Question

Show that for a given forcing frequency $\mathit{\omega }\mathbf{\text{'}}$, the displacement yand the velocity $\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{t}$have their largest amplitude when $\mathit{\omega }\mathbf{\text{'}}\mathbf{=}\mathit{\omega }$.

For a given $\mathit{\omega }$, we have shown in Section 6 that the maximum amplitude of y does not correspond to $\mathit{\omega }\mathbf{=}\mathit{\omega }$. Show, however, that the maximum amplitude of $\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{t}$for a given $\mathit{\omega }$does correspond to $\mathit{\omega }\mathbf{\text{'}}\mathbf{=}\mathit{\omega }$.

State the corresponding results for an electric circuit in terms of $\mathit{L}\mathbf{,}\mathit{R}\mathbf{,}\mathit{C}\mathbf{.}$

Short answer

Answer

For a given forcing frequency $\omega \text{'}$, the displacement y and the velocity $dy/dt$have their largest amplitude when $\omega =\omega \text{'}$is ${\omega }^2={\omega }^{2}2b^2.$.

Step-by-step solution

Given information from question

In the RLC series circuit problem y represents the charge q on the capacitor if the forcing function is the emf and y represent the current $\mathit{I}\mathbf{=}\frac{\mathbf{d}\mathbf{q}}{\mathbf{d}\mathbf{t}}$

The imaginary part of yp

The imaginary part is

${\mathit{y}}_{\mathbf{p}}\mathbf{=}\frac{\mathbf{F}}{\sqrt{{\left({\omega }^2-{\omega }^{\text{'}2}\right)}^{\mathbf{2}}\mathbf{+}\mathbf{4}{\mathbf{b}}^{\mathbf{2}}{\mathbf{\omega }}^{\mathbf{\text{'}}\mathbf{2}}}}\mathit{s}\mathit{i}\mathit{n}\left(\omega \text{'}t-\varphi \right)$

The wide range of topics giving rise to differential equation

The imaginary part is

${\mathit{y}}_{\mathbf{p}}\mathbf{=}\frac{\mathbf{F}}{\sqrt{{\mathbf{(}{\mathbf{\omega }}^{\mathbf{2}}\mathbf{-}{\mathbf{\omega }}^{\mathbf{\text{'}}\mathbf{2}}\mathbf{)}}^{\mathbf{2}}\mathbf{+}\mathbf{4}{\mathbf{b}}^{\mathbf{2}}{\mathbf{\omega }}^{\mathbf{\text{'}}\mathbf{2}}}}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{\omega }\mathbf{\text{'}}\mathbf{t}\mathbf{-}\mathbf{\varphi }\mathbf{)}$

The wide range of topics giving rise to differential equation

The imaginary part of

$y_{\mathrm{p}}=\frac{\mathrm{F}}{\sqrt{{({\mathrm{\omega }}^2-{\mathrm{\omega }}^{\text{'}2})}^2+4{\mathrm{b}}^2{\mathrm{\omega }}^{\text{'}2}}}sin(\mathrm{\omega }\text{'}\mathrm{t}-\mathrm{\varphi })$

The applied force and the solution are out of phase that is their maximum values do not occur at the same time because of the phase angle$\varphi$

The largest amplitude of y occurs if the natural frequency $\omega =\omega \text{'}$.This is called resonance.