Donald R. Davis University of Californio Irvine, 92664
Normal Modes of Vibration A "pendulum" demonstration
Normal modes of vibration can be effectively illustrated with spring-coupled meter sticks1 or with springs and weights which model a molecule.2 Normal modes can also be demonstrated by a simple pendulum-like apparatus which makes especially clear that a complex and often bewildering motion can be decomposed into two simple periodic motions; it also illustrates the consequences of degenerate frequencies. The demonstration is mathematically equivalent to the two-dimensional, one-particle spring system which is used by Barrow to introduce his chapter on the vibrations of polyatomic molecules.3 The mathematical analysis is presented there in detail and is not repeated here. Herzberg4 also discusses the modes of an apparatus similar to this demonstration. The pendulum is best constructed from piano wire (available from hobby shops) with a mass soldered to the bottom end and a l-cm diameter rod soldered to the ton for convenient mounting from a ring stand. A single l-m, I-mm diameter wire with about a 50-g mass illustrates the special case of two degenerate modes- he combined gravitational and elastic restoring force for small displacements from eauilibrium nearlv follows Hook's law. and arbitrary displacements lead to linear periodic motions alwavs of the same freauencv. . - Two-dimensional ellintical or cir&lar paths can be represented as the sum of tGo different linear periodic motions, as can be seen by observing the suspended mass or projecting its shadow from various angles. The motion is simple and has the same period no matter what the observing angle. Hence the normal mode orientations are arbitrary, as is true for the two degenerate bendine" vibrations of linear molecules such as CO?- or OCS. The general case of non-degenerate frequencies requires a suspending wire with a cross section of less symmetry than a circle or square. Such a wire is conveniently constructed by soldering together two wires side-hy-side (see figure). Arbitrary displacements of the suspended mass now lead to restoring forces not directed toward the equilibrium position, and the resulting motion may seem to defy physics, especially if the mass is viewed at eye level. The motion is puzzling partly because the suspendina wire air amears little different than a sinale wire. l ' i e in.itr&ror can separately excite and demonsirate the two mutually perpendicular paths which have dimple periudic morion;i. The force constant5 and frequencies of the 'Onwood, D. P., J. CHEM. EDUC., 46,826 (1969). ZWhitmer, J. C.,J. CHEM. EDUC., 48,134 (1911). 3Barrow. G. M., "Introduction to Molecular Spectroscopy," McCraw-Hill, 1962, p. 116. *Hemberg, G., "Molecular Spectra and Malecular Structure," Vol. 11, D. Van Nostrand Co., 1945, p. 62.
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two modes are clearly distinguishable. A unique feature of this demonstration is that the two normal modes may be clearly singled out even when they are simultaneously excited. As the observer's viewpoint is rotated with respect to the apparatus, the motion is generally complex, but from particular orientations the motion appears simple and has the frequency of one mode or the other. This "visual transformation to normal coordinates" illustrates the mathematical transformations discussed by Barrow and Herzberg. The degenerate-frequency demonstration is not an essential accompanyment of the general case demonstration just described; either can be presented independently. The mentioned shadow projection technique is suggested for large audiences; a visually prominent mass at about eye level is ordinarily quite satisfactory.
Demonstration of the general case of non-degenerate vibrations.
