Masses \(2.0\,\mathrm{kg}\) at \(x=0\) and \(6.0\,\mathrm{kg}\) at \(x=4.0\,\mathrm{m}\) lie on a line. Find \(x_{\mathrm{cm}}\).

A system has total momentum \(30\,\mathrm{kg\,m\,s^{-1}}\) and total mass \(10\,\mathrm{kg}\). Find the center-of-mass speed.

Three masses lie on the \(x\)-axis: \(1.0\,\mathrm{kg}\) at \(0\), \(2.0\,\mathrm{kg}\) at \(3.0\,\mathrm{m}\), and \(3.0\,\mathrm{kg}\) at \(6.0\,\mathrm{m}\). Find \(x_{\mathrm{cm}}\).

A \(12\,\mathrm{kg}\) system has net external force \(18\,\mathrm{N}\) in the \(+x\)-direction. Find \(a_{\mathrm{cm}}\).

A \(2.0\,\mathrm{kg}\) object has velocity \(4\hat{\imath}\,\mathrm{m\,s^{-1}}\), and a \(3.0\,\mathrm{kg}\) object has velocity \(-\hat{\imath}\,\mathrm{m\,s^{-1}}\). Find \(\vec{v}_{\mathrm{cm}}\).

Two masses \(m\) and \(3m\) are released from rest by an internal spring on a frictionless table. If the lighter mass moves right at speed \(6.0\,\mathrm{m\,s^{-1}}\), find the heavier mass velocity.

A \(60\,\mathrm{kg}\) person walks \(3.0\,\mathrm{m}\) forward relative to a \(140\,\mathrm{kg}\) boat. The water exerts negligible horizontal force. How far does the boat move? Interpret the sign.

An isolated two-object system has total mass \(5.0\,\mathrm{kg}\) and center-of-mass velocity \(2\hat{\imath}\,\mathrm{m\,s^{-1}}\). One object has mass \(2.0\,\mathrm{kg}\) and velocity \(5\hat{\imath}+\hat{\jmath}\,\mathrm{m\,s^{-1}}\). Find the other object's velocity.

Two spacecraft modules of masses \(m\) and \(2m\) are initially docked and drifting force-free. An internal actuator separates them by distance \(d\) along the \(x\)-axis. Find each module's displacement from the original center of mass. State the assumption and interpret why the center of mass does not move.

A thin rod of length \(L\) lies on the \(x\)-axis from \(0\) to \(L\), with linear density \(\lambda(x)=\lambda_0(1+\alpha x/L)\). Derive \(x_{\mathrm{cm}}\) and state the allowed range of \(\alpha\) for nonnegative density along the rod.