A point is \(0.40\,\mathrm{m}\) from an axis and turns through \(5.0\,\mathrm{rad}\). Find its arc length.

A point is \(0.25\,\mathrm{m}\) from an axis with \(\omega=12\,\mathrm{rad\,s^{-1}}\). Find its tangential speed.

A point is \(0.30\,\mathrm{m}\) from an axis with \(\alpha=4.0\,\mathrm{rad\,s^{-2}}\). Find tangential acceleration.

A point is \(0.50\,\mathrm{m}\) from an axis with \(\omega=3.0\,\mathrm{rad\,s^{-1}}\). Find centripetal acceleration.

A wheel of radius \(0.30\,\mathrm{m}\) rolls without slipping through \(6.0\,\mathrm{m}\). Find its angular displacement.

A wheel of radius \(0.50\,\mathrm{m}\) rolls without slipping with center speed \(4.0\,\mathrm{m\,s^{-1}}\). Find \(\omega\).

Two points on a rigid wheel are at radii \(0.20\,\mathrm{m}\) and \(0.50\,\mathrm{m}\). If \(\omega=10\,\mathrm{rad\,s^{-1}}\), compare their tangential speeds and centripetal accelerations.

A wheel of radius \(0.40\,\mathrm{m}\) rolls without slipping from rest with constant \(\alpha=3.0\,\mathrm{rad\,s^{-2}}\) for \(5.0\,\mathrm{s}\). Find the center speed and center displacement.

Two pulleys are connected by a non-slipping belt. Pulley 1 has radius \(0.20\,\mathrm{m}\) and starts from rest with \(\alpha_1=6.0\,\mathrm{rad\,s^{-2}}\) for \(4.0\,\mathrm{s}\). Pulley 2 has radius \(0.50\,\mathrm{m}\). Find \(\alpha_2\), \(\omega_2\), and \(\theta_2\), stating the no-slip constraint used.

Two pulleys of radii \(R_1\) and \(R_2\) are connected by a belt that does not slip. Derive the relations between \(\theta_1,\omega_1,\alpha_1\) and \(\theta_2,\omega_2,\alpha_2\), including the sign difference between an open and crossed belt.