For linear damping \(F_d=-bv\), what direction does the damping force point relative to velocity?

A mass \(m=2.0\,\mathrm{kg}\) has damping coefficient \(b=0.80\,\mathrm{kg\,s^{-1}}\). Find \(\beta=b/(2m)\).

If an oscillator is moving in the negative \(x\)-direction, what is the sign of the linear damping force \(F_d=-bv\)?

An oscillator has \(\omega_0=8.0\,\mathrm{rad\,s^{-1}}\) and \(\beta=3.0\,\mathrm{s^{-1}}\). Is it underdamped?

An oscillator has \(\beta=\omega_0\). Does the underdamped cosine form apply?

For \(\omega_0=10\,\mathrm{rad\,s^{-1}}\) and \(\beta=6.0\,\mathrm{s^{-1}}\), find the damped angular frequency \(\omega_d\).

An underdamped oscillator has envelope \(A(t)=A_0e^{-0.40t}\). What fraction of its initial amplitude remains after \(5.0\,\mathrm{s}\)?

A \(1.5\,\mathrm{kg}\) oscillator has damping rate \(\beta=0.60\,\mathrm{s^{-1}}\). Find \(b\).

An underdamped oscillator has \(\omega_0=13\,\mathrm{rad\,s^{-1}}\) and \(\omega_d=12\,\mathrm{rad\,s^{-1}}\). Find \(\beta\).

A \(0.50\,\mathrm{kg}\) oscillator with \(k=32\,\mathrm{N\,m^{-1}}\) has \(b=2.0\,\mathrm{kg\,s^{-1}}\). Find \(\omega_0\), \(\beta\), and decide whether the cosine underdamped form applies.

An underdamped oscillator has \(\beta=0.20\,\mathrm{s^{-1}}\) and period approximately \(1.0\,\mathrm{s}\). What fraction of amplitude remains after \(5\) oscillations?

An underdamped oscillator has \(A_0=0.12\,\mathrm{m}\) and \(\beta=0.25\,\mathrm{s^{-1}}\). Find the time for the amplitude envelope to fall to \(0.030\,\mathrm{m}\).

For \(m=1.0\,\mathrm{kg}\), \(k=25\,\mathrm{N\,m^{-1}}\), find the largest value of \(b\) for which the oscillator is still underdamped.

The amplitude envelope of an underdamped oscillator falls from \(0.080\,\mathrm{m}\) to \(0.050\,\mathrm{m}\) in \(3.0\,\mathrm{s}\). Estimate \(\beta\).

If the displacement amplitude has fallen to \(30\%\) of its initial value, what fraction of the oscillator's mechanical energy remains, assuming energy scales as amplitude squared?

For weak damping where \(\beta\ll\omega_0\), use a first-order binomial argument on \(\omega_d=\sqrt{\omega_0^2-\beta^2}\) to show why the frequency shift is second order in \(\beta/\omega_0\).

Derive the critical damping coefficient \(b_c\) by setting \(\beta=\omega_0\), with \(\beta=b/(2m)\) and \(\omega_0=\sqrt{k/m}\).

An underdamped oscillator has measured damped angular frequency \(\omega_d\) and envelope half-life \(t_{1/2}\). Derive \(\beta\), \(\omega_0\), and \(b\) for a known mass \(m\).

An underdamped oscillator has two measured positive peak displacements \(A_1\) and \(A_2\), separated by \(N\) complete damped periods \(T_d\). Derive \(\beta\) in terms of these measurements.

A mass-spring oscillator with known \(m\) and \(k\) must remain underdamped while its amplitude envelope falls to at most a fraction \(r\) after time \(\tau\). Derive the allowed interval for \(b\), and state when no such underdamped \(b\) exists.