Solving the paradox of the folded falling chain by considering the link transition and link geometry
A folded chain, with one end fixed at the ceiling and the other end released from the same elevation, is commonly modeled as an energy-conserving system in one dimension. However, the analytical paradigm in the existing literature is unsatisfying: The theoretical prediction of the tension at the fixed end becomes infinitely large when the free end reaches the bottom, contradicting the experimental observations. Furthermore, the dependency of the total falling time on the link number demonstrated in numerical simulations is still unexplained. Here, considering the link transition between the two sub-chains and the geometry of each link, we introduce an additional term for the relation of balance of kinetic energy to account for the jump of link velocity at the fold. We derive analytical solutions of the maximal tension as well as the total falling time, in agreement with simulation results and experimental data reported in previous studies. This theoretical perspective extends the classical standard treatment, shows a simple representation of the complicated two-dimensional falling chain system and, in particular, specifies the signature of the chain properties.
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