A body has \(I_{\mathrm{cm}}=0.20\,\mathrm{kg\,m^2}\), mass \(3.0\,\mathrm{kg}\), and axis offset \(d=0.40\,\mathrm{m}\). Find \(I\).

A uniform rod has \(M=2.0\,\mathrm{kg}\) and \(L=1.2\,\mathrm{m}\). Using \(I_{\mathrm{end}}=\frac{1}{3}ML^2\), find its moment of inertia about one end.

In the added term \(Md^2\), what happens to the added moment of inertia if \(d\) doubles?

A solid disk has \(M=5.0\,\mathrm{kg}\), \(R=0.20\,\mathrm{m}\), and \(I_{\mathrm{cm}}=\frac{1}{2}MR^2\). Find \(I\) about a parallel axis \(0.10\,\mathrm{m}\) from the center.

A uniform rod has \(M=3.0\,\mathrm{kg}\) and \(L=2.0\,\mathrm{m}\). Find \(I\) about an axis parallel to its center axis but \(0.50\,\mathrm{m}\) from the center.

An object's moment of inertia about a shifted axis is \(2.0\,\mathrm{kg\,m^2}\). If \(M=4.0\,\mathrm{kg}\) and \(d=0.50\,\mathrm{m}\), find \(I_{\mathrm{cm}}\).

A measured shifted-axis moment of inertia is smaller than \(Md^2\). Explain why that cannot be consistent with the parallel-axis theorem for a real object.

A uniform rod of mass \(2.0\,\mathrm{kg}\) and length \(1.0\,\mathrm{m}\) has a \(0.50\,\mathrm{kg}\) point mass attached at one end. Find \(I\) about an axis through the rod center and perpendicular to the rod.

Two identical disks of mass \(m\) and radius \(R\) are fixed at the ends of a massless bar of length \(L\). The rotation axis passes through the bar midpoint and is parallel to each disk's symmetry axis. Derive \(I_{\mathrm{tot}}\), stating the modeling assumptions.

Two point masses \(m_1\) and \(m_2\) lie on a line at \(x=0\) and \(x=L\). For a rotation axis perpendicular to the line at position \(x\), derive the value of \(x\) that minimizes \(I\), and find the minimum \(I\).