Dynamics of a yoyo using a smartphone gyroscope sensor
Isabel Salinas, Martin Monteiro, Arturo C. Marti, Juan A. Monsoriu
AAnalyzing the dynamics of a yoyo using asmartphone gyroscope sensor
Isabel Salinas , Martin Monteiro , Arturo C. Marti , Juan A.Monsoriu Universitat Politecnica de Valencia, Valencia, Spain Univesidad ORT Uruguay Facultad de Ciencias, Universidad de la Rep´ublica, UruguayE-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected] a r X i v : . [ phy s i c s . e d - ph ] S e p nalyzing the dynamics of a yoyo using a smartphone gyroscope sensor Figure 1.
A yoyo (left panel) and a free body diagram (right).
The yoyo is a traditional toy whose origins can be traced back at least to ancientGreece [9] (Fig. 1). It is made up of two disks joined by an axle and a string woundaround the axle. The players holds one end of the string with their hand by insertingone finger into a small knot and executes several plays. In the simplest version, theplayer lets the toy to fall by the effect of the gravity, spinning and unwinding the string.When the yoyo reaches the lowest position, it bounces back and starts climbing as thestring is wound again in the axle. To counteract the friction effects the player appliessmall pushes, up or down, when the yoyo is climbing or falling respectively.
The physics of the yoyo.
We consider a yoyo, with mass m , hanging from astring and moving only in the vertical direction. The string is considered as weightlessand inextensible. The applied forces are the tension of the string, (cid:126)T , and the weight, nalyzing the dynamics of a yoyo using a smartphone gyroscope sensor m(cid:126)g as indicated in Fig. 1 (right panel) both acting in the vertical direction. Assumingthe y -axis oriented downwards, the second Newton law can be written as ma y = mg − T (1)where a y is the vertical acceleration of the center of mass.We assume that the tension is applied upwards at a perpendicular distance whichcoincides with the radius of the inner cylinder, r , as shown in Fig. 1. Takingcounterclockwise rotations with positive sign, the rotational version of Newton’s secondlaw in the center of mass can be expressed as Iα = ± T r (2)where I is the moment of inertia and α is the angular acceleration, both magnitudesrelatives to an axis through the center of mass and perpendicular to the plane of theyoyo. The positive (negative) sign in the previous equation corresponds to the yoyohanging on the left (right) side of the string. The non-slipping condition of the yoyo,relative to the string, relates the vertical and the angular acceleration a y = αr. (3)The angular acceleration can be obtained solving Eqs. 1-3, α = ± g rr + I/m . (4)Equation 4 links the (constant) angular acceleration to the physical characteristics ofthe yoyo and the gravitational acceleration when the yoyo is either climbing or fallingon the left of the string as shown in Fig. 1. The ± reflects the fact the accelerationdepends on the orientation of the yoyo relative to the string.The expected temporal evolution of the angular velocity deduced from the previousconsiderations is depicted in Fig. 2. In the beginning, the yoyo falls along the right sideof the string (time period 1). At the lowest point, when the string is fully stretched, theyoyo behaves like a physical pendulum during a very brief period of time. In this lapse,the yoyo rotates 180 degrees around the end of the string and the vertical velocity ofthe center of mass and the angular acceleration change their sign. As a consequence ofthe conservation of angular momentum, the angular velocity when leaving is conservedbefore and after the yoyo is in the lowest point. The origin of the impulsive force appliedby the string on the yoyo may be related to the elasticity of the string or to suddendisplacements of the support (or player’s hand). After that, the yoyo starts climbing,but now along the opposite side of the string, decreasing its angular velocity (time period2). When it arrives at the support the angular velocity is null, and the yoyo falls againwith the same angular acceleration, along the left side but now rotating in the oppositesense (time period 3). Finally, during the last period, the yoyo goes upwards again buton the right side (time period 4). The experiment.
Our setup comprised a yoyo and a smartphone (Samsung S8+)with a built-in gyroscope. Since smartphones are usually more voluminous than typicalyoyos we built a home-made yoyo which consists of two methacrylate plates (diameter nalyzing the dynamics of a yoyo using a smartphone gyroscope sensor Figure 2.
Scheme of the angular velocity as a function of time (top panel) and itsorientation relative to the string (bottom panel) in each of the time periods (1 , , , Figure 3.
Experimental setup: home-made yoyo, counterweight and smartphone. . . . . app [10] wasused to visualize and register experimental data and export to a CSV (comma separatedvalues) file. A screenshot of the app displaying the temporal evolution of the angularvelocity during the whole movement can be appreciated (Fig. 4). nalyzing the dynamics of a yoyo using a smartphone gyroscope sensor Figure 4.
Experimental results for the whole temporal evolution of the angularvelocity (upper panel) and a zoom with two windows (bottom panel). In each windowthe yoyo is climbing and falling along one or the other side of the string.
Results.
The yoyo is held with the string wound and then released while thesmartphone registers the angular velocity along several bounces. In Fig. 4 we show thetemporal evolution of the angular velocity during a whole move. At the initial time, theangular velocity is null. When the yoyo is released, it starts going downwards as thestring is unwound and the absolute value of the angular velocity increases. Remarkably,it can be seen that the graph displays several plateaux due to the saturation of thesensor when the angular velocity reaches 20 rad/s (this specific value depends on thesmartphone employed). At successive downwards and upwards travels, the energy isslowly dissipated and the plateau disappears.The overall outlook of the temporal behavior displayed in Fig. 4 can be understoodin the light of the previous discussion. The saw tooth comprises sections withapproximately constant slopes whose (opposite) values correspond to Eq.4. In addition,the changes of slope occur when the yoyo reaches the lowest point and the zeroscorrespond to the passing near the supporting point (or the hand). In the lower panel,two temporal windows are zoomed in, the first as the yoyo travels up, the second as theyoyo travels down, but, as indicating in the previous paragraph, rotating in opposite nalyzing the dynamics of a yoyo using a smartphone gyroscope sensor Figure 5.
The successive screenshots of the smartphone (upper panel) were used toobtain the angular variable ∆ θ ( t ) (bottom panel) and the angular acceleration. directions.Since the rotation is nearly uniformly accelerated, to obtain the angularacceleration, we perform a linear fit, ω = ω + αt , resulting 65 . rad/s in the firstinterval (on the left side of the string) and − . rad/s in the second (in the rightside) showing, absolute value, a minimum discrepancy of 0.06% between both instances.In addition, it can be noticed, in both temporal windows, a small concavity of theangular velocities. The causes of this phenomenon, to be studied elsewhere, could berelated to several effects such as rolling resistance, the non-uniform friction of the threadwith the walls of the yoyo or also the alternation of periods of slipping and rolling relatedto the static and kinetic friction effects.Video analysis provides a useful tool to compare with the experimental resultsobtained from the rotation sensor. The analysis was made from frames extracted fromthe video at 30 fps. On top of Fig. 5 we show 7 consecutive frames in which the yoyo nalyzing the dynamics of a yoyo using a smartphone gyroscope sensor θ = ω ∆ t + 12 α (∆ t ) (5)which results in an angular acceleration α = 64 . rad/s . This value appears very closeto both previously obtained using the smartphone sensor (mean deviation 0 . Conclusion.
In this work we propose a simple experiment using a traditional toy,the yoyo, and a modern device, the smartphone which involves several basics conceptsin mechanics. Thanks to the gyroscope sensor the dynamics of the toy can be accuratelyanalyzed and compared with results obtained from video analysis. The gyroscope sensorprovides the angular velocity and, by means of a linear fit, the angular accelerations canbe also obtained. We analyzed a whole movement of the yoyo and then focus ourattention on two temporal windows. The accelerations in each window are very similarand also coherent with the results obtained analyzing frame by frame the video. It mustbe emphasized that, depending on the altitude of the thrown, the smartphone sensorcould not be able to register all the range of angular velocity values.One important feature of smartphone sensors to bear in mind is the coordinateframe relative to they provide measures. While the accelerometer measures accelerationin a moving frame (relative to the smartphone) the gyroscope measures the angularvelocity with respect to an inertial reference frame [4]. In the present experiment, asthe smartphone –fixed to the yoyo– is rotating, the gyroscope is more appropriate thanthe accelerometer to get a useful measure.Several aspects of this experiment are worth discussing in a classroom activity. Aninteresting starting point is to present the problem to the students, let them discussand predict the evolution of the angular variables and, then, to perform the experimentand compare the prediction with the results. The design and assembly of the yoyois the first stage which involves the mass-balance, the selection of the different piecesand the calculation of their contribution to the moment of inertia of the system. Theexperiment itself requires to set up the sampling frequency of the sensors and to takeinto account their maximum range. And finally, the discussion of the results involves allthe aspects. Other possible classrooms proposals go far beyond the experiment reportedhere and could include the analysis of the mechanical energy, the conservation of angularmomentum and 2-dimensional tricks as the described in Ref.[9]. We estimate that, asthis proposal matches classical physics principles and modern technology, it could be avaluable pedagogical tool to promote students engagement and critical thinking.We thank the Institute of Educational Sciences of the Universitat Polit`ecnica deVal`encia (Spain) for the support of the Teaching Innovation Groups MoMa and e-MACAFI. We also thank the support from the program CSIC
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