The blades of a windmill start from rest and rotate with an angular acceleration of 18.0 rad/s2. At any point on a blade, how much time passes before the magnitude of the tangential acceleration equals the magnitude of the centripetal acceleration? I am not sure how to work this.

Question

The blades of a windmill start from rest and rotate with an angular acceleration of 18.0 rad/s2. At any point on a blade, how much time passes before the magnitude of the tangential acceleration equals the magnitude of the centripetal acceleration?  I am not sure how to work this.

jamesj · Answer

First, what are expressions for the tangential and centripetal accelerations?

anonymous · Answer

centripetal acceleration=v^2/r
$$tangential= r*\alpha$$

jamesj · Answer

write the first one in terms of angular velocity terms

jamesj · Answer

$$ v = \omega r $$

anonymous · Answer

thanks, how do I incorporate time into this?

jamesj · Answer

Well, what is angular velocity as a function of time?
$$ \omega(t) = ...what? $$

anonymous · Answer

oh, ok- I've got it from here, thanks.  I had some brain fart on that part.

jamesj · Answer

Ok.  For the record, if T(t) is the tangential acceleration, C(t) the centripetal, then
$$ T(t) = r\alpha^2 t^2, \ \ \ C(t) = r\alpha $$
which are equal when $ t = 1/\sqrt{\alpha} $.

anonymous · Answer

hey how did you find the function of T and C WITH RESPECT TO TIME?