In steady-state driven motion, does the oscillator move at the driving frequency or its undamped natural frequency?

For light damping, near what angular frequency is the resonance peak?

What prevents the steady-state resonance amplitude from becoming infinite in the driven oscillator model?

A driven oscillator has \(m=0.50\,\mathrm{kg}\) and \(k=18\,\mathrm{N\,m^{-1}}\). Find its natural angular frequency.

For a drive \(F_d(t)=F_0\cos\omega t\), what is the driving period in terms of \(\omega\)?

Using \(A=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2+(b\omega/m)^2}}\), find \(A\) for \(F_0=2.0\,\mathrm{N}\), \(m=1.0\,\mathrm{kg}\), \(\omega_0=5.0\,\mathrm{rad\,s^{-1}}\), \(\omega=3.0\,\mathrm{rad\,s^{-1}}\), and \(b=1.0\,\mathrm{kg\,s^{-1}}\).

What happens to the height and width of the resonance peak when damping is increased?

A very slowly driven spring oscillator has \(F_0=3.0\,\mathrm{N}\) and \(k=150\,\mathrm{N\,m^{-1}}\). Estimate the steady amplitude.

At exact \(\omega=\omega_0=8.0\,\mathrm{rad\,s^{-1}}\), with \(F_0=4.0\,\mathrm{N}\) and \(b=2.0\,\mathrm{kg\,s^{-1}}\), find the steady-state amplitude.

At very low driving frequency, \(\omega\approx0\). Show that the amplitude response reduces to approximately \(F_0/k\).

For fixed \(F_0\), \(m\), and \(\omega_0\), explain why the response is small when the driving frequency is much larger than \(\omega_0\).

For \(m=2.0\,\mathrm{kg}\), \(k=50\,\mathrm{N\,m^{-1}}\), and light damping, estimate the driving frequency in hertz that gives resonance.

At exact \(\omega=\omega_0\), use the amplitude response formula to derive the steady-state amplitude in terms of \(F_0\), \(b\), and \(\omega_0\).

At exact resonance, what damping coefficient \(b\) is needed to keep the amplitude at or below \(A_{\max}\)? Express your answer in terms of \(F_0\), \(\omega_0\), and \(A_{\max}\).

A driven oscillator at exact resonance has \(A=0.25\,\mathrm{m}\), \(b=0.80\,\mathrm{kg\,s^{-1}}\), and \(\omega_0=6.0\,\mathrm{rad\,s^{-1}}\). Find \(F_0\).

Use the amplitude response to show that for \(\omega\gg\omega_0\) and damping not dominant, \(A\) scales approximately as \(1/\omega^2\).

For fixed \(F_0\), \(m\), \(\omega_0\), and drive frequency \(\omega\), derive the value of \(b\) needed to make the steady-state amplitude no larger than \(A_{\max}\). State the condition for such a \(b\) to be necessary.

A driven oscillator is tested at two drive frequencies \(\omega_a\) and \(\omega_b\), with the same \(F_0\). Derive the ratio \(A_a/A_b\) in terms of \(\omega_a,\omega_b,\omega_0,b,\) and \(m\).

Using the amplitude denominator \(D(\omega)=(\omega_0^2-\omega^2)^2+(b\omega/m)^2\), derive the drive angular frequency that maximizes amplitude for the underdamped driven oscillator, and state when this peak frequency is real and nonzero.

A driven oscillator has known \(\omega_0\), but its damping coefficient is unknown. A resonance peak is measured at nonzero angular frequency \(\omega_p\). Using the amplitude-denominator model, derive \(b/m\), and state the consistency condition on \(\omega_p\).