Thermodynamics and Chemistry

APPENDIX G FORCES, ENERGY, AND WORK

G.6 THELOCALFRAME ANDINTERNALENERGY 496

ri 

ri^0

R

ri

Rloc

R^0

x

y

z

x^0

y^0

z^0

bc

rs

b

ut

Figure G.3 Vectors in the lab and local reference frames. Open circle: origin of

lab frame; open triangle: origin of local frame; open square: point fixed in system

boundary at segment; filled circle: particlei. The thin lines are the Cartesian axes

of the reference frames.

local frame is also inertial butUis not equal toEsysand the changeÅUduring a process
is not necessarily equal toÅEsys.
If the origin of the local frame moves with nonconstant velocity in the lab frame, or if the
local frame rotates with respect to the lab frame, then the local frame has finite acceleration
and is noninertial. In this case the motion of particlei in the local frame does not obey
Newton’s second law as it does in an inertial frame. We can, however, define aneffective
net forceFieffwhose relation to the particle’s acceleration in the local frame has the same
form as Newton’s second law:

FieffDmi

dv^0 i

dt

(G.6.1)

To an observer who is stationary in the local frame, the effective force will appear to make
the particle’s motion obey Newton’s second law even though the frame is not inertial.
The net force on particleifrom interactions with other particles is given by Eq.G.3.1:
FiD

P

j§iFijCF

field

i CF

sur

i. The effective force can be written

FieffDFiCFiaccel (G.6.2)

whereFiaccelis the contribution due to acceleration.Fiaccelis not a true force in the sense of
resulting from the interaction of particleiwith other particles. Instead, it is an apparent or
fictitious force introduced to make it possible to write Eq.G.6.1which resembles Newton’s
second law. The motion of particlei in an inertial frame is given bymidvi=dt DFi,
whereas the motion in the local frame is given bymidvi^0 =dtDFiCFiaccel.
A simple example may make these statements clear. Consider a small unattached object
suspended in the “weightless” environment of an orbiting space station. Assume the object
is neither moving nor spinning relative to the station. Let the object be the system, and fix
the local frame in the space station. The local frame rotates with respect to local stars as the
station orbits around the earth; the local frame is therefore noninertial. The only true force
exerted on the object is a gravitational force directed toward the earth. This force explains
the object’s acceleration relative to local stars. The fact that the object has no acceleration
in the local frame can be explained by the presence of a fictitious centrifugal force having