 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ia/paper-4/9c/c/iii/solution

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

For

∣ z ∣ ≪ a

\overset{z}{¨} = - \frac{2 mk}{a ^{3}} z + O (z^{3}) .

ω_{z} = \frac{2 mk}{a ^{3}} .

With

z (0) = z_{0}

and

\overset{z}{˙} (0) = 0

z (t) ≃ z_{0} cos (ω_{z} t),

and its period is

T = 2 π \frac{a ^{3}}{2 mk} .

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