Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 3 / 18B / ii / Solution

At height $z$, $\mathbf{r}\times \mathbf{v}=(-\Omega zx,-\Omega zy,\Omega (x^2+y^2))$. Integrating over a circular cross-section cancels the horizontal components by symmetry. Thus the angular momentum per unit length isThe coefficient is the polar moment of inertia per unit length of the liquid column.