Thermodynamics and Chemistry

CHAPTER 3 THE FIRST LAW

3.7 SHAFTWORK 81

Fstrdz. According to Newton’s second law, the net force on the body equals the

product of its mass and acceleration: .mgCFbuoyCFfricCFstr/ Dmdv=dt.

Solving this equation forFstr, we obtain

FstrD

mgFbuoyFfricCmdv=dt

(3.6.2)

We can therefore express the work in the form

∂wDFstrdzD

mgFbuoyFfricCmdv=dt

dz (3.6.3)

This work can be positive or negative, depending on whether the body is being pulled

up or lowered by the string. The quantity.mdv=dt/dzis an infinitesimal change of

the body’s kinetic energyEk,^14 so that the integral

R

.mdv=dt/dzis equal toÅEk.

The finite quantity of work in a process that starts and ends in equilibrium states, so

thatÅEkis zero, is therefore

wD

Z

∂wD

mgFbuoy

Åz

Z

Ffricdz (3.6.4)

The work has a reversible limit, because the string allows the velocityvto be con-

trolled from the surroundings. Asvapproaches zero from either direction,Ffricap-

proaches zero and the work approaches the reversible limitwD.mgFbuoy/Åz.

(If the fluid is a gas whose density is much smaller than the density of the body,

Fbuoycan be neglected in comparison withmg, and the reversible work can be writ-

tenwDmgÅz.) Ffricand dzhave opposite signs, sowfor a given change of the

work coordinatezis least positive or most negative in the reversible limit.

 The system is the body only. In this case,Fzsuris equal to

FbuoyCFfricCFstr

which

by substitution from Eq.3.6.2is.mgCmdv=dt/. The work is then given by

∂wDFsurdzD.mgCmdv=dt/dz (3.6.5)

For a process that begins and ends in equilibrium states,ÅEkis zero and the finite

work iswDmgÅz, unaffected by the velocityvduring the process. The expressions

for infinitesimal and finite work in the reversible limit are

∂wDmgdz and wDmgÅz (3.6.6)

(reversible gravitational

work of a body)

When we compare Eqs.3.6.3and3.6.5, we see that the work when the system is the
body is greater by the quantity

FbuoyCFfric

dzthan the work when the system is the
combination of body and fluid, just as in the case of the freely-falling body. The difference
in the quantity of work is due to the different choices of the system boundary where contact
forces are exerted by the surroundings.

3.7 Shaft Work

Shaft workrefers to energy transferred across the boundary by a rotating shaft.