All Problems

Mechanical Oscillations

Problem 4.90

A ball of mass m=50 gm=50 \mathrm{~g} is suspended by a weightless spring with stiffness x=20.0 N/mx=20.0 \mathrm{~N} / \mathrm{m}. Due to external vertical harmonic force with frequency ω=25.0 s−1\omega=25.0 \mathrm{~s}^{-1} the ball performs steady-state oscillations with amplitude a=1.3 cm.a=1.3 \mathrm{~cm} . In this case the displacement of the ball lags in phase behind the external force by φ=34π\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.

Reveal Answer
4.90. (a) Q=1/24ω2ω02(ω2−ω02)2tan⁡2φ−1=2.2Q=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=πma2(ω02−A=\pi m a^{2}\left(\omega_{0}^{2}-\right. −ω2)tan⁡φ=6 mJ.\left.-\omega^{2}\right) \tan \varphi=6 \mathrm{~mJ} . Here ω0=x/m‾\omega_{0}=\sqrt{\overline{x / m}}
Prev
Next