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## Homework Statement

## Homework Equations

v = w x r

## The Attempt at a Solution

I got the answer correct after a few tries, but I am little confused on why it has to solved this specific way. I know that v = w x r, so why can I not just say w = v/r? Is it because w is measured from the center of the roller so v = w x r would be v

_{center}, not v

_{top}?

That is how I ended up solving it, but I wanted to make sure I didn't get there "accidentally" or misinterpreted the process.

Using that diagram and concept of similar triangles:

tan θ

_{top}= tan θ

_{center}

S

_{top}/2r = S

_{center}/r

S

_{top}= 2*S

_{center}, S

_{center}= rθ

S

_{top}= 2rθ

d(S

_{top})/dt = v

_{top}= d(2rθ)/dt = 2rθ' = 2rω

_{top}

thus, ω = v/2r

Any reasoning as to why I have to do this would be much appreciated!!