 Solution

ID: past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2014/paper-1/1/d/solution

The new kinetic matrix is diagonal, giving

\ddot{θ} + \frac{g}{4 a} θ = 0, \ddot{ϕ} + \frac{g}{2 a} ϕ = 0, ω_{1}^{2} = \frac{g}{4 a}, ω_{2}^{2} = \frac{g}{2 a} .

For the first mode,

ϕ = 0

: the line between the centres stays vertical, both centres translate together, and

\dot{ψ} = 2 \dot{θ}

. For the second,

θ = 0

: the outer hoop stays still while the inner hoop rolls within it, with

\dot{ψ} = - \dot{ϕ}

. These coordinate motions are independent only at the retained quadratic order.

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