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Physical Pendulum Experiments to Enhance the Understanding of Moments of Inertia and Simple Harmonic Motion

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Physical Pendulum Experiments to Enhance the Understanding of Moments of Inertia and Simple Harmonic Motion Home Search Collections Journals About Contact us My IOPscience Physical pendulum experiments to enhance the understanding of moments of inertia and simple harmonic motion This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2012 Phys. Educ. 47 537 (http://iopscience.iop.org/0031-9120/47/5/537) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 161.24.4.200 The article was downloaded on 30/04/2013 at 19:43 Please note that terms and conditions apply. P APERS www.iop.org/journals/physed Physical pendulum experiments to enhance the understanding of moments of inertia and simple harmonic motion Tim H Richardson and Stuart A Brittle Department of Physics, University of Sheffield, Sheffield, UK E-mail: t.richardson@sheffield.ac.uk Abstract This paper describes a set of experiments aimed at overcoming some of the difficulties experienced by students learning about the topics of moments of inertia and simple harmonic motion, both of which are often perceived to be complex topics amongst students during their first-year university courses. By combining both subjects in a discussion about physical pendula, in which the oscillation time periods for the periodic motion of several objects (a tennis ball, a thin beam, a hoop and a solid disc) are measured and compared, students are able to understand both topics at a higher level and also experience the synergistic effect of combining two or more physics themes in order to accelerate their learning whilst simultaneously raising their motivation. Special attention is given to the ‘ball and stick’ pendulum in which a block of material (treated as a point mass) can be moved along a shaft to create a composite pendulum whose time period exhibits a minimum value at a certain separation between the block and the rotation axis. Introduction subject to understand. The physics of rotation, more generally, is covered only at a very basic Two topics that students in the age group 16–19 level within A-level syllabuses [3]; in the first find challenging, be they A-level students or Level year of university courses, the idea of moment 1 university students, are oscillations and the of inertia is most commonly introduced initially physics of rotation. The former is taught within in the context of point masses separated from an most A-level specifications in the UK [1] in the axis of rotation around which they are revolving. form of simple harmonic motion (SHM) but now At a deeper level, this subject is developed in in a far more qualitative manner than in the 1980s, the context of the rotation of a mass distribution for example; at Level 1 in most university physics which itself defines the geometry (shape) of the programmes worldwide, SHM is covered in the object, and thus provides an appreciation of the form of the simple (point mass) pendulum and fact that the moment of inertia, and therefore the physical pendulum [2]. The latter requires rotational kinetic energy, of an object is dependent an appreciation of the concept of moment of on this mass distribution rather than simply its inertia, which again is perceived as a difficult mass. 0031-9120/12/050537C08$33.00 c 2012 IOP Publishing Ltd P HYSICS E DUCATION 47 (5) 537 T H Richardson and S A Brittle Here, a demonstration lecture is described in which the authors bring these two topics together in order to illustrate practically how the time period of oscillation of various objects (different mass distributions) can be accurately predicted from an understanding of the fundamental physics of the simple and the physical pendulum. From the basic measurement of the oscillation time period of a simple pendulum of length L which consists of a tennis ball tethered to a length of string, the oscillation periods, T, for a swinging rod, hoop and solid disc are predicted. Subsequent measurement of the latter three values of T are found to agree very well with those predicted values, so demonstrating the value of the physics learnt, and how a deeper, synergistic understanding of both topics can be gained. Background to the simple pendulum and the physical pendulum A laboratory simple pendulum can be made most easily using a length, L, of ordinary shop-bought string and a tennis ball. A triple knot tied at the end of the string is forced into a very small slit cut in the rubber of the tennis ball to form the Figure 1. pendulum ‘bob’. The free end of the string is where Tsimple is the oscillation time period of tied to a clamp which forms the pivot or axis of this simple pendulum, which can be expressed in rotation. Figure1 shows a schematic diagram of relation to its length by: the pendulum. s In the figure, the bob is shown with an L Tsimple D 2π : (4) angular displacement, θ, from the vertical; the g restoring force which causes the tennis ball to move towards its equilibrium position is given by Theoretically, this equation is valid when the F , where: pendulum bob is a point mass separated from res the rotation axis by a distance, L. In practice, Fres D −mg sin θ (1) for lengths of L greater than around 50 cm the in which m is the mass of the bob and g is tennis ball acts effectively as a point mass, in the acceleration due to gravity. From Newton’s the context of this experiment. Educationally, it is second law, this restoring force causes the linear important to emphasize that the mass itself of the acceleration such that: pendulum bob does not appear in the expression for its time period. Students initially find this a d2x − mg sin θ D m : (2) difficult concept. dt2 A physical pendulum in which the bob is now This is valid in the case when θ is small so that an object whose mass is distributed over a certain sin θ ∼ θ and x ∼ Lθ and hence: geometry to give the object its characteristic shape can be analysed in a similar manner, as supported d2θ − gθ D L : by figure2. Here, the restoring torque, τ, can be 2 (3) dt written as: Finally, this equation can be compared to the d2φ D − general equation for SHM, D −!2φ, in which τ mg sin θd (5) dt2 ! is the angular frequency given by ! D 2π , d Tsimple in which the distance is shown in figure2. For 538 P HYSICS E DUCATION September 2012 Enhancing the understanding of moments of inertia and simple harmonic motion of moment of inertia is required, which is I D R r2 dM. Reference to texts such as [2] provides the moments of inertia expressions required for the four objects discussed in this paper. Furthermore, if the oscillating body is composed of two of more contributing masses (for example, a metal rod and a metal block through which the rod was placed), then both masses contribute to the total mass, m, and both moments of inertia would contribute to I. This will become important later in the paper. The case of the simple pendulum can be seen to fit the general, physical pendulum analysis. If the tennis ball is treated as a point mass, then 2 its moment of inertia, Iball, is simply mL , but as the pendulum length is also the distance between the tennis ball centre and the rotation axis (pivot), 2 L D d, and so Iball D md . Thus, this leads back to the formula for the oscillation time period of the ball, Tball, which is that of the simple pendulum: s s s md2 d L T D 2π D 2π D 2π : (9) ball mgd g g small angle oscillations,Figure equation 2. (5) reduces to: Furthermore, equation (8) allows the oscilla- τ D −.mgd)θ: (6) tion time period to be predicted for an oscillating From elementary rotational kinematics, the object of any shape, provided its moment of torque causes the angular acceleration via τ D Iα, inertia is known. From the physics education where I is the moment of inertia of the oscillating point of view, this facilitates an excellent way of object and α is the angular acceleration given linking together two topics (moment of inertia 2 and SHM) with a view to providing a less by α D d θ . Thus, the equation for SHM of the dt2 compartmentalized learning experience, and also physical pendulum can be written as: the way in which mathematics—albeit at a simple d2θ level initially—is used as a tool for modelling I D −.mgd)θ (7) dt2 real physical behaviour that can be observed with the intention of putting theory to the test, as will from which the oscillation period for the object become evident later. whose mass is distributed over its own particular geometry or shape is: Oscillation time period measurements for s objects of different shapes I Tphysical D 2π : (8) Most commonly, 16–19 year old students ex- mgd perience as a classroom demonstration only the In equation (8), the mass m is the total mass oscillation of a simple pendulum. Most will of the object and I is the total moment of inertia measure the time taken for several complete of the object. Generally, the moment of inertia is oscillations to occur in order to obtain an accurate P 2 defined as I D imiri for all point masses mi that value for Tsimple. From this value, hopefully they comprise the object. Thus, for a point mass, M, are encouraged to use equation (4) to calculate a revolving around an axis of rotation a distance value for the acceleration due to gravity for the d away, the moment of inertia is simply Md2.
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