What is the difference between centripetal and angular acceleration?

jayadds

So as the title says, what is the difference between centripetal and angular acceleration? I already know that there is a difference in the equations for each of the components but can someone please explain it conceptually? Please use some examples in your explanation.

Nabeshin

Centripetal acceleration simply means a centrally-directed force accelerating an object. As in the case of a ball on a string whirling around, the ball experiences a centripetal acceleration. Angular acceleration, however, is the change in angular velocity of the ball: if it were speeding up in its rotation, it would experience angular acceleration (i.e. if you shorten the string).

jayadds

So from what you're saying, when there is a change in angular velocity of the ball, there is only angular acceleration acting on it? Does centripetal force also apply in this scenario?

Nabeshin

Not necessarily. Change in angular velocity certainly implies angular acceleration, but doesn't rule out there also being a centripetal force. If we go back to my ball on string analogy, the tension of the string is always providing a centripetal force. If I, say, strap a rocket onto the ball and fire it in the direction of motion, it experiences an angular acceleration. Note that as the angular velocity changes, so too does the centripetal force (in this case, the tension in the string) necessary to keep it on a circular trajectory.

haruspex

Let's be clear with the terminology first: a force acts, acceleration is the result.
A body undergoes centripetal acceleration (the acceleration it needs in order to keep moving around some focus) when subjected to a centripetal force (from a string, gravity..).
This applies even when the rate of rotation does not change.
A body undergoes angular acceleration when subjected to an angular force (torque). This would often be a ball/disc/wheel made to spin faster or more slowly, but would also apply if, as Nabeshin says, the orbital rate changes.

In algebra, centripetal acceleration is r.[itex]\omega[/itex][itex]^2[/itex]; angular speed is r.[itex]\omega[/itex]; angular acceleration is [itex]\partial[/itex](r.[itex]\omega[/itex])/[itex]\partial[/itex]t. Either acceleration can be zero (at least briefly) while the other is not.

azizlwl

I wish i could remember these equations

[itex] v=\dot{R}\hat r + Rω\hat θ[/itex]
[itex] a=( \ddot {R}-Rω^2)\hat r +(Rα+2\dot {R}w)\hat θ[/itex]