 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2021-resit/11/b/solution

x_{i} = a_{i} e^{i Ω t}

(2 k - m_{1} Ω^{2} - 2 k - 2 k 2 k - q m_{1} Ω^{2}) (a_{1} a_{2}) = 0.

(2 k - m_{1} Ω^{2}) (2 k - q m_{1} Ω^{2}) - 4 k^{2} = m_{1} Ω^{2} (q m_{1} Ω^{2} - 2 k (1 + q)) .

Ω_{0} = 0, ω = \frac{2 k ( 1 + q )}{q m _{1}} .

The ordinary nonzero frequency is

ω / (2 π)

(1, 1)

and represents a common rotation with no change in spring length. The oscillatory mode shape is

(1, - 1/ q)

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