All Problems Mechanical Oscillations
Problem 4.91
A ball of mass \(m\) suspended by a weightless spring can perform vertical oscillations with damping coefficient \(\beta .\) The natural oscillation frequency is equal to \(\omega_{0}\). Due to the external vertical force varying as \(F=F_{0}\) cos \(\omega t\) the ball performs steady-state harmonic oscillations. Find: (a) the mean power \(\langle P\rangle,\) developed by the force \(F,\) averaged over one oscillation period; (b) the frequency \(\omega\) of the force \(F\) at which \(\langle P\rangle\) is maximum; what is \(\langle P\rangle_{\max }\) equal to?
(a) \(\langle P\rangle=\frac{F_{0}^{2} \beta \omega^{2} / m}{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}\) (b) \(\omega=\omega_{0},\langle P\rangle_{\max }=F_{0}^{2} / 4 \beta m .\)