edited by GEORGEL. GILBERT Denison University Granville. Ohio 43023
Coupled Oscillations SUBMITTEDBY
D. P. Onwood
Indiana-Purdue UniversHy Fort Wayne, IN 46805 CHECKEDBY
Paul Krause
Central Arkansas Conway, AR 72032 Unlverslty of
Preparation
Two pendulums of suitable length and period are mounted so as to produce a coupling between their motions in narallel olanes. This mav be done bv mounting a length of stnmg cord hurizontallg and tightly between two rigid supourts. Su.iuend twu idtmtical umdulum bobs from that cord. at puints equidis~anrfrom and cloie ro the center, withequal Irnmhs uf the same material usine: slio-nroof knots. rAnuth. er procedure may be found in J. fhem. ~ d u c1969,46,826.) Demonstration The system should be set up so that the pendulums appear, to the observers, to be in eclipse when a t rest in the reference plane. One may then show the two normal modes of the system, in vertical planes perpendicular to the reference plane, by 1. Displacing the bobs in the same sense and to the same extent
before releasing them simultaneously. Note that the two bobs move with the same frequency (f,),which may be determined by timing a suitable number of periods. 2. Displacing the bobs in opposite senses and the same extent hefore simultaneous release. Observe that once wain the two bobs amin move with the same frequenev (f?)but that this is hiaher than before
ence is a measure of the connline between the two modes, . and, if it is indistinguishable from zero, the vertical cords must be moved closer to the center to increase that cou~linrr. . . (As a guide to representative dimensions, a 1-yard "horizontal" cord with 16-in. verticals, 5 in. apart, supporting filled 12-07.. beverage bottles of the slim kind, and using a domestic twine, yielded the following values: fl, 37/min; f2, 40/min; f3,3 cycles in 65 s.) It can be helpful to review, as a preliminary, the essentials of the theorv of small vibrations. and to sueeest that the "" superposition of simple vibrations may produce motions that mav" anoear .. verv comnlicated unless described in terms of their components. This demonstration does not of course show molecular vibrations per se, nor is it the first of its kind. It is rather an accessible illustration of the way in which the motions of coupled masses may appear to be very complicated unless analyzed in terms of their normal modes, and i t may serve as an introduction to this concept in, for example, molecular vibrations. One might suggest that invisible atoms in their characteristic force fields will also have normal modes, and that the actual oscillations of a molecule could be a mixture of modes. The generality of this concept may be suggested by demonstratine beats. the acoustical analoe. Two tunine forks of similar fr&uencies FLand FZshouli be excited-simulta(F, - F7). (A neouslv. The beats mav be heard a t afreauencv . related phenomenon is involved in the "phantom" sound, sometimes heard from a wind instrument, far below the fundamental of the instrument.)
The properties of these modes now established, another excitation is demonstrated by 3. Dis~laeineonebobonlv. ,. while theother is held inits rest nosition umil simuitaneow release. This pn,ducez a "mixed" \,ihrarion. whit h often ~ r ~ w ~5tronl: k e i merest in the d,servert. The I
