All Problems

Mechanical Oscillations

Problem 4.93

A uniform horizontal disc fixed at its centre to an elastic vertical rod performs forced torsional oscillations due to the moment of forces Nz=NmN_{z}=N_{m} cos wt. The oscillations obey the law φ=\varphi= =φmcos⁡(ωt−α).=\varphi_{m} \cos (\omega t-\alpha) . Find: (a) the work performed by friction forces acting on the disc during one oscillation period; (b) the quality factor of the given oscillator if the moment of inertia of the disc relative to the axis is equal to II.

Reveal Answer
(a) A=−πφmNmsin⁡αA=-\pi \varphi_{m} N_{m} \sin \alpha (b) Q=(cos⁡α+2ω2Iφm/Nm)2−12sin⁡αQ=\frac{\sqrt{\left(\cos \alpha+2 \omega^{2} I \varphi_{m} / N_{m}\right)^{2}-1}}{2 \sin \alpha}.
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