A wheel has \(I=0.50\,\mathrm{kg\,m^2}\) and \(\omega=10\,\mathrm{rad\,s^{-1}}\). Find its rotational kinetic energy.

A \(2.0\,\mathrm{kg}\) point mass is \(0.30\,\mathrm{m}\) from an axis. Find its moment of inertia about that axis.

If a rigid body's angular speed doubles while \(I\) stays fixed, what happens to rotational kinetic energy?

A thin hoop has \(M=3.0\,\mathrm{kg}\), \(R=0.40\,\mathrm{m}\), and \(\omega=5.0\,\mathrm{rad\,s^{-1}}\). Find \(K_{\mathrm{rot}}\).

A solid disk rolls without slipping with \(M=2.0\,\mathrm{kg}\) and center speed \(3.0\,\mathrm{m\,s^{-1}}\). Use \(I_{\mathrm{cm}}=\frac{1}{2}MR^2\) to find total kinetic energy.

Two point masses are fixed to a light frame: \(1.0\,\mathrm{kg}\) at \(0.20\,\mathrm{m}\) and \(2.0\,\mathrm{kg}\) at \(0.50\,\mathrm{m}\) from the axis. If \(\omega=4.0\,\mathrm{rad\,s^{-1}}\), find \(K_{\mathrm{rot}}\).

A thin hoop and a solid disk have the same \(M\), \(R\), and \(\omega\). Compare their rotational kinetic energies and explain the difference.

A solid sphere with \(I_{\mathrm{cm}}=\frac{2}{5}MR^2\) rolls without slipping from rest down a vertical height of \(1.0\,\mathrm{m}\). Find its center speed at the bottom, ignoring losses.

A solid disk and a thin hoop with the same mass and radius roll without slipping from rest down the same vertical height \(2.0\,\mathrm{m}\). Find both bottom speeds and state which arrives faster under the rolling-energy model.

A rigid body with \(I_{\mathrm{cm}}=\kappa MR^2\) rolls without slipping from rest through vertical height \(h\). Derive \(v_{\mathrm{cm}}\) at the bottom and interpret how increasing \(\kappa\) changes the result.