Masses \(3.0\,\mathrm{kg}\) and \(5.0\,\mathrm{kg}\) are at \(x=-1.0\,\mathrm{m}\) and \(x=2.0\,\mathrm{m}\). Find \(x_{\mathrm{cg}}\).

Where is the center of gravity of a uniform straight rod of length \(L\) in a uniform gravitational field?

Three equal point masses are at \(x=-2.0\,\mathrm{m}\), \(x=1.0\,\mathrm{m}\), and \(x=4.0\,\mathrm{m}\). Find \(x_{\mathrm{cg}}\).

Masses \(1.0\,\mathrm{kg}\), \(3.0\,\mathrm{kg}\), and \(2.0\,\mathrm{kg}\) are at \(x=0\), \(x=2.0\,\mathrm{m}\), and \(x=6.0\,\mathrm{m}\). Find \(x_{\mathrm{cg}}\).

A uniform \(3.6\,\mathrm{m}\) board has mass \(9.0\,\mathrm{kg}\). A \(4.0\,\mathrm{kg}\) toolbox is placed \(0.60\,\mathrm{m}\) from the left end. Find \(x_{\mathrm{cg}}\) from the left end.

A rectangular object has its center of gravity \(0.18\,\mathrm{m}\) from the left edge. Its base extends from \(x=0\) to \(x=0.30\,\mathrm{m}\). Is it stable on a horizontal table?

A uniform square plate of side \(a\) has a small circular hole cut out with its center at \(x=3a/4\), \(y=a/2\). Explain qualitatively how the hole shifts the center of gravity.

A uniform \(5.0\,\mathrm{m}\) beam of mass \(8.0\,\mathrm{kg}\) carries a \(2.0\,\mathrm{kg}\) sensor at \(x=0.50\,\mathrm{m}\) and a \(6.0\,\mathrm{kg}\) battery at \(x=4.0\,\mathrm{m}\). Find \(x_{\mathrm{cg}}\) and state whether it lies left or right of the midpoint.

Two thin uniform rods are joined at right angles at one end. One rod lies along the \(x\)-axis with length \(a\); the other lies along the \(y\)-axis with length \(b\). Assuming the same linear density, derive the center-of-gravity coordinates and interpret the result when \(a=b\).

A rectangular block of width \(b\) along a slope and height \(h\) normal to the slope rests on an incline. Derive the critical incline angle for tipping about the downhill edge, state the assumptions, and interpret how changing \(b/h\) affects stability.