A cylinder whose cross section is represented below is placed on an inclined plane. I would like to determine the maximum slope of the inclined plane so that the cylinder does not roll. The mass centre (CM) of the cylinder is at a distance r from the central axis. The cylinder consists of a cylindrical shell with mass $m_1$ and a smaller cylinder with mass $m_2$ placed away from the axis and rigidly attached to the larger cylinder. What is the influence of friction? Is it possible to establish the law of the movement? I think that the piece may roll upwards until it stops.
The figure was copied from Projecto Ciência na Bagagem -- Cilindro desobediente
EDIT: Depending on the initial conditions is it possible to find the highest point the cylinder rolls to, before stopping?
EDIT2: From Institute and Museum of the History of Science -- Cylinder on inclined plane [another cylinder]
"When placed on the inclined plane, [another] cylinder tends to roll upward, coming to a halt at a well-determined position."