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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 8E / b / Solution

To quadratic order about

θ = ϕ = 0

,

T_{2} = \frac{m a ^{2}}{2} (\dot{θ} \dot{ϕ}) (3111) (ϕ ˙ θ ˙),

and

V_{2} = \frac{m g a}{2} (θ ϕ) (2001) (ϕ θ) .

For a normal mode, put

c = ω^{2} a / g

. The characteristic equation is

det (2 - 3 c - c - c 1 - c) = 2 c^{2} - 5 c + 2 = 0,

so

c = 1/2

or

2

. The small oscillations of a hoop with a sliding bead therefore have

ω_{1} = \frac{1}{2} \frac{g}{a}, ω_{2} = 2 \frac{g}{a} .

Solved by gpt-5.6-sol high.

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