A loudspeaker cone of mass is mounted on a suspension of stiffness and has a damping…

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations · Simple, damped and driven oscillations

Mathematical Methods for Physics, Ch Second-order linear equations: damping, driving and resonance

Problem

A loudspeaker cone of mass m=0.020 kgm = 0.020\,\mathrm{kg} is mounted on a suspension of stiffness k=8000 N m−1k = 8000\,\mathrm{N\,m^{-1}} and has a damping constant giving γ=50 s−1\gamma = 50\,\mathrm{s^{-1}}. It is driven by a force F(t)=F0cos⁡ωtF(t) = F_0\cos\omega t with F0=0.40 NF_0 = 0.40\,\mathrm{N}. ((a)) Obtain the complex amplitude A~\tilde{A} of the steady-state displacement x(t)=Re[A~e−iωt]x(t) = \mathrm{Re}[\tilde{A}e^{-i\omega t}], and from it ∣A~∣|\tilde{A}| and the phase lag ϕ\phi at ω=0.95 ω0\omega = 0.95\,\omega_0. ((b)) Compute ∣A~∣|\tilde{A}| and ϕ\phi at ω=ω0\omega = \omega_0. ((c)) Compute ∣A~∣|\tilde{A}| and ϕ\phi at ω=1.05 ω0\omega = 1.05\,\omega_0, and comment on the trend in amplitude and phase across the three frequencies.

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