What is the optimal radius for stability in a system of two semicircles?

[Numeriprimi]

Yesterday I was at the Physics Olympiad. For one example I didn't do. I am ashamed! :-( And I want to know how to do it. Please help. It is like complicated homework, so I give it here.

We have two semicircles (as pictured) - http://fykos.cz/rocnik26/obrazky/s5u3_zadani.png

The lower semicircle has a radius r, the upper semicircle has a radius R. r is constant. The system must be stable. What must be R? How to determine R? If we shifted the upper semicircle, with which period will oscillate?

Thanks for your ideas. And sorry for my bad English.

Numeriprimi

 

[TSny]

Hello. Please show some attempt at a solution (or at least state some general ideas that you think would help solve this problem).

 

[Numeriprimi]

A stable system must be characterized by the fact that even after a few oscillations must return to the same position. The upper semicircle can not be too big - large oscillations and fell. Equally can not too small. Some golden middle... I think about it still and if I thought something better yesterday, I didn't ask about it :-(

 

[TSny]

Stable equilibrium means that if you displace the upper semicircle a small amount from the initial position and let it go, then the forces (or torques) acting on the semicircle will cause the semicircle to move back toward the initial position.

 

[Numeriprimi]

Hmmm, vela, thanks for infraction... :-( However, I'm fifteen years, I know just the basics of oscillation and I can not know any great procedure ... And that is why I am asking here, in English - it is really hard for me! In my country isn't a physics forum. Tasks at school and competitions of physics are a big difference! So, explain it to me that someone? When I got negative points?