Understanding Angular Velocity Calculations for Rotating Objects

AI Thread Summary
The discussion focuses on the calculation of angular velocity for rotating objects, specifically addressing the relationship between linear velocity (v), angular velocity (w), and radius (r). The equation v = w x r is emphasized, with clarification that w should be measured from the instantaneous center of rotation (IC). The confusion arises from whether w can be derived as v/r, which is not applicable in this context due to the need for measuring from the IC. The correct application leads to the formula v = 2rω, where 2r represents the distance from the bottom to the top of the roller. Understanding these principles is crucial for accurately solving problems related to angular motion.
Nikstykal
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Homework Statement


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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 vcenter, not vtop?
That is how I ended up solving it, but I wanted to make sure I didn't get there "accidentally" or misinterpreted the process.
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Using that diagram and concept of similar triangles:
tan θtop = tan θcenter
Stop/2r = Scenter/r
Stop = 2*Scenter, Scenter = rθ
Stop = 2rθ
d(Stop)/dt = vtop = d(2rθ)/dt = 2rθ' = 2rωtop
thus, ω = v/2r

Any reasoning as to why I have to do this would be much appreciated!
 
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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.
 
Ahh, i forgot, when using v = w x r you have to always measure from from IC unless using relative velocity equation. Thank you!
 
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