Pairwise moment-of-inertia identity
= Pairwise moment-of-inertia identity
{title2=$I=\frac1{2M}\sum_{j<k}m_jm_kr_{jk}^2$}
For positive <masses> in their <centre of mass> frame, $\sum_{j<k}m_jm_k|\boldsymbol r_j-\boldsymbol r_k|^2=M\sum_km_kr_k^2$. Thus the scalar inertia $I=\tfrac12\sum_km_kr_k^2$ equals $(2M)^{-1}\sum_{j<k}m_jm_kr_{jk}^2$. The identity compares total spatial extent with pairwise separations.