Page 176 - Fisika Terapan for Engineers and Scientists

376 CHAPTER 12 Rotation of a Rigid Body

QUESTION 2: The wheel of a bicycle rolls on a flat road. Is the angular velocity con-
stant if the translational velocity of the bicycle is constant? Is the angular acceleration
constant if the translational acceleration of the bicycle is constant?
QUESTION 3: A grinding wheel accelerates uniformly for 3 seconds after being turned
on. In the first second of motion, the wheel rotates 5 times. In the first two seconds of
motion, the total number of revolutions is:
(A) 6 (B) 10 (C) 15 (D) 20 (E) 25

12.4 MOTION WITH TIME-DEPENDENT
ANGULAR ACCELERATION

The equations of angular motion for the general case when the angular acceleration is
a function of time are analogous to the corresponding equations of translational motion
discussed in Section 2.7. Such equations are solved by integration. Integral calculus
was discussed in detail in Chapter 7, and we now revisit the technique of integration
of the equations of motion for the case of angular motion. To see how we can obtain
kinematic solutions for nonconstant accelerations, consider the angular acceleration
d dt. We rearrange this relation and obtain
d dt

We can integrate this expression directly, for example, from the initial value of the angu-
lar velocity at time t 0, to some final value at time t (the integration variables
0
are indicated by primes to distinguish them from the upper limits of integration):

t
d
dt
0
0
0 t
dt
(12.21)
0
This gives the angular velocity as a function of time:

0
angular velocity for time-dependent t dt
(12.22)
angular acceleration
0

Equation (12.22) enables us to calculate the angular velocity as a function of time for
any angular acceleration that is a known function of time.
The angular position can be obtained in a similar manner:

df dt
f t
df
dt
0
f 0
0
angular position for time-dependent f f t dt
(12.23)
angular velocity
0