Page 171 - Fisika Terapan for Engineers and Scientists

12.2 Rotation about a Fixed Axis 371

body. It is interesting to focus on one of the particles and evaluate its translational y
speed and acceleration as it moves along its circular path around the axis of rotation of
v
the rigid body. If the particle is at a distance R from the axis of rotation (see Fig. 12.7),
then the length along the circular path of the particle is, according to the definition of P
angle, Eq. (12.1),
R s
s fR (12.9) O x

Since R is a constant, the rate of change of s is entirely due to the rate of change of
,so

ds df
R (12.10) FIGURE 12.7 The instantaneous transla-
dt dt tional velocity of a particle in a rotating rigid
body is tangent to the circular path.
Here ds/dt is the translational speed v with which the particle moves along its circu-
lar path, and d /dt is the angular velocity ; hence Eq. (12.10) is equivalent to

v R (12.11) translational speed in circular motion

This shows that the translational speed of the particle along its circular path around
the axis is directly proportional to the radius: the farther a particle in the rigid body is
from the axis, the faster it moves. We can understand this by comparing the motions
of two particles, one on a circle of large radius R , and the other on a circle of smaller
1 Translational speeds
radius R (see Fig. 12.8). For each revolution of the rigid body, both of these particles are proportional to y
2
complete one trip around their circles. But the particle on the larger circle has to travel radial distances.
v 1
a larger distance, and hence must move with a larger speed.
For a particle at a given R, the translational speed is constant if the angular veloc- v 2
ity is constant.This speed is the distance around the circular path (the circumference)
v 3
divided by the time for one revolution (the period), or
R 2 R 1
R R 3 3
2pR
v (constant speed) (12.12) O x
T

Since 2 T 2 f , Eq. (12.12) can be obtained from Eq. (12.11).
If v is changing, it also follows from Eq. (12.11) that the rate of change of v is
proportional to the rate of change of :
FIGURE 12.8 Several particles in a rigid
dv d body rotating about a fixed axis and their
R velocities.
dt dt
A rate of change of the speed along the circle implies that the particle has an acceler-
ation along the circle, called a tangential acceleration. According to the last equa-
tion, this tangential acceleration is

a R (12.13) tangential acceleration
tangential
Note that, besides this tangential acceleration directed along the circle, the parti-
cle also has a centripetal acceleration directed toward the center of the circle. From
Section 4.5, we know that the centripetal acceleration for uniform circular motion is

2
v
a (12.14)
centripetal
R