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The Spring-Mass Experiment As a Step from Oscillationsto Waves: Mass and Friction Issues and Their Approaches

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The Spring-Mass Experiment As a Step from Oscillationsto Waves: Mass and Friction Issues and Their Approaches The spring-mass experiment as a step from oscillationsto waves: mass and friction issues and their approaches I. Boscolo1, R. Loewenstein2 1Physics Department, Milano University, via Celoria 16, 20133, Italy. 2Istituto Torno, Piazzale Don Milani 1, 20022 Castano Primo Milano, Italy. E-mail: [email protected] (Received 12 February 2011, accepted 29 June 2011) Abstract The spring-mass experiment is a step in a sequence of six increasingly complex practicals from oscillations to waves in which we cover the first year laboratory program. Both free and forced oscillations are investigated. The spring-mass is subsequently loaded with a disk to introduce friction. The succession of steps is: determination of the spring constant both by Hooke’s law and frequency-mass oscillation law, damping time measurement, resonance and phase curve plots. All cross checks among quantities and laws are done applying compatibility rules. The non negligible mass of the spring and the peculiar physical and teaching problems introduced by friction are discussed. The intriguing waveforms generated in the motions are analyzed. We outline the procedures used through the experiment to improve students' ability to handle equipment, to choose apparatus appropriately, to put theory into action critically and to practice treating data with statistical methods. Keywords: Theory-practicals in Spring-Mass Education Experiment, Friction and Resonance, Waveforms. Resumen El experimento resorte-masa es un paso en una secuencia de seis prácticas cada vez más complejas de oscilaciones a ondas en el que cubrimos el programa del primer año del laboratorio. Ambas oscilaciones libres y forzadas son investigadas. El resorte-masa se carga posteriormente con un disco para introducir la fricción. La sucesión de pasos es: determinación de la constante del resorte tanto por la ley de Hooke y la ley de oscilación frecuencia-masa, reducción del tiempo de medición, y representación de curvas de fase y resonancia. Todos los controles cruzados entre cantidades y leyes se realizan aplicando las normas de compatibilidad. Se discute la masa no despreciable del resorte y los problemas específicos de física y enseñanza derivados de la fricción. Se analizan las formas de onda complejas generadas en los desplazamientos. Se describen en términos generales los procedimientos utilizados en el experimento para mejorar la capacidad de los alumnos para manejar los equipos, para elegir adecuadamente los aparatos, para poner de manera crítica la teoría en práctica, y para practicar el tratamiento de los datos con métodos estadísticos. Palabras clave: Teoría-prácticas en la educación experimental Masa-Resorte, Fricción y Resonancia, Formas de onda. PACS:01.40.Di, 01.50.Qb ISSN 1870-9095 I. INTRODUCTION context, our mass and spring experiment is strictly connected to the development towards waves. The spring-mass oscillator experiment is used worldwide in The course organization is based on the idea of creating first year laboratory courses because it is physically and groups of about twenty students at a time in a laboratory, pedagogically rich, and its apparently simple contents lead working together as in a research group studying a certain students to face the gap between theory and practice, to topic. As such, the group head guides the experimental program and master experimental techniques and to treat research project: he presents the theory in a few lectures, data statistically. In our course, the spring-mass is the describes the test strategy, defines steps and, very second step in a sequence of six experiments, as follows: importantly, creates the occasions for discussions at crucial the pendulum, vertical oscillations with the spring-mass, points. In the laboratory sessions sub-groups of three spring-mass with two degrees of freedom, horizontal students are helped by assistants. oscillations with two masses and three springs, vibrations in Given that our first year students usually do not have a an elastic string, vibrations in a gas tube and the ripple- background in practical physics, we focus our attention tank. The idea is to evolve from the single oscillator to only on the simplest damped harmonic oscillator during the waves, given their importance in many fields such as theory lessons which preceed the practical sessions. In the electronics, accelerators, quantum-mechanics, etc. In this laboratory, students start with the most common spring and Lat. Am. J. Phys. Educ. Vol. 5,No. 2, June2011 409 http://www.lajpe.org I. Boscolo, R. Loewenstein mass apparatus as described in so many textbooks. We try The SHM equations describing the free and forced as much as possible to make it behave as a textbook motion are reported for convenience. damped simple harmonic oscillator. This means that the 푑 2푥(푡) 퐶 푑푥 (푡) 푘 spring is assumed massless and friction is assumed velocity + + 푥 푡 = 0, (1) dependent. Nevertheless, neither assumption complies with 푑푡 2 푚 푑푡 푚 our reality. The discussion about the gap between simple 푑 2푥(푡) 퐶 푑푥 (푡) 푘 퐹0 theory and rather more complex reality is postponed to the + + 푥 푡 = 푓 푡 = sin 휔 , (2) 푑푡 2 푚 푑푡 푚 푚 moment when this is noticed by students in the laboratory sessions. Then we bring up the various oscillating modes, whose solutions are so students notice that the initial mathematical model was inadequately simpler than reality. SHM is indeed far from 1 푥 푡 = 푥 푒−훾푡 sin 휔′ 푡 훾 = , (3) simple. 0 휏 The spring force and the weight act in the spring-mass vertical system making the physics somewhat sophisticated, 푥 푡 = 퐴(휔) sin 휔푡 + 휑(휔 ), (4) if not very-complex [1, 2]. The real spring-mass behavior cannot be treated as a textbook SHM; many authors discuss where how, in practice, we must consider factors such as the 2 torsional force [3], possible standing waves in the spring ′2 푘 퐶 2 2 휔 = − = 휔0 − 훾 , (5) [4], loaded vertical oscillations [5, 6, 7] and multi-mode 푚 2푚 oscillations [8, 9], the mass of the spring if non-negligible 푐표푠푡 퐴 휔 = 2 2, (6) with respect to the object mass attached to it [2, 8, 9]. 휔0−휔 −훾 The spring-mass system can have a behaviour which is acceptably near a SHM only if proper ratios spring constant 2훾휔 tan 휑 = − 2 2. (7) /mass, k/m, and spring constant/spring length, k/ℓ, are 휔0−휔 chosen [5, 9]. In addition, the choice of the mass m is connected to the damping constant γ and also to the In these equations, F0 and ω are amplitude and frequency of resonance curve full width half maximum, FWHM; the applied force, 훾 = 퐶/(2푚) is the damping constant, C moreover, it should match the driving oscillator sensitivity. is the friction coefficient and φ is the phase difference The main objects of discussion which appear during the between the applied force and the position x(t) and m is the lab work are: (1) the relevant action of the spring mass to mass attached to the spring. The damping time is 휏 = 1/훾. be considered in the comparison of the two results on the In the above equations, the gravitational force mg is spring constant obtained from the Hooke’s law and the opposed by the spring force 푘ℓ0. frequency-mass oscillation law; (2) the damping of a After that, when they come to practicals, students system whose behavior depends also on the non-linear assemble the apparatus and follow their plan after an initial coupling between longitudinal and transverse oscillations; guiding plenary discussion. Before starting each sequence (3) the motion at the stat-up composed by the forced and of measurements, we test the apparatus with a few values self oscillations. far apart from each other and compare practical results with The approach to treat and to discuss choices, problems theory. Assistance is given to help slower groups keep the and discrepancies between theory and experiment with pace. At regular intervals data of all groups are collected on students are presented. All choices about the system are the board for statistical treatment, discussion and discussed with students during the experimental work. conclusions. This procedure helps to show how to spot Moreover, students have to try to understand and explain systematic errors and casual mistakes. several unexpected experimental observations; therefore, In the first part of the experiment students observe and they look deeply into the gap between theory and pratice gather data directly, see Fig. 1, in the second part they work and discuss it in small groups and, after that, in plenary on line with electronic data systems, as shown in Fig. 2. sessions. Students start with the test of Hooke’s law by doing a set of force-distance measures and by doing in succession a linear fit with those data pairs to find the spring constant. Springs II. THE TEACHING PLAN, ORGANIZATION are long enough for precise measurement of force and distance. The same springs are used to check the relation AND SETUPS 2 휔 = 푘/푚. The resonance frequency ω0 is measured via the period T with a chronometer. Then students do the linear fit The theory of free and forced oscillations is presented in 2 about 10 hours of lectures. In the first three we present the of the data pairs(푚, 푇 ) to get the spring theory of free oscillations for the pendulum and springs. constant.Compatibility between the two measured spring We also give guidelines to plan the very first experiments constants k is analyzed. Then students make a rough that follows suit. In the other six or seven hours we discuss measure of decay time τ and, re-set the system for the damped oscillations and guide students towards planning resonance experiment section, they take measurements to the second spring-mass experiment.
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