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

At a point

(0, 0, z)

, the two particles are at equal distance

a^{2} + z^{2}

. Their force components in the rotating

x y

-plane cancel, while their

z

-components add. Hence a particle initially moving along the

z

-axis remains on it, and

m_{3} \overset{z}{¨} = - \frac{k m _{3} m z}{( a ^{2} + z ^{2} ) ^{3/2}} - \frac{k m _{3} m z}{( a ^{2} + z ^{2} ) ^{3/2}} .

Cancelling

m_{3}

gives

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

Solved by gpt-5.6-sol high.

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