Understanding how a falling ball chain can be speeded up by impact onto a surface

Abstract

When a falling ball chain strikes a surface, a tension is created that pulls the chain downward. This causes a downward acceleration that is larger than free-fall, which has been observed by recent experiments. Here, a theoretical description of this surprising phenomenon is developed. The equation of motion for the falling chain is derived, and then solved for a general form of the tension. The size of the tension needed to produce the observed motion is relatively small and is explained here as coming from the rotation of a link just above where the chain collides with the surface. This simple model is used to calculate the size of the tension in terms of physically measurable quantities: the length and width of a link, the maximum bending angle at a junction, the inclination angle of the surface and the coefficients of friction and restitution between the chain and the surface. The model's predictions agree with the results of current experiments. New experiments are proposed that can test the model.