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## Homework Statement

Landau&Lifshitz Vol. Mechanics, p101 Q1

Find the moment of inertia of a collinear set of molecules

## Homework Equations

I=sum m_i*(r_i^2)

## The Attempt at a Solution

r_i= r_i'-r_cm where r_cm=sum m_i*r_i/sum m_i

I=sum m_i r_i^2

I= 1/M sum m_i m_j (r_i' - r_cm)^2

I=1/M sum m_i m_j (r_i'^2 -2r_i' r_cm + r_cm^2)

The central term will reduce to zero by definition of the COM leaving

I=1/M sum m_i m_j (r_i'^2 + r_cm^2)

The answer of 1/M sum m_i m_j r_ij is given, where r_ij is the distance between the ith and jth particle, and the summation is over i not equal to j, but I'm not sure how to simplify this into that expression- I have attempted it a few different ways but none have yet to work. Any help would be much appreciated

Thanks in advance :)