= Solution
In a frame centered at the fixed point or centre of mass, a rotating material point has velocity $\dot r=\omega\times r$. The identity $|\omega\times r|^2=|\omega|^2|r|^2-(\omega\cdot r)^2$ converts the <rotational kinetic energy> into $T=\tfrac12\omega_iI_{ij}\omega_j$, where the <inertia tensor> is
$$
\boxed{I_{ij}=\int\rho(r)(|r|^2\delta_{ij}-r_ir_j)\,d^3r.}
$$
Likewise integrating $r\times(\omega\times r)$ gives the <angular momentum> $\boxed{L=I\omega}$. These expressions refer to rotational motion about the chosen origin.
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