[QUE/ME-14002]

Show that the principal moments of inertia of a linear chain consisting of two kinds of atoms \(a,b\), with origin chosen to coincide with the centre of mass, is given by \[I_1=I_2=\frac{1}{M}\sum_{i\ne j}m_im_j \,d_{ij}^2, \qquad I_3=0.\] where the summation includes each pair of \(i,j\) atoms once and \(d_{ij}\)is the distance between atoms in the pair and \(M\) is the total mass. Verify that this gives correct answer for a triatomic molecule.