Tension in rope coiling on a rotating plane

Abstract

Coiling of rope, fed from a height onto a rotating plane, progresses through a sequence of shapes, a hypotrochoid to an epitrochoid to a circle as the frequency of the plane increases. Feeding velocity controls the rate of length deposition on a plane and frequency of the plane controls the rate of length transfer from a contact point, where rope first touches a plane. Secondary loops of a hypotrochoid or an epitrochoid are formed when the deposition rate is faster than the transfer rate. When these two rates are comparable, secondary loops disappear and the shape returns to a circle like in rope coiling on a static plane. In a reference frame co-rotating with rope, the Coriolis and centrifugal forces act only at the contact point, not extending to the portion of rope far above a rotating plane. For a small deflection of rope, tension is inferred from the equations of motion by using the radius and frequency of a primary loop measured in experiments. Tension changes continuously at both the hypotrochoid–epitrochoid transition and the epitrochoid–circle transition, reminiscent of the features of a second-order phase transition.