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slimnexus
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Homework Statement
Question comes from this book http://www.anselm.edu/internet/physics/cbphysics/index.html
In the section on rotational energy it gives the formula K[itex]_{rolling}[/itex] = K[itex]_{translational}[/itex]+K[itex]_{rotational}[/itex] which it then expands to K[itex]_{rolling}[/itex] = [itex]\frac{1}{2}[/itex]mv[itex]^{2}[/itex] + [itex]\frac{1}{2}[/itex]Iω[itex]^{2}[/itex]. Then it states that when an object that is rolling without slipping completes one rotation it travels a distance equal to its circumference so far so good. But then it says to divide both sides of the equation by 2πr, and on the left side we should have the speed of the object and on the right side its angular velocity. So I should end up with v = rω. But the author doesn't show any intermediate steps and I am not following this, I don't want to move on until I understand what's going on here.
So my question is how is v = rω derived from the earlier stated equation. Intermediate steps would be much appreciated. Thanks for any help you can provide.