Newtonian Analysis of a Folded Chain Drop

Abstract

Consider a chain of length L that hangs in a U shape with end A fixed to a rigid support and free end E released from rest starting from the same initial height (call it y = 0) as A. Figure 1 sketches the chain after end E has fallen a distance y. Points O and A are assumed to be close enough to each other and the chain flexible enough that the radius of curvature r at the bottom point C can be taken to be negligibly small (compared to the length of the chain). The problem is to compare the speed of descent v(y) = dy/dt of the free end E of the chain to the speed vfree(y)=2gy of a free-falling point mass that has descended the same distance y. If v(y) > vfree (y) for all y > 0, then, in a race to fall any arbitrary distance Y (where 0 < Y < L), the chain end E will always beat a simultaneously released point mass, because the fall time t for E will be shorter than tfree for the point mass, t=∫0Ydyv(y)<tfree=∫0Ydyvfree(y).(1)Experimentally, this outcome of the race is observed in a “Veritasium” video.

A Lagrangian approach to bungee jumping

“…But the factor ½ in the speed of the bend of the hanging chain is easily overlooked (e.g. in [8]; see also [9]) and is from physics point of view not easily underpinned. Wong and Yasui [10] have shown that the factor ½ is required for energy conservation and that the mass transfer at the fold of the chain must take place elastically at the mean velocity when the falling part of the folded chain and the part of the chain that hangs still are both conservative systems.…”

Unfolding the dynamics of free-falling folded chain: Experiments and simulations

“…Due to momentum balance, the falling chain tip accelerates faster than the gravitational acceleration 𝑔 as the moving part becomes smaller over time [9]. Due to this counter-intuitive fact, falling chains have attracted the research community's attention in recent years, [10,11]. To this end, several experimental and numerical efforts have been reported in the literature.…”

Understanding and enhancing the impact-induced tension of a falling chain

“…Well-known examples of this are the crack of a whip 5,8,9 or the fall of a folded chain. 3,4,6,7,10,25 In the last few years, two new motion-enhancing systems have been discovered, occurring when a chain is lifted off of or laid down onto a surface. The former is responsible for the chain fountain, [14][15][16][17][18][19][20][21]23,24 where a chain leaps out of a beaker as it falls to the ground.…”