A wheel starts from rest with \(\alpha=2.0\,\mathrm{rad\,s^{-2}}\) for \(3.0\,\mathrm{s}\). Find \(\omega\).

A wheel turns through \(10\,\mathrm{rad}\) in \(2.0\,\mathrm{s}\) at constant angular velocity. Find \(\omega\).

Convert \(5.0\) revolutions to radians.

A wheel has \(\omega_0=4.0\,\mathrm{rad\,s^{-1}}\), \(\alpha=3.0\,\mathrm{rad\,s^{-2}}\), and \(t=5.0\,\mathrm{s}\). Find \(\omega\) and \(\Delta\theta\).

A rotor slows from \(20\,\mathrm{rad\,s^{-1}}\) to \(5.0\,\mathrm{rad\,s^{-1}}\) in \(3.0\,\mathrm{s}\). Find \(\alpha\) and \(\Delta\theta\).

A disk starts from rest and has \(\alpha=4.0\,\mathrm{rad\,s^{-2}}\). Find its angular speed after turning through \(18\,\mathrm{rad}\).

A wheel has \(\omega_0=2.0\,\mathrm{rad\,s^{-1}}\) and \(\alpha=3.0\,\mathrm{rad\,s^{-2}}\). Find the positive time when it has turned through \(20\,\mathrm{rad}\).

A rotor decelerates uniformly from \(50\,\mathrm{rad\,s^{-1}}\) to \(10\,\mathrm{rad\,s^{-1}}\) with \(\alpha=-5.0\,\mathrm{rad\,s^{-2}}\), then coasts for \(6.0\,\mathrm{s}\) at \(10\,\mathrm{rad\,s^{-1}}\). Find the total angular displacement.

A rotor starts and ends at rest after turning through \(100\,\mathrm{rad}\). It accelerates at \(+4.0\,\mathrm{rad\,s^{-2}}\) for the first phase and at \(-4.0\,\mathrm{rad\,s^{-2}}\) for the second phase. Find the total time and peak angular speed, and justify where the switch occurs.

A rotation through angle \(\Theta\) must start and end at rest. The angular acceleration magnitude cannot exceed \(\alpha_{\max}\). Derive the minimum time and peak angular speed, stating the acceleration schedule assumed.