The mysterious spinning cylinder—Rigid-body motion that is full of surprises

Abstract

We explore the steady-state rotational motion of a cylinder on a flat horizontal surface from a pedagogical perspective. We show that the cylinder's inclination angle depends on its rotational velocity in a surprisingly subtle manner, including both stable and unstable solutions as well as a forbidden region with no (real) solutions. Moreover, the cylinder's behavior undergoes a qualitative change as the aspect ratio decreases below a critical value. Using a high-speed video, we measure the inclination angle as a function of rotation speed and demonstrate good agreement with the theoretical predictions. All aspects of the analysis are well within the capabilities of undergraduate students, making this an ideal system to explore in courses or as an independent project.

A hemispherical tippe top

“…The quantity X P is zero when the hemisphere is inclined at an angle where G is directly above the contact point, in which case G reaches its maximum height and the torque about G is then zero. That type of behaviour has previously been observed with cylinders precessing in a steady state [14].…”

Complete inequivalence of nonholonomic and vakonomic mechanics: rolling coin on an inclined plane

“…Rotational dynamics is full of surprises and rich in examples that violate our intuitive expectations, cases in point being the tippe top [40], the spinning cylinder [41], the asym-FIG. 2: Typical trajectories of the rolling coin's center of mass according to nonholonomic mechanics for the initial conditions (38).…”