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Mechanical Oscillations

Problem 4.91

A ball of mass mm suspended by a weightless spring can perform vertical oscillations with damping coefficient β.\beta . The natural oscillation frequency is equal to ω0\omega_{0}. Due to the external vertical force varying as F=F0F=F_{0} cos ωt\omega t the ball performs steady-state harmonic oscillations. Find: (a) the mean power ⟨P⟩,\langle P\rangle, developed by the force F,F, averaged over one oscillation period; (b) the frequency ω\omega of the force FF at which ⟨P⟩\langle P\rangle is maximum; what is ⟨P⟩max⁡\langle P\rangle_{\max } equal to?

Reveal Answer
(a) ⟨P⟩=F02βω2/m(ω02−ω2)2+4β2ω2\langle P\rangle=\frac{F_{0}^{2} \beta \omega^{2} / m}{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}} (b) ω=ω0,⟨P⟩max⁡=F02/4βm.\omega=\omega_{0},\langle P\rangle_{\max }=F_{0}^{2} / 4 \beta m .
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