Page 175 - Fisika Terapan for Engineers and Scientists

12.3 Motion with Constant Angular Acceleration 375

The cable supporting an elevator runs over a wheel of radius 0.36 m
EXAMPLE 6 Upward acceleration
0.36 m (see Fig. 12.11). If the elevator begins from rest and a equals tangential
2
ascends with an upward acceleration of 0.60 m/s , what is the angular accelera- acceleration of wheel.
tion of the wheel? How many turns does the wheel make if this accelerated motion
lasts 5.0 s? Assume that the cable runs over the wheel without slipping.
a
SOLUTION: If there is no slipping, the speed of the cable must always coincide
with the tangential speed of a point on the rim of the wheel. The acceleration
2
a 0.60 m/s of the cable must then coincide with the tangential acceleration of
a point on the rim of the wheel:

a a R (12.20)
tangential
where R 0.36 m is the radius of the wheel. Hence

a 0.60 m/s 2 2
1.7 radians/s
R 0.36 m
According to Eq. (12.18), the angular displacement in 5.0 s is
1 2
f f t t
0
0
2
2
1
0 1.7 radians/s (5.0 s) 2 FIGURE 12.11 Elevator supported by
2
a cable that runs over a rotating wheel.
21 radians
Each revolution comprises 2 radians; thus, the number of turns the wheel makes is
f f 0 21 radians
number of turns 3.3 revolutions
2p 2p

PROBLEM-SOLVING TECHNIQUES ANGULAR MOTION

The solution of kinematic problems about angular velocity contact with the rim of the wheel does not slip,the translational
and angular acceleration involves the same techniques as the speed of this body equals the tangential speed of the contact
problems about translational velocity and translational accel- point at the rim of the wheel; that is, v R and a R.
eration in Chapter 2. You might find it useful to review the Keep in mind that although some of the equations in this
procedures suggested on page 50. chapter remain valid if the angular quantities are expressed
Sometimes a problem contains a link between a rotational in degrees, any equation that contains both angular quanti-
motion and a translational motion, such as the link between ties and distances (e.g., v R) is valid only if the angular
the rotational and translational motions of the wheels of an quantity is expressed in radians. To prevent mistakes, it is
automobile (see Example 4) or the link between the transla- safest to express all angular quantities in radians; if degrees
tional motion of the elevator cable and the rotational motion are required in the answer, convert from radians to degrees
of the wheel over which it runs (Example 6). If the body in after completing your calculations.

✔ Checkup 12.3

QUESTION 1: Consider a point on the rim of the wheel shown in Fig. 12.11, (instan-
taneously) at the top of the wheel. What is the direction of the centripetal acceleration
of this point? The tangential acceleration?