Equilibrium and movement of a cylinder with asymmetric mass centre on an inclined plane

The effect of friction is to make the cylinder roll down the ramp rather than slide.

To find an equilibrium angle, use virtual work.

If $\phi$ changes by a small amount $d\phi$, as the cylinder rolls, then everything goes down a little (neglecting at first the small interior cylinder's upward movement) because you're moving down the ramp. You move $R d\phi$ down the ramp, and lose elevation $\sin \Phi R d\phi$. The total work done by gravity is $(m_1 + m_2) g \sin\Phi R d\phi$

On the other hand, the interior cylinder rises with respect to the center of the big cylinder by an amount $(R - R_2) d\phi$. The work done by gravity on the little cylinder is $g m_2 (R-R_2) d\phi$.

Equilibrium is achieved when these are equal, so

$m_2 (R - R_2) = (m_1 + m_2) R \sin\Phi$

or

$\sin\Phi = \frac{m_2(R-R_2)}{(m_1+m_2)R} = \frac{r}{R}$