Question on Rotational Dynamics of Yoyo

[fchen720]

Homework Statement

Please see attached picture.

Homework Equations

Shown in picture

The Attempt at a Solution

It's a textbook example that has a solution.
At the middle right hand side of the picture, they say "From Eq... Vcm = Rw"
(Where 'cm' is centre of mass)
I don't understand why this is true.

Previously in the textbook they talked about rolling without slipping, where they mentioned a rubber tire rolling on cement. They reasoned that the velocity of the point of contact is zero relative to the road.
I.E Vcontact_point-rel-road = 0
Then Vcm-rel-road = Vcm-rel-contact_point + Vcontact_point-rel-road = Vcm-rel-contact_point = R * w
(Where w is angular speed and R is radius)
I understand all of that.

But I question the soundness of applying that to the example in question, because the yoyo string would be analogous to the road, but the road didn't move and the string does. And since the string is being pulled up, wouldn't Vcm be not as simple as just R*w? Wouldn't I have to account for the speed of the string?

And furthermore, looking at (10.14) in the picture, if my T force is large enough to equal Mg, would Vcm not just equal zero? And looking at (10.15), such a large T force would cause a large angular velocity.
So then Vcm =/= R*w

Is the something wrong with my reasoning?

 

[haruspex]

the string is being pulled up

I don't see anything to suggest that. Looks like the top of the string is being held fixed.

 

[fchen720]

I'm face palming right now... Thank you so much.

 

[haruspex]

I'm face palming right now... Thank you so much.

ok!