A rigid body of four particles of masses m, 2m, 3m, 4m, respectively are situated at the points (a, a, a), (a, -a, -a), (-a, a, -a), (-a, -a, a) and connected together by a light framework.
(a) Find the inertia tensor at the origin and show that the principle moments of inertia are 20ma², and (20 ± 2√5)ma².
(b) Find the principle axes and verify that they are orthogonal.
Hint: Use the following expression for the Inertia tensor of n particles of masses $m_k$, k = 1, 2, ...n
$I_{ij} = \sum_{k=1}^{n} m_k (\delta_{ij}r_k^2 - x_{ki}x_{kj})$
where $r_k = (x_{k1}, x_{k2}, x_{k3})$.
