- #1
Physgeek64
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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 :)