The problem of the thrown string

Abstract

ULAM¹ reports that “there was another problem which Fermi wanted to study but which we somehow never came to formulate well or to work on. He said one day, ‘it would be interesting to do something purely kinematical. Imagine a chain consisting of very many links, rigid, but free to rotate around each other. It would be curious to see what shapes the chain would assume when it was thrown on a table, by studying purely the effects of the initial energy and constraints, no forces’.” Synge² posed the problem of estimating the average D̄ of the rectilinear distance D between the ends of a flexible string of length L when thrown down at random on a table. Later³ he published conceptual models and some experiments which together increased the challenge (recently restated by Kingman⁴): the models yielded values of D̄/L equal either to 0 or 1 unity, whereas in three experimental series by different throwers of different strings, the values obtained were 0.32 (100 throws), 0.34 (50 throws) and 0.38 (60 throws). (Only for the last of these was the value of D̄²/L² also recorded, at 0.18.) Synge wrote³: “the agreement of the results is rather striking and one might hope to explain the numerical value (say 1/3) on theoretical grounds....I hoped that someone would suggest a way of treating directly a system with an infinite number of degrees of freedom, but perhaps no such way exists”. Here we propose a solution along the desired lines by kinematical means.