# Angular acceleration

Radians per second squared
Unit systemSI derived unit
Unit ofAngular acceleration

In physics, angular acceleration refers to the time rate of change of angular velocity. It is measured in units of angle per unit time squared (which in SI units is radians per second squared), and is usually represented by the symbol alpha (α). By convention, positive angular acceleration indicates an increase in counter-clockwise rotation or decrease in clockwise rotation, while negative angular acceleration indicates an increase in clockwise rotation or decrease in counterclockwise rotation. In three dimensions, angular acceleration is a pseudovector.[1]

For rigid bodies, angular acceleration must be caused by a net external torque. However, this is not so for non-rigid bodies: For example, a figure skater can speed up her rotation (thereby obtaining an angular acceleration) simply by contracting her arms and legs inwards, which involves no external torque.

## Mathematical definition

The angular acceleration vector is defined as:

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

where ${\displaystyle {\boldsymbol {\omega }}}$ can be either the orbital or the spin angular velocity vector, depending on whether ${\displaystyle {\boldsymbol {\alpha }}}$ is the orbital or spin angular acceleration vector.

### Equation of Motion for a Point Particle

The orbital angular acceleration of a point particle α can be connected to the applied torque τ by the following equation:

${\displaystyle {I{\boldsymbol {\alpha }}}={\boldsymbol {\tau }}-{\frac {d{I}}{dt}}{\boldsymbol {\omega }}}$,

where I is its moment of inertia.

The above relationship indicates that, unlike the relationship between force and acceleration, the orbital angular acceleration need not be directly proportional or even parallel to the torque. However, in the special case where the distance to the origin does not change with time, the torque does turn out to be proportional and parallel to the angular acceleration (with the constant of proportionality being the moment of inertia of the particle).