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Resonance for Mechanical Oscillator

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https://texercises.com/exercise/resonance-for-mechanical-oscillator/

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Exercise:
The differential s for a driven series LC oscillator with damping and a driven mass on a spring with linear drag are formally equivalent: ddotqt + omega_e^ qt+ delta_e dotqt Q_cosomega t ddotyt + omega_m^ yt+ delta_m dotyt A_cosomega t with the angular frequencies for the undamped system omega_e fracsqrtLC labelome quad textrmand quad omega_m sqrtfrackm and the damping constant delta_e fracRL labeldee quad textrmand quad delta_m fracbetam Using the analogy between mechanical and electrical quantities derive the expressions for the resonance frequency of the mechanical oscillator. Calculate the numerical value for a mass mO an elastic constant kO and a drag coefficient beO.

Solution:
The displacement yt of the mass corresponds to the charge qt on the capacitor which is proportional to the capacitor voltage v_Ct. We know that the amplitude of the capacitor voltage has a maximum for omega_R sqrtomega_^-delta^ For the mechanical system we find omega_R omRF sqrtfrackm-fracbe^timesm^ omR approx resultomRP The angular frequency of the undamped system is omega_ omudF sqrtfrackm omud approx resultomudP

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Damping & Driven Oscillation by by

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Branches	Forced Oscillations
Tags	damping, driven oscillation, lc oscillator, mass on spring
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Difficulty	(2, default)
Points	0 (default)
Language	(English)
Type	Calculative / Quantity
Creator	by
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