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10-8 WORK AND ROTATIONAL KINETIC ENERGY 283

Also, we can calculate the work with a rotational equivalent of Eq. 10-50,

(work, rotation about fixed axis), (10-53)

wheretis the torque doing the work W, and uiandufare the body’s angular
positions before and after the work is done, respectively. When tis constant,
Eq. 10-53 reduces to

Wt(ufui) (work, constant torque). (10-54)

The rate at which the work is done is the power, which we can find with the rota-
tional equivalent of Eq. 10-51,

(power, rotation about fixed axis). (10-55)

Table 10-3 summarizes the equations that apply to the rotation of a rigid body
about a fixed axis and the corresponding equations for translational motion.

Proof of Eqs. 10-52 through 10-55

Let us again consider the situation of Fig. 10-17, in which force rotates a rigid
body consisting of a single particle of mass mfastened to the end of a massless
rod. During the rotation, force does work on the body. Let us assume that the
only energy of the body that is changed by is the kinetic energy. Then we can
apply the work – kinetic energy theorem of Eq. 10-49:

KKfKiW. (10-56)

Using and Eq. 10-18 (vvr), we can rewrite Eq. 10-56 as

(10-57)

From Eq. 10-33, the rotational inertia for this one-particle body is Imr^2.
Substituting this into Eq. 10-57 yields

which is Eq. 10-52. We derived it for a rigid body with one particle, but it holds for
any rigid body rotated about a fixed axis.
We next relate the work Wdone on the body in Fig. 10-17 to the torquet
on the body due to force. When the particle moves a distance F dsalong its
:

K^12 Ivf^2 ^12 vi^2 W,

K^12 mr^2 vf^2 ^12 mr^2 vi^2 W.

K^12 mv^2

F

F :

:

F

:

P

dW

dt

tv

W

uf

ui

tdu

Table 10-3 Some Corresponding Relations for Translational and Rotational Motion

Pure Translation (Fixed Direction) Pure Rotation (Fixed Axis)

Position x Angular position u
Velocity vdx/dt Angular velocity vdu/dt
Acceleration adv/dt Angular acceleration adv/dt
Mass m Rotational inertia I
Newton’s second law Fnetma Newton’s second law tnetIa
Work W F dx Work W tdu
Kinetic energy K 21 mv^2 Kinetic energy K 21 Iv^2
Power (constant force) PFv Power (constant torque) Ptv
Work – kinetic energy theorem WK Work – kinetic energy theorem WK