Disk with a Hole Rolling in a Semicircular Well

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The center of mass of a disk with a hole is offset from the disk's geometric center, so that as the disk rolls in a circular well its motion is irregular.


This Demonstration simulates the motion of a disk of radius with a hole of radius at a distance from the center. Lagrangian mechanics are used to get the equation of motion for the animation and to compute the traces of the center of mass and the geometric centers of the disk and the hole.


Contributed by: Erik Mahieu (April 2014)
Open content licensed under CC BY-NC-SA



Consider a disk of mass and radius . The disk has a hole with radius at a distance from its center. The mass cut out by the hole is .

The potential energy of the disk plus hole is .

The kinetic energy of the system is , where is the moment of inertia of the disk plus hole.

The Lagrangian gives the equation of motion: .

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