Download CCR 29: Reduced Mass

CLASSICAL CONCEPT REVIEW 29
Reduced Mass
To understand the concept of reduced mass, consider the simple case of an isolated
system consisting of two particles with masses m1 and m2 attracted toward each other
by a central force such as gravity or the Coulomb force. Being isolated, the center of
mass (CM) of the system moves at constant velocity and, without any loss of generality, we can locate the coordinate origin at the center of mass (see Figure RM-1). That
allows us to write
m 1r1 + m 2r2 = 0
RM-1
Referring to Figure RM-1, we see that the position of m1 relative to m2 is then
r = r1 - r2
RM-2
Rewriting Equation RM-1 as r2 = - 1m 1 >m 2 2r1 and substituting into Equation
RM-2, we have
r = r1 +
m1
m1
r = a1 +
br m2 1
m2 1
RM-3
The equations of motion (Newton’s second law) for m1 and m2 can then be written as
m1
d 2r1
r
= F = F1r2 r
dt 2
RM-4a
and
d 2r2
r
= F = F1r2 RM-4b
r
dt 2
where F1r2 is the magnitude of the force acting on each particle due to the other
particle when they are separated by a distance r.
Using the relation between r1 and r given by Equation RM-3, we can write the
equation of motion for m1 given by Equation RM-4a in either of two ways:
m2
r
r2
m2
r1
CM
m1
RM-1 Particles m1 and m2 are attracted toward each other by a central field. CM is the center
of mass of the two-particle system and the origin of their coordinate vectors r1 and r2.
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TIPLER_CCR.indd 82
Tipler: Modern Physics 6/e
Perm fig.: CCR51, New fig.: RM-01
First Draft: 2011-10-10
2nd Pass: 2011-10-24
23/11/11 5:46 PM
Classical Concept Review 29
d 2r1
r1
= F1ar1 2 2
r1
dt
where a = 1 + 1m 1 >m 2 2. Or we can write
m1
m
d 2r
r
= F1r2 r
dt 2
83
RM-5
RM-6
where
m =
m1
m1
m1 m2
=
=
a
1 + 1m 1 >m 2 2
m1 + m2
RM-7
The quantity m is called the reduced mass. Equation RM-6 describes the motion of m1
relative to m2; that is, we can consider m2 to be the center of mass (and the coordinate
origin) if we substitute the reduced mass m for m1. Notice that, if m1 V m2 as is the
case, for example, for the Earth-Sun system, then m1 1 m2 m2 and m m1. Also,
if m1  m2  m, as for example in the hydrogen molecule, then m = m>2.
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