When a falling chain strikes a surface, it can accelerate downwards faster than free-fall. This counterintuitive effect occurs when a tension is created in the chain above where it strikes the surface. The size of this tension, and how it is produced, depend on the type of chain used. For a chain made of rods that are slightly tilted from horizontal, the impact-induced tension is readily observable. Here are reported experimental observations on such a falling chain for two different situations: when the chain strikes an inclined surface, and when the chain's mass density decreases with height. It is found that both of these arrangements can increase the downward acceleration. To quantitatively describe these observations, a theoretical model is developed. The model successfully predicts the chain's position and velocity, even when the top end approaches the surface, without any free parameters. The model also predicts that uniform rods are practically the best for producing large tensions.

1.
Comte de Buquoy,
Exposition d'un Nouveau Principe Général de Dynamique, dont le Principe des Vitesses Virtuelles n'est qu'un Cas Particulier
(
Paris
,
1815
).
2.
S. D.
Poisson
, “
Mouvement d'un Syst`eme de Corps, en supposant les masses variables
,”
Bull. Sci. Soc. Philomat.
60
,
60
62
(
1819
).
3.
J.
Miller
, “
An extension of the falling chain problem
,”
Am. J. Phys.
19
,
383–384
(
1951
).
4.
J.
Satterly
, “
Falling chains
,”
Am. J. Phys.
19
,
383
384
(
1951
).
5.
B.
Bernstein
,
D. A.
Hall
, and
H. V.
Trent
, “
On the dynamics of a bull whip
,”
J. Acous. Soc. Am.
30
,
1112–1115
(
1958
).
6.
M. G.
Calkin
and
R. H.
March
, “
The dynamics of a falling chain: I
,”
Am. J. Phys.
57
,
154–157
(
1989
).
7.
M.
Schagerl
,
A.
Steindl
,
W.
Steiner
, and
H.
Troger
, “
On the paradox of the freely falling folded chain
,”
Acta Mech.
125
,
155
168
(
1997
).
8.
A.
Goriely
and
T.
McMillen
, “
Shape of a Cracking Whip
,”
Phys. Rev. Lett.
88
,
24430145
(
2002
).
9.
T.
McMillen
and
A.
Goriely
, “
Whip waves
,”
Physica D
184
,
192–225
(
2003
).
10.
C. W.
Wong
and
K.
Yasui
, “
Falling chains
,”
Am. J. Phys.
74
,
490
496
(
2006
).
11.
E.
Hamm
and
J. C.
Geminard
, “
The weight of a falling chain, revisited
,”
Am. J. Phys.
78
,
828
833
(
2010
).
12.
A.
Grewal
,
P.
Johnson
, and
A.
Ruina
, “
A chain that speeds up, rather than slows, due to collisions: How compression can cause tension
.”
Am. J. Phys.
79
,
723
729
(
2011
);
A.
Grewal
,
P.
Johnson
, and
A.
Ruina
, “
Erratum
,”
Am. J. Phys.
79
,
981
(
2011
).
13.
A.
Grewal
,
P.
Johnson
, and
A.
Ruina
, “
A chain that pulls itself onto the table it falls on
,” <http://ruina.tam.cornell.edu/research/topics/fallingchains/index.html> (
2011
).
14.
J. A.
Hanna
and
H.
King
, “
An instability in a straightening chain
,” e-print arXiv:1110.2360 [physics.flu-dyn] (
2011
).
15.
J. A.
Hanna
and
C. D.
Santangelo
, “
Slack dynamics on an unfurling string
,”
Phys. Rev. Lett.
109
,
134301
(
2012
).
16.
S.
Mould
, “
Self-siphoning beads
,” <http://stevemould.com/siphoning-beads/> (
2013
).
17.
J.
Blundell
, “
Rope siphon
,” <https://www.youtube.com/watch?v=X7CXzjFVUHQ> (
2013
).
18.
J. S.
Biggins
and
M.
Warner
, “
Understanding the chain fountain
,”
Proc. R. Soc. A
470
,
20130689
(
2014
).
19.
J. S.
Biggins
, “
Growth and shape of a chain fountain
,”
EPL
106
,
44001
(
2014
).
20.
E. G.
Virga
, “
Chain Paradoxes
,”
Proc. Roy. Soc. A
471
,
2173
(
2015
).
21.
J.
Pantaleone
, “
A quantitative analysis of the chain fountain
,”
Am. J. Phys.
85
,
414
421
(
2017
).
22.
N. A.
Corbin
,
J. A.
Hanna
,
W. R.
Royston
,
H.
Singh
, and
R. B.
Warner
, “
Impact-induced acceleration by obstacles
,”
New J. Phys.
20
,
053031
(
2018
).
23.
J.
Pantaleone
and
R.
Smith
, “
A bullet-block experiment that explains the chain fountain
,”
Phys. Teach.
56
,
294
297
(
2018
).
24.
E. G.
Flekkoy
,
M.
Moura
, and
K. J.
Maloy
, “
Mechanisms of the flying chain fountain
,”
Front. Phys.
6
,
84–90
(
2018
).
25.
C. E.
Mungan
, “
Newtonian analysis of a folded chain drop
,”
Phys. Teach.
56
,
298
301
(
2018
).
26.
J.
Pantaleone
, “
Understanding how a falling ball chain can be speeded up by impact onto a surface
,”
Proc. Roy. Soc. A.
475
,
2223
(
2019
).
27.
M.
Denny
, “
A uniform explanation of all falling chain phenomena
,”
Am. J. Phys.
88
,
94–101
(
2020
).
28.
Wikipedia on variable mass systems, <
https://en.wikipedia.org/wiki/Variable-mass_system>.
29.
R.
Resnick
,
D.
Halliday
, and
K.
Krane
,
Physics
(
Wiley
,
New York
,
2002
), Vol.
1
, pp.
149
152
.
30.
References 11 and 22 describe the impact-induced tension using the parameter γ =1-α. See Ref. 26 for why.
31.
See supplementary material at https://doi.org/10.1119/10.0002361 for details of the derivations, a discussion of link lift-off from a horizontal surface, its similarities with link lay-down, and for additional photographs and experimental details.
32.
Tracker is a free video analysis and modeling tool for physics education
, available at <physlets.org/tracker/>.
33.
D. E.
Stewart
, “
Rigid-body dynamics with friction and impact
,”
SIAM Rev.
42
,
3
39
(
2000
).
34.
D.
Stoianovici
and
Y.
Hurmuzlu
, “
A critical study of the applicability of rigid body collision theory
,”
J. Appl. Mech.
63
,
307
316
(
1996
).
35.
W. J.
Stronge
, “
Rigid body collisions with friction
,”
Proc. Roy. Soc. A
431
,
168
181
(
1990
).

Supplementary Material

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