# Angular acceleration

## Definition

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

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

## Formula

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

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

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

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

## Relationship with Constant Torque

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

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

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

## Angular Kinematics

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

${\displaystyle \theta =\omega _{0}\,t+{\frac {1}{2}}\alpha \,t^{2}}$

in which ${\displaystyle \omega _{0}\,}$ is the angular velocity at the beginning of the time period ${\displaystyle t\,}$

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

in which case the angular velocity at the beginning ${\displaystyle \omega _{0}\,}$ is "zero"

${\displaystyle \theta =\omega _{t}\,t-{\frac {1}{2}}\alpha \,t^{2}}$

in which ${\displaystyle \omega _{t}\,}$ is the angular velocity at the end of the time period ${\displaystyle t\,}$