 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iv/6/b/ii/solution

The rod and disc contribute restoring torques

- m g (ℓ /2) sin θ

and

- M g d sin θ

, respectively. Using i in the rotational equation of motion gives

\ddot{θ} = - \frac{g}{ℓ} \frac{\frac{1}{2} + μ δ}{\frac{1}{3} + μ ( δ ^{2} + α ^{2} /2 )} sin θ .

Only the disc's center of mass enters its gravitational torque; its finite radius enters its moment of inertia. The formula is also regular at

M = m

, although the source happens to exclude that value.

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