The graph below represents the relationship between the k-values, as predicted in the theory section, and our observed frequencies for a rod length of 43.2 cm. Theoretically, the slope of the line in a log-log graph should be one-half (k4~freq.2), as opposed to the 0.4545 obtained below. Rod lengths of 43.2 cm, 34 cm, and 28 cm were the only rod lengths that gave us enough data to make accurate calculations of the power relation between the theoretical k-values and our observed frequencies. The average power dependency was calculated to be 0.4572, corresponding to a 8.9% error. The r/(E*I) ratio can be calculated from the y-intercept data. In the case of the 43.2 cm length rod, the magnitude of the ratio is 0.338.
One very interesting aspect that was observed was a very linear relation between the position of the nodes and the length of the wire. The graphs of this data set are given below, along with the equations of the trend line and the R2 values.
Another relation that needed to be investigated was the length dependence of the resonant frequencies. As can be seen from the log-log graph below, there appears to be an inverse-square dependency between the two.
We now move onto determining if a one-dimensional form of Chladni's law is valid for this experiment. Below is the log-log plot of the number of nodes that appear at each respective enharmonic frequency. If Chladni's law held, then we would expect a slope of two on this line (squared relationship). However, we observe a slope of 2.8059.
The data and results of our experimentation with the hoop of wire can be found on its page.
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