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shows the values Bm, the relative amplitude corresponding

to the oscillation mode m, for m from 1 to 25 and 0.025.The graph in Fig. 3 shows the calculated values of hori-

zontal deflection Yof the free of end of the chain (X0) as

a function of dimensionless time . The results were calcu-

lated with 0.025. The remaining parameters,VandL, are

absorbed into the dimensionless variables. Results for largervalues of show the same time between kicks, but withincreased amplitude and increased duration of the kicks.

Note that the free end kicks whenever is an integer mul-tiple of 4 and this corresponds to tbeing an integer multiple

of 4L/g since t/L/g). Thus the solutions of the ODEmodel and PDE model are in close agreement in predictingthe time between kicks. The PDE model also accurately pre-dicts the successive directions of the kicks as described in theIntroduction.

To obtain quantitative data on the motion of the chain, wemade a movie at 30 frames per second. We then played themovie one frame at a time, input the horizontal deflectioninto a PC, and graphed the results. These are shown in Fig. 4for a 4-ft chain. The vertical axis is labeled on the left withtime in seconds and on the right with dimensionless time.The horizontal axis is the horizontal deflection of the free

end in inches. Note that for a 4-ft chain, the experimentalround-trip time is 1.41 s and this agrees with both the ODEmodel and the PDE model to within a few hundredths of asecond. The general shape of the graph of the experimentalresults Fig. 4 is in agreement with the PDE solutionFig. 3.

Figure 5 shows the calculated waveforms of the entirechain, as snapshots for six values of dimensionless time dur-ing the first round trip, that is, graphs ofYvs Xfor six valuesofin the range from 0 to 4. Alpha was chosen to be 0.025

for these snapshots. The deflection of the bottom link (X

0) corresponds to that shown in Fig. 3. Also it is clear that

the wave does not retain the same shape as it travels alongthe chain.

FINAL REMARKS

The ODE model can be used in an introductory Physics orCalculus course. The experiment is easy to set up, and therequired background is available after the students havecompleted introductory wave theory and integration. Weused #16 single steel jack zinc plate chain from the localhardware store. The chain was attached to a ceiling tile bracewith a bent paper clip. To see some details of the motionwithout a camera, we illuminated the chain with the lightfrom an overhead projector and observed the chains shadowon a white board. The initial equilibrium position of the

shadow was marked on the white board.The PDE model makes a nice capstone project for students

who have studied solutions of partial differential equationsusing separation of variables. Additional results can be ob-tained when the initial velocity is assumed to be a Dirac deltafunction, and, in this case, the computations are similar tothose above.

Another variation in the initial conditions is to give thechain an initial deflection with zero initial velocity. For ex-

ample if the initial deflection is a line given by Yk(X

1) , then a series expansion analogous to Eq. 7 can befound, and for this case the series is dominated by the first

Fig. 2. Relative amplitude,B m, for the first 25 modes of oscillation.

Fig. 3. PDE solution for the deflection of the free end versus time with

0.025.

Fig. 4. Experimental data from the movie showing the deflection of the free

end versus time.

766 766Am. J. Phys., Vol. 68, No. 8, August 2000 Herb Bailey

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term. Thus the oscillation is quite close to the fundamental

mode. This is an easy initial condition to impose experimen-tally, and the period of oscillation will be close to the fun-damental period.

A surprising result is that the ODE model predicts the timebetween kicks within a few hundredths of a second, in spiteof the fact that the wave does not even come close to retain-ing its form as it travels up and down the chain, as wasassumed for this model. Perhaps one could find a more real-istic model that would lead to the same differential equation.

Because of the antiquity of the hanging chain problem, itwas surprising to me and to the editor and referees of thisJournal that none of us is aware of any previous discussionof the subjects treated in this paper. Although such a discus-sion may exist somewhere in the literature, a search of avail-

able mechanics texts and the pages of this and other journalshave not turned up any such discussion.

ACKNOWLEDGMENTS

The author thanks Professor Mike Moloney and ProfessorCharles Joenathan for their help with the motion pictures. Ialso thank the referees for their valuable suggestions.

1J. Satterly, Some experiments in dynamics. Am. J. Phys. 1 8, 411413

1950.2I. Freeman, Propagation of a transverse pulse along a hanging chain,

Phys. Teach. 15, 5451977.3

J. Cannon and S. Dostrovsky, The Evolution of Dynamics: VibrationTheory from 1687 to 1742 Springer-Verlag, New York, 1981, pp. 78,

5357.4H. Lamb, Higher Mechanics Cambridge U.P., London, 1929, pp. 225226.

5E. Routh, The Advanced Part of a Treatise on Dynamics of a System of

Rigid Bodies Dover, New York, 1955, p. 404.6J. P. McCreech, T. L. Goodfellow, and A. L. Seville, Vibration of a

hanging chain of discrete links, Am. J. Phys. 43, 6466481975.7D. A. Levinson, Natural frequencies of a hanging chain, Am. J. Phys.

45, 680681 1977.8A. Western, Demonstration for observing J0(x) on a resonant rotating

vertical chain, Am. J. Phys. 48, 54561980.9C. Wylie,Advanced Engineering MathematicsMcGrawHill, New York,

1975, 4th ed., pp. 388427.10C. Edwards and D. Pinney, Differential Equations and Boundary Value

ProblemsPrenticeHall, Englewood Cliffs, NJ, 1999, 2nd ed., pp. 578,712713.11C. Trantner,Bessel Functions with Some Physical Applications Hart, New

York, 1969, pp. 2729.

E-MAIL

Ive simply reached the point where I cant answer my E-mail, nice or not. If somebody really

wants to reach me, Im listed in the phone book in Oakland. Send me a letter. If its a nastyenough letter, I wont respond to it either.

Matthew L. Wald, Conversations with Clifford Stoll, New York Times, April 30, 1995.

Fig. 5. Calculated waveforms with0.025 for various values of time.

767 767Am. J. Phys., Vol. 68, No. 8, August 2000 Herb Bailey