 Solution

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

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

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}} .

 New to topics? Read the docs here!