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:
or the angle between the vertical and the line connecting the center of the disk with the center of the ring;
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 the radii of the disk and ring:
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:
are the moments of inertia of the ring and disk.
The equations of motion can be derived from the Euler–Lagrange equations:
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.