In (6.38), show that for a given forcing frequency ω', the displacement y and the velocits d y /

step 1 Hello everyone here our equation is D square y by d t square plus 2b D y by d t plus omega squar

In (6.38), show that for a given forcing frequency ω', the...

In $(6.38)$, show that for a given forcing frequency $\omega$ ', the displacement $y$ and the velocits $d y / d t$ have their largest amplitude when $\omega=\omega^{\prime}$. For a given $\omega$, we have shown in Section 6 that the maximum amplitude of $y$ does nos correspond to $\omega^{\prime}=\omega .$ Show, however, that the maximum amplitude of $d y / d t$ for a given $\theta$ does correspond to $\omega^{\prime}=\omega$ State the corresponding results for an electric circuit in terms of $L, R, C$.

In $(6.38)$, show that for a given forcing frequency $\omega$ ', the displacement $y$ and the velocits $d y / d t$ have their largest amplitude when $\omega=\omega^{\prime}$.

For a given $\omega$, we have shown in Section 6 that the maximum amplitude of $y$ does nos correspond to $\omega^{\prime}=\omega .$ Show, however, that the maximum amplitude of $d y / d t$ for a given $\theta$ does correspond to $\omega^{\prime}=\omega$
State the corresponding results for an electric circuit in terms of $L, R, C$.

Key Concepts

Electrical Resonance in RLC Circuits

An RLC circuit, consisting of an inductor (L), resistor (R), and capacitor (C), serves as an electrical analog to a mechanical oscillatory system. In these circuits, resonance occurs when the inductive reactance equals the capacitive reactance, leading to a minimal total impedance and maximal current amplitude. This situation corresponds to a natural resonant frequency given by ? = 1/?(LC), mirroring the condition in the mechanical system where maximum velocity amplitude is achieved when the forcing frequency matches the natural frequency.

Forced Oscillations

Forced oscillations refer to systems that are driven by an external periodic force. In such systems, the equation of motion includes a forcing term, leading to a steady-state response that depends on the frequency of the applied force. The response of the system, in terms of displacement and velocity, may not always have their maxima at the system’s natural frequency, especially in systems with damping or other complexities.

Resonance

Resonance is the phenomenon where a system exhibits a maximum response amplitude when the forcing frequency coincides with a key frequency of the system. In forced oscillations, while the displacement may not reach its maximal amplitude at resonance due to phase differences and energy storage factors, the velocity (first derivative of displacement) reaches its maximum when the forcing frequency matches the resonant condition, ensuring optimal energy absorption.

Phase Relationships in Oscillatory Systems

The phase lag between the forcing term and the response plays a crucial role in determining which component of the response (displacement or velocity) is maximized. For a given forcing frequency, the displacement and its derivative (velocity) can have differing phase relationships with the external force, leading to distinct conditions for their maximum amplitudes. Specifically, while the maximum displacement may not occur exactly at the resonant condition, the maximum velocity does because it best captures the instantaneous rate of energy transfer.

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