We had very little time to investigate the more visually stunning form of our vibrating wire experiment, the wire hoop, due to time constraints. We were able to draw some useful conclusions as well as postulate several possibly fruitful avenues of further investigation from what little time we did spend with the hoop.
In our experiments, we positioned the hoop vertically. We first vibrated the hoop with the mechanical vibrator clamped to the lab bench. This produced longitudinal vibrations in the hoop, and we were able to observe a few lower order resonance states. For the bulk of the experiment, we had the vibrator clamped to a ring stand (as shown in Apparatus), so we observed transverse vibrational states.
We observed resonance at eight frequencies, each producing an odd number of nodes. This is to be expected, as the apparatus forces one of the nodes to be at the clamped point of the ring, where excitation occurs. Representing the resonance frequency as a function of the number of nodes yields the following:
Notice how the resonance frequency is related to the square of the number of nodes. This is reminiscent of Chladni's Law, f ~ (m + 2n)^2. Since Chladni experimented with two dimensional round plates, there are two node numbers "m" and "n." Our hoop can be thought of as a one dimensional object (not in Cartesian coordinates, surely, but in polar coordinates), so we would expect one node number.
Further investigation into this could take many directions. One possibility is that the hoop behaves in the same way as the outer edge of the Chladni plate, or possibly any section of the plate that is at a constant radius. An investigation into the similarities and differences between this experiment and Chladni's Law should prove rewarding. For example, one difference between the Chladni experiment and the hoop is the interesting visual effect caused by exceeding the elastic limit of the wire as shown on the pictures page.
The hoop can also be used as a demonstration for the Bohr model of the hydrogen atom. Each resonant state is frequency dependent, as shown above, just as a photon must have a specific excitation frequency to cause the electron to jump from a lower energy level to a higher. An investigation into the mathematical relationship between the Bohr model and our resonance hoop would provide more depth to the demonstration. The demonstration could then be used not only to introduce the idea of energy levels as it is now, but to aid in the explanation of those energy levels later on in the introductory physics course.
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