a given concept while students take notes and ask ques- tions. In the case of CRP-based instruction, students’ agency is Resistance on doing specific sociopolitical issues a large part of the learning process.17 Student agency focuses Given the potentially polarizing nature of sociopolitical on students being active while shaping their learning.18 This can be done through activities, problem-solving sessions, or topics, and the very limited experience many have in the presentations. One of the major challenges teachers who are application of physics to address social issues, there may be new to using CRP face is adjusting to changes in practice.19 262 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

resistance when teachers attempt to discuss or make reference many cases, teachers are hesitant out of fear of causing contro- to them. As an example, Sarah had worked on a lesson to ex- versy and/or trouble among students, community members, amine the physics involved in the then-Presidential adminis- and staff.23 As they attempt to enact CRP with underrepre- tration’s border wall, as she knew this was a topic student had sented students, physics teachers must be willing to take risks expressed interest in. However, after speaking to her depart- in addressing “uncomfortable” ideas. Physics teachers must ment head, she chose not to use it, as it was deemed “too con- be aware there are systemic inequities many students expe- troversial.” While English and language arts classrooms have rience if they want to enact instruction that speaks to those for years been productive sites for students wrestling with inequities. These backgrounds can be based on race, class, problems from their lived experiences, this has not been the and/or gender depending on circumstance. In many cases, norm for physics. This notion that physics is “acultural” and students enter the classroom with challenges from home that divorced from everyday concerns is not atypical and is seen in can impact their approach to learning inside the classroom. a great deal of literature related to physics instruction.23 Sar- CRP maintains that teachers should be non-judgmental and ah’s difficulties speak to the need for cultivating a recognition inclusive of the backgrounds of their students to be effective for the benefits of students grappling with sociopolitical issues learners.5 in the context of physics, not only on the part of teachers and students but also their administrators. Culture is not a monolith There is a challenge in defining what is a “cultural norm” Conclusion There were positives to take from Sarah’s attempt to use and “cultural artifact.” Every year students enter the classroom with different experiences, norms, interests, and values, which CRP. Sarah had successes in shifting her practice; however, yield a multicultural dynamic. This bodes difficulties for there were challenges in developing curricula and supporting physics teachers who are trying to establish relationships with agency among her students. We expected Sarah to encounter students and develop creative teaching methods that connect challenges as she worked to shift her pedagogy. What we did culturally with them. The only way to tackle this issue is to be not expect was how the relationships Sarah created with stu- willing to take risks in discussions, lessons, assessments, and dents impacted their receptiveness to a pedagogical change other forms of practice. and influenced willingness to have dialogue on sociopolitical concerns. It is important to acknowledge CRP does not “live” Unfortunately, there is a vagueness that teachers have when at the level of the lessons but extends a teacher’s concerns describing and developing pedagogy that aligns with the CRP outside of the classroom. Indeed, it is important that teachers framework. This can be possibly due to the fluidity of describ- become familiar with students’ outside concerns and interests, ing culture. Physics teachers need to understand there is not a as this knowledge can provide essential insights for CRP in singular approach to teaching physics in a culturally relevant physics. Going forward, we plan to further research how phys- manner. Within cultures, there are many aspects that can be ics teachers can improve their use of CRP in their classrooms. unpacked and examined, which can help physics teachers Based on findings from our observations of Sarah, to enact a connect with their students, ultimately improving their learn- culturally relevant approach to physics, there are many aspects ing. teachers need to be aware of. Below is a description of major factors physics teachers need to be aware of. There will be challenges in connecting physics ideas to students’ concepts Relationships are essential for teachers who want to do CRP The framework of CRP has a large emphasis on cultural connections and sociopolitical concerns. For physics teachers One of the things that was apparent in our observations of to use CRP they need to be keen on social political concerns Sarah was her strong relationships with students. Given that that impact the physics community. There are many issues, students enjoyed being in Sarah’s classroom, she had com- from energy, force, momentum, electricity, nuclear energy, fort in creating new, innovative approaches to instruction. If optics, that can be used to make connections to students’ cul- teachers have difficulty in establishing relationships, it can tural norms. In order for physics teachers to use CRP, they will be difficult to take pedagogical risks. Sarah's success in using need to understand how community concerns that can relate CRP was largely based on her relationships with students. to physics can be introduced in the classroom. It is important Also, Sarah's understanding of students’ strengths helped in to acknowledge that not all students will be receptive to in- building relationships and developing lessons. For example, struction that is informed by CRP, particularly in early efforts. building relationships was done through learning students’ Teachers and administrators will need to understand that this names, being attentive to interests, and supporting learning new approach to instruction will be accompanied by tension and extracurricular activities. among students and their teachers—as is the case with any new instructional design. Willingness to create an open forum on issues of race, class, and privilege Implications Results from Sarah’s teaching have implications on how Teachers, particularly science teachers, struggle as they at- tempt to address sociopolitical concerns in the classroom. In physics teachers can improve their instruction for under- represented students. An examination of how teachers learn THE PHYSICS TEACHER ◆ Vol. 60, April 2022 263

about students’ culture, connect it to physics ideas, and adjust of student-teacher relationships. We realized that the ease at to changes in pedagogy need to be done. This study has impli- which a teacher can shift and revise their practice is highly cations for teachers who want to build upon students’ cultural dependent on the relationships they cultivate with students. capital in their classroom instruction. CRP is more than just teaching moves, it is more about the re- lationships in the classroom, which is based on how you view • Physics teachers will need to develop a skill for yourself and others. We also did not consider the malleability identifying the cultural capital students bring to of CRP, meaning there is not a singular approach or “one size the classroom to make connections to physics fits all” method to use it. Physics teachers cannot view CRP as ideas: In order to effectively use CRP, physics teachers will a curriculum, blueprint, or script that is transferred between need to develop a skill of identifying “cultural norms” and classes. Teachers have to see CRP as an evolving pedagogy making connections to physics ideas. Teachers will have to that is continuously revised. Lastly, we were not aware of the understand many aspects of their students and have the will- difficulty teachers and students have in shifting their peda- ingness to develop skills on making connections to physics gogy from teacher-centered to student-centered. Given that topics. Every year students bring in different skills, knowl- most teachers are accustomed to a particular style, coupled edge, and abilities that can be used by instructors to make with students accustomed to receiving a standard approach to connections to physics ideas. CRP calls for teachers to be instruction, there will be challenges for students and teachers more ambitious with their teaching practices, as they work to to adapt to change in pedagogy. identify students' cultural capital and leverage it in instruc- tion.1This will involve students having an active role in the • Physics teachers will need to show patience when learning process. shifting instruction: Having patience is key when using CRP. Physics teachers will need to understand that there will • Physics teachers should create their own unique be challenges and mistakes made when attempting to shift approach to using CRP: Given there is no “official blue- their pedagogy: they will need to be open to critique and ad- print” for CRP, physics teachers who intend to use it will justment to changes. Also, this study shows the importance of need to examine the framework but develop their own skills collaboration and school support on the development of CRP. and practices. Teaching is more of an art than a blueprint, so We recommend providing professional development on how teachers will have to be creative on which approach best suits to help physics teachers develop constructivist-based prac- them depending on the classroom in which they teach. Differ- tices and how to impact student learning outcomes as a result ent classes bring in different personalities and approaches to of making connections between students’ culture and physics engage in learning. Having practical examples of CRP is ben- ideas. If schools grow the professional development of their eficial for teachers and teacher educators who want to learn teachers, it will improve their culturally relevant teaching how to use it in the physics classroom. In many cases, there practices. This can be done through providing opportunities will be limited access to examples, which calls for more studies for teachers to do communities of practice. on teachers who want to use CRP. This will help physics teach- ers understand the details in relationship, communication, Professional development in CRP should happen in a co- lesson planning, and revising curricula. herent and consistent way to help teachers in their classrooms. This professional development will need to be extensive, as Even though Sarah had comfort in discussing sociopoliti- physics teachers need to be given a chance to learn, critically cal topics, it does not imply that physics teachers need to have assess, and reflect on their attempts to implement CRP in the similar racial and class backgrounds as their students to use classroom. We suggest that this professional development CRP. What it does suggest is that teachers must be willing to needs to be structured around ways to recognize, value, and allow students to show their vulnerabilities on sociopolitical capitalize on the cultural capital students bring into the class- concerns that impact them. room. Teachers should also use community connections to create a further understanding of students’ background and • Physics teachers will need to cultivate a commu- the concerns, interests, and experiences they bring with them nity of practice among administrators/districts/ into physics. If we are to broaden representation of students of community more supportive of these efforts: It color in physics, developing teachers’ awareness and abilities is important for physics teachers to work with others who to enact CRP will be essential. are willing to use CRP to develop practices that will best suit themselves and their students. Given that CRP is a cul- References ture-based pedagogy, there are various approaches that can 1. G. Ladson-Billings, “Toward a theory of culturally relevant ped- be used to teach students. Initially, our understanding of CRP was focused on creating fun lessons and activities for students agogy,” Am. Educ. Res. J. 32 (3), 465–491 (1995). that incorporated their culture into curricula. We had an as- 2. H. R. Milner, “Culturally relevant pedagogy in a diverse urban sumption that you can just create curricula that highlighted student cultural artifacts and that would be enough to engage classroom,” Urban Rev. 43 (1), 66–89 (2011). your students. What we did not consider was the importance 3. G. Ladson-Billings, “Yes, but how do we do it?”: Practicing Culturally Relevant Pedagogy,” in City Kids, City Schools: More Reports from the Front Row, edited by W. Ayers, G. Ladson-Bill- ings, G. Michie, and P. Noguera (The New Press, New York, 264 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

2008), pp. 162–177. 14. O. Lee and A. Luykx, Science Education and Student Diversity: 4. K. A. Morrison, H. H. Robbins, and D. G. Rose, “Operation- Synthesis and Research Agenda (Cambridge University Press, 2006). alizing culturally relevant pedagogy: A synthesis of class- room-based research,” Equity Excell. Educ. 41 (4), 433–452 15. J. S. Rosenberg, “Doing physics while Black” (2019), https:// (2008). www.mindingthecampus.org/2019/09/17/doing-physics-while 5. S. Brown-Jeffy and J. E. Cooper, “Toward a conceptual frame- black/. work of culturally relevant pedagogy: An overview of the con- ceptual and theoretical literature,” Teach. Educ. Q. 38 (1), 65–84 16. American Physical Society, “Physics degrees by race/ethnicity” (2011). (2018), https://www.aps.org/programs/education/statistics/ 6. P. B. Baker and Lee Woodham Digiovanni, “Narratives on cul- degreesbyrace.cfm. turally relevant pedagogy: Personal responses to the standard- ized curriculum,” Curr. Issues Educ. 8, 112 (2005). 17. J. Arnold and D. J. Clarke, “What is ‘agency’? Perspectives in 7. A. Johnson, “A model of culturally relevant pedagogy in phys- science education research,” Int. J. Sci. Educ. 36 (5), 735–754 ics” AIP Conf. Proc. 2109, 130004-1-1003-4 (2019). (2014). 8. C. Mathis, S. Southerland, and M. Akubo, “A Study of Select Physics Teachers’ Beliefs on Implementing Culturally Relevant 18. J. Esposito and A. N Swain, “Pathways to social justice: Urban Practices in the Classroom,” presented at the American Asso- teachers' uses of culturally relevant pedagogy as a conduit for ciation of Physics Teachers Winter Meeting, San Diego, CA teaching for social justice,” Penn GSE Perspect. Urban Educ. 6 (2018). (1), 38–48 (2009). 9. S. Traweek, Beamtimes and Lifetimes (Harvard University Press, 2009). 19. D. Stroupe, “Examining classroom science practice communi- 10. A. R. Daane, S. R. Decker, and V. Sawtelle, “Teaching about ties: How teachers and students negotiate epistemic agency and racial equity in introductory physics courses,” Phys. Teach. 55, learn science‐as‐practice,” Sci. Educ. 98 (3), 487–516 (2014). 328–333 (Sept. 2017). 11. J. G. Hill, “Education and Certification Qualifications of De- 20. C. C. Johnson, “The road to culturally relevant science: Ex- partmentalized Public High School-Level Teachers of Core ploring how teachers navigate change in pedagogy,” J. Res. Sci. Subjects: Evidence from the 2007-08 Schools and Staffing Teach. 48 (2), 170–198 (2011). Survey,” Statistical Analysis Report, NCES 2011-317 (National Center for Education Statistics, 2011). 21. M. Tekkumru‐Kisa, M. K. Stein, and C. Schunn, “A framework 12. H. Siegel, “Multiculturalism, universalism, and science educa- for analyzing cognitive demand and content‐practices integra- tion: In search of common ground,” Sci. Educ. 86 (6), 803–820 tion: Task analysis guide in science,” J. Res. Sci. Teach. 52 (5), (2002). 659–685 (2015). 13. S. Nieto, Language, Culture, and Teaching: Critical Perspectives (Routledge, 2001). 22. G. Ladson-Billings, “Culturally relevant pedagogy 2.0: aka the remix,” Harvard Educ. Rev. 84 (1), 74–84 (2014). 23. D. Paris, “Culturally sustaining pedagogy: A needed change in stance, terminology, and practice,” Educ. Res. 41 (3), 93–97 (2012). Florida State University, Physics, Tallahassee, FL 32306; [email protected] THE PHYSICS TEACHER ◆ Vol. 60, April 2022 265

Free Software Resources for Teaching AC RC Circuits Minjoon Kouh, Drew University, Madison, NJ The importance of introducing computational ap- for further advanced topics that deal with other frequen- proaches early and actively in science education is cy-dependent systems or phenomena, such as Fourier analy- widely acknowledged among educators and scientists. sis, resonance, spectroscopy, etc. Many great ideas have been put forward and implemented Using any of the following software resources, students by advocates and organizations such as Partnership for Inte- can learn how to characterize a temporal signal in terms of its gration of Computation into Undergraduate Physics (https:// frequency and amplitude by producing and visualizing sinu- www.compadre.org/PICUP) and the Science Education Re- soidal function. Then, students can observe how an RC cir- source Center at Carleton College (https://serc.carleton.edu/ cuit takes in a sinusoidal input and produces a phase-shifted teaching_computation/index.html).1-3 According to “Phys21: and amplitude-attenuated output that depends on the input Preparing Physics Students for 21st-Century Careers,” a re- frequency. Students can easily experiment with different cent report by the Joint Task Force on Undergraduate Physics simulation parameters (R, C, and input frequency) and may Programs, physics students benefit from learning how to compare with the theory of RC circuits quantitatively. represent physics concepts in multiple ways: mathematically, The first resource is an interactive simulation from the experimentally, and computationally. In his best-selling book on career advice, Cal Newport argues that quickly and ef- fectively learning how to wield difficult tools is a valuable skill for success.4 This article introduces a few peda- gogical approaches and free computa- tional resources that can supplement the traditional theory-driven lectures and hands-on experiments for teaching RC circuits. The first resource (see Fig. 1) is an interactive simulation from the widely known PhET project. When stu- dents’ access to electronics lab supplies and equipment is limited (due to remote learning during a pandemic, for exam- ple), these software packages may even (a) substitute for in-person electronics labs. Some software resources are very intui- tive and beginner-friendly, while others are versatile and require more technical proficiency and sophistication. Some can be accessed just with an internet browser, while others require an instal- lation on a user’s computer. The topic of an RC circuit appears in most introductory physics textbooks and continues to be an important part of a typical physics curriculum. Built only with a resistor and a capacitor connected in series with an alternating voltage source, an RC circuit is a great example of how a collection of simple components can generate a rich set of interesting behaviors. It is one of the canonical examples in physics and math (b) for illustrating how a complex system Fig. 1. Screenshots of software resources. Each example contains a plot that shows the may be described and analyzed with a sinusoidal voltage trace from a simulated RC circuit. The output of the circuit shows the differential equation. The filtering effect attenuated amplitude and shifted phase compared to the input signal. (a) PhET Interactive of an RC circuit is a good stepping stone Simulations from University of Colorado Boulder (\"Circuit Construction Kit: AC\"). (b) Multisim Live from NI. (c) VirtualScope. (d) Spreadsheet. (e) GNU Octave. 266 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 DOI: 10.1119/5.0038131

widely known PhET project at the University of Colorado simulations, such as Multisim Live from NI (https://multisim. Boulder.5,6 The “Circuit Construction Kit: AC” is available as com) and CircuitLab (https://circuitlab.com). Just like the a new HTML5 simulation, which has the advantage of being PhET simulations, these simulators can be accessed with an compatible with a large variety of devices, at the following internet browser, but require user accounts (free accounts link: https://phet.colorado.edu/sims/html/circuit-construc- are available). They require deeper electronics knowledge tion-kit-ac/latest/circuit-construction-kit-ac_en.html. A user and familiarity with circuit diagrams. For example, a proper can easily and quickly construct an RC circuit by dragging placement of electrical ground is often necessary to obtain a and dropping the circuit components. The advantages of this sensible result. There are many more circuit elements, such as resource are that it is very intuitive and simple to use and that diodes, transistors, and operational amplifiers, and the range it can be accessed with any modern internet browser. There- of circuit parameters is very wide. Other useful features in- fore, it has a low entry-barrier for students and instructors. clude: export of circuit schematics, measurements and visual- However, the range of the simulation parameter is limited. ization of voltage/current at multiple points in the circuit, AC The frequency range of the AC voltage is between 0.2 and 2.0 frequency sweep, etc. Hz only, which is just enough to observe the basic operation of an RC circuit. The third software resource is VirtualScope created by Katherine Matthews (http://expphys.com/public/scope_intro. The second resource is a family of more advanced online htm). It needs to be installed on a computer along with the LabVIEW Runtime engine. One of the most advantageous (c) features of this software is that a user can work with an os- cilloscope and a signal generator in a realistic graphical user interface. Students can, therefore, become familiarized with these two important pieces of electronics equipment, by turn- ing knobs, pressing the switches, and observing the output of an RC circuit on a virtual oscilloscope. Before students touch a real oscilloscope or a function generator, they can practice on these virtual devices, so that they would not feel as overwhelmed or afraid of breaking an expensive piece of equipment. The next two software resources are multipurpose compu- tational tools, whereas the previous three are more specific to the field of physics and electrical engineering. The fourth soft- ware resource is a spreadsheet (e.g., Microsoft Excel or Google Sheets), which is a widely used, versatile tool for data analysis and plotting. The instructor can build upon students’ prior familiarity with spreadsheets and illustrate the concept of nu- merical integration with the RC circuit as an example. From the basic theory, the following differential equation holds for an RC circuit: (d) . Using the last equation, the amount of charge (e) accumulated on the capacitor can be calculated by integrating dQ/dt in time, and then the voltage across the capacitor can be found by dividing the amount of charge by its capacitance. Table I shows the formulas that can be entered into a spreadsheet to perform numerical integration. The variables R, C, and w in the formula need to be replaced by specific values. Column C con- tains the key step of numerical integration, where THE PHYSICS TEACHER ◆ Vol. 60, April 2022 267

The accuracy improves as t becomes infinitesimal. This Z_C = 1/(j*w*C); pedagogical approach has the advantage of helping students to Z_R = R; deeply examine the foundational idea of calculus: differentia- dt = 0.01; tion and integration with infinitesimal time steps. t = [0:dt:15]; The fifth resources are scientific/technical computing plat- V_in = exp(j*w*t-j*pi/2); forms like MATLAB and its open source alternative GNU V_C = V_in * Z_C/(Z_R+Z_C); Octave, which are widely and professionally used.7 Octave can V_R = V_in * Z_R/(Z_R+Z_C); be freely downloaded from http://www.octave.org. With some plot(t,V_in,'k-',t,V_C,'c-',t,V_R,'r-'); basic knowledge of programming, one could implement the xlabel('Time (sec)'); same numerical integration that was illustrated with a spread- ylabel('Voltage (Volts)'); sheet program. One could also use a built-in function for solv- legend('Input Voltage','Voltage across ing a differential equation (such as ode45). However, after a C','Voltage across R'); short introduction to the mathematics of complex numbers, The top three lines define the circuit parameters (capaci- one could simulate an RC circuit with complex impedances, tance, resistance, and frequency of the driving voltage source). as shown in the following code sample. The next three lines define the impedances of R and C as com- plex numbers. The next five lines of code calculate the voltage C = 1/10^6; % 1 microfarad capacitance. across the capacitor and the resistor, illustrating how solving R = 10^6; % 1 megaohm resistance. the differential equation turned into an algebraic calculation w = 1; thanks to the usage of complex numbers. The next four lines j = sqrt(-1); create a plot of voltage vs. time in three different colors (black, cyan, and red for the source, capacitor, and resistor). Table I. Setup of a spreadsheet for simulating an AC RC circuit via numerical These lines of code are fairly readable and students integration. The values of R, C, w, and dt need to be specified. are exposed to a script for scientific analysis and AB C D visualization. 1 Time Vin Q dQ/dt = –Q/(RC) + Vin/R As listed above, there are new pedagogical ap- 2 0 = SIN(w*A2) 0 = -C2/(R*C) + B2/R proaches and modern computational resources 3 = A2+dt = SIN(w*A3) = C2+D2*dt = -C3/(R*C) + B3/R that can be adopted to cover the traditional topic of an RC circuit.8 Table II is a quick summary of the presented resources. The PhET interactive sim- 4 = A3+dt = SIN(w*A4) = C3+D3*dt = -C4/(R*C) + B4/R ulation is intuitive but may seem elementary for some students. Resources like Multisim Live and ... VirtualScope can be a great stepping stone for more ... ... ... advanced studies in electronics. Spreadsheet and Octave/MATLAB have the advantage that they are Table II. Comparison of five computational resources in terms of their entry versatile tools and used extensively in other disci- barriers, audiences, and versatility and power. plines for numerical analysis. With these two latter tools, students can also simulate and observe how Options Beginner's Entry Barriers Target Versatility an RC circuit removes high-frequency noise. One PhET Audiences and Power could inject noise by adding random numbers to Very Low (drag-and-drop Low the input with a built-in function like rand() and Multisim or interface, browser access) High school and plot the corresponding smooth output, as shown in CircuitLab first-year college High Fig. 2. Medium (drag-and-drop students (advanced VirtualScope interface, but requires circuits can This article is not an exhaustive list of available understanding of circuit Physics or engi- be built) circuit simulators. There are a plethora of other Spreadsheets diagrams) neering students Medium excellent options produced by many educators and High (requires understanding (Year 2-4 in col- organizations, such as the Open Source Physics Octave or of hardware components lege) High Project of the AAPT ComPADRE Digital Library MATLAB and lengthy software instal- (https://www.compadre.org/osp/), Java Applets by lation) Physics or engi- Very high Paul Falstad (http://falstad.com/circuit/), Medium neering students HTML5 simulations by Andrew Duffy (http://phys- (Year 2-3 in col- ics.bu.edu/~duffy/HTML5/RLC_circuit.html), just High (requires installation lege) to name a few. and learning how to code) Students majoring Which other core curricular topics in physics in various disci- can be given a modern makeover with computa- plines that involve tional tools? By complementing and supplementing data analysis the traditional pedagogical approaches, these new resources help our students to learn better and Physics, engi- neering, math, computer science students (Year 2-4 in college) 268 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

Fig. 2. A plot from a spreadsheet that illustrates the low-pass References filtering and noise-reducing behavior of an RC circuit. The noise 1. R. Allain, “You physics teachers really ought to teach numerical was added with the following formula: (rand()-1)*0.5 calculations,” Wired (March 4th, 2017). understand deeper difficult topics. Students benefit from 2. M. D. Caballero, J. Burk, J. M. Aiken, and B. D. Thoms, “Inte- having multiple modes of thoughts and expressions: words, diagrams, mathematical formulas, hands-on experiments, grating numerical computation into the Modeling Instruction codes, and simulations.3 curriculum,” Phys. Teach. 52, 38 (Jan. 2014). 3. P. Nelson, “Epilogue,” in Physical Models of Living Systems (W. H. Freeman and Company, 2015). 4. C. Newport, Deep Work: Rules for Focused Success in a Distract- ed World (Grand Central Publishing, 2016). 5. N. Finkelstein, W. Adams, C. Keller, P. Kohl, K. Perkins, N. S. Podolefsky, S. Reid, and R. LeMaster, “When learning about the real world is better done virtually: A study of substituting computer simulations for laboratory equipment,” Phys. Rev. ST Phys. Educ. Res. 1, 010103 (2005). 6. C. E. Wieman, W. K. Adams, P. Loeblein, and K. K. Perkins, “Teaching physics using PhET simulations,” Phys. Teach. 48, 225 (April 2010). 7. A. Azemi and C. Stook, “Utilizing MATLAB in undergraduate electric circuits courses,” Proc. Frontiers Educ. Conf. 2, 599–602 (1996). 8. W. S. M. Sanjaya, D. Anggraeni, A. Sambas, and R. Denya,“Nu- merical method and laboratory experiment of RC circuit using Raspberry Pi microprocessor and Python interface,” J. Phys. Conf. Ser. 1090, 012015 (2018). Drew University, Physics Department, 36 Madison Ave., Madison, NJ 07940; [email protected]  STEPUP PHYSICS TOGETHER PHYSICS TOGETHER STEP UP is a national community of physics teachers, researchers, and professional societies. We design high school physics lessons to empower teachers, create cultural change, and inspire young women to pursue physics in college. If half of the high school physics teachers encourage just one more female student to pursue physics as a major, a historic shift will be initiated — female students will make up 50% of incoming physics majors. Are you a high school physics teacher, or do you know a high school physics teacher? Join the STEP UP community to download the curriculum and help recruit teachers to the movement. STEPUPphysics.org THE PHYSICS TEACHER ◆ Vol. 60, April 2022 269

Video Analysis of an Oscillating Cantilever for Introductory Laboratories Blane Baker, Maggie Sherer, Ben Mossinghoff, and Will Laycock, William Jewell College, Liberty, MO Awood cantilever of length ~2.5 m is driven into according to resonance using the hand as a simple driver. Vid- eo recordings of these oscillations are analyzed to d = ( 02 – 2)0.5. (3) determine experimental second harmonic (n = 2) damped resonance frequencies. These frequencies are compared to Motions of simple harmonic oscillators often occur due to theoretical ones, obtained from measurements of elastic application of periodic driving forces, produced by external moduli, damping properties, and physical dimensions of the drivers. When these forces vary harmonically with time [e.g., cantilever. Experimental frequencies agree with theoretical F0 cos ( t)], a simple harmonic oscillator can be driven into ones to within their respective uncertainties. Experiments resonance. Maximum amplitude (resonant) oscillations occur described here can be incorporated seamlessly into intro- when the driving frequency matches one of the natural fre- ductory college physics laboratories, following discussions quencies of the oscillator. of simple harmonic oscillators (including damping). These kinds of experiments are useful for developing students’ Like other oscillating systems, flexible cantilevers can be skills in designing and troubleshooting experiments, analyz- driven into resonance by applying periodic driving forces. ing video recordings, and modeling data using computers. Studies show that resonance frequencies depend on a cantile- The ease of set up and simple apparatus also make these ver’s elastic properties, density, and physical dimensions. For activities ideal for remote learning such as during the recent transverse oscillations, a flexible cantilever has (undamped) COVID-19 pandemic. resonance frequencies given by n = cn2 [Eh2/(12 L4)]0.5, (4) Background and previous work where c1 = 1.875, c2 = 4.694, E is the elastic modulus, h is the Many natural and human-constructed systems exhibit thickness of the cantilever beam in the direction of motion, oscillations, characterized by back-and-forth motions about is the mass density, and L is the length of the cantilever.1 points of equilibrium. Examples range from vibrations of Flexible cantilevers exhibit damping, so their observed res- molecules to swaying of bridges and buildings. A special onance frequencies are slightly lower than predicted for the kind of oscillatory motion—called simple harmonic mo- undamped case, as discussed above. tion—occurs when systems experience restoring forces that are proportional to their displacements from equilibrium. Previous studies have determined damping constants for an underdamped harmonic oscillator,2 obtained kinematic When a simple harmonic system (oscillator) is un- quantities for magnetically damped systems,3 analyzed ener- damped due to the absence of dissipative forces such as gy of damped oscillators,4 and examined structural stiffness friction, motions are purely oscillatory in time. Thus, the po- of 3D-printed cantilevers.5 In work related to that described sition x of the system as a function of time t is given by here, resonances of saw-blade cantilevers driven by electro- magnetic coils have been studied6 and vibrations of car anten- x(t) = A cos ( 0t), (1) nas have been analyzed mathematically.7 where A represents the amplitude of the oscillations and 0 Various resonance experiments are available for introduc- is the natural angular frequency associated with undamped tory college physics laboratories; however, the ones described oscillations. here offer several distinct advantages. Experiments proposed require only simple equipment and a straightforward setup When a system is damped due to dissipative forces such so that they can be done in traditional laboratory classes or as friction or drag, motions decay with time. For the case in remote settings. The ability to conduct experiments remotely which one or more complete oscillations occur before the is especially important when students cannot attend in per- motions decay to zero, the system is said to be underdamped; son. In addition, given the simplicity described, students have its position is given by increased opportunities to develop their own experimental methods and to troubleshoot problems. The ability to design x(t) = Ae– t cos ( dt), (2) and troubleshoot experiments is crucial for improving stu- dents’ laboratory skills. Experiments here also focus on com- where d is the damped angular frequency and is the paring experimental results to predictions from theory. To damping coefficient. The damping coefficient determines determine theoretical resonance frequencies, students must the time over which oscillations decay; it is equivalent to the perform a second experiment to evaluate the stiffness of the inverse of the time required for the oscillations to decay to cantilever. Performing different kinds of experiments (reso- e-1. When an oscillator experiences damping, its resonance nance and stiffness here) is how scientists pursue their work, frequency decreases in comparison with the undamped case thus students gain valuable experience as practicing scientists. 270 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 DOI: 10.1119/5.0024016

Laboratory assignment and summary of procedure The basic equipment for this laboratory assignment consisted of flexible wood cantilevers, iPads, meter sticks, tape measures, cal- ipers, masses, mass hangers, and mass scales. The flexible cantilevers (with dimensions given in Table I) were purchased at a local hard- ware store as unfinished pine wood screen trim. Other equipment was already part of our general laborato- ry. Students were familiar with the CAPSTONE® software from pre- Fig. 2. Maximum displacement during four consecutive cycles at the antinode near the hand vs. time for free vibrations for one of our vious experiments, and instructors trials. Exponential fits of these data give an average damping coeffi- were available to help students as cient of 4.09 0.47 rad/s for three trials. questions arose. At the beginning of the labora- The damping coefficient was found experimentally by tory exercise, instructors showed using the same or a similar video, as described above. In this students the basic equipment. Then case, however, the cantilever was driven into resonance; then we reviewed the theory of damped the driving force (exerted by the hand) was reduced to zero oscillators (see section above) and while the cantilever was still held by the hand. Free vibration Fig. 1. Resonant showed students how to drive the of the cantilever was video recorded; maximum displacements (n = 2 transverse mode) system, as depicted in Fig. 1, to find during four consecutive cycles (at the antinode near the hand) oscillations of a thin the experimental damped resonance were measured over time as the displacements decayed. Expo- cantilever with two frequency d,exp. We also discussed nential fits of these data yielded an average of 4.09 rad/s for antinodes clearly visi- and demonstrated how to observe three trials with a standard deviation of 0.47 rad/s; see Fig. 2 ble. The experimental damping of the cantilever during for a plot of the data for a representative trial. resonance frequency free vibration to obtain and how Next, the theoretical undamped frequency 2,theory of to determine the elastic modulus E the cantilever was found from elastic properties of the wood d,exp is determined by experiment (see Fig. 3). From E cantilever, its density, and its physical dimensions. Physical from video analysis of and , students found a theoretical dimensions were obtained by direct measurements; the den- these oscillations for damped resonance frequency sity was found from direct measurements and determination comparison with theo- d,theory and compared it with the of the mass-to-volume ratio. The elastic modulus E was found ry. The wood cantile- experimental value d,exp. Students by supporting the cantilever at two points along its length, ver is oriented vertically performed the measurements and hanging a known mass at the middle, and measuring the ver- to eliminate sagging effects due to gravita- tional forces. reported their findings (along with estimated uncertainties tical deflection (see Fig. 3). The elastic modulus was then associated with d,theory and d,exp). A general summary of computed from the procedure is described below. The experimental n = 2 damped resonance frequency E = Fl3/(4 wh3), (5) d,exp of the cantilever was determined from video analysis. where F is the load, l is the separation between supports, w is The cantilever was held at one end and driven (shaken) by width of the cantilever, and h is its thickness. From 2,theory the hand, as depicted in Fig. 1. By varying the driving fre- and , students determined a theoretical damped resonance quency, resonant oscillations for the n = 2 mode were gen- erated and then recorded via smartphones or iPads (which were supplied to students in our setting). Once the data were recorded, they were imported into CAPSTONE® software for analysis. (Both the n = 1 and n = 2 modes can be attained easily by driving frequencies produced by the hand, so ana- lyzing the n = 1 mode is possible, too.) To determine d,exp, oscillations at the antinode near the hand were monitored and timed over 20 complete cycles. For comparison, oscillations of the hand (the driver) were ana- lyzed in the same way with no differences observed. Thus, Fig. 3. Experimental setup for determining the elastic modulus E of the driving and damped resonance frequencies are equal, as the wood cantilever. The cantilever is supported at two locations expected when resonance occurs. separated by a distance l, a load F is applied at the middle of the cantilever, and the deflection under the load is measured. THE PHYSICS TEACHER ◆ Vol. 60, April 2022 271

Table I. Quantities to determine the elastic modulus E and the- and experimental values to within their respective uncertain- oretical damped resonance frequency d,theory of a wood canti- ties. Experiments described here offer several advantages over lever for comparison with its experimental damped resonance traditional resonance experiments. They require very little frequency d,exp. equipment and, therefore, are easily adaptable to virtual or re- mote learning environments. They also offer convenient ways Quantity Numerical value to observe and analyze underdamped oscillations, measure damping coefficients via video analysis and modeling, and en- Mass of cantilever M 0.155 kg gage students in developing skills in experimental design and troubleshooting. Length of cantilever L 2.44 m References Thickness h 0.00643 m 1. Joseph W. Tedesco, William G. McDougal, and C. Allen Ross, Width w 0.0187 m Structural Dynamics: Theory and Dynamics (Addison-Wesley Longman, Inc., Menlo Park, CA, 1999), p. 450. Load F to determine elastic modulus 2.45 N 2. Michael C. LoPresto and Paul R. Holody, “Measuring the damp- ing constant for underdamped harmonic motion,” Phys. Teach. Separation between supports l to get the 2.14 m 41, 22 (Jan. 2003). elastic modulus 3. Peter F. Hinrichsen, “Acceleration, velocity, and displacement for magnetically damped oscillations,” Phys. Teach. 57, 250 Deflection of the cantilever under load F 0.104 0.015 m (April 2019). 4. Tommaso Corridoni, Michele D’Anna, and Hans Fuchs, Elastic modulus E 12. 2.0 GPa “Damped mechanical oscillator: Experiment and detailed ener- Density 528 kg/m3 gy analysis,” Phys. Teach. 43, 88 (Feb. 2014). 5. Lawrence N. Virgin, “Comparative structural stiffness: Exploit- Theoretical undamped resonance frequen- 32.7 2.6 rad/s ing 3D-printing,” Am. J. Phys. 88, 1049 (Dec. 2020). cy 2, theory 0.47 rad/s 6. Michael Liebl, “Saw blades and resonance,” Phys. Teach. 52, 282 (May 2005). Damping coefficient (from computer fit) 4.09 7. Ronald Newburgh and G. Alexander Newburgh, “Finding the equation for a vibrating car antenna,” Phys. Teach. 38, 31 (Jan. Theoretical damped resonance frequency 32.4 3.2 rad/s 2000). d,theory William Jewell College; [email protected] Experimental damped resonance frequen- 28.6 3.5 rad/s cy d,exp (from video) frequency d,theory [see Eq. (3)] and compared it to the exper- imental value d,exp. Data and analysis Data to determine the elastic modulus E and d,theory along with d,exp are given in Table I. The elastic modulus E is found experimentally, as described above (see Fig. 3). The undamped resonance frequency 2,theory is found from the quantities E, h, , and L [see Eq. (4)]; the value obtained here is 32.7 2.6 rad/s. The theoretical damped resonance fre- quency d,theory is determined from the undamped resonance frequency 2,theory and the damping coefficient [see Eq. (3)]; the value for d,theory obtained here is 32.4 3.2 rad/s. For comparison, determination of d,exp from 20 complete oscillations yields 28.6 3.5 rad/s. Errors are estimated by de- termining how measurement uncertainties impact computed quantities such as E and 2. Conclusions Flexible wood cantilevers are ideal systems for demonstrat- ing resonance and analyzing resonant oscillations in introduc- tory college physics laboratories. Experimental resonance fre- quencies are found by monitoring video-recorded motions of such a cantilever as it is driven by the hand at various frequen- cies. In addition, the damping coefficient is determined by measuring decreases in maximum displacement (vs. time) for a freely oscillating cantilever and fitting those data to decaying exponential functions. Using the obtained, the n = 2 theo- retical resonance frequency for the damped cantilever system is calculated. Results show agreement between the theoretical 272 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

Using a Smartphone Pressure Sensor as Pitot Tube Speedometer Dominik Dorsel, Sebastian Staacks, Heidrun Heinke, and Christoph Stampfer, RWTH Aachen University, Institute of Physics I and II, Aachen, Germany As smartphones have become a part of our everyday life, their sensors have successfully been used to allow fluid flow direcƟon data acquisition with these readily available devices in a variety of different smartphone-based school experi- stagnaƟon pressure staƟc pressure ments.1-4 Such experiments most commonly take advantage of the accelerometer and gyroscope. A less frequently used sensor in smartphone-based experiments is the pressure sen- sor or barometer.5-7 Pressure sensors in smartphones can im- liquid fluid prove indoor navigation, for example in multi-story shopping Fig. 1. The Pitot tube in a cylinder with a moving fluid. The left malls. In a popular smartphone experiment, the barometer is opening of the Pitot tube is aligned against the flow direction used to determine the current altitude in an elevator with the and exposed to the stagnation pressure. The other end of the barometric height formula.8 Along with accelerometer data Pitot tube perpendicular to the fluid flow direction is exposed and by deriving the height data to calculate the velocity, z(t), to the static pressure only, resulting in a height difference of the vre(ta)l,taimnde.a9(t) plots can be generated and shown to students in two fluid columns in the manometer. In this article, an experiment to measure air velocity with off staƟc pressure the barometer of a smartphone is presented utilizing the con- cept of a Pitot tube. For the experiment, the app phyphox is used, which has a set of useful features for performing smart- phone-based experiments.9,10 Both the data analysis tools in- on stagnaƟon pressure cluded in the app phyphox and the remote access function to smartphones in experimental setups are used in the presented experiment. This experiment is also an example for designing customized experiments within phyphox that can be realized by any user via a web-based editor.9 In addition, it is shown how an external Bluetooth pressure sensor can be used to ex- tend an experiment, which is used here to measure the speed Fig. 2. To determine the wind velocity, the static and stagnation pressure are measured with a smartphone, while the ventilator of a vehicle. Such an integration of external Bluetooth Low is switched off or after the ventilator is turned on, respectively. Energy sensors into smartphone experiments became avail- fluids. Air can be considered as incompressible for velocities able with phyphox version 1.1.0. smaller than 30% of the speed of sound.12 The temperature dependence of the fluid density ρfluid can be taken into ac- The Pitot tube count by measuring the air temperature T and using the Fluid flow velocities can be measured with a Pitot tube11 known function ρair (T) for dry air, leading to (Fig. 1). One opening of the Pitot tube points against the flow (3) direction where it is subject to stagnation pressure. The other opening is perpendicular to the flow direction, which in con- where p is the absolute s (p = pstatic) and Rs = 287 J/(kg .K) trast makes the static pressure accessible. Original Pitot tubes pressure is the specific gas constant. contain water, which ascends to different heights due to the pressure difference. Experimental setup with internal pressure sensor The pressure difference depends on the fluid velocity and In the first experiment, we determine the speed of wind can be expressed by the Bernoulli equation: produced by a wind machine. Therefore, the static and stag- nation pressure have to be measured. The static pressure is (1) measured with the internal pressure sensor of the smartphone while the wind machine is switched off. Afterwards the wind Rearranging the terms allows calculating the velocity from the machine is turned on and the same setup measures the stag- pressure difference: nation pressure. Unlike the original Pitot tube, the pressure (2) is measured with the smartphone’s capacitive pressure sensor instead of the liquid fluid. The measurement process is illus- trated in Fig. 2, while a photo of the experimental setup can be In general, the Bernoulli equation is valid for incompressible seen in Fig. 3. DOI: 10.1119/5.0025899 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 273

Fig. 3. A picture of the experimental setup for measuring wind be presumed to be available, external sensors support equal speed. opportunities in performing student experiments. The use of external sensors can also reduce possible risks damaging As described in Ref. 5, phyphox allows a customized in- smartphones in experiments. In addition, it facilitates the app analysis of the measured sensor data and a user-designed support of experimenting students if sensors with identical display of its results. This can be created by any user and has specifications are used in contrast to a possible broad variety also been used for the shown experiment. The resulting phy- of sensors in students’ smartphones in the widespread BYOD phox experiments are not available in phyphox by default, but approach (BYOD—bring your own device). can easily be shared with other users, for example via a QR code to be scanned in phyphox (see Fig. 4). In our case of a Pitot tube experiment, using external sen- sors offers the additional possibility to improve the accuracy Figure 4 shows the two screens “Measurement” and “Set- of the experiment by using two pressure sensors to measure tings” in the designed phyphox experiment. At the bottom of the static and stagnation pressure simultaneously. This the screen “Measurement,” the current value of the measured becomes important in all experimental settings where a time- pressure is shown. As long as the wind machine is switched dependent variation of the static pressure cannot be neglect- off, this is the static pressure. By using the button “SET STAT- ed. An example is using a Pitot tube to determine the velocity IC PRESSURE,” the current numerical pressure value is set to of a vehicle. Since streets are mostly not perfectly even, the the static pressure, which will be used for calculation accord- static pressure is not constant anymore due to small variations ing to Eq. (2). The air density, which is also required in Eq. (2), in altitude. Hence, it is mandatory to measure stagnation and is calculated from the room temperature according to Eq. (3). static pressure simultaneously. The room temperature can be set in the settings tab, as shown in the right screenshot of Fig. 4. Here, the pressure sensor of a Texas Instruments SensorTag CC2650 has been used as an external sensor measuring the If the wind machine is switched on, the current pressure stagnation pressure. At the same time, the static pressure value shows the stagnation pressure. It is given in both the is measured with the internal pressure sensor of the smart- lower graph of the pressure sensor’s raw data and its current phone. The phyphox experiment has been adapted to the numerical value. These data can be converted to the air veloc- modified experimental setup. The customized experiment can ity by using Eq. (2) (see the top of the screen “Measurement” be added to phyphox by scanning the QR code in Fig. 5. on the left side of Fig. 4). Providing an appropriate choice of the experimental conditions, the upper graph shows mean- The sensor module has been placed in a wide-necked ingful velocity values. Exemplary measurement results are bottle and held out of the window of a car while the car was given in Fig. 4. In this case, the wind machine was able to pro- in motion. The determined driving speed via the Pitot tube duce an air velocity of up to 10 m/s. can be verified by comparing it to the velocity measured by a global navigation satellite system (GNSS), i.e., the GPS posi- Experimental setup with external pressure tion of the smartphone. An exemplary measurement can be sensors seen in Fig. 5. The measured driving velocity from the Pitot tube in the top graph is noisier than the GPS-generated data, Since version 1.1.0, the app phyphox can not only readout but considering the simplicity of the experimental setup it fits internal smartphone sensors but also external sensors via very well to the measured GPS data. In this way, it validates Bluetooth Low Energy (BLE). Thus, the described experiment the Bernoulli equation and demonstrates its applicability for can also be performed with a smartphone without an inter- velocity measurements in racing cars and aircrafts. nal barometer by placing an external pressure sensor in the experimental setting in Fig. 2 in place of the smartphone. As Conclusion one can see in the sensor database “phyphoxDB,”13 roughly An experiment has been presented to measure the wind 49% of the “phyphoxDB” contributors have a pressure sensor available. Since the database is filled on a voluntary basis by speed produced by a wind machine with a smartphone’s ba- phyphox users with their personally available sensors, the da- rometer. By integrating external sensors, this experiment can tabase is not necessarily a representative sample for students. be simply extended to measure the headwind while driving a However, as it clearly indicates that pressure sensors cannot car, which can be used to determine a car’s velocity. The ve- locity measured by the Pitot tube fits very well to the velocity determined from GPS data. The experiments show both the possibility to design cus- tomized experiments by using the phyphox web-based editor and the extended experimental options by integrating exter- nal sensors into smartphone-based experiments. This opens a wide range of new experiments with smartphone-based data acquisition and analysis, which pursue a variety of education- al goals not only for physics teaching but for teaching science in general. 274 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

Fig. 4. Screenshots of a taken measurement. The calculated velocity is shown on References top of the left screenshot. The graph below shows the measured raw data for the 1. S. Staacks, S. Hütz, H. Heinke, and C. Stamp- stagnation pressure. The values of the current pressure and the static pressure are shown below. To set the static pressure before the wind machine is turned on, the fer, “Simple time-of-flight measurement of button “SET STATIC PRESSURE” is used. The room temperature is required for the the speed of sound using smartphones,” Phys. calculation of the air density. It can be set in the settings tab, as shown in the middle Teach. 57, 112 (Feb. 2018). screenshot. The experiment can be added to phyphox with the QR code on the right. 2. S. Staacks, S. Hütz, H. Heinke, and C. Stamp- fer, “Advanced tools for smartphone-based Fig. 5. Screenshot for the measurement of car velocity. The top graph shows the experiments: phyphox (2018),” https://arxiv. results of the measurement with a Pitot tube. The bottom graph presents the mea- org/pdf/1804.06239. sured velocity by GPS. The results by the Pitot tube are noisier but fit overall very well 3. J. Kuhn and P. Vogt, “Smartphones as ex- with the GPS data. The customized experiment can be added to phyphox by scanning perimental tools: Different methods to the QR code in the figure. determine the gravitational acceleration in Acknowledgment classroom physics by using everyday devic- es,” Eur. J. Phys. Educ. 4, 16–27 (2013). The Federal Ministry of Education and Research supports the 4. J. Chevrier, L. Madani, S. Ledenmat, and A. project “Lehrerbildung Aachen” (LeBiAC2, FKZ 01JA1813) of Bsiesy, “Teaching classical mechanics using the RWTH Aachen University in the context of their funding smartphones,” Phys. Teach. 51, 376 (Sept. program “Qualitätsoffensive Lehrerbildung” (Phase 2). As 2013). part of LeBiAC2 we develop and evaluate the use of Blue- 5. M. Monteiro, P. Vogt, C. Stari, C. Cabeza, and tooth-based data acquisition by students during their educa- A. C. Marti, “Exploring the atmosphere us- tion to become teachers. ing smartphones,” Phys. Teach. 54, 308–309 (May 2016). 6. S. Macchia and R. Vieyra, “A simple wind tunnel to analyse Bernoulli’s principle,” Phys. Educ. 52, 13004 (2017). 7. S. Macchia, “Analyzing Stevin’s law with the smartphone barometer,” Phys. Teach. 54, 373 (Sept. 2016). 8. M. Monteiro and A. Martí, “Using smart- phone pressure sensors to measure vertical velocities of elevators, stairways, and drones,” Phys. Educ. 52, 15010 (2016). 9. S. Staacks, S. Hütz, H. Heinke, and C. Stamp- fer, “Advanced tools for smartphone-based experiments: phyphox,” Phys. Educ. 53, 045009 (2018). 10. C. Stampfer, H. Heinke, and S. Staacks, “A lab in the pocket,” Nat. Rev. Mater. 5, 169–170 (2020). 11. H. Pitot, “Description d'une machine pour mesurer la vitesse des eaux courantes et le sillage des vaisseaux,” Histoire de l'Académie royale des sciences avec les mémoires de mathématique et de physique tirés des regis- tres de cette Académie, (1732), pp. 363–376. 12. P. Balachandran, Fundamentals of Compress- ible Fluid Dynamics (PHI Learning, 2006). 13. See http://www.phyphox.org/sensordb, “Sen- sor Database,” accessed Feb. 2022. RWTH Aachen University, Institute of Physics I and II, Aachen, Germany; [email protected] aachen.de THE PHYSICS TEACHER ◆ Vol. 60, April 2022 275

Effectively Illustrating Nature’s Magic with Magic F. D. Becchetti, University of Michigan, Ann Arbor, MI There is an ongoing challenge with STEM education: Problem: While many of these demonstrations and making physics, math, and science, in general, inter- revelations reinforce scientific concepts or specific phe- esting, understandable, and retentive for college sci- nomena presented in a conventional lecture, many are ence and non-science majors, K-12 students, and the public. limited in scope and A/V impact. As noted, they often If not imparting detailed knowledge, at least one would like demonstrate simple classical physics phenomena (kine- to introduce important concepts that will be remembered, matics, optics, sound, E&M). Hence, although interest- appreciated, and hopefully would be pursued in more detail ing, they often are limited in the “wonder” factor that by audience members. One solution: as noted by Socrates, is key to retention.2-7 However, Prof. Clint Sprott at the “Wisdom begins in wonder.” Indeed, magic as a form of University of Wisconsin and Paul Hewitt, among others, wonderment dates back to Socrates and even earlier. One of have developed physics demonstrations, some including tnhaeldfiSrsctomt,1awgiacsbpouobklsis, hTehde Discoverie of Witchcraft by Regi- magic, intended to invoke audience interest in phys- in 1584, predating publication of ics.11,12 Many of these demonstrations can be effective many science texts. In this paper the author, based on recent for use in classroom instruction or public lectures when, research, advocates using special forms of magic to both as noted, primarily illustrating classical physics. amaze and teach, and in particular to illustrate the wonders of modern physics, i.e., Nature’s magic, but with connections The approach described here is a combination of A and B also to classical physics. with important changes: Past and proposed methods of incorporating C) Adopt or develop special magic effects that illustrate magic in science presentations primarily modern physics concepts, but not necessarily with the physics as the working basis for the magic. Psy- Here is a summary of some past methods of incorporat- chological research using advanced brain imaging and ing magic in STEM education: other methods on the use of magic for instruction has shown that magic encourages critical thinking, problem A) Use conventional magic (linking rings, cups and balls, solving, peer-to-peer interaction, and other useful learn- cut and restore rope, card predictions, etc.) to first get ing skills.2-7 It also can impart a sense of well-being in the attention and interest of the audience. Follow this participants.2-7 Also, most of the magic used in A) and with explanations of a specific science topic, usually B) is of human construct. Nature’s magic (i.e., modern unrelated to the magic performed. This typically has physics) is beyond human imagination and far more been done by amateur or semi-professional magicians amazing than humans could develop, with the possible who also teach science courses or give science-based exception of Albert Einstein. As shown in the examples, public talks. the magic will not be limited to simple effects and we are not necessarily explaining how the magic works. Hence Problem: Studies2-7 have shown that magic used in professional-level magic, including large stage magic, this way, with the possible exception of chemical- or can be utilized for maximum impact. math-based magic, can be counterproductive. Rather than being attentive to the subsequent science being It has been verified that audience members will better explained, many students or audience members will retain scientific material if the magic is embedded as part of instead be thinking about the magic and how it was the magic presented.2,3 Hence, the material should illustrate done, basically “tuning out” the remaining lecture. specific and interesting science concepts integrated with the Thus, the science may have minimal long-term reten- magic presented. Especially, as noted, such a presentation can tion as, often, the magic doesn't illustrate and reinforce serve to introduce many aspects of modern physics, “Nature’s the science being discussed. magic,”13,14 that may otherwise seem difficult to present. This can generate wonder and appreciation of modern science, B) Present more or less conventional science teaching but, as will be shown, in many cases important connections to demonstrations as “magic” (bowling ball pendulum, classical physics can also be included. This approach has been action-reaction cannon, optical illusions with mir- used by the author and others9,10,15 with good success. This rors, harmonic motion, Ampere’s law, hair-raising includes college lectures to both science and non-science ma- Van de Graaff demonstration, etc.). Alternately, as jors, public lectures, and specialized talks, e.g., banquet talks pioneered by Martin Gardner and others,8-10 one can and AAPT presentations. At the University of Michigan-Ann demonstrate a magic trick that utilizes some particular Arbor, magic also is used as part of a highly successful high physics principle, using that to get the attention of the school summer camp program developed by the author and audience and introduce a specific physics phenome- colleagues.16 This session typically is fully enrolled, usually non. The Physics Teacher has many examples of this, es- has a wait list, and often also attracts teachers to sit in. Like- pecially as related to optics, E&M, acoustics, and fluids, wise, college classes at other schools employing magic effec- i.e., mainly classical physics. 276 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 DOI: 10.1119/10.0009992

tively are usually in high demand, especially among non- ing card deck, gimmicked dice, Svengali deck or pad, special science majors.15 book tests, magic squares, etc.), use of magician’s choice and Once students are motivated, they often will read assigned multiple outs, levitation techniques, vanish and production material not covered in detail in lectures more readily, so that methods, optical illusions, sleight of hand, handcuff and the lecture time needed for the magic can be accommodated. other escapes, mentalism techniques, and other well-known Some students may then often be interested in learning magic methods.9,10,17-22 Many of these are based on stage magic and, as noted, that can be beneficial in developing good learn- used by famous vaudeville magicians.23,24 Again, the object is ing skills and a sense of well-being. Likewise, successful teach- to use special magic illustrating science concepts to generate ing often involves an instructor who is interested in the course wonder, produce long-term retention, and inspire audience material and in preparing good lectures. Incorporating magic members to pursue the topics introduced in more detail, in- in lectures can help with that goal. cluding, if students, more detailed course material, homework problems, etc. Some examples of appropriate science-related As indicated, the method proposed here is different than magic Background information revelation methods that explain the physics behind a partic- ular trick or demonstration.8-12 Instead, we generally don’t As noted, to be most effective the magic performed should explain the methods employed but instead leave that as a chal- have some direct link to the science being presented and il- lenge for the audience. As noted, using magic in lectures can lustrate the science in a memorable fashion. This often can be make these fun and interesting for the instructor, especially if done by adopting or modifying well-known magic routines to some of the magic is changed from year to year, so new effects introduce specific science topics or by using standard magic are learned and added as appropriate. techniques to develop new science-related effects. This in- Some examples of the magic the author has adopted or cludes the use of “forcing” techniques (e.g., employing a forc- developed are demonstrated in a public-lecture video as part of the highly successful Saturday Morning Physics Table I. Examples of physics topics and related magic physics topic-related public lecture series for the public at the Univer- magic. sity of Michigan-Ann Arbor.13 This YouTube Gravity and the forces of Levitation effects; happy/sad ball drop; series includes lectures on a wide range of topics Nature in modern physics and science. Van de Graaff levitation; magnetic globe levitation; In many cases minor but important changes floating ball (Fig.1); eddy current magic (drop tube) Quarks and the Standard Six card prediction matrix; gimmicked dice; magic in a standard magic routine can be made to in- Model troduce a specific physics topic. In other cases, slate prediction effect (Fig.3); math magic (e.g., well-known techniques, as noted for example magic squares) forcing a specific card or number, levitation using magnets or thread and other methods, use of op- Particle-wave duality and Needle through balloon (Fig. 4); card frame pene- tical illusions, and employing gimmicked appara- QM tration; selected card transpositions; handcuff and tus (handcuffs, liquid pitcher, vanish box, magic other escapes; gimmicked guillotine Nuclear reactions and Clock and color-changing chemical magic; color- slate, marked or “forcing” cards, gimmicked transmutations dice, etc.), are utilized. These and other basic changing silks; index of refraction matching and methods can usually be adopted to develop new, Nuclear decay vanish science-related magic routines.13,14 Chemical clock reactions (various lifetimes); math- As a member of a magic society, the author based magic; barrier penetration effects has taken a pledge not to reveal in detail the magic techniques employed, especially if using Handiness, chirality, and Polarized light magic; twisted arm illusion; related commercial magic, to non-magicians. But most stereo isomers optical illusions of the techniques, methods, routines, and appara- tus discussed here are described in standard ref- E=mc2; matter-antimatter Vanish box and flash paper; other vanishing effects erence books,17-24 DVDs, online sources, or are Time dilation and the twin Time predictions and magic (gimmicked watches paradox and clocks; chemical clock reactions) Equivalence principle, Limp/stiff rope; optical magic (modified lenses or otherwise available commercially. The latter in- bending of light, and grav- mirrors) clude instructional videos that can be purchased itational lensing from various magic suppliers25-34 or are available Mind reading and mentalism; electronics-based online for purchase and download. Another use- E&M waves; electricity in magic; book tests; remote vision effects ful resource includes members of local chapters brain and body Gimmicked liquid pitcher; chemical clock and color- of national magic societies (below), who usually changing reactions or effects will be quite willing to share information. In ad- Phase transitions Scientific observations Optical illusions; floating head illusion dition, they often have swap meets where gently Most topics Book tests with relevant books; modified card magic; used magic apparatus, DVDs, and books can be mentalism; and ... ?? purchased at modest cost. A local magic store of THE PHYSICS TEACHER ◆ Vol. 60, April 2022 277

course is another good resource. In some cases, special appa- vious if a particular magic effect or technique can be adopted ratus can be custom built by commercial vendors.25-28 This for this purpose. is especially useful if non-gimmicked, duplicate apparatus is Levitation and the basic forces of Nature also used (see below). A college, high school, DIY public, or home workshop can often be utilized to build apparatus from In Fig. 1 we show a standard levitation effect, a floating ball plans readily available from various books and other sourc- that can be manipulated to do some amazing movements. In es.25 any case there are many levitation effects that can be done. All can be used to introduce the forces of Nature and the fact that Some specific examples gravity is by far the weakest of these, hence it’s easy to levitate The introductory material objects. This can then lead to a discussion of Galileo’s free-fall experiments and the strange nature of gravity, i.e., acceler- When the author introduces himself for the first time to an ation of an object independent of its mass. What is limit as audience (students or public), it is stated that the author may mass goes to zero,13 i.e., does light fall due to gravity?? So, is at times assume the role of a magician, “The Great Frederico,” gravity a conventional force?? and as a magician may lie, cheat, deceive, and for example use secret collaborators in the audience. Alternately, the author (a) (b) may at other times be presenting material as a well-respected Fig. 2. (a) Adelson’s checker illusion and (b) the Ochi illusion. physics professor, hence will tell the truth, never deceive, and Squares A and B in (a) are the same shade of grey, and nothing always acknowledge any collaborators. The audience is then of course is actually moving in the Ochi illusion. CAUTION: Don’t challenged to decide which persona is appropriate at a given look at these if such illusions cause vision and other problems. time during the presentation.13 Seeing is believing?? Likewise, the primary difference between magicians and There are many interesting optical illusions35-37 that can be scientists is noted before a public lecture: magicians know what they are doing, i.e., the outcome of a magic effect is linked to science. In particular, how reliable are human obser- known, for example appearing to saw an assistant in half. In vations? As seen in the illusions shown in Fig. 2 and Refs.13 contrast, scientists do experiments with often unknown or and 14, one cannot always trust what one “sees” or is reported surprising outcomes, the latter often welcomed as they can seen by others. There are many examples of false discoveries represent new discoveries. in science, including the infamous case of N-rays.38 The de- bunking of that discovery, although it had been supposedly Table I lists a few examples of the physics topics and re- verified by others, led to the disgrace and eventual suicide of lated magic one can use in a lecture. Several of the examples the scientist who had first claimed the discovery. The lesson: are demonstrated in one of the author’s talks for the general scientists must use instruments for observations, and these public.13 Suitable video clips from that presentation are used in several of the figures shown here. Table I includes only a small selection of various science-related magic that has been developed (or adopted) and used by the author and colleagues in various presentations. More examples are discussed in the extended version of the paper and other material posted as a supplement online.14 Once the basic premise is accepted, i.e., that the magic should illustrate the science to be presented, it usually is ob- Fig. 1. Author demonstrates floating ball illusion illus- Fig. 3. Audience or student selected quark combina- trating the weak nature of gravity. tions and author’s + prediction (uus=usu= +; Table II). 278 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

Table II. Some elementary particles and their 3-quark constituents Particle Quarks a) Charge (e) Neutron udd 0 Proton uud +1  – dds -1 ––– – dss -1 ––– 0 uss 0  + uus +1 a) u=up quark (charge +2/3 e); d= down quark (charge= -1/3 e); s= strange quark (charge = -1/3 e) must be carefully calibrated to produce objective, reproduc- ible data independent of a particular human operator. Quarks and the particle zoo Fig. 4. Needle through balloon effect illustrating particle-wave In Fig. 3 we illustrate a mentalism effect using a gimmicked duality and quantum barrier penetration as a wave, e.g., alpha decay and solid-state electronics. Needle magically then pad, a variation of Corinda’s “slate test”18 as performed by becomes a “particle” (solid needle) when a spectator tries the Max Maven and other contemporary magicians. Spectators effect (on sides) and bursts the balloon. select an unseen set of three cards each with a quark marked on the card (up, down, strange, charmed, bottom, top). Magi- this (Fig. 4), by claiming that the needle represents a subatom- cally the chosen cards have been predicted (usu) together with ic particle that can be either a classical particle or a wave, and the particle they make, in this case,13 Σ+. As example, Table if a wave, it can penetrate the balloon material (the barrier an- II shows some simple three-quark systems suitable for this alog) without breaking it as is demonstrated. But surprisingly, effect. This is highly effective magic and illustrates the use of it then acts as a classical particle and bursts the balloon with professional-level magic without the constraint that the meth- good A/V impact when an audience member tries it. (The bal- od employed must be based on a specific physics phenome- loon is secretly turned to a thinner section on the side of the non. The magic effect serves to illustrate a physics concept, balloon.) Penetration effects using a card in a frame, sword in this case that just a few quarks (and their antiquarks) can through neck apparatus, rope through body, and similar mag- make a whole “zoo” of elementary subatomic particles. Amaz- ic can also be adapted for this. This can lead to a discussion of ingly, however, one needs to introduce fractional charges and, electrons moving in solids as waves, the advent of solid-state even stranger, these have never been observed alone. Another electronics, and all the practical applications (computers, cell effect, among many possible illustrating this and the Standard phones, GPS, etc.). Likewise, penetration effects as noted can Model, utilizes a six-card special matrix (representing the six serve to illustrate long-lived nuclear decay and the use of such basic quarks) with a special key embedded in the cards.13,14 decay to determine the age of geological and archaeological formations, human and animal remains (e.g., dinosaurs), Particle-wave duality dating the time scale and forms of human evolution, the path Particle-wave duality is another aspect of Nature’s magic of human migrations, and dating other important, interesting events.39,40 with important practical applications. At the atomic and nuclear level, particles such as electrons, protons, neutrons, Enhancing the magic for wonder and retention alpha particles, etc. can act either as classical particles or as The above are only a few illustrative examples employing waves with wave-like properties. Just as a beam of laser light will create a diffraction pattern passing through fine powder magic to effectively illustrate Nature’s magic, with a few addi- (e.g., lycopodium), a beam of electrons passing through a thin tional examples given in Table I, in the related video,13 and in foil of atoms will create a similar diffraction pattern, thus act- the supplementary material.14 ing as a wave, as can readily be demonstrated in a lecture with an electron diffraction apparatus. The electron microscope, Most effects are more effective if a “swap out” method is key in the study of viruses, is another important manifestation used to reinforce the magic, making the magic even more puz- of electrons acting as waves. Likewise, subatomic particles zling.14 This hopefully keeps the audience engaged well after acting as waves can penetrate a classically forbidden potential the presentation (“How did he/she do that?”), thus increasing energy barrier, as happens in nuclear decay, e.g., alpha de- the “wonder” factor. This is done by using duplicate props, cay.39,40 Long-lived nuclear decays are one of the main tools one gimmicked, one not. It can include some custom props for determining the age of objects, such as Earth (4.5 billion made by magic equipment suppliers,25-28 or modifications years old39,40). The needle through balloon effect is a good illustration of THE PHYSICS TEACHER ◆ Vol. 60, April 2022 279

made to a standard gimmicked prop, by removing or making Conclusions inoperable the gimmick portion of the prop. As example, one Modern psychological research supporting the author’s can alter a gimmicked pitcher so it’s an identical appearing but now normal pitcher that can be used and then swapped out and others’ anecdotal evidence suggests that employing spe- to be later casually left out for inspection. Likewise, one can cially developed or modified science-related magic to intro- have two sets of books for book tests, one gimmicked with an duce an audience to modern science, i.e., Nature's magic, can answer key, another normal, and similarly for gimmicked card be very effective. In addition, it can make teaching science decks, modified Rubik's cubes, and other props.17-22 Students more engaging and interesting for instructors, an important and audience members, many at times in groups of several factor for effective teaching. Nature is magical, but like all people, will often come up after a lecture to inspect any props magic once the secrets are revealed, it may no longer be as casually left out. But of course, the props left out will appear magical but still very amazing and a source of wonder. to be normal, which they are, making the previous magic even more amazing and memorable. Acknowledgments The author thanks Ramon Torres-Isea and UM undergrad- Science of Magic Association (SoMA)41 uate student John Dunne (class of 2020) for their assistance. Related to the topic of this paper, a special magic society has Thanks also to the referees for their comments and sugges- tions. Many useful discussions with SoMA and SAM mem- been created (SoMA) consisting of magicians, neurologists, bers are also acknowledged. psychologists, and other scientists dedicated to studying how magicians (and Nature) deceive human observers. In the case References of magicians, what are the most effective moves to divert the 1. American Museum of Magic, attention of the audience, e.g., as measured by eye-tracking studies.4 A classic example of this is the gorilla in the room42 http://americanmuseumofmagic.com/ . and related tests, where most observers will not report seeing a 2. S .A. Moss, M. Irons, and M. Boland, “The magic of magic: The “gorilla” come and go into a room of people!4,42 Likewise, how reliable then are eyewitnesses, for example at a crime scene effect of magic tricks on subsequent engagement with lecture where eyewitness identification can be a matter of life or death material,” Br. J. Educ. Psychol. 87, 324 (2017). (and has been) for those accused of a serious crime? 3. J. L. Lin et al., “The effects of combining inquiry-based teaching with science magic on the learning outcomes of a friction unit,” Other magic societies and resources J. Baltic Sci. Educ. 16, 218 (2017). In addition to SoMA, there are two major magic societies 4. A. S. Barnhart and S. D. Goldinger, “Blinded by magic: Eye-movements reveal the misdirection of attention,” Front. that can provide many resources needed for incorporating Psychol. 5, 1461-1 (2014). magic in instructional material. The primary U.S.-based 5. Gustav Kohn, Experiencing the Impossible: The Science of Magic group is the Society of American Magicians (SAM),43 whose (MIT Press, Cambridge, MA, 2019). first president was Houdini. The other major group, which is 6. Stephen Macknik and Susana Martinez-Conde, Sleights of a large worldwide association, is the International Brother- Mind: What the Neuroscience of Magic Reveals about Our Every- hood of Magicians (IBM).44 SAM in particular has many local day Deceptions (H. Holt, New York, 2010). chapters, and members of those chapters can provide resourc- 7. Science of Magic Association (SoMA), Bibliography 2018 up- es and advice as needed to develop and critique science-relat- date, https://scienceofmagicassoc.org/. ed magic developed. 8. Martin Gardner, Magic for the Class (1941), Martin Gardner’s Science Magic (Dover Magic Books, New York, 1997). As noted, additional material from the author can be 9. Marshall Ellenstein, “Magic and physics,” Phys. Teach. 20, 104 found in the online supplementary material for this paper.14 (Feb. 1982). Likewise, there are many online websites that offer discussion 10. Robert Friedhoffer, Magic Tricks, Science Facts (Franklin Watts, groups related to magic. A public encyclopedic website devot- New York,1990); More Magic Tricks, Science Facts (Franklin ed to magic with extensive historical and other information, Watts, New York,1993). MagicPedia (http://geniimagazine.com/wiki/index.php/ 11. C. Sprott, The Wonders of Physics (U. Wisconsin, Madison, WI), Main_Page), is a good resource for magic-related material.45 http://sprott.physics.wisc.edu/wop.htm. Most magic effects and science topics mentioned in this paper 12. Paul G. Hewitt, Best of Physics Alive (Educational Innovations, are described in more detail on the internet and in particular Bethel, CT). in Wikipedia and MagicPedia,45 and an extensive compendi- 13. Fred Becchetti, “Nature's Magic,” Fred Becchetti – Saturday um of “book test” methods used for mentalism and remote vi- Morning Physics – 04/02/11, sion effects has recently been published.46 Fortunately, some YouTube, https://www.youtube.com/watch?v=VcVvBtWPRS4 . brick-and-mortar magic shops still exist and are usually run 14. Readers can view the material at TPT Online, http://dx.doi. by magicians anxious to help others. Finally, the author may org/10.1119/10.0009992, under the Supplemental tab. be contacted to provide more details and other information 15. Steven C. Okulewicz, Hofstra University, private communica- about specific magic effects. tion. 16. Michigan Math and Science Scholars, https://sites.lsa.umich.edu/mmss/ . 17. T. Annemann, Practical Mental Effects (Dover Publications, New York, 1983). 280 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

18. T. Corinda, 13 Steps to Mentalism (D. Robbins and Co., Cranbury, Physics teachers... NJ, 1996). get your students registered for 19. The Original Tarbel Lessons in Magic (Magicmakers, New York, the preliminary exam in the U.S. 2011). Physics Team selection process. 20. Joshua Jay, Magic: The Complete Course (Workman Publishing, New All physics students are encouraged to York, 2008). participate in the American Association of Physics Teachers’ Fnet=ma Contest! 21. Max Maven, Prism: The Color Series (Hermetic Press, Seattle, WA, 2005). The Fnet=ma Contest is the United States Physics Team selection process that leads to participation 22. Nicholas Einhorn, The Ultimate Compendium of Magic Tricks (Her- in the annual International Physics Olympiad. The mes House, London, UK, 2009). U.S. Physics Team Program provides a once-in- a-lifetime opportunity for students to enhance 23. Harry Blackstone, Blackstone’s Modern Card Tricks and Secrets of their physics knowledge as well as their creativity, Magic (Garden City Publishing, Garden City, NY, 1941). leadership, and commitment to a goal. 24. J. Dunniger, Dunniger’s Complete Encyclopedia of Magic (Vanishing For more information, visit: Magic), https://www.vanishingincmagic.com/. https://aapt.org/physicsteam/PT-landing.cfm 25. Abbott’s Magic Company, https://www.abbottmagic.com/. ® 26. FAB Magic, https://fabmagic.com/. 27. Viking Magic, https://www.vikingmagic.com/. 28. Daytona Magic, https://www.daytonamagic.com/. 29. Tannen’s Magic Shop, http://www.tannens.com/ . 30. Penguin Magic, https://www.penguinmagic.com/. 31. Trick Supply, https://tricksupply.com/. 32. Fun, Inc.-Royal Magic, https://www.funinc.com/ . 33. Magic, Inc., https://www.magicinc.net/. 34. Cobra Magic Technologies, https://www.cobramagic.com/. 35. Robert Friedhoffer, Magic and Perception: The Art of Fooling the Senses (Franklin Watts, New York, 1996). 36. Susana Martinez-Conde and Stephen Macknik, Champions of Illu- sion: The Science Behind Mind-Boggling Images and Mystifying Brain Puzzles (Scientific American, New York, 2017). 37. Matthew Tompkins, The Spectacle of Illusion: Deception, Magic and the Paranormal (Distributed Art Publishers, London, UK, 2019). 38. “September 1904: Robert Woods debunks N-rays,” APS News 16 (8), 1 (2007) 39. “Clocks in Rocks,” Hyperphysics, http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/clkroc.html . 40. “\"How Old: The Physics of Dating Artifacts,\" Fred Becchetti – Saturday Morning Physics - 03/11/06, YouTube, https://www.youtube.com/watch?v=_T6zkRhuwac . 41. Science of Magic Association (SoMA), https://scienceofmagicassoc.org/. 42. Christopher Chabris and Daniel Simons, The Invisible Gorilla (Broadway Paperbacks, New York, 2009), http://www.theinvisiblegorilla.com . 43. The Society of American Magicians (SAM), https://www.magicsam.com/ . 44. The International Brotherhood of Magicians (IBM), https://www.magician.org/ . 45. MagicPedia, http://geniimagazine.com/wiki/ . 46. Jim Kleefeid, The Book Test Book (Coda Maxphin Publishing, Cleve- land, OH, 2020). Frederick Becchetti is professor of physics, emeritus, at the University of Michigan, Ann Arbor, MI; [email protected] THE PHYSICS TEACHER ◆ Vol. 60, April 2022 281

Design of a Compact Camera Obscura María Jesús Sánchez, Julia Gil, and José Manuel Vaquero, Universidad de Extremadura, Mérida, Spain The camera obscura is a well-known optical device in In order to build a useful compact camera, both the size of the form of a closed box with a hole in one of its walls the camera and the diameter of the light entrance hole must through which light rays pass, forming an inverted be taken into account. To optimize the size of the compact image of the external objects on the opposite wall, as can be camera obscura and the number of light paths within it, the seen in Fig. 1(a). Despite the simplicity of its basic design, maximum and minimum diameters of the Sun that must be they have been widely used by scientists and artists.1-3 In projected inside it have been taken into account. The diameter particular, dark cameras have been used throughout history of the projected Sun has been calculated very simply from Fig. to measure Earth-Sun distance. To do this, ancient scientists 3, where the Sun, the hole through which light enters (A), and used cameras of enormous dimensions4 to accurately measure the diameter of the projected image DS are represented. From the variations in the apparent diameter of the Sun that depend the angular diameter of the Sun and the focal length f, we on the Earth-Sun distance. This is because the farther the have light travels inside the camera, the larger the projected image (1) will be [Fig. 1(b)]. In this work, the construction of a compact dark camera for educational purposes is presented so that, having reduced dimensions, it allows the images to be very Accordingly, the diameter of the projected image will be large. In this way, it can be used to measure the apparent varia- (2) tions in the diameter of the Sun’s image that varies throughout where s = θ⁄2. the year depending on the Sun-Earth distance. For this, a system of flat mirrors inside the camera has been used. Thus, The values for the maximum and minimum angular radius sunlight travels a great distance within the camera obscura of the Sun (can be consulted in any volume of astronomical through successive reflections (Fig. 2). ephemerides) are sMAX = 16'18'' and sMIN = 15'45''. Calcula- The compact camera obscura is a closed box with a small tions have been made for different values of f and a good solu- hole at the top, through which sunlight enters (point A) so tion is to take f = 4.5 m, since with this value the maximum that the image is projected at the bottom (area B) (Fig. 2). To and minimum projected diameters that can be observed are achieve a greater light path, a system of flat mirrors has been respectively: placed inside it, forming an angle of 45º with respect to the horizontal, and, therefore, the image of the solar disc will be DMAX = 2 · 4500 mm · tan 16' 18''= 38.41 mm projected on the opposite wall of the chamber (area B). The DMIN = 2 · 4500 mm · image of the Sun obtained by the camera will be visualized by tan 15' 45'' = 37.11 mm. means of a piece of graph paper of medium grammage placed where the image of the Sun is formed, through which the size of the image will be observed and can be measured. (a) (b) Fig. 2. Basic design of our camera obscura. Sunlight Fig. 3. Geometrical descrip- enters through A and is projected onto B. Dimensions tion of a camera obscura. Fig. 1. (a) Basic schema of a camera are expressed in millimeters. obscura. (b) A larger camera produces a larger image. 282 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 DOI: 10.1119/5.0029800

k = 2.44. Using the Rayleigh criterion, the estimate is k = 3.66. Taking f = 4.5 m and = 550 nm, we will have Fig. 4. A photograph of the interior of our compact And making the average for both values, we have d = (dAiry + camera obscura. dRayleigh) ⁄ 2 = 2734 µm; accordingly we chose a bit size of 2.5 mm to make our hole for the sunlight. The difference of 1.3 mm is considerable, taking into account that the variations between the maximum and minimum angle By way of conclusion, it can be said that a compact camera diameters are very small. With these calculations, we have built obscura has been designed, built, and tested that allows mea- a 50-cm long dark chamber, where the light makes nine tours, suring the variation of the solar diameter throughout the year. so that the inner system of the dark chamber is made up of 16 It can be used by students from all academic levels. mirrors placed as indicated in Fig. 4. References Another important aspect to consider is the size of the inlet 1. R. E. Alley, “The camera obscura in science and art,” Phys. hole. If the light entry hole is too small, an unfocused image will form due to diffraction phenomena. Conversely, if the Teach. 18, 632 (Dec. 1980). hole is too large, light from a single point on the object reaches 2. T. B. Greenslade, “The opaque projector: The inverse of the many points on the image, also causing an image to be out of focus.5 To calculate the correct diameter of the hole through camera obscura,” Phys. Teach. 49, 241–241 (April 2011). which sunlight enters, we can use the following equation: 3. M. Young, “The pinhole camera: Imaging without lenses or (3) mirrors,” Phys. Teach. 27, 648–655 (Dec. 1989). 4. J. L. Heilbron, The Sun in the Church: Cathedrals as Solar Obser- where d is the aperture diameter of the camera, f is the dis- tance light travels, is the wavelength, and k is a constant. vatories (Harvard University Press, 2001). tTwhoerreefiesrneoncdeevfianleudesc.6r-it7eBriaosnedfoornthAeirvya’sludeisokf,kw. We ceacnaensttiamkeate 5. G. Morales, J. Perkins, H. Pomfrey, and M. J. Ruiz, “Accurate pinhole camera apertures using insect pins,” Phys. Educ. 54 (2), 025002 (2019). 6. L. A. Turner, “Resolving power and the theory of the pinhole camera,” Am. J. Phys. 8, 112–115 (Feb. 1940). 7. R.W. Lambrecht, and C. Woodhouse, Way Beyond Mono- chrome: Advanced Techniques for Traditional Black & White Photography, 2nd ed. (Elsevier, Oxford, 2011). Gender Bias in Physics: Learn more! An International Forum https://genderbias.compadre.org Learning from Experiences of Gender Bias in Physics THE PHYSICS TEACHER ◆ Vol. 60, April 2022 283

An Arduino Investigation of the Temperature Dependence of the Speed of Sound in Air Calin Galeriu, Mark Twain International School, Bucharest, Romania and matching the reading from the sensor to the reading on a ruler. This zero position is located somewhere in the middle, The determination of the speed of sound in air is a where the physical transmitter and receiver are located inside classical experiment, usually performed with a res- the cylindrical structures. It turns out that we don’t have to ac- onance tube apparatus. The measured value can be tually trace this line (the zero mark) on the HC-SR04 sensor. checked against Eq. (1), which describes the temperature Because in the experiment presented here the distance d does dependence of the speed of sound in dry air. A modern im- not change, all we have to do is use the HC-SR04 sensor to plementation of this speed of sound investigation1,2 uses an measure the echo time t for one given temperature T C, calcu- Arduino Uno microcontroller board, an HC-SR04 ultrason- late the distance d for this temperature, ic distance sensor, and a DS18B20 temperature sensor. The distance sensor’s transmitter produces a burst of eight ultra- sonic rectangular pulses that travel through the air, reflect (4) on an object placed in front at a distance d, and then return to the sensor’s receiver after an echo time t. This Arduino and then use this distance d in all the calculations (3) of the investigation, unfortunately, is harder to perform than one speed of sound at other temperatures. Equally well we could might expect after a first reading of Ref. 1 or 2. In this article use an average of such distance measurements. we discuss some sources of experimental errors that can complicate this laboratory activity, and we describe some important steps that must be included in the data collection and analysis procedure, in order to obtain successful results every time. The theoretical speed of sound in dry air is given by the formula3 ( 1) where TK is the temperature in kelvins, and TC is the tem- perature in degrees Celsius. A first order approximation of Eq. (1) gives3 v ≈ ( 331.4+0.6 TC ) m/s . (2) Fig. 1. The HC-SR04 and the DS18B20 sensors. The experimental speed of sound is determined as What can we say about the error in the measurement of the echo time t? Surprisingly, for the same distance d and at (3) the same temperature TC, the readings from the HC-SR04 sensor are not constant. Usually there is an echo time value where d is the distance traveled by the ultrasonic wave from that shows up more often, with some other values that show the transmitter to the wall, equal to the distance traveled up less often. What echo time are we supposed to use for the from the wall back to the receiver. In a typical experiment calculation of the speed of sound? The easiest thing to do is the temperature might change from about 10 oC to about 20 to select just one echo time value,2 presumably the one that oC, and the speed of sound changes from about 337 m/s to shows up more often, and ignore the other echo time values. about 343 m/s. The relative variation of the speed is less than Another option is to calculate the average of these echo time 2%, and therefore the distance d and the echo time t have to values, for a given large number of echo time measurements. be measured with great accuracy. A third option is to calculate the speed of sound for each echo The experimental errors related to the time measurement, and then calculate the average of these HC-SR04 sensor speed values.1 Students are familiar with the process of calcu- lating the average of a set of experimental values, with the goal What can we say about the error in the measurement of of decreasing the random errors affecting the measurements, the distance d? First of all, the HC-SR04 datasheet4 is not and the averaging performed in Ref. 1 seems reasonable. This very clear about how the distance d should be measured. The averaging process is fully justified when we have a Gaussian HC-SR04 sensor rises about 12 mm above the printed circuit distribution of errors, with the experimental values spread to board, as seen in Fig. 1, with a transmitter and a receiver of the left and to the right of the true value. But do the measured cylindrical shape. In the absence of a clear mark on the sen- echo time values always follow a Gaussian distribution? sor, one should start by finding the zero position, like in a calibration procedure, by placing the sensor in front of a wall The pattern followed by the measured echo time values emerged in full light when we reflected the 284 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 DOI: 10.1119/10.0009993

Fig. 2. A sample of 800 echo time values measured with wall. As shown in Fig. 3, the measured echo time values are a rectangular piece of cardboard placed at 30 cm in again separated by gaps. front of the HC-SR04 ultrasonic sensor. Arduino inter- rupts are enabled. We obtained the most clear pattern when the 800 echo times were again measured with the wall at 70 cm in front of Fig. 3. A sample of 800 echo time values measured with the HC-SR04 sensor, but this time with the Arduino inter- a wall at 70 cm in front of the HC-SR04 ultrasonic sen- rupts disabled. As shown in Fig. 4, the spread of the measured sor. Arduino interrupts are enabled. values around the echo time levels is greatly reduced. Fig. 4. A sample of 800 echo time values measured How can we understand this pattern? The key is to realize with a wall at 70 cm in front of the HC-SR04 ultrasonic that the separation between the observed echo time levels sensor. Arduino interrupts are disabled. is about 25 s, which corresponds to a frequency of 40 kHz. This is exactly the ultrasonic frequency at which the HC- burst of ultrasonic pulses on a rectangular piece of soft and SR04 sensor emits the eight rectangular pulses. Although the not very flat cardboard (an envelope), a material on which the electronic circuit of the HC-SR04 sensor is not described in ultrasonic pulses did not reflect very well. The surface area of its datasheet,4 we can assume that the rectangular pulses have 38.0 cm × 26.5 cm = 1007 cm2 was less than the 0.5 m2 recom- a cumulative effect. As a screen capture from a digital oscil- mended in the HC-SR04 datasheet,4 but this was compensated loscope shows,5 the voltage on the echo pin of the HC-SR04 for by the fact that the piece of cardboard was at a relatively sensor changes from 0 V to 5 V right after the burst of eight short distance of only 30 cm. As shown in Fig. 2, the echo ultrasonic pulses is sent out. This voltage stays high while the time values belong to a set of quasi-discrete levels, resembling ultrasonic pulses travel to the target, reflect off the target, and quantum energy levels. then return to the ultrasonic sensor. After a small number of these pulses are received back, the voltage on the echo pin of A similar pattern was obtained when we increased the dis- the HC-SR04 sensor changes from 5 V to 0 V, and this is how tance to 70 cm and we reflected the burst of ultrasonic pulses the echo time is determined. When the reflected pulses are on a wall in front of the HC-SR04 sensor. The greater distance received unattenuated, the ultrasonic sensor measures the was compensated for by the large, flat, and hard surface of the correct echo time. What happens when some of the pulses are received attenuated, or missed altogether? In this case more pulses are needed, in order for the ultrasonic sensor to reach that threshold at which the echo pin voltage changes. If only one extra pulse is needed, then the reported echo time will be 25 s longer. At 20 oC, this corresponds to an additional 4.3 mm in the calculated distance. Unlike the special situations discussed above, for a large, flat, and hard surface (a wall) not too far away from the HC- SR04 sensor (at about 30 cm), the measured echo time values usually reveal only two echo time levels, or only one level. The averaging of all the echo time values is justified only in this latter case. Other factors can affect the performance of the HC-SR04 sensor. If the sensor is placed directly on the floor, or on a desk, some ultrasonic pulses could scatter back because of the roughness of the horizontal surface. For this reason we rec- ommend the mounting of the HC-SR04 sensor at some height above the floor or desk, as shown in Fig. 5. Another issue is the size of the reflecting object. The HC- SR04 datasheet4 mentions that, in order to achieve the best performance, the reflecting object must present a flat smooth surface of no less than 0.5 m2. For this reason we recommend to use a wall. Smaller reflecting objects will result in longer echo time values, with corresponding shorter calculated speed of sound values.2 We have also noticed that the HC-SR04 sensor is very sen- sitive to air currents in the room. We cannot use fans in order to vary the temperature,1 the air has to be completely still. For this reason we recommend to collect the experimental data in a room with closed windows and doors. THE PHYSICS TEACHER ◆ Vol. 60, April 2022 285

Fig. 6. The wiring of the electronic circuit. tines are serviced only during some of the echo time measure- ments. This being said, one should not rush to disable the Arduino interrupts everywhere. In order for the micros(), millis(), and delay() Arduino functions to work properly, the timer inter- rupts must be enabled. Fig. 5. The experimental setup. The experimental investigation of the speed of sound vs. temperature The experimental errors related to the Arduino interrupts For the actual experiment presented here we have turned off the heat during winter, and we have allowed the room tem- In order to measure the echo time, an electronic device perature to slowly decrease from 25.31 oC to 10.31 oC during (Arduino Uno, digital oscilloscope, etc.) has to find out for three cold, cloudy days. We recommend to do this investi- how long the voltage on the echo pin of the HC-SR04 ultra- gation in winter not only because of the large temperature sonic sensor stays high. The Arduino code relies on the variation that can be obtained in this way, but also because the pulseIn() function for this measurement. The pulseIn() speed of sound depends on humidity, and Eq. (1) applies only function (implemented in the wiring_pulse.c file) calls the to dry air. When the outside temperature is below the freezing countPulseASM() function (implemented in the point, the air inside is mostly dry. wiring_pulse.S file), which counts a number of CPU clock cycles. The countPulseASM() function, written in assembly The experimental setup, shown in Figs. 1 and 5, consists language, runs a while loop during which a counter variable of an Arduino Uno microcontroller board, an HC-SR04 ul- named width is continuously incremented for as long as the trasonic distance sensor, and a DS18B20 temperature sensor voltage on the specified Arduino pin stays high. In the Ardui- together with its 4.7 k pull-up resistor. The breadboard no 1.8.13 Integrated Development Environment the with the electronic components is mounted up high on some wiring_pulse.c and wiring_pulse.S source files are found in Lego bricks, at a distance of 30 cm in front of a wall. Keep in the hardware\\arduino\\avr\\cores\\arduino folder. mind that for an accurate temperature measurement the black DS18B20 sensor must be kept out of direct sunlight. The fact that the Arduino interrupts are detrimental to the accuracy of the pulseIn() function is mentioned in the As shown in Fig. 6, the GND pins of the sensors are con- wiring_pulse.c file. During an Arduino interrupt the width nected to an Arduino GND pin, and the VCC pins of the sen- variable stops from being incremented, and as a result the sors are connected to the Arduino 5 V pin. The trig pin of the number of CPU clock cycles gets underreported. To avoid this ultrasonic sensor is connected to digital pin 2, and the echo behavior one must disable the Arduino interrupts, and this is pin is connected to digital pin 3. The data pin of the tempera- done by placing the critical code in between noInterrupts() ture sensor is connected to digital pin 12. A detailed descrip- and interrupts() instructions. tion of the HC-SR04 sensor, and the Arduino code that mea- sures the echo time, are given in Ref. 6. A detailed description By comparing the echo times from Figs. 3 and 4, we dis- of the DS18B20 sensor and the Arduino code that measures cover that, when the Arduino interrupts are enabled, two the temperature are given in Ref. 7. The DS18B20 temperature undesired things happen: the reported echo times are slightly sensor has an accuracy of ±0.5 oC.8 shorter (1.3%) and slightly more spread around the echo time levels. This is because some Interrupt Service Routines are The Arduino program9 used in our experimental inves- serviced during all of the echo time measurements (timer tigation has two parts. In the first part, the setup() function, interrupts fit this behavior) and some Interrupt Service Rou- the resolution of the temperature sensor is set to 12 bits. In the second part, the loop() function, the experimental data is collected and partially analyzed. About every two minutes, one temperature measurement and 800 echo time measure- ments are taken. The average of all 800 echo time values and 286 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

Fig. 7. The average echo time as a function of temperature. Fig. 8. The experimentally determined speed of sound as a We compare the average of all 800 echo time values (solid function of temperature (black circles) matches very well red circles) with that of the short echo time values (open the theoretical formula (red line). black circles). mined values and the theoretical prediction is excellent. the average of the short echo time values are calculated. The The calibration procedure that gave us the distance [Eq. short echo times are selected by the condition: echo time— shortest echo time < 12 s. For each measured temperature TC (4)] also helps us understand why the experimental error with the program also calculates the distance dshort traveled by the which the DS18B20 sensor measures the temperature does ultrasonic wave, as derived from Eqs. (3) and (1): not affect the excellent match seen in Fig. 8. The experimental speeds are given by . (5) (6) where tshort is the average of the short echo time values. These calculated distances dshort are needed during the subsequent while the theoretical speeds are given by data analysis. .(7) In Fig. 7 we show the average echo time as a function of temperature. The solid red circles represent the average of all The ratio t /t is very close to one, accounting for the less than 800 echo time values, while the open black circles represent 2% relative variation of the speed of sound. A small offset the average of the short echo time values. We realize that calculating the average of all 800 echo time values can be mis- T in the measurement of the T C and TC temperatures will leading. In the 10 to 21 oC temperature range, the red circles vertically shift both Eqs. (6) and (7) on the graph with approx- go up and down according to the random ways in which the imately the same amount. This argument holds true not only echo time levels are populated. On the other hand, the black when the distance d is the outcome of one measurement, but circles display a consistent pattern. There are two categories of also when it is the average of several measurements made at black circles that we see in the graph. We have the data points different temperatures. for which no ultrasonic pulses are lost; these are the black circles below, with a reported dshort distance of about 30.1 cm. Looking at Eq. (6) we also notice that, when both t and t And we have the data points for which one ultrasonic pulse are reduced by 1.3%, the ratio t /t does not change. This ex- is lost; these are the black circles above, with a reported dshort plains why we can perform this speed of sound investigation distance of about 30.5 cm. The red and the black circles over- with the Arduino interrupts enabled, and still obtain good lap in the 21 to 25 oC temperature range where all, or almost results. all, of the 800 measured echo times belong to the same echo time level. Unfortunately, these are exactly the data points that Conclusions have to be discarded, due to the lost ultrasonic pulses. The Arduino investigation of the speed of sound presented We select only the data points for which no ultrasonic puls- here serves several purposes. While the main goal is to verify es are lost in at least one of the 800 echo time measurements, experimentally the theoretical formula in Eq. (1), we also use the data points with a reported dshort distance of about this integrated STEM activity to teach students about elec- 30.1 cm. For this restricted set of data points, the average tronics and computer programming. Especially important of the reported dshort distances is 30.144 cm. Next, using is the idea that one should not treat an electronic sensor as a the average of the short echo time values as the time t, and black box, but instead try to understand its inner workings. In the average distance of 30.144 cm as the distance d, we cal- particular, the echo time values reported by the HC-SR04 sen- culate the speed of sound according to Eq. (3). In Fig. 8 we sor do not display a Gaussian distribution, and as a result we show the variation of this measured speed of sound with simply cannot average all of these values in order to decrease temperature, together with the theoretical curve given by the random errors. What is needed is the average of the short Eq. (1). The agreement between the experimentally deter- echo time values, those on the lowest echo time level. THE PHYSICS TEACHER ◆ Vol. 60, April 2022 287

The author is very much indebted to the three anonymous reviewers for their very valuable comments. The manuscript has greatly benefitted from their feedback. One of the review- ers mentioned the need to disable the Arduino interrupts, a very important implementation detail. References 1. Marcelo Dumas Hahn, Frederico Alan de Oliveira Cruz, and Paulo Simeão Carvalho, “Determining the speed of sound as a function of temperature Using Arduino,” Phys. Teach. 57, 114–115 (Feb. 2019). 2. M. Oprea and Cristina Miron, “Didactic experiments for de- termining the speed of sound in the air,” Romanian Reports in Physics 68 (4), 1621–1640 (2016). 3. Sound Speed in Gases, http://hyperphysics.phy-astr.gsu.edu/ hbase/Sound/souspe3.html. 4. HC-SR04 Datasheet, https://cdn.sparkfun.com/datasheets/Sen- sors/Proximity/HCSR04.pdf. 5. PC Services, Ultrasonic Distance Sensing Using HC-SR04, http://www.pcserviceselectronics.co.uk/arduino/Ultrasonic/. 6. Calin Galeriu, Scott Edwards, and Geoffrey Esper, “An Ardui- no investigation of simple harmonic motion,” Phys. Teach. 52, 157–159 (March 2014). 7. Calin Galeriu, “An Arduino investigation of Newton’s law of cooling,” Phys. Teach. 56, 618–620 (Dec. 2018). 8. DS18B20 Datasheet, https://datasheets.maximintegrated.com/ en/ds/DS18B20.pdf. 9. The Arduino program echo_time_temp.ino is provided as a supplemental online material and can be found at TPT Online, http://dx.doi.org/10.1119/10.0009993, under the Supplemental tab. Calin Galeriu teaches physics and chemistry at Mark Twain International School. He has a BS degree in physics from the University of Bucharest, an MA degree from Clark University, and a PhD degree from Worcester Polytechnic Institute. Mark Twain International School, Strada Erou Iancu Nicolae 89-93, Voluntari, IF, Romania; [email protected] 288 THE PHYSICS TEACHER ◆ Vol. 60, April 2022

STEM Education of Kinematics and Dynamics Using Arduino A. Çoban, Yeditepe University, Faculty of Arts and Sciences, Department of Physics, Ataşehir, Istanbul, Turkey M. Erol, Dokuz Eylül University, Education Faculty of Buca, Department of Mathematics and Science Education, Buca, İzmir, Turkey Ever increasing technological progress opens novel op- portunities concerning educational activities, and the Knowing the masses of m1, m2, and mr measured with pre- ability to use the technology effectively is one of the cision scales and also G1, G2, and Gr, the corresponding ac- 21st century’s most demanding skills.1,2 The Partnership Fo- celerations can theoretically be estimated. The distance-time relation is then given by the kinematics equation of rum for 21st-Century Skills (P21) states that no organization (4) can achieve satisfying results without using technology and therefore the use of technology particularly in schools should be at the highest level.3 Some novel teaching methods have re- where x0 denotes the initial position, v0 denotes the initial ve- cently emerged based on technological developments, such as locity, and a denotes the acceleration of the body. STEM education, which is based on the integration of science, technology, engineering, and mathematics.4 In this study, the Experimental setup motion of a body connected to a pulley system is analyzed by The experimental components required are a fixed pulley, using an Arduino microprocessor and also related mathemat- a moving pulley, an ultrasonic distance sensor, an Arduino ical equations. Arduino is a programming platform on which Uno, a computer, two objects with certain masses, and con- various electronic elements can be connected.5 There are var- nection cables, as shown in Fig. 1. ious Arduino compatible sensors (sound, light, distance, tem- The experimental setup employed throughout the work is perature, etc.) that can be coded for their intended use. Due to composed of two standing rods, one fixed pulley, one movable its cost effectiveness, this technological instrument can be used pulley, two masses, and a complete Arduino system including as a measurement tool in many physics experiments.6-12 In the computer. A photo of the whole setup is shown in Fig. 2. this work, the time-varying data of the position of the moving The Arduino sys- object is determined using the HC-SR04 ultrasonic distance tem is set to measure sensor (see Fig. 1). The HC-SR04 distance sensor can deter- distance d between mine the distance of the obstacle from the sensor when the the moving object transmitted signal is reflected from an obstacle and returned m2 and the Arduino to the sensor,13 and the Arduino IDE computer program is distance sensor. The used to load the corresponding codes into the Arduino and to distance sensor is retrieve the incoming data.14 positioned vertically below the mass m2, Theory connected to the The theoretical resolution is managed under two basic as- moving pulley, and sumptions: firstly, the rotational inertia of the pulleys will be the base area of the ignored since their mass is small, and secondly, the friction of m2 body is enlarged the system is ignored since the ropes and pulleys are carefully by using a piece Fig. 1. Photo of the components. lubricated before the operation. of paper to ensure Newton’s second law of dynamics, Fnet = ma , can be rewrit- more accurate mea- ten for the fırst body, m1, as follows (see Fig. 2): surements. There are a transmitter and a G1 – T = m1 a1. (1) receiver side by side Similarly, the equation of motion for m2 can be expressed as on the distance sen- follows: sor. The signal sent from the transmitter 2T – (G2 + Gr) = (m2 + mr)a2, (2) is reflected from the where T denotes the tensile force on the string, G1 is the weight obstacle and comes of m1, G2 is the weight of m2, and Gr denotes the weight of the to the receiver. At moving pulley. Additionally, it is well known that x1 = 2x2 , this point, the path hence, relating the accelerations of the two bodies, a1 = 2a2 can followed by the sig- be written. In the following resolution, a1 = 2a and a2 = a are nal is not straight, selected, and then it is straightforward to obtain the accelera- but conical with tion as respect to the sensor. Fig. 2. Photo of the experimental setup The piece of paper including the two pulleys, two masses, (3) and complete Arduino system. DOI: 10.1119/10.0009994 THE PHYSICS TEACHER ◆ Vol. 60, April 2022 289