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: $\alpha =\underset{\mathrm{\Delta }t\to 0}{\lim }\frac{\mathrm{\Delta }\omega }{\mathrm{\Delta }t}$

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 $t$