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A student wrote: "I was looking at your solution for problem 8 in chapter 13. I am confused as
to how you derived the equation that you used. I could find no reference of
it in the book or other resources. Thank you for your help."

$a_R = \omega^2 r$ comes from equation 8-6 on page 202. It's a different way of writing centripetal acceleration. The other way of writing it is $a_R = \dfrac{v^2}{r}$, but with a substitution for $v$ as $v = \dfrac{\omega}{r}$, you end up with $a_R = \omega^2 r$.

Hope that helps,
Mr. Dychko