282
CHAPTE R 10 ROTATION
Work and Rotational Kinetic Energy
As we discussed in Chapter 7, when a force F causes a rigid body of mass m to accelerate along a coordinate axis, the force does work W on the body. Thus, the
body’s kinetic energy
can change. Suppose it is the only energy of the
(K ϭ
1
2 mv
2
)
10-8 WORK AND ROTATIONAL KINETIC ENERGY
After reading this module, you should be able to . . .
10.29 Calculate the work done by a torque acting on a rotating body by integrating the torque with respect to the angle of rotation.
10.30 Apply the work–kinetic energy theorem to relate the
work done by a torque to the resulting change in the rotational kinetic energy of the body.
10.31 Calculate the work done by a constant torque by relating the work to the angle through which the body rotates.
10.32 Calculate the power of a torque by finding the rate at
which work is done.
10.33 Calculate the power of a torque at any given instant by
relating it to the torque and the angular velocity at that instant.
● The equations used for calculating work and power in rotational motion correspond to equations used for translational
motion and are
and
P ϭ
dW
dt
ϭ tv.
W ϭ ͵
u f
u i
t du
● When t is constant, the integral reduces to
W ϭ t(u f Ϫ u i ).
● The form of the work – kinetic energy theorem used for
rotating bodies is
⌬K ϭ K f Ϫ K i ϭ
1
2 Iv f
2 Ϫ
1
2 ⌱v i
2 ϭ W.
Learning Objectives
Key Ideas
body that changes. Then we relate the change ⌬K in kinetic energy to the work W
with the work – kinetic energy theorem (Eq. 7-10), writing
(work – kinetic energy theorem).
(10-49)
For motion confined to an x axis, we can calculate the work with Eq. 7-32,
(work, one-dimensional motion).
(10-50)
This reduces to W ϭ Fd when F is constant and the body’s displacement is d.
The rate at which the work is done is the power, which we can find with Eqs. 7-43
and 7-48,
(power, one-dimensional motion).
(10-51)
Now let us consider a rotational situation that is similar. When a torque
accelerates a rigid body in rotation about a fixed axis, the torque does work W
on the body. Therefore, the body’s rotational kinetic energy
can
change. Suppose that it is the only energy of the body that changes. Then we
can still relate the change ⌬K in kinetic energy to the work W with the
work – kinetic energy theorem, except now the kinetic energy is a rotational
kinetic energy:
(work – kinetic energy theorem).
(10-52)
Here, I is the rotational inertia of the body about the fixed axis and v i and v f are
the angular speeds of the body before and after the work is done.
⌬K ϭ K f Ϫ K i ϭ
1
2 Iv f
2 Ϫ
1
2 ⌱v i
2 ϭ W
(K ϭ
1
2 I␻
2
)
P ϭ
dW
dt
ϭ Fv
W ϭ ͵
x f
x i
F dx
⌬K ϭ K f Ϫ K i ϭ
1
2 mv f
2 Ϫ
1
2 mv i
2 ϭ W