 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ib/paper-3/18b/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

z

r \times v = (- Ω z x, - Ω zy, Ω (x^{2} + y^{2}))

. Integrating over a circular cross-section cancels the horizontal components by symmetry. Thus the angular momentum per unit length is

M = ρ Ω e_{z} \int_{0}^{2 π} \int_{0}^{a} r^{2} r d r d θ = \frac{π ρ a ^{4} Ω}{2} e_{z} .

The coefficient is the polar moment of inertia per unit length of the liquid column.

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