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Bonding, Structure and Solid-State Chemistry$

Mark Ladd

Print publication date: 2016

Print ISBN-13: 9780198729945

Published to Oxford Scholarship Online: May 2016

DOI: 10.1093/acprof:oso/9780198729945.001.0001

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(p.403) A8 Reduced Mass

(p.403) A8 Reduced Mass

Source:
Bonding, Structure and Solid-State Chemistry
Author(s):

Mark Ladd

Publisher:
Oxford University Press

The reduced mass is an effective inertial mass of a system, considered here as a two-particle problem in classical mechanics. Let the particles of masses m1 and m2 be at position vectors r1 and r2 with respect to an origin at the centre of mass C; r is the linear distance between m1 and m2 (Fig. A8.1).

A8 Reduced Mass

Fig. A8.1 Two-particle system of masses m1 and m2 vibrating about the centre of mass C.

Then:

(A8.1)
r=r1+r2

and

(A8.2)
m1r1=m2r2

Now:

(A8.3)
r2=rr1

Hence:

(A8.4)
m1r1=m2(rr1)

so that

(A8.5)
r1=m2r/(m1+m2)

and

(A8.6)
r2=m1r/(m1+m2)

The moment of inertia I of the system is given by

(A8.7)
I=m1r12+m2r22=m1m2rm1+m22+m2m1rm1+m22=m1m22r2(m1+m2)2+m12m2r2(m1+m2)2=m1m2(m1+m2)r2(m1+m2)2
(A8.8)
I=m1m2r2(m1+m2)=μr2

(p.404) where μ‎‎ is the reduced mass of the system:

(A8.9)
μ=m1m2(m1+m2)

A similar expression arises for the moment of inertia of a diatomic molecule, in which case m1 and m2 are the masses of the two atoms and r is the length of the bond between them.