Angular Acceleration v. Centripetal Acceleration What is the difference between angular acceleration and centripetal acceleration? Don't they both apply to circular motion?
 A: angular acceleration:$\frac {d\omega} {dt}$ the change of angular velocity with respect to time
centripetal acceleration: $\omega^2r$ if the orbital is circular.  The component of linear acceleration of a object that causes it to have a angular motion. 
A: Angular acceleration $\vec{\alpha}$ is defined as $$\vec{\alpha}=\frac{d\vec{\omega}}{dt}$$
$\vec{\omega}$ is the angular acceleration of the particle.
Centripetal acceleration in circular motion is given as
$$\vec{a}=-\omega^2r\vec{e_{r}}$$
$\vec{e_{r}}$ is the unit vector along radial direction.
A: In general, angular acceleration ($\vec a$) is the rate of change of angular velocity $\vec \omega$ w.r.t. time i.e. $$\vec a=\frac{d\vec \omega}{dt}$$
An object in curvi-linear (non-circular) motion can have angular acceleration
While the centripetal acceleration causes a body to rotate in a circular path & its magnitude is $$a_c=r\omega^2$$
Arc length traversed by the body with tangential velocity $v$ in time $dt$ on a circular path of radius $r$ is given as
$$rd\theta=vdt$$$$ \frac{d\theta}{dt}=\frac{v}{r}$$
$$\omega=\frac{v}{r}$$
$$v=r\omega$$
In circular motion, the differential change in tangential velocity: $$dv=vd\theta$$
$$\frac{dv}{dt}=v\frac{d\theta}{dt}$$
$$a_c=r\omega\cdot \omega$$
$$a_c=r\omega^2$$
In vector form $$\color{blue}{\vec{a_c}=- r\omega^2\hat{r}}$$
Where $\hat{r}$ is unit radial vector.
