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 Incoming links: Transverse stiffness of a stretched spring

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 / b / Solution 2026-10-20

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The transverse stiffness of a stretched spring follows because a spring of length

2 l

develops transverse force

T_{0} Δ y / (2 l)

to first order. Its extension changes only at second order, so the leading transverse stiffness comes from its existing tension, not from that extension change. Here

T_{0} / (2 l) = k /2

, giving

m (\ddot{Y}_{1} \ddot{Y}_{2}) = - \frac{k}{2} (2 - 1 - 1 2) (Y_{1} Y_{2}), ω_{y, +} = k / (2 m), ω_{y, -} = 3 k / (2 m) .

(1)

The mode vectors are again

(1, 1)

and

(1, - 1)

. Treating the springs as unstretched would incorrectly remove this linear restoring force.

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