9781118230725.pdf

An angle defined in this way is measured in radians(rad) rather than in

revolutions (rev) or degrees. The radian, being the ratio of two lengths, is a

pure number and thus has no dimension. Because the circumference of a circle of

radiusris 2pr, there are 2pradians in a complete circle:

(10-2)

and thus 1 rad57.30.159 rev. (10-3)

We do notresetuto zero with each complete rotation of the reference line about

the rotation axis. If the reference line completes two revolutions from the zero

angular position, then the angular position uof the line is u 4 prad.

For pure translation along an xaxis, we can know all there is to know

about a moving body if we know x(t), its position as a function of time.

Similarly, for pure rotation, we can know all there is to know about a rotating

body if we know u(t), the angular position of the body’s reference line as a

function of time.

Angular Displacement

If the body of Fig. 10-3 rotates about the rotation axis as in Fig. 10-4, changing the

angular position of the reference line from u 1 tou 2 , the body undergoes an

angular displacementugiven by

uu 2 u 1. (10-4)

This definition of angular displacement holds not only for the rigid body as a

whole but also for every particle within that body.

Clocks Are Negative.If a body is in translational motion along an xaxis, its

displacementxis either positive or negative, depending on whether the body is

moving in the positive or negative direction of the axis. Similarly, the angular dis-

placementuof a rotating body is either positive or negative, according to the

following rule:

1 rev 360 

2 pr

r

 2 p rad,

260 CHAPTER 10 ROTATION

An angular displacement in the counterclockwise direction is positive, and one in

the clockwise direction is negative.

Checkpoint 1

A disk can rotate about its central axis like a merry-go-round. Which of the following

pairs of values for its initial and final angular positions, respectively, give a negative

angular displacement: (a) 3 rad,5 rad, (b) 3 rad,7 rad, (c) 7 rad,3 rad?

The phrase “clocks are negative” can help you remember this rule (they certainly

are negative when their alarms sound off early in the morning).

Angular Velocity

Suppose that our rotating body is at angular position u 1 at time t 1 and at

angular position u 2 at time t 2 as in Fig. 10-4. We define the average angular velocity

of the body in the time interval tfromt 1 tot 2 to be

(10-5)

whereuis the angular displacement during t(vis the lowercase omega).

vavg

u 2 u 1

t 2 t 1



u

t

,