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A folded chain, with one end fixed at the ceiling and the other end releasedfrom the same elevation, is commonly modeled as an energy-conserving system inone-dimension. However, the analytical paradigms in previous literature isunsatisfying: The theoretical prediction of the tension at the fixed endbecomes infinitely large when the free end reaches the bottom, contradicting tothe experimental observations. Furthermore, the dependence of the total fallingtime on the link number demonstrated in numerical simulations is stillunexplained. Here, considering the horizontal kinetic energy and the geometryof each link, we derived analytical solutions of the maximal tension as well asthe total falling time, in agreement with simulation results and experimentaldata reported in previous studies. This theoretical perspective shows a simplerepresentation of the complicated two-dimensional falling chain system and, inparticular, specifies the signature of the chain properties.

Solving the paradox of the folded falling chain by considering horizontal kinetic energy and link geometry