The Falling Chain

Luu ChauKayla 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•We neglect individual link oscillations to further explain the down-pulling force created on the fold•This force creates an equal & opposite reactive force pointing downward, adding to the gravitational 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 free-falling 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