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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.