Disk with a Hole Rolling in a Semicircular Well

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.


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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 .
is the angular position of the disk in the well and is the angle of rotation of the disk. With the no-slip constraint:
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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