Rolling of asymmetric disks on an inclined plane

TL;DR

This paper investigates the dynamics of asymmetric disks rolling down an inclined plane, emphasizing the importance of choosing an inertial point to correctly derive the equations of motion, correcting common misconceptions.

Contribution

It provides a straightforward method for deriving the correct equations of motion for asymmetric rolling bodies on inclined planes, clarifying the role of the chosen reference point.

Findings

Correct equation of motion obtained using an inertial reference point

Clarification of misconceptions about contact point analysis

Demonstration of the method's applicability to asymmetric disks

Abstract

In a recent papers, Turner and Turner (2010 {\em Am. J. Phys.} {\bf 78} 905-7) and Jensen (2011 {\em Eur. J. Phys.} {\bf 32} 389-397) analysed the motion of asymmetric rolling rigid bodies on a horizontal plane. These papers addressed the common misconception that the instantaneous point of contact of the rolling body with the plane can be used to evaluate the angular momentum $\mathbf L$ and the torque $\boldsymbol\tau$ in the equation of motion $d\mathbf L/dt = \boldsymbol\tau$. To obtain the correct equation of motion, the "phantom torque" or various rules that depend on the motion of the point about which $\mathbf L$ and $\boldsymbol\tau$ are evaluated were discussed. In this paper, I consider asymmetric disks rolling down an inclined plane and describe the most basic way of obtaining the correct equation of motion; that is, to choose the point about which $\mathbf L$ and $\boldsymbol\tau$ are evaluated that is stationary in an inertial frame.