Angular acceleration

Definition
Angular acceleration is a vector whose magnitude is defined as the change in angular velocity in unit time.

It is in $$rad\cdot s^{-2}$$ in SI unit.

Formula
Analogous to translational acceleration, $$a=\frac{dv}{dt} $$, angular acceleration has the defining formula:

$$\alpha=\frac{d\omega}{dt}$$

in which $$d\omega\,$$ represents an instantaneous change in angular velocity,which takes place in  $$dt\,$$, a short flitting time.

Equivalently, think about the limiting case: $$\alpha= \lim_{\Delta t\rightarrow 0}\frac{\Delta\omega}{\Delta t}$$

Relationship with Constant Torque
The angular acceleration of an fixed-axis-object is proportional to the net torque applied.

$$\alpha=\frac{\tau}{I}$$

in which $$I\,$$ is the Moment of Inertia of the object.

Angular Kinematics
When an rotation has constant angular acceleration $$\alpha\,$$, the angle displacement $$\theta\,$$ covered in a given time  $$t\,$$ is given by an equation that is strikingly similar to the equation for displacement under constant acceleration.

$$\theta=\omega_0\,t+\frac{1}{2}\alpha\,t^2$$

in which $$\omega_0\,$$ is the angular velocity at the beginning of the time period $$t\,$$

$$\theta=\frac{1}{2}\alpha\,t^2$$

in which case the angular velocity at the beginning $$\omega_0\,$$ is "zero"

$$\theta=\omega_t\,t-\frac{1}{2}\alpha\,t^2$$

in which $$\omega_t\,$$ is the angular velocity at the end of the time period $$t\,$$