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# Deriving formula for centripetal acceleration from angular velocity

Deriving formula for centripetal acceleration in terms of angular velocity. using linear speed formula.

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• I understand that centripetal acceleration is what causes change in the constant velocity direction, which altogether allows for circular motion. So is there a specific magnitude of centripetal acceleration required for every specific constant velocity value for an object to go in a circle?

And how does the change in the magnitude of centripetal acceleration affect motion? Since tangential velocity is what accelerates the object going around in a circle, would changing centripetal acceleration magnitude simply affect the path of the object? Example being if it is insufficient, an object then will follow something like an elliptical trajectory or even a straight-line path because acceleration is no longer enough to maintain circular motion.
• Yes, if a an object wants go in a circle of certain radius at a certain speed, there is a certain centripetal acceleration that it must attain. If it does not have enough centripetal acceleration it will just spiral in. If it has too much, it will fly outwards.
• Derivation of centripetal acceleration
• ...So is the centripetal acceleration directly or inversely proportional to the radius of the circular path, since ac=v^2/r=w^2*r.In the first scenario, ac is inversely proportional to the radius, whereas, in the second scenario, ac is directly proportional to the radius of the circular path. Please enlighten me.
• Since v^2/r is centripetal acceleration, and v = (omega) x r, doesn't centripetal acceleration equal (omega) x v?

(centripetal acceleration is equal to (v/r) x r, and v/r = omega
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• I do not understand what is meant by "and we typically view radians as an angle but if you think of it as an arc length, a radian you could view it as how many radii in length am I completing per second? And so, if I multiply that times the actual length of the radii, then you can get a sense of well, how much distance am I covering per second? "
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• The length of one radian is the radius, so when he is talking about radian per second, you can think of it as radii (radiuses) per second in terms of speed. When you multiple radii per second with the radius, you get distance per second.
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• what is the difference between angular acceleration (alpha) and centripetal acceleration?
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