Three identical thin rods, each of length L and mass m, are welded perpendicular to one another as shown in figure. The assembly is rotated about an axis that passes through the end of one rod and is parallel to another. The moment of inertia of this structure about this axis is

The moment of the rod on the y axis about the y axis itself is essentially zero (axis through centre, parallel to rod) because the rod is thin. The moments of the rods on the x and z axes are each $\mathrm{I} = \frac{1}{12}{\mathrm{ML}}^2$ (axis through centre, perpendicular to rod) from the table in the chapter.

The total moment of the three rods about the y axis (and about the CM) is

${\mathrm{I}}_{\mathrm{CM}} = {\mathrm{I}}_{\mathrm{on} \mathrm{x}-\mathrm{axis}}+{\mathrm{I}}_{\mathrm{on} \mathrm{y} \mathrm{axis}}+{\mathrm{I}}_{\mathrm{on} \mathrm{z} \mathrm{axis}}$

      = $\frac{1}{12}{\mathrm{ML}}^2+0+\frac{1}{12}{\mathrm{ML}}^2 = \frac{1}{6}{\mathrm{mL}}^2$

For the moment of the rod-combination about the axis of rotation, the parallel-axis theorem gives

$\mathrm{I} = {\mathrm{I}}_{\mathrm{CM}}+3\mathrm{m}{\left(\frac{\mathrm{L}}{2}\right)}^2 = [\frac{1}{6}+\frac{3}{4}]{\mathrm{mL}}^2$

  = $[\frac{2}{12}+\frac{9}{12}]{\mathrm{mL}}^2 = \frac{11}{12}{\mathrm{mL}}^2$