Mechanics dual disc problem

[Arthur]

Two discs of same radius and thickness but different density are joined to make one disc. it is then placed on an incline with high coefficient of friction. At what ratio of densities will the disc not move? (ρ1 > ρ2)

[Alisher (Alisher Yerkebayev)]

the center of mass of the disc and point of contact should lie in one vertical line

[Arthur]

but how can we proceed from this, could explain a bit more?

[Alisher (Alisher Yerkebayev)]

First you need to calculate the position of the center of mass of the semi-disc using \displaystyle x_C = \frac{1}{m}\int x dm, then you find the center of mass of these two semi-discs. The angular position of the disc can be varied, but the ratio \rho_1/\rho_2 should be high enough you could find an extreme angle at which center of mass can lie in the same line with the contact point

i’ve got an answer \rho_1/\rho_2 = 12.23

[Arthur]

Thanks!