Past exam of the options I course of the University of Oxford / 2004 / H / H1

The inertia tensor is $I_{ij}=\int (r^2{\delta }_{ij}-x_i{x}_j)dm$. The mixed moments vanish by reflection in $x$ or $y$. Every diagonal integrand is even in $z$, so its integral over the upper hemisphere is half its integral over the corresponding full ball of mass $2M$. A uniform ball's coordinate moments are each $(2M)a^2/5$, obtained by spherical symmetry and $⟨r^2⟩=3a^2/5$. Hence the hemisphere has each coordinate second moment $M{a}^2/5$ andThe tensor parallel axis theorem gives $I_O=I_C+M(|d{|}^{2}I-d{d}^t)$ with $d=(0,0,3a/8)$. ThereforeTranslation along the symmetry axis changes the two transverse moments but leaves the axial one unchanged.