Angular acceleration and linear acceleration

[sys.fail]

For a disk in the x-y plane that is rotating about the z-axis which travels through its center of mass, how does the angular acceleration relate to the linear acceleration of a particle on the body? Is the direction and the magnitude both affected? How do we calculate these in vector form? I would greatly appreciate it if someone would enlighten me about this.

 

[Galileo]

The angular velocity is related to the linear velocity by:
$$\omega = \dot \theta = \frac{v}{r}$$

Taking the time derivative of both sides and using that r is independent of time:

$$\alpha = \ddot \theta = \frac{a}{r}$$

The direction is always pointing towards the axis of rotation.

 

[sys.fail]

Thanks for replying, but would there be a tangential component? And if alpha=a/r, how is it that the linear acceleration is maintained constant?

 

[Doc Al]

For a rotating object undergoing an angular acceleration, a point on that object will have both a radial and tangential component of linear acceleration:
$$a_r = \omega^2 r$$
$$a_t = \alpha r$$