Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 17 / b

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- Solution b

Solution

Each proposed displacement has second derivative

\overset{x}{¨}_{i} = - ω^{2} a_{i} cos ω t

. Matching the coefficients of the common time factor in the three equations gives

m ω^{2} a_{1} m ω^{2} a_{2} m ω^{2} a_{3} = k (2 a_{1} - a_{2}), = k (- a_{1} + 2 a_{2} - a_{3}), = k (- a_{2} + 2 a_{3}) .

These algebraic relations hold for the entire oscillation, including the isolated times when the common cosine vanishes.

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