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10-1 ROTATIONAL VARIABLES 261

Figure 10-4The reference line of the rigid body of Figs. 10-2 and 10-3 is at angular position
u 1 at time t 1 and at angular positionu 2 at a later time t 2. The quantity u(u 2 u 1 ) is the
angular displacement that occurs during the interval t(t 2 t 1 ). The body itself is not
shown.

x

y

O Rotation axis

θ 1 θ^2

Δ θ

At t 2

At t 1

Reference line

This change in the angle of the reference line

(which is part of the body) is equal to the angular

displacement of the body itself during this

time interval.

The (instantaneous) angular velocityv, with which we shall be most con-
cerned, is the limit of the ratio in Eq. 10-5 as tapproaches zero. Thus,

(10-6)

If we know u(t), we can find the angular velocity vby differentiation.
Equations 10-5 and 10-6 hold not only for the rotating rigid body as a whole
but also for every particle of that bodybecause the particles are all locked
together. The unit of angular velocity is commonly the radian per second (rad/s)
or the revolution per second (rev/s). Another measure of angular velocity was
used during at least the first three decades of rock: Music was produced by vinyl
(phonograph) records that were played on turntables at “ ” or “45 rpm,”
meaning at or 45 rev/min.
If a particle moves in translation along an xaxis, its linear velocity vis either
positive or negative, depending on its direction along the axis. Similarly, the angu-
lar velocity vof a rotating rigid body is either positive or negative, depending on
whether the body is rotating counterclockwise (positive) or clockwise (negative).
(“Clocks are negative” still works.) The magnitude of an angular velocity is called
theangular speed,which is also represented with v.

Angular Acceleration

If the angular velocity of a rotating body is not constant, then the body has an an-
gular acceleration. Let v 2 andv 1 be its angular velocities at times t 2 andt 1 ,
respectively. The average angular accelerationof the rotating body in the interval
fromt 1 tot 2 is defined as

(10-7)

in which vis the change in the angular velocity that occurs during the time
intervalt. The (instantaneous) angular accelerationa, with which we shall be
most concerned, is the limit of this quantity as tapproaches zero. Thus,

(10-8)

As the name suggests, this is the angular acceleration of the body at a given in-
stant. Equations 10-7 and 10-8 also hold for every particle of that body.The unit of
angular acceleration is commonly the radian per second-squared (rad/s^2 ) or the
revolution per second-squared (rev/s^2 ).

alim

t: 0

v

t



dv

dt

.

aavg

v 2 v 1

t 2 t 1



v

t

,

3313 rev/min

3313 rpm

vlim

t: 0

u

t



du

dt

.