Page 174 - Fisika Terapan for Engineers and Scientists

374 CHAPTER 12 Rotation of a Rigid Body

12.3 MOTION WITH CONSTANT

ANGULAR ACCELERATION

We will now examine the kinematic equations describing rotational motion for the
special case of constant angular acceleration; these are mathematically analogous to the
equations describing translational motion with constant acceleration (see Section 2.5),
and they can be derived by the same methods. In the next section, we will develop an
alternative method, based on integration, for obtaining the kinematic equations describ-
ing either angular or translational motion for the general case of accelerations with
arbitrary time dependence.
If the rigid body rotates with a constant angular acceleration , then the angular
velocity increases at a constant rate, and after a time t has elapsed, the angular veloc-
ity will attain the value
constant angular acceleration:
, , and t t (12.17)
0
where is the initial value of the angular velocity at t 0.
0
The angular position can be calculated from this angular velocity by the arguments
used in Section 2.5 to calculate x from v [see Eqs. (2.17), (2.22), and (2.25)]. The
result is

constant angular acceleration: 1 2
0
, , and t f f t t (12.18)
0
2
Furthermore, the arguments of Section 2.5 lead to an identity between acceleration,
position, and velocity [see Eqs. (2.20)–(2.22)]:

constant angular acceleration: 1 2 2
2
0
0
, , and (f f ) ( ) (12.19)
Note that all these equations have exactly the same mathematical form as the equations
of Section 2.5, with the angular position taking the place of the position x, the angu-
lar velocity taking the place of v, and the angular acceleration taking the place of
a. This analogy between rotational and translational quantities can serve as a useful
mnemonic for remembering the equations for rotational motion. Table 12.2 displays
analogous equations.

TABLE 12.2 ANALOGIES BETWEEN TRANSLATIONAL
AND ROTATIONAL QUANTITIES

dx df
v S
dt dt
dv d
a S
dt dt
v v at S t
0 0
1
1
x x v t at 2 S f f t t 2
0 0 2 0 0 2
2
1
2
2
1
2
a(x x ) (v v ) S (f f ) ( )
0 2 0 0 2 0