On the paradox of the free falling folded chain

Schagerl, Martin; Steindl, Alois; Steiner, Wolfgang; Tröger, H.

doi:10.1007/bf01177305

Cited by 46 publications

(64 citation statements)

References 3 publications

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“…It was realized in [38] that conservation of energy actually requires that τ + equals the tension, τ − , acting at the bight on the resting arm. As also pointed out in [44,45], different constitutive choices for these tensions would grant a full host of evolutions for the folded chain, including the one where the free arm falls freely [46]. The problem, however, is that we are not at liberty of choosing constitutive laws for τ + and τ − , at least as long as we model chains as inextensible strings, for which the tension in an internal reactive force to be determined.…”

Section: (I) Early Historymentioning

confidence: 99%

Chain paradoxes

Virga

2015

Proc. R. Soc. A.

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For nearly two centuries, the dynamics of chains have offered examples of paradoxical theoretical predictions. Here, we propose a theory for the dissipative dynamics of one-dimensional continua with singularities which provides a unified treatment for chain problems that have suffered from paradoxical solutions. These problems are duly solved within the present theory and their paradoxes removed-we hope.

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Section: (I) Early Historymentioning

confidence: 99%

Chain paradoxes

Virga

2015

Proc. R. Soc. A.

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“…1 The results of their measurements 15 came as a surprise to them because they had concluded by theoretical arguments that the chain fell only as fast as g, and that the total mechanical energy was not conserved. 20,21 In these earlier papers, the authors rejected Hamel's energy conserving solution, 14 and claimed that there was dissipation caused by the inelastic but momentum-conserving impacts at the fold of the chain.…”

Section: 13mentioning

confidence: 99%

“…15 to conclude that the motion of the falling chain is non-unique, because "it is important to note that for the folded string itself there exist more solutions which fulfill the balance of linear momentum (but do not conserve the mechanical energy). "…”

Section: 13mentioning

confidence: 99%

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Falling chains

Wong

Yasui

2006

American Journal of Physics

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The one-dimensional fall of a folded chain with one end suspended from a rigid support and a chain falling from a resting heap on a table is studied. Because their Lagrangians contain no explicit time dependence, the falling chains are conservative systems. Their equations of motion are shown to contain a term that enforces energy conservation when masses are transferred between subchains. We show that Cayley's 1857 energy nonconserving solution for a chain falling from a resting heap is incorrect because it neglects the energy gained when a transferred link leaves a subchain. The maximum chain tension measured by Calkin and March for the falling folded chain is given a simple if rough interpretation. Other aspects of this falling folded chain are briefly discussed.

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“…Strnad [8] described a theoretical model of a bungee jump that takes only the mass of the bungee rope into account. The first phase of bungee jumping can also be related to other phenomena such as the dynamics of a falling, perfectly flexible chain suspended at one end and released with both ends nearby to each other at the same vertical elevation [9][10][11][12][13][14]. Experiments, numerical simulations, and analytical models discussed in the literature (also for discrete models of chains) point at the paradoxical phenomenon that the tip of a freely falling, tightly folded chain with one end suspended from a rigid support moves faster than a free falling body under gravity.…”

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confidence: 99%