question_answer

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

A) \[\frac{7}{12}m{{L}^{2}}\]

B) \[\frac{11}{14}m{{L}^{2}}\]

C) \[\frac{5}{15}m{{L}^{2}}\]   

D) \[\frac{11}{12}m{{L}^{2}}\]

Correct Answer: D

Solution :

[d] The moment of the rod on the y axis about the y axis itself is essentially zero (axis through axis centre,' parallel to rod) because the rod is thin. The moments of the rods on the x and z axes are each \[I=\frac{1}{12}M{{L}^{2}}\] (Axis through centre, perpendicular to rod) from the table in the chapter. The total moment of the three rods about the axis (and about the CM) is \[{{I}_{CM}}={{I}_{on\,x\,axis}}+{{I}_{on\,y\,axis}}+{{I}_{on\,z\,axis}}\] \[=\frac{1}{12}M{{L}^{2}}+0+\frac{1}{12}M{{L}^{2}}=\frac{1}{2}m{{L}^{2}}\] For the moment of the rod-combination about the axis of rotation, the parallel-axis theorem gives \[I={{I}_{CM}}+3m{{\left( \frac{L}{2} \right)}^{2}}=\left[ \frac{1}{6}+\frac{3}{4} \right]m{{L}^{2}}\] \[=\left[ \frac{2}{12}+\frac{9}{12} \right]m{{L}^{2}}=\frac{11}{12}m{{L}^{2}}\]