 Solution

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

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

(- \dot{Ω} y - Ω^{2} x, \dot{Ω} x - Ω^{2} y, 0)

, so

\frac{p _{x}}{ρ} = (α + Ω^{2}) x + (β + \dot{Ω}) y, \frac{p _{y}}{ρ} = (γ - \dot{Ω}) x + (δ + Ω^{2}) y, p_{z} = 0.

(β + γ) /2

, ensuring integrability. Thus

p (x, y, t) = p_{0} (t) + \frac{ρ}{2} [(α + Ω (t)^{2}) x^{2} + (β + γ) x y + (δ + Ω (t)^{2}) y^{2}] .

Pressure is determined up to the spatially uniform function

p_{0} (t)

. The specified force has no vertical component; if gravity were additionally included in the body force, the usual hydrostatic term

- ρ g z

would also appear. It is not implied by the displayed force here.

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