Line integral of friction forces

marellasunny

I was just performing the usual wine glass experiment where I rub my wet finger across the rim and I hear the resonant frequencies from the water column-due to the friction produced from my finger rubbing.
Today I learnt about "conservative" forces from my professor.Since,frictional forces come under the category of non-conservative forces(which I gather means they depend on the path followed),shouldn't the sound produced by the water column vary upon the direction I choose to move my finger i.e C.W direction should produce one tone and A.C.W direction another tone??

1.But,practically,I get the same tone when I rub the finger in both clockwise and anti-clockwise directions.But,by the concept of non-conservative forces and [tex]\oint\!F.dr[/tex],shouldn't I get different tones for the C.W and the A.C.W directions?(since the work done by the frictional forces would vary for the diffferent paths I chose?)Maybe,resonance has something to explain for this,if so could you please elaborate?
2.Can I take the closed path integral for non-conservative forces?It seems illogical since there might be infinite points between many different paths.

Please help!Thanks.