 Solution

ID: past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/section-b/b1/a/solution

Let

X_{1}, X_{2}

be longitudinal displacements. Hooke's law gives

m (\ddot{X}_{1} \ddot{X}_{2}) = - k (2 - 1 - 1 2) (X_{1} X_{2}) .

The symmetric and antisymmetric eigenvectors are respectively

(1, 1)

and

(1, - 1)

, so

ω_{x, +} = k / m, ω_{x, -} = 3 k / m .

Ordinary frequencies are these values divided by

2 π

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