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# degree per second squared

The degree per second squared (symbolized deg/s 2 or deg · s -2 ) is an alternative unit of angular (rotational) acceleration magnitude, which 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 degrees 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 degrees 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 20 deg/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 20 deg/s 2 , pointing toward the viewer in line with the wheel's axle. But if rate of rotation were decreasing at 20 deg/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 20 deg/s 2 in the opposite direction, that is, away from the viewer in line with the wheel's axle.

This was last updated in September 2005

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