# The Falling Chain - Full

```The Falling Chain
Luu Chau
Kayla Chau
Jonathan Bernal
Question: What falls faster?
• What falls faster?
– The end of a vertically hanging folded chain
– A free falling object (tennis ball)
Physical Experiment
• Camera takes multiple
pictures in a given time
increment
•After first flash from
camera, detector switches
open the circuit
•Circuit gives charge to
magnets holding a steel
ring (object) and end of
chain
•As steel ring and end of
chain fall, camera takes
multiple pictures, marking
position
a: End of chain
b: Steel Ring
(object)
c: Mathematical
model of a
freefalling object
End of Chain Wins
• On a physical level, the end of a chain falls
faster than a free falling object
• A down-pulling force at the fold of the chain is
created giving the chain extra pull as it falls
Chain Fold
•Close-up representation
of the fold in a falling
chain
oscillations to further
explain the down-pulling
force created on the fold
•This force creates an
equal & opposite reactive
force pointing downward,
force
Mathematical Level
•Chain divided into parts:
-Falling section of chain (La)
-Motionless section of chain
(Lb)
•As time goes on
-La will decrease
-Lb will increase
Chain Equations
•By assuming
that energy is
conserved, we
can come up with
equations for
velocity,
acceleration, and
time
Free Falling Object
•We assume no air
resistance when modeling
this experiment on Matlab
Object Equations
•We will use these
equations to
model the freefalling object
Work Cited
• M. Schagerl, A. Steindl, W. Steiner, and H. Troger, “On the paradox of the
free falling folded chain,” Acta Mech. 125, 155-168 1997.
• W. Tomaszewski and P. Pieranski, “Dynamics of ropes and chains I. The fall
of the folded chain,” New J. Phys. 7, 45-61 2005.
• W. Steiner and H. Troger, “On the equations of motion of the folded
inextensible string,” Z. Angew. Math. Phys. 46, 960-970 1995.
• M.G. Calkin and R. H. March, “The dynamics of a falling chain I,” Am. J.
Phys. 57, 154-157 1989.
• T. McMillen and A. Goriely. “Shape of a Cracking Whip,” Phys. Rev. Lett,
88(24) 2002
```

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