Angular acceleration

## Definition

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

It is in  r a d ⋅ s − 2 {\displaystyle rad\cdot s^{-2}} in SI unit.

## Formula

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

α = d ω d t {\displaystyle \alpha ={\frac {d\omega }{dt}}}

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

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

## Relationship with Constant Torque

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

α = τ I {\displaystyle \alpha ={\frac {\tau }{I}}}

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

## Angular Kinematics

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

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

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

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

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

θ = ω t t − 1 2 α t 2 {\displaystyle \theta =\omega _{t}\,t-{\frac {1}{2}}\alpha \,t^{2}}

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