Experimental analysis of a physical pendulum with variable suspension point
Martin Monteiro, Cecilia Stari, Cecilia Cabeza, Arturo C. Marti
EExperimental analysis of a physical pendulum withvariable suspension point
Mart´ın Monteiro , Cecilia Stari , Cecilia Cabeza , Arturo C.Mart´ı Universidad ORT, Montevideo, Uruguay Instituto de F´ısica, Facultad de Ingenier´ıa‘”, Universidad de la Rep´ublica, Uruguay Instituto de F´ısica, Facultad de Ciencias, Universidad de la Rep´ublica, UruguayE-mail: [email protected]
Abstract.
A physical pendulum with variable point of suspension (and, as anoutcome, variable inertia moment) is experimentally analysed. In particular, the periodof the small oscillations as a function of position of the suspension point is measuredusing three different methods: a smartphone used both as an independent tool or asa data-logger and commercial photo-gate. The experimental results are successfullycompared with theoretical calculations based on the addition of inertia moments andthe Steiner theorem. a r X i v : . [ phy s i c s . e d - ph ] S e p xperimental analysis of a physical pendulum with variable suspension point Physical pendulum . Concepts about inertia and torque are of paramountimportance in almost all introductory Physics courses in high school and introductoryuniversity level. In particular, one of the typical examples is the physical pendulum(also known as compound pendulum) which consists of a rigid body that can freelyrotate around a horizontal axis through a fixed center of suspension. The period of thesmall oscillations, T , depends on the mass M , the distance from the suspension pointto the center of mass R and the inertia moment I as T = 2 π (cid:115) IM gR (1)where g is the gravitational acceleration.In the experiment proposed here, the period of the small oscillations of a physicalpendulum whose point of suspension, and then, its inertia moment can be controlledis experimentally analyzed using different modern technologies [1, 2, 3, 4]. As itinvolves key concepts in classical mechanics and can be readily implemented virtually inany Physics laboratory, the present experiment could encourage students’ interest andmotivation to experiment by themselves. Experimental implementation.
The experimental setup, depicted in Fig. 1,consists of a rigid metallic bar with equispaced holes and a smartphone. As there areseveral possible suspension points, the radius of gyration and the inertia moment dependon the selected point. The dimensions of the bar, with holes made at points separated auniform distance of 1 . L = 1 .
199 m and w = 0 .
024 m, and the mass M = 0 . m = 0 . L s = 0 .
135 m and w s = 0 .
068 cm. The distance from the suspension point, O , to the center of mass ofthe bar, C , is indicated with z , while the distance from C to the center of mass of thesmartphone is z s .Using Eq. 1 and the geometrical characteristics of the systems the period of thephysical pendulum in the regime of small oscillations can be written as T = 2 π (cid:115) I (( M + m ) z + mz s ) g (2)where the inertia moment, I is obtained as the sum of the contributions from the barand the smartphone I = I bar + I s . Applying the well-known Steiner theorem the inertiamoments can be expressed as I bar = 112 M (cid:16) L + w (cid:17) + M z (3)and I s = 112 m (cid:16) L s + w s (cid:17) + m ( z + z s ) (4) Methods and analysis.
The period of physical pendulum was measured by threedifferent experimental methods: • Phyphox: direct measurement on the smartphone screen using the
Pendulum toolof the
Phyphox app . From a temporal series of the angular velocity (about 30s) xperimental analysis of a physical pendulum with variable suspension point Figure 1.
Experimental setup consisting of a bar with equispaced suspension pointsand a smartphone. The labels are described in the text. provided by the gyroscope sensor, the app automatically calculates the period bymeans of an autocorrelation function (similar to a FFT but taking the correlationbetween maximums of several cycles). The advantage is that it is a direct measurefor which only the smartphone with the free app is needed. • Vernier: direct measurements using a LabQuest interface and an optical barrier.The advantage is that direct values are obtained even if there is cell phone mountedon the bar. The disadvantage is that it is necessary to dispose of a commercialinterface or some equivalent home-made system. • Androsensor: smartphone as a data-logger and data processing on a PC. In thisprocedure, a temporal series of the gyroscope sensor (z axis) was taken for 10 s, andsaved in the smartphone using the
Androsensor app . All generated .csv files (onefor each suspension point) were exported to the PC and analyzed using a simplesoftware package (in this case,
Scilab ) and, by means of a sinusoidal fit the periodof oscillation is obtained. This method can be useful for experimental courses inwhich students must develop data management techniques and implement signalprocessing algorithms in different languages such as Python or Matlab.Experimental results using the three methods (symbols) and the theoreticalprediction (continuous model), plotted in Fig. 2, display a great agreement. The root-mean-square deviations with respect to the theoretical model, also indicated in thefigure caption, reveal that the three methods, each one with pros and cons are valid xperimental analysis of a physical pendulum with variable suspension point Figure 2.
Experimental results (symbols) and model (continuous line) for the periodof the small oscillations as a function of the distance of the suspension point to thecenter of the bar. The root-mean-square deviation with respect to the theoreticalmodel are: 5 . . . and possible implementations of the present experiment. It is also worth noting thatthe period displays a minimum for a particular distance to the center of the bar. Thispoint could be also the object of an interesting classroom discussion. To sum up, wepresented a simple experiment that involves relevant concepts of classical mechanicsusing moderns technologies that can be readily implemented in a Physics laboratory. Acknowledgements.
We acknowledge financial support from CSIC (UdelaR,Uruguay) and Programa de Desarrollo de las Ciencias Basicas (Uruguay).
References [1] M. Monteiro, C. Cabeza, and A. C. Mart´ı. Exploring phase space using smartphone accelerationand rotation sensors simultaneously.
European Journal of Physics , 35(4):045013, 2014.[2] M. Monteiro, C. Cabeza, and A. C. Mart´ı. Rotational energy in a physical pendulum.
The PhysicsTeacher , 52:561, 2014.[3] M. Patrinopoulos and C. Kefalis. Angular velocity direct measurement and moment of inertiacalculation of a rigid body using a smartphone.
The Physics Teacher , 53(9):564–565, 2015.[4] I. Salinas, M. Gimenez, J. Monsoriu, and J. Sans. Demonstration of the parallel axis theoremthrough a smartphone.