Mass Moment Determination Using Compound Pendulum

Available online at www.ilcpa.pl International Letters of Chemistry, Physics and Astronomy 3 (2013) 85-100 ISSN 2299-3843 Mass Moment Determination Using Compound Pendulum Zdzisław Pluta, Tadeusz Hryniewicz* Koszalin University of Technology, Raclawicka 15-17, PL 75-620 Koszalin, Poland *E-mail address: „People fall down due to the truth from time to time but most of them rise and walk off quickly allegedly nothing happened” Winston Churchill ABSTRACT This work has been performed to verify the existent knowledge on determination of the mass moment. For the experiment, a compound pendulum was used. The motivation to undertake these studies were experimental results indicating a big discrepancy in mass moments between the values coming from calculations using the definition formula and these obtained from the experiment. In relation to the axial moment the relative error equals 23.6%, whereas regarding the polar moment the error reached 56.4%. Considering the reason of that discrepancy we could find the existent theory not to be adequate. The theory is then considered in view of verifying first the mathematical pendulum and next the physical/ compound pendulum theory. The consideration has been focused on the description of accelerated motion cycle of both pendulums as it was enough to solve the problem. A source differential equation, which serves to solve any quantum phenomena, was used in the study. Then the course of creation of detailed characteristics of the phase of mathematical pendulum accelerated motion is presented as the basis to derive formula on the mass moment of a compound pendulum. At the end this new adequate theory was verified showing the relative error to be less than one per cent. Keywords: Mathematical pendulum; Compound pendulum; Mass moment; Angular acceleration; Inertia moment; Time constant; Gravitation; Pendulum vibration period International Letters of Chemistry, Physics and Astronomy 3 (2013) 85-100 1. INTRODUCTION In his „Logic of scientific discovery” Popper [1] writes: „In the history of science, always theory, not experiment, idea and not observation, open the way to a new knowledge; I am also convinced that just the experiment saves us from making for nowhere; experiment allows getting out of old ruts and calls for finding new ways/paths.” The essential meaning of experiment is also accented in [2], indicating the mind to cooperate with the matter and the scientist’s intellect with techniques and technology. If the knowledge is inadequate then it has to be excluded and replaced by a new one adequate theory. Maybe the discrepancies between the theory and practice are noticed or even experienced but, for the time being, ignored to walk off quickly as if nothing happened [3]. Authors of the work used to penetrate the existent knowledge to find numerous and clear discrepancies from the reality. The cognitive actions and trials are undertaken to describe the studied phenomena in an adequate way [4]. This work, concerned with a new adequate description of mathematical pendulum and compound pendulum, is the example of this approach. It is worthy admitting that the thesis of inadequacy of the existent descriptions of these pendulums was formulated after an in-depth analysis of the variable motion of material bodies. The experimental results are delivered to confirm this thesis. Also a proper description of kinetics of the two pendulums is presented. The idea to undertake this subject was born much earlier and now it was confirmed by the experimental results. This experiment was to determine the mass moment of a material body. At the beginning, however, before the development of the thesis, let us clarify the notion of mass moment. It exists in science as the inertia mass moment or just inertia moment, without a necessary strictness. This last notion does not specify if it is concerned on surface or mass. Anyway, it is not about the inertia moment as its essence is quite different. It is worthy presenting analytical form of this physical magnitude, then MB  J  (1) The inertia moment M B is equal to the product of the inertia moment J and the angular acceleration  ? That does not make any sense! A cardinal terminological error has occurred; the rule of univocality [5]. That rule means the name to be referred to one notion; one cannot use ambiguous multivocal names meaning different notions (homonyms). Therefore it is justified to differentiate the names of these magnitudes, i.e. M B and J. This first symbol determines then the inertia moment, whereas the second one – the mass moment. The inertia moment M B is equal to the product of the mass moment J and the angular acceleration  , and it is the right reading of the formula (1) on the inertia moment of a material body. 2. EXPERIMENTAL The subjected material body is presented in Fig. 1, with the mass moment determined by an experiment. This is the ring of the measured mass m equaling to 74 g, that is 0.074 kg. The outer diameter of the ring equals D = 94 mm, or 0.094 m, with the inner diameter d = 84 mm, or 0.084 m. 86 International Letters of Chemistry, Physics and Astronomy 3 (2013) 85-100 Fig. 1. Characteristics of the studied ring A known formula on the polar mass moment J 0 , that is the mass moment against the axis coming through the solid centre, takes the following form: J0   m 2 R  r2 2  (2) where R determines the outer radius, and r  the inner radius of the ring, then R  D : 2; r  d :2. Substituting the results of measurements characterizing that ring, with the magnitudes m = 0.074 kg; R = 0.047 m; r = 0.042 m, one obtains J0    0.074 0.0472  0.0422  0.147  10 3 kg.m2 2 And now let us derive the mass moment J1 , that is this kind of moment against the axis coming through the periphery of the ring’s hole (see Fig. 1). That moment is determined by a Steiner theorem, namely J1  J 0  mr 2 (3) where r is the distance between the considered axes, i.e. 0-0 and 1-1. That distance is equal to the inner radius of the ring. 87 International Letters of Chemistry, Physics and Astronomy 3 (2013) 85-100 By substituting the values of the magnitudes: J0 = 0.147×103 kg.m2; m = 0.074 kg; r = 0.42 m, to the formula (3), one obtains J 1  0.147  10 3  0.0740.042  0.278  10 3 kg.m2 2 Thus the process of determination of the moments J 0 and J1 could be considered as completed. The moments have been determined and the aim achieved. However, in fact it is just the beginning of the titled problem: now it should be referred to the pendulums, mathematical and a compound one. In the further course of actions an essential cognitive problem will appear. It was the experimental-analytical method for the moments determination, referred to a simple symmetric material body. The accuracy of this method fist of all depends on the accuracy of the component measurements of physical magnitudes. Formulae used for calculations of the mass moments cannot be (...truncated)