An oscillator has \(a=-16x\), with SI units understood. Find \(\omega\).

A \(0.50\,\mathrm{kg}\) mass is attached to a \(200\,\mathrm{N\,m^{-1}}\) spring. Find \(\omega\).

A motion has acceleration \(a=-25x\), with SI units understood. Does it satisfy the SHM acceleration condition, and what is \(\omega\)?

For \(x=A\cos(\omega t)\), where is the acceleration largest in magnitude?

A simple harmonic oscillator has \(A=0.040\,\mathrm{m}\) and \(\omega=15\,\mathrm{rad\,s^{-1}}\). Find \(a_{\max}\).

A \(0.20\,\mathrm{kg}\) mass on a spring has \(k=80\,\mathrm{N\,m^{-1}}\). Find its period.

A spring oscillator has \(A=0.10\,\mathrm{m}\) and \(\omega=12\,\mathrm{rad\,s^{-1}}\). Find its maximum speed.

A spring has \(k=45\,\mathrm{N\,m^{-1}}\), and the attached mass oscillates with \(\omega=3.0\,\mathrm{rad\,s^{-1}}\). Find the mass.

If the mass on a spring is increased by a factor of \(9\), by what factor does the SHM period change?

For \(x=0.080\cos(6t+\pi/3)\,\mathrm{m}\), find \(x(0)\), \(v(0)\), and \(a(0)\).

For \(x=A\cos(\omega t)\), find the first positive time when the oscillator passes through equilibrium and state the sign of its velocity then.

A mass-spring oscillator starts at \(x=0\) moving in the positive direction with speed \(v_{\max}\). Write \(x(t)\) in cosine form.

A \(0.40\,\mathrm{kg}\) oscillator has period \(0.50\,\mathrm{s}\). Find the spring constant.

A \(0.20\,\mathrm{kg}\) mass on a horizontal spring is displaced \(0.060\,\mathrm{m}\) and released from rest. The spring constant is \(50\,\mathrm{N\,m^{-1}}\). Write \(x(t)\).

For \(x=A\cos(\omega t+\phi)\), derive expressions for \(v_{\max}\) and \(a_{\max}\) in terms of \(A\) and \(\omega\).

A \(0.30\,\mathrm{kg}\) mass is attached to a horizontal spring. It is observed to have acceleration \(1.8\,\mathrm{m\,s^{-2}}\) toward equilibrium when displaced \(0.060\,\mathrm{m}\). Find \(k\) and \(T\).

A mass-spring oscillator has \(x(0)=0\), \(v(0)=-0.80\,\mathrm{m\,s^{-1}}\), and \(\omega=20\,\mathrm{rad\,s^{-1}}\). Find \(A\) and a cosine-form phase \(\phi\).

An oscillator obeys \(m\ddot{x}+kx=0\). It has \(x(0)=x_0\) and \(v(0)=v_0\). Derive \(A\) and \(\phi\) for \(x=A\cos(\omega t+\phi)\), with \(\omega=\sqrt{k/m}\), and state how the quadrant of \(\phi\) is chosen.

A particle has measured motion \(x(t)=C\cos\omega t+D\sin\omega t\). Show that it satisfies \(a=-\omega^2x\), then express its amplitude in terms of \(C\) and \(D\).

A spring oscillator is required to have period \(T\) and maximum acceleration no greater than \(a_{\max}\). Derive the largest allowed amplitude in terms of \(T\) and \(a_{\max}\), and state whether mass or spring constant enters this amplitude bound directly.