The phenomenology of behavior of the jerked-string system (Fig. 1 in the paper) depends, in our simplistic model, on the static loading parameter gamma = mg/T0. That is, when this loading is sufficiently small, the analysis predicts that the lower string can break after the mass has 'bounced' one or more times on the elastic upper string. This behavior is clearly seen in our Fig. 3, which assumes the limit of negligible loading. Our Fig. 4 attempted to show (schematically) the disappearance of these 'anomalous zones' when the loading (gamma) is not negligible.
We realized that Fig. 4, and the arguments discussing it, did not provide a transparent explanation, but it was the best we could do at the time that we submitted our paper. Following welcome suggestions from John Wallingford (P.O. Box 168, Cashiers, NC 28717), we think we now have a much clearer way of illustrating the situation.
The explicit time dependence of our Figs. 3 and 4 is not of particular interest. The important concern is the boundary condition between zones, which is expressed by Eq. (13). With a bit of algebra, one can eliminate the time t, and the string tensions Tlow and Tup, from Eqs. (8), (9), and (13) (note that 'Eq. 13' is actually two equations). The result expresses the zone-boundary value(s) of the jerk-parameter alpha in terms of the loading-parameter gamma and the other (presumably fixed) parameters of the system.
In terms of the normalized jerk variable X = (alpha) / [T0*sqrt(k/m)], this result is:
(Note that the mathematics has the form of the 'sinc' function of diffraction theory.) We can now plot the loading-parameter gamma versus the normalized jerk-parameter X, as shown in the following figure. We believe that this figure shows more clearly the multivalued behavior of the system when gamma is small.
In the final paragraph of our paper we describe an alternative model ("Model 2") by which the displacement of the bottom end of the lower string grows linearly with time: let's say, xlower = (beta)*t (as opposed to the force growing linearly). The coefficient beta now becomes the 'jerk'-parameter (actually it represents a constant velocity, suddenly turned on at t=0; the true jerk of the end of the string is a delta-function). As noted in the paper, this Model makes things a little messier because the spring constant k of the lower string brings in another free parameter. But it's reasonable to assume equal lengths of the two strings (same Young's modulus, hence same k-values). With this additional 'special' assumption, the time behavior (analogous to Fig. 3 for our original "Model 1") looks like the following figure:
In this case we choose the new normalized jerk variable as X2 = (beta) / [T0*sqrt(8/mk)]. Performing an elimination of variables among Eqs. (13) and the analogs to Eqs. (8)-(9) for this case, we obtain the result:
Computationally, choose an array of values of the argument of the sine (let's call it z). Then compute:
The results are shown in the following figure:
The normalization of the horizontal axis was chosen so that the intercepts are numerically the same as in the figure above for "Model 1"--i.e., at 1/(n*pi)--although this equalization doesn't mean much since the 'jerk-parameters' alpha and beta of the two Models are different kinds of quantities.
We're hoping that our paper may encourage some enterprising students to attempt experimentally to see the lower string breaking 'on the bounce' in one of the higher-order zones. Alternatively, one could substitute for our Eq. 2 (Fig. 2) an analytical model with smooth curvature as the tension approaches the point of failure (chosen to approximate empirical data for real strings), and carry out numerical solutions by computer of the resulting nonlinear equation of motion (replacing our Eqs. 3-4). Presumably the results would be qualitatively like those of the 'simplistic' model, with some numerical variation and probably faster attenuation or suppression of the higher-order zones. In the real world, a controlled mechanical drive mechanism is likely to be of finite (nonzero, noninfinite) mass, giving a driving condition somewhere between the cases we have called Model 1 and Model 2.
We will be interested to hear from anyone who carries this project further.
mheald@oursquare.com or
Mark A. Heald, P.O. Box 284, Pleasant Hill, TN 38578
PS added 12/11/97: We have just discovered a prior analysis of this problem:
P. LeCorbeiller, A classical experiment illustrating the notion of 'jerk', Am.J.Phys. 13, 156 (1945).
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