R radius of circular disk with mass m with initial velocity v and an angular velocity w comes into contact with the ground as shown in the figure. Friction is not negligible, so both the velocity and angular velocity of the disk changes. What is the critical velocity (V_C) when the comes to instantaneous rest (in terms of angular velocity w_c and R)
click below link for figure.

Attachment 48739
https://drive.google.com/file/d/0B3v...w?usp=drivesdk

We won't do homework for you. Please try your hands at this, post back with your attempt, and we'll help you if you are stuck. I suggest you start with basic equations such as and .

what theory is applied when this critical position?
Is centre of mass's resultant velocity is zero? ( V_c=Rw_c​ )

Over some period of time the angular velocity slows to zero due to friction forces. While that is occurring the velocity V is also slowing, due to that same friction force. You are asked to determine the change in velocity that results from friction forces necessary to cause omega to become zero. At that instant the disc is sliding along thee surface but not rotating.

Question is at critical position, at this instant linear velocity v_c and angular velocity w_c given in the question (not zero).

Please restate the question, precisely as it was originally posed to you. The question as first posed was: "What is the critical velocity (V_C) when the comes to instantaneous rest (in terms of angular velocity w_c and R)" I assume the word "disk" was left out by mistake (should be "....when the disk comes to instantaneous rest ..."). You haven't defined what "instantaneous rest" means, but given that you asked about velocity at that point it would appear that perhaps w is zero but ve is not. But from your latest post it seems the question should have been: "what is the critical position when the disk comes to instantaneous rest," which I would assume means the point when both v and w are zero. Is that correct?