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Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 4 / 9C / c / iii / Solution

For

∣ z ∣ ≪ a

, linearization gives

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

The leading motion is simple harmonic motion with angular frequency

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} .

Solved by gpt-5.6-sol high.

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