Understanding Forces and Motion in a Translating and Rotating Ring

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The discussion revolves around a physics problem involving a translating and rotating ring, detailing its mass, speeds, and friction coefficient. The user attempts to solve the problem using equations for forces and moments but becomes confused about the signs of angular acceleration and linear acceleration. Clarification is sought on why angular acceleration is negative while linear acceleration is positive, and the relationship between linear velocity and angular velocity is questioned. The user references a scanned solution from a book for guidance. Understanding the distinction between tangential and angular components is crucial for resolving these queries.
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1)Problem statement
A translating and rotating ring of mass 1 kg, angular speed of 500 rpm, and translational speed of 1 m/s is placed on a horizontal surface. The coefficient of friction between the ring and the surface is 0.35.

Outside radius is 3 cm.

This is a problem from a FE review manual, thanks guys.

2)Attempt at a solution.
ΣFx=ma
ΣFy=0 therefore N=mg
From summation of forces in the x direction,a=g/m
ΣM=0,(µmgR)= α I
I=3MR^2
α=µg /(3R)
v= v_0-at
ω= ω_0- αt
This is where I get lost.
 
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Ok, this is a scanned solution of the problem according to the book. It's problem 13.

http://img218.imageshack.us/f/002kfv.jpg/

What I don't understand is why is angular acceleration negative, and a is positive. Shouldn't they both be negative? Also, how did they get v = romega, isn't romega the tangential component?
 

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