Abstract
This work investigates the dynamics of a one-dimensional homogeneous harmonic chain on a horizontal table. One end is anchored to a wall; the other (free) end is pulled by an external force. A Green's function is derived to calculate the response to a generic pulling force. As an example, I assume that the magnitude of the pulling force increases with time at a uniform rate β. If the number of beads and springs used to model the chain is large, the extension of each spring takes a simple closed form, which is a piecewise-linear function of time. Under an additional assumption that a spring breaks when its extension exceeds a certain threshold, results show that for large β the spring breaks near the pulling end, whereas the breaking point can be located close to the wall by choosing small β. More precisely, the breaking point moves back and forth along the chain as β decreases, which has been called “anomalous” breaking in the context of the pull-or-jerk experiment. Although the experiment has been explained in terms of inertia, its meaning can be fully captured by discussing the competition between intrinsic and extrinsic time scales of forced oscillation.
Highlights
The pull-or-jerk experiment is commonly used to demonstrate Newton’s laws of motion: As depicted in Fig. 1, a ball of mass m is hung from a ceiling by a string, whose tension is denoted as Tup
I have investigated the dynamics of a harmonic chain which is anchored to a wall at one end and subject to an external force at the other end
With the simple failure behavior assumed in Eq (31), anomalous breaking3–5 is still a theoretical possibility in this many-body system when driven by a ramp force F / t
Summary
If the number of beads and springs used to model the chain is large, the extension of each spring takes a simple closed form, which is a piecewise-linear function of time. Under an additional assumption that a spring breaks when its extension exceeds a certain threshold, results show that for large b the spring breaks near the pulling end, whereas the breaking point can be located close to the wall by choosing small b.
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