 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017/9/c/solution

\frac{g}{l} I + \frac{k}{m} 1 - 1 0 - 1 2 - 1 0 - 1 1 .

v_{1} = \frac{1}{3} (1, 1, 1), v_{2} = \frac{1}{2} (1, 0, - 1), v_{3} = \frac{1}{6} (1, - 2, 1),

0, 1, 3

. Adding the gravitational term therefore gives

ω_{1} = \frac{g}{l}, ω_{2} = \frac{g}{l} + \frac{k}{m}, ω_{3} = \frac{g}{l} + \frac{3 k}{m} .

The uniform mode leaves both springs unstretched. The other two modes deform the springs and so have higher frequencies.

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