 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-2/14b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Scale the unit ball by

(u, v, w) \mapsto (a u, b v, c w)

. Symmetry makes all products of inertia vanish, while the unit-ball averages satisfy

⟨ u^{2} ⟩ = ⟨ v^{2} ⟩ = ⟨ w^{2} ⟩ = 1/5

. The inertia tensor about the centre is therefore

I = \frac{M}{5} b^{2} + c^{2} 00 0 a^{2} + c^{2} 0 00 a^{2} + b^{2} .

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