Solution
= 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 is
$$
\mathbf M=\rho\Omega\mathbf e_z\int_0^{2\pi}\int_0^ar^2r\,dr\,d\theta
=\boxed{\frac{\pi\rho a^4\Omega}{2}\mathbf e_z.}
$$
The coefficient is the polar moment of inertia per unit length of the liquid column.