A purely geometrical method of determining the location of a smartphone accelerometer
AA purely geometrical method of determining the location of asmartphone accelerometer
Christopher Isaac Larnder ∗ Department of Physics, John Abbott College,St-Anne-de-Bellevue QC, Canada H9X 3X8
Abstract
In a paper ( posthumously ) co-authored by Isaac Newton himself , the primacy of geometricnotions in pedagogical expositions of centripetal acceleration has been clearly asserted. In thepresent paper we demonstrate how this pedagogical prerogative can inform the design of an exper-iment involving an accelerometer-equipped smartphone rotating uniformly in a horizontal plane.Specifically, the location of the sensor itself within the body of the smartphone will be determinedusing a technique that is purely geometrical in nature, relying on nothing more than the notion thatcentripetal accelerations are centrally-pointing. The complete absence of algebraic manipulationsobliges students to focus exclusively on the development of their geometrical reasoning abilities. Inparticular, it provides a healthy challenge for those algebraically-accomplished students for whomequations, calculations and data tables represent a means of avoiding a direct confrontation withthe imposing spectre of material that is otherwise purely conceptual in nature. a r X i v : . [ phy s i c s . e d - ph ] M a r . INTRODUCTION A recent paper determined accelerometer positions across a wide range of host devicesincluding not only smartphones but also tablets and dedicated accelerometry devices. Theauthors applied a simple but rigorous mathematical technique relying on the vector form ofthe equation for the centripetal acceleration (cid:126)a , viz. (cid:126)a = − ω (cid:126)R, (1)in which (cid:126)R is the position vector of the sensor with respect to a coordinate system whoseorigin coincides with the pivot point. Its determination relies on the values of (cid:126)a and ofthe angular velocity ω . Earlier attempts , reviewed in the same paper, are moregeometrical in nature, but still rely on the numerical value of ω and a computation basedon the scalar form of the same equation, viz. a = ω R. (2)Rewriting Eq. 1 as a ˆ a = − ω R ˆ R (3)demonstrates its separability into two equations, one involving only magnitudes (Eq. 2) andthe other involving purely geometrical quantities expressed as unit vectors, viz.ˆ a = − ˆ R (4)Since the position vector (cid:126)R points from the pivot point to the sensor, the accelerationvector points from the sensor back towards the pivot point: It is a centrally-pointing vector.Whereas previous work has relied on Eq. 1 or Eq. 2, we rely only on the purely geometricrelationship expressed in Eq. 4.For most students this introductory discussion should be avoided altogether, as it per-tains mostly to notation. Their attention should be exclusively occupied with the centrally-pointing property of the acceleration vector and its implications in the context of the exper-iment. 2 I. METHODS AND RESULTS
The apparatus, a photo of which is presented in Fig. 1, consists of a 3D-printed rectan-gular frame mounted on a flat disk, which in turn is mounted onto a turntable using anadditional 3D-printed fitting . The dimensions of the frame are such that a standard 8.5 x11 - inch piece of paper fits snugly within it. Students are expected to arrive in class with anaccelerometer app already downloaded onto their smartphones . They initiate a recordingwith the app, place it in one of the four corners, and set the frame spinning at a constantangular velocity . The data is transferred to a PC and average values for the accelerationcomponents are obtained.The two acceleration components are drawn on the frame paper, from the center outwards,and used to form a vector indicating the direction of acceleration, as depicted in Fig. 2. Fora turntable at 78 rpm, a scale factor of 1.0 m/s = 1 cm will ensure all drawn vectors fiton the sheet of paper. This is a good point to remind students that the acceleration vector,when drawn from the true (but as-yet unknown) sensor position outwards, must i) pointtowards the center and ii) have the same angle as the vector they have just drawn. Withthese points in mind , have them try to identify positions at which the sensor could havebeen located. These constitute candidates for the true position of the accelerometer sensorwithin the body of the phone. The astute student will realize that there are multiple suchpoints, that the collection of all such points forms a line, and that an extension of this linewill intersect the origin: it is a radial line and is the unique radial line associated with thegiven acceleration vector. It follows immediately that this radial line can be established ina rigorous manner by simply extending the acceleration vector backward through the areain which the phone had been placed, as illustrated in Fig. 3.The smartphone is then moved consecutively to each of the other quadrants and theexperiment is repeated. The students, having beforehand traced and cut out a rectangularoutline of their phone, place this outline at each of the corners and trace each radial lineover it. Figs.4, 5 and 6 illustrate one step in this cumulative procedure. The accumulationof 4 radial lines, as in Fig. 7, provides a means of visually estimating both a position andan uncertainty in the position value.The only way of directly verifying the accuracy of the result would be to invite students todismantle their smartphones and locate the sensor on the circuit board. Given the general3eluctance to take the spirit of scientific inquiry to such a level, we suggest a lower-riskalternative that involves browsing for the smartphone model on ”teardown” sites thatdocument the dismantling process and identify the components. Fig. 8 depicts a circuitboard obtained from such a site, scaled to fit appropriately into the smartphone outline. III. CONCLUSIONS
The procedure, as advertised, relies entirely on geometrical reasoning. We do not rely onthe value of ω nor on the ratio of the acceleration components. There is no calculation ofangles using the inverse tangent function. Even the scale ratio, involving a 1:1 relationship,is chosen so as to eliminate the need for calculations. Thus freed of a category of activitythat often occupies a considerable portion of the time spent in lab, teachers can afford tostructure the lab activity in such a way as to lead students towards discovering some of theprocedural steps themselves .More importantly, the absence of numeric activities means that students are left withnothing to contemplate but the radially-inward lines that they themselves have constructedfrom the acceleration components of their own phone. They have nothing to gain from aconsideration of the algebraic relationship of (scalar) Eq. 2 and everything to gain fromthe geometric relationship of ( unit-vector) Eq. 4. We believe such an approach producesfavorable conditions for the emergence of the geometric insight that is so critical to anappreciation of centripetal acceleration. IV. ACKNOWLEDGEMENTS
The author thanks colleagues Brian Larade, Etienne Portelance, Margaret Livingstone,Bruce Tracy and Hubert Camirand for helpful suggestions and in-class evaluations of thislab activity; and the Engineering Technologies department at John Abbott College for ex-tensive 3D-print technical support. This work was funded in part by Qu´ebec’s