 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-2/7/ii/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

In a breathing mode, each displacement is proportional to its equilibrium position. A representative vector is

q_{b} = (- 1, 1, 1, 1, - 1, - 1, 1, - 1)^{T} .

Multiplying by the stiffness matrix gives

K q_{b} = 4 k q_{b}

. Thus

q \propto q_{b}, ω^{2} = \frac{4 k}{m}, ω = 2 \frac{k}{m} .

Positive and negative amplitudes are uniform expansion and contraction about the center.

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