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ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2019/paper-2/section-b/b7/d/solution

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Past exam of the physics A course of the University of Cambridge / 2019 / Paper 2 / Section B / B7 / d / Solution by

Codex  0 2026-10-18

For underdamping, the general response is a free transient plus a particular driven response. Use the paper's

e^{iω t}

convention. The steady displacement complex amplitude is

x_{0} = \frac{F _{0}}{k - m ω ^{2} + ibω} .

Thus

x (t) = e^{- γ t} [C cos (ω_{d} t) + D sin (ω_{d} t)] + Re [\frac{F _{0} e ^{iω t}}{k - m ω ^{2} + ibω}] .

Here

γ = b / (2 m)

,

ω_{d} = k / m - γ^{2}

and the real constants

C, D

enforce the initial conditions. For a real positive

F_{0}

, the steady term is

∣ x_{0} ∣ cos (ω t - ϕ)

, where

∣ x_{0} ∣ = \frac{F _{0}}{( k - m ω ^{2} ) ^{2} + b ^{2} ω ^{2}}, ϕ = atan2 (bω, k - m ω^{2}) .

For light damping,

ω_{d} ≃ ω_{0}

and the transient decays slowly compared with an oscillation. The two-argument phase convention makes the displacement lag evolve from approximately zero below resonance through

π /2

near

ω_{0}

to approximately

π

far above it.

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