The rolling body paradox: an oscillatory motion approach

Abstract

By means of an example that could be used to stimulate discussion among students, we deal with different fundamental ideas related to rolling motion and give several didactic suggestions on the subject. In addition, using numerical calculations, we study the motion described by a spool when the force applied is directed constantly towards a fixed point. In general, the resulting motion is oscillatory. When small oscillations are considered, the spool behaves as a simple harmonic oscillator. In order to show the oscillatory behaviour predicted by the numerical calculation, we have designed an experimental device similar to the one considered in the theoretical model, in which the oscillatory motion was observed. Resumen. Por medio de un ejemplo que podría ser útil para estimular discusiones entre los alumnos, estudiamos algunas ideas fundamentales relacionadas con el movimiento de rodadura, y proponemos diversas sugerencias didácticas sobre este problema. Además, utilizando cálculo numérico, estudiamos el movimiento descrito por un carrete cuando la fuerza aplicada está dirigida constantemente hacia un punto fijo. En general, el movimiento resultante es oscilatorio. Si se consideran oscilaciones pequeñas, el sistema se comporta obviamente como un oscilador armónico simple. Con objeto de ilustrar el comportamiento oscilatorio que se obtiene a partir de los cálculos numéricos, hemos diseñado un dispositivo experimental similar al considerado en el modelo teórico en el que hemos observado el movimiento oscilatorio descrito por el carrete.

Non-slipping domains of a pulled spool

“…This behavior of the system is summarized in the parameter plot (figure 8). The limits of the gap between the two non-slipping domains can be derived from equation (8)…”

“…Around this angle the friction should not cause any problem, since in the neighborhood of θ m2 the spool moves in a non-slipping domain. The same idea can also be applied to the oscillating spool [8]. Using, for example, the conditions shown in figure 7 (γ = 0.5) and considering a ratio of = 0.…”

Static properties of a spool on an incline: an IBSE lecture proposal

“…with T given by equation (5). By now considering that static equilibrium cannot be attained in the absence of friction, so that the requirement N > 0 in equation ( 11) is a necessary condition, we need to solve a set of inequalities which also includes equation (12).…”

“…ground by exerting a tension on a string wrapped around the inner hollow cylinder of radius r connected to two discs of radius R, as shown in figure 1. This system had already been fully studied in the literature [3][4][5]. In this rather theatral performance, the instructor pretended to be walking with a doggie kept on a leash.…”

A sphere rolling on the inside surface of a cone

“…Many subtle points of dynamics and of mathematical techniques for studying the behaviour of physical systems can be learned by studying the motion of rigid bodies. On this matter see, for example, [3,4,5,6,7,8,9,10]. Part of the interest of today comes from the insight that can be gained on the behaviour of spinning asteroids or artificial satellites and, furthermore, rigid body motion can be chaotic [11,12,13].…”