A wheel rolls without slipping with \(v_{\mathrm{cm}}=6.0\,\mathrm{m\,s^{-1}}\) and \(R=0.30\,\mathrm{m}\). Find \(\omega\).

A rolling disk has \(\omega=12\,\mathrm{rad\,s^{-1}}\) and \(R=0.20\,\mathrm{m}\). Find \(v_{\mathrm{cm}}\).

A wheel rolls to the right without slipping. If counterclockwise is positive, what is the sign of \(\omega\)?

A rolling object has \(a_{\mathrm{cm}}=2.4\,\mathrm{m\,s^{-2}}\) and \(R=0.40\,\mathrm{m}\). Find \(|\alpha|\).

A rolling hoop has \(M=1.5\,\mathrm{kg}\), \(R=0.25\,\mathrm{m}\), and \(v_{\mathrm{cm}}=4.0\,\mathrm{m\,s^{-1}}\). Find its total kinetic energy using \(I_{\mathrm{cm}}=MR^2\).

A solid cylinder rolls without slipping at \(3.0\,\mathrm{m\,s^{-1}}\). Its mass is \(2.0\,\mathrm{kg}\). Find its total kinetic energy using \(I_{\mathrm{cm}}=\frac12MR^2\).

A solid sphere rolls without slipping down an incline with \(a_{\mathrm{cm}}=3.5\,\mathrm{m\,s^{-2}}\) and \(R=0.10\,\mathrm{m}\). Find the angular acceleration magnitude and state the rotation sense if it rolls downhill to the right.

A rolling body has \(K_{\mathrm{trans}}=18\,\mathrm{J}\) and \(K_{\mathrm{rot}}=9\,\mathrm{J}\). Find the total kinetic energy and the fraction in rotation.

A solid cylinder rolls without slipping down an incline of angle \(\theta\). Use \(I_{\mathrm{cm}}=\frac12MR^2\) to derive \(a_{\mathrm{cm}}\).

A hoop, a solid cylinder, and a solid sphere roll without slipping from rest down the same vertical height. Rank their final speeds and justify using energy.

A cylinder of mass \(M\), radius \(R\), and moment of inertia \(I_{\mathrm{cm}}\) is pulled horizontally by a force \(F\) applied through its centre while it rolls without slipping on a rough horizontal surface. Derive \(a_{\mathrm{cm}}\), the static friction force, and the minimum \(\mu_s\) required.

A spool of mass \(M\), outer rolling radius \(R\), hub radius \(b\), and moment of inertia \(I_{\mathrm{cm}}\) rests on a rough floor. A horizontal string leaves the top of the hub and is pulled to the right with force \(F\). Assuming rolling without slipping, derive \(a_{\mathrm{cm}}\), determine the direction of static friction, and give the condition for zero friction.