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270 CHAPTER 10 ROTATION

whereadv/dt. Caution:The angular acceleration ain Eq. 10-22 must be

expressed in radian measure.

In addition, as Eq. 4-34 tells us, a particle (or point) moving in a circular path

has a radial componentof linear acceleration,arv^2 /r(directed radially inward),

that is responsible for changes in the directionof the linear velocity. By substi-

tuting for vfrom Eq. 10-18, we can write this component as

(radian measure). (10-23)

Thus, as Fig. 10-9bshows, the linear acceleration of a point on a rotating rigid

body has, in general, two components. The radially inward component ar(given

by Eq. 10-23) is present whenever the angular velocity of the body is not zero.

The tangential component at(given by Eq. 10-22) is present whenever the angu-

lar acceleration is not zero.

ar

v^2

r

v^2 r

:v

Checkpoint 3

A cockroach rides the rim of a rotating merry-go-round. If the angular speed of this

system (merry-go-roundcockroach) is constant, does the cockroach have (a) radial

acceleration and (b) tangential acceleration? If vis decreasing, does the cockroach

have (c) radial acceleration and (d) tangential acceleration?

and radial accelerations are the (perpendicular) compo-

nents of the (full) acceleration.

Calculations: Let’s go through the steps. We first find the

angular velocity by taking the time derivative of the given

angular position function and then substituting the given

time of t2.20 s:

v (ct^3 ) 3 ct^2 (10-25)

3(6.39 10 ^2 rad/s^3 )(2.20 s)^2

0.928 rad/s. (Answer)

From Eq. 10-18, the linear speed just then is

vvr 3 ct^2 r (10-26)

3(6.39 10 ^2 rad/s^3 )(2.20 s)^2 (33.1 m)

30.7 m/s. (Answer)

du

dt



d

dt

a:

Sample Problem 10.05 Designing The Giant Ring, a large-scale amusement park ride

We are given the job of designing a large horizontal ring
that will rotate around a vertical axis and that will have a ra-
dius of r33.1 m (matching that of Beijing’s The Great
Observation Wheel, the largest Ferris wheel in the world).
Passengers will enter through a door in the outer wall of the
ring and then stand next to that wall (Fig. 10-10a). We decide
that for the time interval t0 to t2.30 s, the angular posi-
tionu(t) of a reference line on the ring will be given by

uct^3 , (10-24)

withc6.39 10 ^2 rad/s^3. After t2.30 s, the angular
speed will be held constant until the end of the ride. Once
the ring begins to rotate, the floor of the ring will drop away
from the riders but the riders will not fall—indeed, they feel
as though they are pinned to the wall. For the time t2.20 s,
let’s determine a rider’s angular speed v, linear speed v,an-
gular acceleration a, tangential acceleration at, radial accel-
erationar, and acceleration.

KEY IDEAS

(1) The angular speed vis given by Eq. 10-6 (vdu/dt).
(2) The linear speed v(along the circular path) is related to
the angular speed (around the rotation axis) by Eq. 10-18
(vvr). (3) The angular acceleration ais given by Eq. 10-8
(adv/dt). (4) The tangential acceleration at(along the cir-
cular path) is related to the angular acceleration (around
the rotation axis) by Eq. 10-22 (atar). (5) The radial accel-
erationaris given Eq. 10-23 (arv^2 r). (6) The tangential

a:

u

a

ar

at

(a) (b)

Figure 10-10(a) Overhead view of

a passenger ready to ride The

Giant Ring. (b) The radial and

tangential acceleration compo-

nents of the (full) acceleration.