The radian per second squared (symbolized rad/s ^{2} or rad · s ^{-2} ) is the unit of angular (rotational) acceleration magnitude in the International System of Units ( SI ). This is the rate of change of angular speed or velocity. The angular acceleration vector also has a direction component that can be defined in either of two senses: counterclockwise or clockwise.

The average angular acceleration magnitude can be obtained by evaluating an object's instantaneous angular speed (in radians per second) at two different points *t* _{1} and *t* _{2} in time, and then dividing the distance by the span of time *t* _{2} - *t* _{1} (in seconds). Suppose the instantaneous angular speed at time *t* _{1} is equal to *u* _{1} , and the instantaneous angular speed at time *t* _{2} is equal to *u* _{2} . Then the average angular acceleration magnitude *b* _{avg} (in radians per second squared) during the time interval [ *t* _{1} , *t* _{2} ] is given by:

*b* _{avg} = ( *u* _{2} - *u* _{1} ) / ( *t* _{2} - *t* _{1} )

Instantaneous angular acceleration magnitude is more difficult to intuit, because it involves an expression of rotational motion over an "infinitely short" interval of time. Let *p* represent a specific point in time. Suppose an object is in rotational motion at about that time. The average angular acceleration magnitude can be determined over increasingly short time intervals centered at *p* , for example:

[ *p* -4, *p* +4]

[ *p* -3, *p* +3]

[ *p* -2, *p* +2]

[ *p* -1, *p* +1]

[ *p* -0.5, *p* +0.5]

[ *p* -0.25, *p* +0.25]

.

.

.

[ *p* - *x* , *p* + *x* ]

.

.

.

where the added and subtracted numbers represent seconds. The instantaneous angular acceleration magnitude, *b* _{inst} , is the limit of the average angular acceleration magnitude as *x* approaches zero. This is a theoretical value, because it can be obtained only by inference from instantanous speed values determined at the starting and ending points of progressively shorter time spans.

In the complete sense, angular acceleration is a vector quantity having direction as well as magnitude, and representing the rate of change of angular velocity. Suppose, for example, that a wheel's rate of rotation is increasing at 0.5 rad/s ^{2} in the counterclockwise sense; this might be the case for a car or truck moving from right to left (relative to the viewer) with increasing speed. This would produce an angular acceleration vector with a magnitude of 0.5 rad/s ^{2} , pointing toward the viewer in line with the wheel's axle. But if the rate of rotation were decreasing at 0.5 rad/s ^{2} in the counterclockwise sense (the same car or truck slowing down while moving from right to left), the angular acceleration vector would have a magnitude of 0.5 rad/s ^{2} in the opposite direction, that is, away from the viewer in line with the wheel's axle.

Also see angular acceleration , radian , degree per second squared , SI , and Table of Physical Units .

*This was last updated in*September 2005

*Posted by:*Margaret Rouse

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