Wavelength variation of a standing wave along a vertical spring Dylan Welsch and Blane Baker, William Jewell College, 500 College Hill, Liberty, MO 64068 and-driven resonance can be observed readily in Ha number of mechanical systems including thin boards, rods, strings, and springs. In order to show such behavior in the vertical spring pictured in Fig. 1, a sec- tion of spring is grasped at a location about one meter from its free end and driven by small, circular motions of the hand. At driving frequencies of a few hertz, a dra- matic standing wave is generated. One of the fascinating features of this particular standing wave is that its wave- length varies along the length of the spring. Variation in wavelength along a vertical spring at resonance can be understood by treating the system as a medium under tension. A medium under tension T produces a mechanical wave whose speed v depends upon the expression where represents mass per unit length of the medium. In this particular case, tension along the length of the spring increases from bottom to top due to the fact that more mass is supported as the spring is traversed in this direction. Here tension can be expressed mathemati- cally as where T is the tension at the top of Fig. 1. Standing wave generated by hand- max driven resonance of a vertical spring. Due the suspended spring, x is the distance from the free end to systematic changes in tension and mass to a point of interest, and L represents the total length per unit length along the length of the spring, of the suspended spring. From these relationships, both the standing wave exhibits variation in wave- speed and wavelength depend on length. given that the driving frequency is fixed for a particular Linear fits to these data confirm that the proposed mode of oscillation. model describes the standing wave along the vertical Besides tension, mass per unit length of a vertical spring. In addition to wavelength variation observed spring also varies due to the fact that the spring expands for this particular standing wave, other wave phenom- increasingly along its length from bottom to top in or- ena in physics show similar behavior. Such changes der to support the length of the spring below it. Given are observed when light crosses the boundary between this variation, wavelength is expected to be propor- materials with different indices of refraction or when tional to quantum-mechanical particles encounter potential en- ergy barriers. In classroom or laboratory settings, this demon- stration is particularly useful when studying standing Taking both of these effects into account predicts that waves. Instructors can begin by showing students the wavelength should vary according to demonstration and asking them to record observations. As an additional activity, students can propose expla- where c is a constant that depends upon factors such nations for observed variations in wavelength given as maximum tension and frequency. To quantify the their knowledge of mechanical standing waves. Finally, standing wave depicted in Fig. 1, values of wavelength instructors can draw analogies with other phenomena of several sections along the spring were determined by such as wavelength variations across boundaries be- direct measurement and plotted against tween materials with different indices of refraction. DOI: 10.1119/1.5025291 The Physics Teacher ◆ Vol. 56, March 2018 155 Trick of the Trade
The Physics Teacher – American Association of Physics Teachers
Published: Mar 1, 2018
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