Understanding Angular Velocity Calculations for Rotating Objects

[Nikstykal]

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!

 

[tommyxu3]

That's for the centro of the motion is at the bottom of the roller. Your formula should be applied as $v=2r\cdot\omega,$ where $2r$ is the distance from the bottom to the top.

 

[Nikstykal]

Ahh, i forgot, when using v = w x r you have to always measure from from IC unless using relative velocity equation. Thank you!