As a second example of solving for the motion and constraint forces,
we considered a disk rolling on a fixed incline plane without
slipping. We counted fully unconstrained degrees of freedom for the
two extended objects -- disk and incline plane - in three
dimensions. For each extended object we have 6 degrees of freedom --
3 describing translation of the center of mass, and 3 describing
rotation about the center of mass. So the system naively has 12
degrees of freedom. We then reduced that to 1 degree of freedom with
our constraints -- fixed incline plane, disk translation along the
(infinitely thin) incline plane, rotational motion remaining in the
plane of the incline plane, and rolling without slipping. We found a
proper coordinate, derived the Lagrangian, and obtained and solved
E-L. We found E-L just as for simple translation down an incline
plane, but with an effective mass 
, where 
for the disk. This makes sense; rotational motion
eats up some of our total energy, making less available to go into
translational kinetic energy -- effectively increasing the inertia of
the rolling disk with respect to translation.
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KB