All Problems Mechanical Oscillations
Problem 4.90
A ball of mass \(m=50 \mathrm{~g}\) is suspended by a weightless spring with stiffness \(x=20.0 \mathrm{~N} / \mathrm{m}\). Due to external vertical harmonic force with frequency \(\omega=25.0 \mathrm{~s}^{-1}\) the ball performs steady-state oscillations with amplitude \(a=1.3 \mathrm{~cm} .\) In this case the displacement of the ball lags in phase behind the external force by \(\varphi=\frac{3}{4} \pi\) Find: (a) the quality factor of the given oscillator; (b) the work performed by the external force during one oscillation period.
4.90. (a) \(Q=1 / 2 \sqrt{\frac{4 \omega^{2} \omega_{0}^{2}}{\left(\omega^{2}-\omega_{0}^{2}\right)^{2} \tan ^{2} \varphi}-1}=2.2\) (b) \(A=\pi m a^{2}\left(\omega_{0}^{2}-\right.\) \(\left.-\omega^{2}\right) \tan \varphi=6 \mathrm{~mJ} .\) Here \(\omega_{0}=\sqrt{\overline{x / m}}\)