Simulating a Chain Sliding off a Desktop

Abstract

Consider a chain consisting of n+ 1 point-like particles of the same mass (equal to 1, by a choice of units), connected by n massless, perfectly flexible, inelastic links of equal length (also equal to 1). The chain is laid on a table top, straight and perpendicular to the edge. Then the first particle is pulled (together with the rest of the chain) gently over the edge of the table. This causes the chain to start sliding down, due to gravity (also of unit magnitude), in a frictionless manner [1]. Let us assume now that k particles have already left the table, and that their positions are defined by k angles j1, j2, ..., jk by which the first k links deviate from the vertical, and by s, the distance of the last particle to have left the table edge (jk is thus the angle of the hanging part of the corresponding link; the rest of it still lies flat on the table). Collectively, these k + 1 variables are known as generalized coordinates [2], as they fully specify the position of every particle. Now, using rectangular coordinates with the origin at the table’s edge, the x axis oriented vertically downward, and the y axis pointing horizontally, away from the table, we can compute the corresponding x and y coordinates of each particle by