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J.-C. Géminard

    The problem of bodies falling in a gravitational field is so old that it is difficult to imagine anything new being added to it. However, the development of simulation methods has led to the analysis of a few interesting cases that are difficult to analyze analytically. The dynamics of a falling chain is among them.

    In the first variation of the falling chain problem considered here, the chain is initially attached at both ends to a horizontal support. Then, as one of the ends is released, the chain begins to fall. The case in which the horizontal separation Dx between the ends of the chain is zero, that is, the chain is tightly folded, has an analytical solution. We describe the results of experiments in which the ends of the chain of length L are initially located at the same level but their horizontal separation Dx is variable. We were able to record the entire shape of the consecutive conformations of the falling chain. From the recorded conformations, we are able to extract quantitative data for the time dependence of the velocity and acceleration of the chain tip. In addition, we formulated the complete equations of motion for the chain and integrated them numerically, arriving at a quantitative comparison between the experimental and numerical results [Tomaszewski, 2006]. We complemented this study by measurements of the horizontal and vertical component of the force applied by the chain at the fixed end [Géminard, 2008].

Left: Sketch of the experimental setup. Right: Successive conformations of the falling chain versus time. The left end of the chain remains attached to the frame, while the right end is free to fall due to gravity. In (b), (c), and (d), white lines have been sketched on the photographic sequence to connect the free falling end of the chain to a freely falling mass or the last five images before the maximum extension of the chain. The length L=1.022 m, the time between successive images is 1/50 s, and the initial separation between the chain ends is (a) Dx=1.019 m (b) 0.765 m (c) 0.510 m and (d) 0.255 m.

    In the second variation of the falling chain problem considered here, a vertically hanging chain is released from rest and falls due to gravity on a scale pan. We discuss the various experimental and theoretical aspects of this classic problem. Careful time-resolved force measurements allow us to determine the differences between the idealized problem and its implementation in the laboratory. We observe that, in spite of the upward force exerted by the pan on the chain, the free end at the top falls faster than a freely falling body. Because a real chain exhibits a finite minimum radius of curvature, the contact at the bottom results in a tensional force, which pulls the falling part downward [Hamm, 2010].

Left: Experimental situation - The chain is falling vertically on a scale pan. During the fall, the apparent weight is predicted to be three times the deposited weight. Right: Snapshots of the falling ball chain with the simultaneous drop of a steel ball. The free end of the chain (dark vertical line) falls faster than the steel ball (dark point on the right-hand side). The chain falls directly on a flat surface; note the formation of a compact heap at the bottom. Images are taken at intervals of 40 µs.

The motion of a freely falling chain tip,
W. Tomaszewski, P. Pieranski, and J.-C. Géminard, Am. J. Phys. 74 (2006) 776-783.

The motion of a freely falling chain tip. Force measurements,
J.-C. Géminard and L. Vanel, Am. J. Phys. 76 (2008) 541-545.

The weight of a falling chain, revisited
E. Hamm and J.-C. Géminard, Am. J. Phys. 78 (2010) 828-833.