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372 CHAPTER 12 Rotation of a Rigid Body

With v R, this becomes
2
centripetal acceleration a R (12.15)
centripetal
The net translational acceleration of the particle is the vector sum of the tangen-
tial and the centripetal accelerations, which are perpendicular (see Fig. 12.9); thus, the
Centripetal and tangential
accelerations are perpendicular. magnitude of the net acceleration is
y
2 2
a (12.16)
net 2a tangential a centripetal
a tan
Although we have here introduced the concept of tangential acceleration in the
a net P
context of the rotational motion of a rigid body, this concept is also applicable to the
a cent
translational motion of a particle along a circular path or any curved path. For instance,
O x consider an automobile (regarded as a particle) traveling around a curve. If the driver
steps on the accelerator (or on the brake), the automobile will suffer a change of speed
as it travels around the curve. It will then have both a tangential and a centripetal
acceleration.

FIGURE 12.9 A particle in a
rotating rigid body with an angular The blade of a circular saw is initially rotating at 7000 revolu-
acceleration has both a centripetal EXAMPLE 3
tions per minute.Then the motor is switched off, and the blade
acceleration a centripetal and a tangential
acceleration a . The net instan- coasts to a stop in 8.0 s. What is the average angular acceleration?
tangential
taneous translational acceleration a net SOLUTION: In radians per second, 7000 rev/min corresponds to an initial angu-
is then the vector sum of a centripetal lar velocity 7000 2 radians min, or
and a tangential . 1
7000 2p radians
2
7.3 10 radians/s
1
60 s
The final angular velocity is 0. Hence the average angular acceleration is
2
2
¢ 1 0 7.3 10 radians/s
2

¢t t t 8.0 s 0
2 1
91 radians/s 2

An automobile accelerates uniformly from 0 to 80 km/h in 6.0 s.
EXAMPLE 4
The wheels of the automobile have a radius of 0.30 m. What is
the angular acceleration of the wheels? Assume that the wheels roll without slipping.
SOLUTION: The translational acceleration of the automobile is
R
v v 0 80 km/h (80 km/h) (1000 m/1 km) (1 h/3600 s)
a
R t 6.0 s 6.0 s
v
3.7 m/s 2
Since wheel rolls without
slipping, tangential speed R
equals the ground speed v. The angular acceleration of the wheel is related to this translational accel-
eration by a R, the same relation as Eq. (12.13). We can establish this rela-
tionship most conveniently by viewing the motion of the wheel in the reference
FIGURE 12.10 Rotating wheel of the
automobile as viewed in the reference frame frame of the automobile (see Fig. 12.10). In this reference frame, the ground
of the automobile. The ground moves is moving backward at speed v, and the bottom point of the rotating wheel is
toward the left at speed v. moving backward at the tangential speed R. Since the wheel is supposed to