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282 CHAPTER 10 ROTATION

Work and Rotational Kinetic Energy

As we discussed in Chapter 7, when a force Fcauses a rigid body of mass mto ac-

celerate along a coordinate axis, the force does work Won the body. Thus, the

body’s kinetic energy (K^12 mv^2 )can change. Suppose it is the only energy of the

10-8WORK AND ROTATIONAL KINETIC ENERGY

After reading this module, you should be able to...

10.29Calculate the work done by a torque acting on a rotat-
ing body by integrating the torque with respect to the an-
gle of rotation.
10.30 Apply the work–kinetic energy theorem to relate the
work done by a torque to the resulting change in the rota-
tional kinetic energy of the body.

10.31Calculate the work done by a constanttorque by relat-

ing the work to the angle through which the body rotates.

10.32Calculate the power of a torque by finding the rate at

which work is done.

10.33Calculate 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 rota-
tional motion correspond to equations used for translational
motion and are

and P

dW

dt

tv.

W

uf

ui

t du

●When tis constant, the integral reduces to

Wt(ufui).

●The form of the work – kinetic energy theorem used for

rotating bodies is

KKfKi^12 Ivf^2 ^12 vi^2 W.

Learning Objectives

Key Ideas

body that changes. Then we relate the change Kin 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 xaxis, we can calculate the work with Eq. 7-32,

(work, one-dimensional motion). (10-50)

This reduces to WFdwhenFis 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 Kin 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,Iis the rotational inertia of the body about the fixed axis and viandvfare

the angular speeds of the body before and after the work is done.

KKfKi^12 Ivf^2 ^12 vi^2 W

(K^12 I^2 )

P

dW

dt

Fv

W

xf

xi

Fdx

KKfKi^12 mvf^2 ^12 mvi^2 W