Thermodynamics and Chemistry

APPENDIX G FORCES, ENERGY, AND WORK

G.9 ROTATINGLOCALFRAME 502

x # y

z

x^0

y^0

ri

zi

b

ei

Figure G.5 Relation between the Cartesian axesx,y,zof a lab frame and the axes

x^0 ,y^0 ,zof a rotating local frame. The filled circle represents particlei.

wherezcmis the elevation of the center of mass in the lab frame. The quantitymgÅzcmis
the change in the system’s bulk gravitational potential energy in the lab frame—the change
in the potential energy of a body of massmundergoing the same change in elevation as the
system’s center of mass.
The third term on the right side of Eq.G.8.6can be shown to be zero when the local
frame is a cm frame. The derivation uses Eqs.G.6.4andG.8.5and is as follows:

Å

X

i

iaccel

!

D

X

i

Z

Fiacceldri^0 D

X

i

Z

mi

dvcm

dt

dri^0

D

Z^ X

i

mi

dri^0

dt

!

dvcmD

Z^ X

i

miv^0 i

!

dvcm (G.8.11)

The sum

P

imiv
0
iin the integrand of the last integral on the right side is zero (Eq.G.8.4)
so the integral is also zero.
With these substitutions, Eq.G.8.6becomesÅUÅEsysD^12 mÅ

v^2 cm

mgÅzcm.
SinceÅUÅEsysis equal towwlabwhen the local frame is nonrotating (Eq.G.7.1), we
have
wwlabD^12 mÅ

v^2 cm

mgÅzcm (G.8.12)

G.9 Rotating Local Frame

A rotating local frame is the most convenient to use in treating the thermodynamics of
a system with rotational motion in a lab frame. A good example of such a system is a
solution in a sample cell of a spinning ultracentrifuge (Sec.9.8.2).
We will make several simplifying assumptions. The rotating local frame has the same
origin and the samezaxis as the lab frame, as shown in Fig.G.5. Thezaxis is vertical
and is the axis of rotation for the local frame. The local frame rotates with constant angular
velocity!Dd#=dt, where#is the angle between thexaxis of the lab frame and the