Angular Quantities

Angular measure

We have already discussed angular velocity when we studied uniform circular motion (UCM) and found that any object moving in a circle has a centripetal acceleration directed toward the center of that circle.

However, let us refresh our memory for a moment, . . .

How fast is something turning or rotating or revolving? We call this its angular velocity and give it the symbol , the lower case Greek letter omega. It may have units of revolutions per minute or revolutions per second or, better yet, radian per second.

Most of us are comfortable measuring an angle in degrees. But measuring an angle in radians is usually more useful.

While we can measure the angle in degrees using a protractor, there is another way to measure or define this angle . This -- or any other -- angle can me defined as the ratio of the arc length s to the radius r; that is,

Such a definition means this measure of the angle has no dimensions. We have taken an arc length, measured in something like meters, and divided it by a radius, also measured in something like meters. However, since we expect dimensions or units, we call this unit of angular measure a radian. You may think of radians as added for cosmetic purposes! How large is a radian?

Consider a complete circle. We would describe a complete circle as having an angle of 360o. In terms of radians, a complete circle has an arc length equal to its circumference, s = C = 2 r.

What does all this have to do with uniform circular motion? The arc length traveled by a point on a rotating object is equal to the radius of that point multiplied by the angle through which it has rotated,

s = r

when is measured in radians

That means the linear speed of a point on a rotating object is equal to the radius of that point multiplied by the angular speed with which it is rotating, provided is measured in radians per time. That is,

v = r

provided is measured in radians per time.

More on Angular Velocity

Angular Acceleration

We have already written

v = r

for the linear velocity of a point at a distance r from the axis of rotation. We could also call this linear velocity the tangential velocity. In a similiar manner, we can write

at = r

for the tangential acceleration at of a point located a distance r from the axis of rotation on an object that has an angular acceleration . The angular acceleration is measured in radians per second per second or 1/s2.
ToC

Rotational Kinematics
Return to ToC, Ch10, Rotation about a Fixed Axis

(c) Doug Davis, 2001; all rights reserved