The radian per second (symbolized rad/s or rad/sec) is the Standard International ( SI ) unit of angular (rotational) speed. This quantity can be defined in either of two senses: average or instantaneous.

Average angular speed is obtained by measuring the angle in radian s through which an object rotates in a certain number of seconds, and then dividing the total angle by the time. If *u* _{avg} represents the average angular speed of an object (in radians per second) during a time interval *t* (in seconds), and the angle through which the object rotates in that time is equal to *q* (in radians), then:

*u* _{avg} = *q* / *t*

Instantaneous angular speed is more difficult to intuit, because it involves an expression of 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 speed can be measured 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 speed, *u* _{inst} , is the limit of the measured average speed as *x* approaches zero. This is a theoretical value, because it cannot be obtained except by inference from measurements made over progressively shorter time spans.

It is important to realize that angular speed is not the same thing as angular velocity. Angular speed is a scalar (dimensionless) quantity, while angular velocity is a vector quantity consisting of angular speed and direction in the form of a rotational sense (clockwise or counterclockwise). We might say the earth rotates at at 7.272 x 10 ^{-5} rad/s, and this tells us its angular speed. Or we might say the earth rotates at at 0.00007272 (7.272 x 10 ^{-5} ) rad/s counterclockwise relative to the sun as viewed from above the north geographic pole; this tells us the earth's angular velocity. As with angular speed, we might specify either the average angular velocity over a period of time, or the instantaneous angular velocity at an exact moment in time.

In the case of the earth, the instantaneous angular velocity is essentially constant, and it can be said that *u* _{inst} = *u* _{avg} . In the case of a moving car or truck, the angular speeds of the wheels, axles, and drive shaft often change, and the statement *u* _{inst} = *u* _{avg} is, in general, not true.

Also see radian , second , radian per second squared , and Standard International ( SI ) System of Units.

*This was last updated in*March 2011

*Posted by:*Margaret Rouse

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