Using these holes, the rod can be hanged from a pivot, changing the distance between the CM and this point. The experiment proposed to the students consists on measuring the period of oscillations of an aluminum stick of length L = 1.4830 ± 0.0005 m and mass M = 0.9216 ± 0.0001 kg, through which circular holes, separated by a distance of approximately 5 cm, have been drilled. Finally, we present our conclusions in Sec. In section 5, once we have solved the problem with the acceleration of gravity, we take advantage of the experiment to analyze the variation of the radius of gyration of the physical pendulum. 4 we present different hypotheses raised and tested to explain the discrepancy in the measurement of g. 3 we present the results obtained when the same analysis performed by the students is applied to a data set collected under the same experimental conditions present in the classroom. 2 we present the experimental setup used, discarding directly some of the explanations argued by the students. This article is organized as follows: in Sec. We present here our own investigation of the oscillatory motion of the physical pendulum, trying to figure out why such discrepancies appear in our experimental setup. In order to explain this discrepancy between their result and the known value of g, students give all kind of explanations: non-validity of the small angle approximation in their experiment, the use of a non-uniform rod (due to the holes drilled on it), the difficulty in finding the exact position of the CM, the presence of air currents in the laboratory producing unexpected effects, the oscillations of the rod not happening in a plane, to mention just a few. The weighted average of the results is shown in the graph, with its boundaries indicated by the red vertical lines. The full dynamical equation for the physical pendulum without energy dissipation can be written as:ĭistribution of the results for the measurement of the acceleration of gravity g by a sample of fifty groups. A variation is obtained when a smaller metallic bar is attached (at different points) to the stick, modifying the position of the CM and the moment of inertia. The rigid body consists of a metallic stick of length L through which holes have been drilled at various positions, in such a way that the rod can be hanged from different points, changing the distance between the CM and the pivot point. The mathematical techniques required to analyze the oscillatory motion of the physical pendulum require the use of Taylor series to approximate a second order nonlinear differential equation to a linear one and the solution of differential equations with constant coefficients using characteristic equations, leading to the transformation of complex exponentials to real-valued functions.įinally, from the experimental physicist perspective, the physical pendulum requires to configure the experiment in order to test the validity of the theoretical approximation, data taking skills and processing, the identification of diverse sources of uncertainty, the ability to choose appropriately the independent and dependent variables to make a plot showing their relationship, the statistical analysis required to fit a model to the data, and the ability to evaluate its reliability (which could lead to the observation of faults in the previous steps).įor some years now, students at our university have been performing the physical pendulum experiment with some success. Blackburn, The pendulum: a case study in physics (Oxford University Press, Oxford, 2005). The main advantages of this experiment are its simplicity, low cost and wealth of both physical and mathematical concepts.įrom the theoretical physics point of view, the physical pendulum can be used to understand the behavior of systems acted on restoring forces proportional to the displacement from the equilibrium point, the concept of a rigid body and the determination of its center of mass (CM) and moment of inertia (and the radius of gyration), the use of Steiner's theorem, the effect of damping, among others G.L. Brittle, Physics Education 47, 537 (2012).], which is generally used to get a measurement of the gravitational acceleration g. Weltin, American Journal of Physics 32, 267 (1964). The analysis of the oscillatory behaviour of physical systems is a fundamental part of physics courses at the undergraduate level, where simple harmonic oscillators are studied, both from the theoretical and the experimental point of view.Ī very common experiment in fundamental physics courses, which has been used throughout the years, consists on the analysis of the oscillatory motion of a physical pendulum H.
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