Question on ap physics c 2012 free response

[RoboNerd]

Homework Statement

The problem is attached in the file "selection(1).pdf"
The explanations/solutions are attached in "selections.pdf" which is below

[Problem image - added by moderator to make it visible in-thread]

Looking at how they solved part C, I have no idea why they say that the linear speed is equal to R * omega when the slipping stops. When the slipping stops, it is just rotating without any sliding, and it is stuck in place [not moving with translational motion], so why do they say that the velocity = R * omega?

Homework Equations

V = R*omega

The Attempt at a Solution

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I can photograph my attempts if you all want to see them, but my problem focuses on theory more.

I have two equations from part b. the one for the v of the ring and the angular velocity of the ring... two different things. Why do they feel that they can set these two equations equal to each other? This does not make sense as when slipping stops, I have rotational but no translational motion.

if V = R * omega, and my V is zero, then my omega has to be zero, but this is not the case as seen in the problems statement.

Could anyone please help me understand this?

Thanks so much!

 

[haruspex]

If the ring is moving without sliding it must be rolling. As it rolls, at each instant, the point on the ring that is in contact with the ground is stationary. That is what is meant by not sliding. Thus that point is the instantaneous centre of rotation. From that, we deduce that the linear velocity of the ring's centre is rω.

 

[RoboNerd]

Thanks a lot! That makes sense.