The weight of a falling chain, revisited

Abstract

A vertically hanging chain is released from rest and falls due to gravity on a scale pan. We discuss the various experimental and theoretical aspects of this classic problem. Careful time-resolved force measurements allow us to determine the differences between the idealized and its implementation in the laboratory problem. We observe that, in spite of the upward force exerted by the pan on the chain, the free end at the top falls faster than a freely falling body. Because a real chain exhibits a finite minimum radius of curvature, the contact at the bottom results in a tensional force which pulls the falling part downward

Chain paradoxes

“…This solution was already common knowledge by the early 1950s [33] and also received some experimental confirmation, mixed with the voicing of some concerns as to the validity of the assumption that the chain falls with acceleration g, or freely, as one might also say [34]. 13 It was indeed shown experimentally by Hamm & Géminard [10] that a chain does not fall freely on a supporting plane, but with an acceleration greater than g. This naturally led them to envision the existence of a dynamical force pulling the chain down, which can only be exerted by the solid surface on which the chain is accumulating, and for which an explicit representation was also proposed, based on a model for the growth of the accumulating coil [10]. As remarked in [9], this force is for all intents and purposes a tension exerted by the supporting surface, pulling downward the falling chain.…”

“…Making use of (5.10), it is a simple matter to show thaṫ 25) which coincides with (13) of [10]. Sinceẏ 0, (5.25) requires that for the total dissipation to be non-negative (which is the same as requiring that W s + W * + 0), f must obey the inequality f …”

“…10 When a chain is described as a one-dimensional, inextensible string, as we did above, and a shock propagating through it is envisaged as a localized mass transfer with rate λ|ṡ 0 | associated with the energy supply W s , (4.3) directly suggests taking the rate of change of T aṡ 4) so that (4.2) then justifies (4.1), after setting…”

“…which coincides with (5) of [10], provided that f is set equal to the parameter γ introduced there. Accordingly, by (5.6), (5.4a) and (5.1), we can write the tension τ − exerted on the falling chain at x 0 as τ…”

“…Comparing our approach to the problem of a falling chain with the one followed in both [9,10], by far the most complete available in the literature, we may say that the latter essentially amount to assuming (5.11) as a law for the tension τ − , which I resist calling 'constitutive', as it does not represent a direct material property, whereas in the approach followed here (5.11) is a consequence of (5.1), which in turns derives from (2.25) and (2.27), the latter being a genuine constitutive law.…”