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.
1.
Chain Drop Experiment
,”
YouTube
, https://www.youtube.com/watch?v=1erU-Cwcl2c.
2.
M. G.
Calkin
and
R. H.
March
, “
The dynamics of a falling chain I
,”
Am. J. Phys.
57
,
154
157
(
Feb.
1989
).
3.
J. S.
Miller
, “
An extension of the falling chain problem
,” and
J.
Satterly
, “
Falling chains
,” letters to the editor,
Am. J. Phys.
19
,
383
384
(
Sept.
1951
).
4.
W. A.
Heywood
,
H.
Hurwitz
, and
D. Z.
Ryan
, “
Whip effect in a falling chain
,”
Am. J. Phys.
23
,
279
280
(
May
1955
).
5.
C. E.
Mungan
, “
The challenge of explaining why
,”
Phys. Teach.
54
,
388
389
(
Nov.
2016
).
6.
C. A.
de Sousa
and
V. H.
Rodrigues
, “
Mass redistribution in variable mass systems
,”
Eur. J. Phys.
25
,
41
49
(
Jan.
2004
).
7.
The appendix is available at TPT Online, https://doi.org/10.1119/1.5033873 , under the Supplemental tab.
8.
W.
Steiner
and
H.
Troger
, “
On the equations of motion of the folded inextensible string
,”
Z. Angew. Math. Phys.
46
,
960
970
(
Nov.
1995
).
9.
E.
Hamm
and
J.-C.
Géminard
, “
The weight of a falling chain revisited
,”
Am. J. Phys.
78
,
828
833
(
Aug.
2010
).
10.
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
(
July
2011
) with a Supplemental Simulation on p. 981 of the Sept. 2011 issue.
11.
W.
Tomaszewski
and
P.
Pieranski
, “
Dynamics of ropes and chains I: The fall of the folded chain
,”
New J. Phys.
7
, (
45
),
1
17
(
Feb.
2005
).
12.
W.
Tomaszewski
,
P.
Pieranski
, and
J.-C.
Géminard
, “
The motion of a freely falling chain tip
,”
Am. J. Phys.
74
,
776
783
(
Sept.
2006
).
13.
J.-C.
Géminard
and
L.
Vanel
, “
The motion of a freely falling chain tip: Force measurements
,”
Am. J. Phys.
76
,
541
545
(
June
2008
).
14.
M.
Schagerl
,
A.
Steindl
,
W.
Steiner
, and
H.
Troger
, “
On the paradox of the free falling folded chain
,”
Acta Mech.
125
,
155
168
(
March
1997
).
15.
D.
Kagan
and
A.
Kott
, “
The greater-than-g acceleration of a bungee jumper
,”
Phys. Teach.
34
,
368
373
(
Sept.
1996
).
16.
H.
Biezeveld
, “
The bungee jumper: A comparison of predicted and measured values
,”
Phys. Teach.
41
,
238
241
(
April
2003
).
17.
C. W.
Wong
and
K.
Yasui
, “
Falling chains
,”
Am. J. Phys.
74
,
490
496
(
June
2006
).
18.
When points A and O in Fig. 1 are far apart, sections AB and DE will initially be exact catenaries [as in
F.
Behroozi
, “
In praise of the catenary
,”
Phys. Teach.
56
,
214
217
(
April
2018
)] only for a uniform string rather than for a chain having discrete links.
19.
See Fig. 2 in
T.
Pritchett
,
R. C.
Nelson
,
T. J.
Creamer
, and
B. G.
Oldaker
, “
Does an ideal wheel really rotate about its instantaneous point of contact?
Phys. Teach.
36
,
167
170
(
March
1998
).
20.
The top fold in a chain fountain can likewise be thought of as going around an ideal pulley. See
J.
Pantaleone
, “
A quantitative analysis of the chain fountain
,”
Am. J. Phys.
85
,
414
421
(
June
2017
).
21.
The tangential acceleration of the entire chain in the co-moving frame is a/2, which is the average in the lab frame of the acceleration a of section DE and the zero acceleration of section AB.
22.
The other three forces and accelerations introduced in connection with Fig. 3 (namely mg, a/2, and ac) have zero net tangential component and so do not appear in Eq. (2).
23.
24.
P.
Hewitt
, “
Chain drop revisited
,”
Phys. Teach.
54
,
329
(
Sept.
2016
) and 439 (Oct. 2016).
25.
P.
Krehl
,
S.
Engemann
, and
D.
Schwenkel
, “
The puzzle of whip cracking uncovered by a correlation of whip-tip kinematics with shock wave emission
,”
Shock Waves
8
,
1
9
(
Feb.
1998
).
26.
Conservation of mechanical energy for the falling chain from the instant before release to the instant shown in Fig. 1 is expressed as 2(λgL2)L4=12(λgLy2)v2λgLy2(y+Ly4)λgL+y2(y+L+y4), which rearranges into Eq. (10).

Supplementary Material

AAPT members receive access to The Physics Teacher and the American Journal of Physics as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.