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6m_lab4 - Physics 6M Experiment#4 Standing Waves on a...

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1 Physics 6M Experiment #4: Standing Waves on a Guitar String Introduction In the preceding experiment we observed all of the fundamental properties of standing waves in a medium. In this lab we will see how these properties are exhibited in a guitar or piano string. In such an instrument, the string is fixed at both ends, so the pattern of standing waves you will see will be the “node at each end” pattern from last week’s lab. You saw last week that waves an your Shive machine had a propagation speed v which was independent of their frequency. This is true not only for the Shice machine but for many types of waves in a wide variety of media: the wave speed depends on properties of the medium itself, but is almost perfectly independent of frequency. For example, transverse waves on a string propagate with a speed that depends on the tension in the string (higher tension -> faster wave speed) and on the density of the string (heavier string -> slower wave speed), but not on the wavelength or frequency: μ S T v = where T S is the tension in the string (in Newtons) and µ is the linear mass density – in other words, the mass per unit length of the string -- expressed in kg/m. Prelab questions #1 and #2 explore the idea of linear mass density, and how it is related to the more familiar value ρ , which is mass per unit volume . Experiment #1. Fundamental Properties of a Stretched String In this lab we do not have a direct way to observe the wave speed v on the guitar string; the speed is simply too fast for us to try something like we did last week, watching a pulse travel along the string and timing its motion. Instead, we can observe the wave speed indirectly by exploring the relationship between wavelength and frequency of standing waves on the string, and using the equation v = λ f . As we showed in the previous laboratory, a medium that is bound at both ends can sup- port standing waves with wavelengths " n = 2 L / n , w here n = 1,2,3, L and thus with frequencies
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2 f n = v / " n = nv /2 L , where n = 1,2,3, L Therefore, the spectrum of the modes for the stretched string will be μ S m T L m f 2 = Our first experiment will be to verify that this formula works for the fundamental mode, m = 1; i.e. , f 1 = v /2 L . To do this, we will use the setup shown in figure 1.
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