An oscillator completes \(18\) cycles in \(9.0\,\mathrm{s}\). Find its frequency.

An oscillator has period \(0.25\,\mathrm{s}\). Find its frequency.

An oscillator has frequency \(0.50\,\mathrm{Hz}\). Find its period.

An oscillator has frequency \(5.0\,\mathrm{Hz}\). Find its angular frequency.

The maximum displacement from equilibrium is \(0.12\,\mathrm{m}\). What is the amplitude and what is the peak-to-peak displacement?

An oscillator has angular frequency \(12\,\mathrm{rad\,s^{-1}}\). How much phase does it advance in \(0.50\,\mathrm{s}\)?

A periodic displacement is \(x(t)=0.30\cos(8\pi t)\,\mathrm{m}\). Find \(A\), \(\omega\), \(f\), and \(T\).

A sinusoidal oscillator has \(A=0.050\,\mathrm{m}\) and \(T=0.80\,\mathrm{s}\). Write one possible displacement function that starts at maximum positive displacement.

A motion has period \(1.6\,\mathrm{s}\). How many complete cycles occur in \(12.8\,\mathrm{s}\)?

Write a displacement model for an oscillator with amplitude \(0.20\,\mathrm{m}\), period \(4.0\,\mathrm{s}\), and \(x(0)=0.20\,\mathrm{m}\).

For \(x(t)=A\cos(\omega t+\phi)\), what phase constant \(\phi\) makes \(x(0)=0\) and the initial velocity negative?

For \(x(t)=A\cos(\omega t)\), find the first positive time when \(x=-A\).

Two oscillations have the same amplitude and period. One is \(x_1=A\cos\omega t\), and the other reaches maximum displacement a quarter period later. Write \(x_2(t)\) in cosine form.

An oscillator has \(x(0)=-A/2\) and \(v(0)>0\). Find \(\phi\) for \(x=A\cos(\omega t+\phi)\), with \(0\leq\phi<2\pi\).

Two identical sinusoidal oscillators have period \(T\). The second reaches every feature \(0.15T\) later than the first. Find its phase lag in radians.

An oscillator is modeled by \(x(t)=A\cos(\omega t+\phi)\). It has \(x(0)=A/2\) and \(v(0)<0\). Find \(\phi\) in the range \(0\leq\phi<2\pi\).

An oscillator has \(A=0.060\,\mathrm{m}\), \(T=2.0\,\mathrm{s}\), \(x(0)=0.030\,\mathrm{m}\), and \(v(0)>0\). Construct \(x(t)=A\cos(\omega t+\phi)\).

A sensor records successive positive maxima at \(t=0.30\,\mathrm{s}\), \(1.10\,\mathrm{s}\), and \(1.90\,\mathrm{s}\). The maximum displacement is \(0.040\,\mathrm{m}\). Construct a cosine model \(x(t)=A\cos(\omega t+\phi)\), choosing \(0\leq\phi<2\pi\), and state the timing assumptions.

A sinusoidal displacement has a positive-going equilibrium crossing at \(t=0.20\,\mathrm{s}\) and the next negative-going equilibrium crossing at \(t=0.70\,\mathrm{s}\). Its amplitude is \(0.090\,\mathrm{m}\). Construct \(x(t)=A\cos(\omega t+\phi)\), choosing \(0\leq\phi<2\pi\).

Two signals are \(x_1=A\cos(\omega t)\) and \(x_2=A\cos(\omega t+\phi)\). Derive all time shifts \(\Delta t\) for which \(x_2(t)=x_1(t+\Delta t)\), and explain why the answer is not unique.