# 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 $t\,$