A system consisting of two pendula attached to an air cart is mathematically analyzed, and the coefficient of restitution is shown to pass through a deep minimum. The solution to the small angle equation of motion is transcendental and provides an exercise in graphical methods for the beginning mechanics student

Choate, L.

Dennis, James C.

Gruebel, R.

English

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Stephen F. Austin State University SFA ScholarWorks Faculty Publications Physics and Astronomy 1971 A Variable Coeficient of Restitution Experiment on a Linear Air Track R. Gruebel James C. Dennis L. Choate Follow this and additional works at: https://scholarworks.sfasu.edu/physicsandastronomy_facultypubs  Part of the Physics Commons Tell us how this article helped you. A Variable Coeficient of Restitution Experiment on a Linear Air TrackR. Gruebel, J. Dennis, and L. ChoateCitation: American Journal of Physics 39, 447 (1971); doi: 10.1119/1.1986174View online: https://doi.org/10.1119/1.1986174View Table of Contents: https://aapt.scitation.org/toc/ajp/39/4Published by the American Association of Physics TeachersARTICLES YOU MAY BE INTERESTED INCollision of a Ball with a Stationary OscillatorThe Physics Teacher 58, 241 (2020); https://doi.org/10.1119/1.5145467Experiment for Measuring the Coefficient of RestitutionAmerican Journal of Physics 26, 386 (1958); https://doi.org/10.1119/1.1996166A simple model for inelastic collisionsAmerican Journal of Physics 76, 1071 (2008); https://doi.org/10.1119/1.2970052Model of the behavior of solid objects during collisionAmerican Journal of Physics 44, 671 (1976); https://doi.org/10.1119/1.10353Instructional Uses of the Computer: General Fourier SynthesisAmerican Journal of Physics 39, 450 (1971); https://doi.org/10.1119/1.1986175Investigation of magnetic damping on an air trackAmerican Journal of Physics 74, 974 (2006); https://doi.org/10.1119/1.2232645A,Ar.1. 1 VIRTUAL SUMMER MEETING April 1971 A Variable Coeficient of Restitution Experiment on a Linear Air Track R. GRUEBEL J. DENNIS L. CHOATE Stephen F. Austin State University Nacogdoches, Texas 75961 (Received 13 January 1970; revision 6 July 1970) A system consisting of two pendula attached to an air cart is mathematically analyzed, and the coefficient of restitution is shown to pass through a deep minimum. The solution to the small angle equation of motion is transcendental and provides an exercise in graphical methods for the beginning mechanics student. A recent article in another journal1 described a simple but interesting lossy collision experiment on a linear air track. The experimental-theoretical correlation needed for a good undergraduate lab experiment was missing. In this article a simple system is discussed which has been mathematically analyzed and has attracted a great deal of atten-tion from advanced undergraduates. This exercise is illustrative of a number of real research problems met in the physics laboratory where a straight-forward mathematical treatment of complex physical phenomena yields a simple result which can readily be compared to physical observations. The student is afforded the opportunity to use graphical methods fo~ solving transcendental equations such as are encountered in the analysis of boundary value problems in quantum me-chanics and classical wave phenomena.2 The system referred to here consists of two pendula attached to an air cart. As it1 the previously mentioned arrangement, the coefficient of restitu-tion passes through a deep minimum. The coefficient of restitution, the ratio of the velocity after collision to the velocity before, is measured as a function of pendula mass (m1) when the cart ( mass m2) plus penduls, undergo a collision with an air track bumper spring. The spring constant must be small enough to allow the completion of a quarter-cycle of pendulum swing before the cart and spring separate. There-fore the restriction 2T2 > Ti> T2 is placed on the system, i.e., the pendulum should be in the second quarter cycle of its motion during separation. Consider the motion of the arrangement when the pendulum is set into motion relative to the cart following the collision: The collision is inelastic with respect to the cart velocity to a degree dependent on the momentum of the pendula at the instant of separation. When m1<<m2 and m1rvm2, the pendulum has very little effect on the cart rebound[ velocity. The latter region is characterized by small in-phase oscilla-tions of the pendula. The out-of-phase oscillations in the intermediate region produce a minimum in the cart rebound velocity and hence in e, as seen in Fig. 2. DERIVATION The Lagrangian of the system is and the small angle equatiollls of motion are where m2 is the cart mass, l is the pendulum length, and k the spring constant [see Fig. 1]. After rearranging terms, Eqs. (2) and (3) become (4) (5) AJP Volume 39 I 447 Gruebel, Dennis and Choate ----r y• i-----X ____ J Fm. 1. Cart-pendulum-bumper spring system. The actual experimental arrangement had two pendula for reasons of convenience. where w1 and w2 are defined by (6) Assuming a solution of the form x=A1[sin(a1t) - (ai/a2) sin(a"t) J + (vo/a2) sin(a2t), (12) 0=B1[sin(a1t) - (ai/a,) sin(a2t) J + (vo/a,l) sin(a2t). (13) From Eq. (8) we have (14) At 0=0max, 0=0 and (15) where t', the time when 0=0, is found by substi-tuting Eq. (15) into Eq. (13) and solving (13) for its maximum value. The latter is maximized x =A sin(at), 0=B sin(at) (7) at t' =1r/2w1. Thus leads to (8) The solutions are where a1 and a2 represent the modulated fre-quencies of the pendula and spring, respectively. In terms of the a;, the general solutions become At t = 0, :i; = v0 and 0 = v0/l, where Vo is the velocity of the cart-pendulum system just before collision with the spring. Therefore at t = 0 (U) Solving from A2 and B, in terms of A1 and B1 and substituting into Eq. (10), the solutions are 448 / April 1971 and X Vo (16) To solve for the coefficient of restitution the time t" when x = 0 on the rebound must be determined, or from Eq. (17), sin(a1t") = (a1/a2) X [l-wz2/ (w12-a12 ) J sin(a2t"). (18) 0 100 200 300 400 500 600 M1_(Qf'Urn} Fm. 2. Coefficient of restitution (x/vo at x =0 for cart) as a function of pendula mass for wz2 = 100, m2 = 550 g, and l = 20 cm. The solid line represents theoretical values. Note that at m1 =600 g, e> 1. The above relationship is to be solved graphically for t". Finally, the coefficient of restitution, x/vo, is e=e0[(1-R) cos(a2t")+R cos(a1t")], (19) where e0 is the coefficient of restitution with m1=0, and (20) Coefficient of Restitution Experiment DISCUSSION A comparison of Eq. (19) with experimental results is shown in Fig. 2. The natural spring frequency was measured by kinetic-potential conversion in a collision with m1 = 0. Agreement lessens as m1 increases due to increasing non-linenrity of the spring. 1 .T. L. Stnll, Phys. Teacher 7, 225 ( 1969). z L. M. Clendenning, Amer. J. Phys. 36, 879 (1968). AJ P Volume 39 / 449 

A Variable Coeficient of Restitution Experiment on a Linear Air Track

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A Variable Coeficient of Restitution Experiment on a Linear Air Track

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