9853

Disk Rolling inside a Rotating Ring

This Demonstration shows a disk rolling, without slipping, inside a rotating ring.
The system has two degrees of freedom: the angles of rotation of the ring and of the center of the disk within the ring. A brake shoe can stop the ring, reducing the system to only one degree of freedom: a disk rolling inside a stationary ring.

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DETAILS

A steel ring of mass is supported at the bottom by two frictionless Teflon rollers. This ring can be rotated with an initial angular speed . Inside the ring, a steel disk of mass is released from angular position with an angular speed . The friction between the ring and disk is sufficient to avoid sliding.
Lagrangian mechanics are used to extract the equations of motion for this system with two degrees of freedom:
1. or the angle between the vertical and the line connecting the center of the disk with the center of the ring;
2. or the angle of rotation of the ring around its center.
Since no slipping is allowed between the disk and the ring, or the angle of rotation of the disk around its center can be calculated from and and the radii of the disk and ring: , where is the inner radius of the ring and is the radius of the disk.
The Lagrangian of the system is the difference between the kinetic and potential energies:
,
where and are the moments of inertia of the ring and disk.
The equations of motion can be derived from the Euler–Lagrange equations:
,
,
resulting in:
,
.
In order to better verify the rotations of ring and disk, you may have to slow down the animation to eliminate stroboscopic effects. Run the bookmark "sliding check" at different speeds to convince yourself.
See also a similar Demonstration by Christoph Meyer: "Solid Cylinder Rolling in a Turnable Hollow Cylinder".
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