 Solution

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

sin θ = θ + O (θ^{3})

reduces ii to

\ddot{θ} + ω^{2} θ = 0

, where

ω = \frac{g}{ℓ} \frac{\frac{1}{2} + μ δ}{\frac{1}{3} + μ ( δ ^{2} + α ^{2} /2 )} .

The coefficient is positive, so the solution is simple harmonic motion,

θ (t) = C cos ω t + D sin ω t

. Here

ω

f = \frac{ω}{2 π}, T = \frac{2 π}{ω} .

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