Rigid body kinetic energy+ constraints (upper level classical mechanics)

[fluidistic]

okay

NOW let's visualise:

fisrt i forgot 'Wc' . Just forget that it exists in the question.

i only know Wcm and Vcm:

NOW according to pure rolling Vcm = Wcm * a;

now the only twist is that this cyllinder is rolling not on a plane but in cyllindrical surface

We Find time period(Tc) of the cyllinder = 2*pi*(R-a)/Vcm

THIS GIVES Wc as::

Wc = 2*pi/Tc

= Vcm/(R-a)

=Wcm *a/(R-a)

I understand everything if I assume that Vcm = Wcm * a. Could you explain a bit more how do you get it?
You considered a cylinder of radius a rolling over a plane (with rolling and not slipping?)? In those case I don't understand the meaning of $\omega _{CM}$. I mean the cylinder rotates around an axis that passes through the center of mass over a plane. Since the center of mass does not rotate, omega of center of mass is worth 0... What am I misunderstanding?

 

[darkxponent]

i took Wcm is not the angular velocity of centre of mass. It is angular velocity of the cyllinder about its centre of mass and similarly Wc is the angular velocity if the cyllinder about the centre of the cyllindrical surface