Question

Four thin metal rods, each of mass $M$ and length $L$, are welded to form a square. The moment of inertia of the composite structure about a line which bisects any two opposite rods is:

Answer 1

The MI of the rod AB($I_1$ )and CD($I_2$ ) about the axis PQ( as shown in the figure ) are same.
$I_1=I_2=\frac{M{L}^2}{12}$
The MI of the rod BC($I_3$ )and AD($I_4$ ) about the axis PQ( as shown in the figure ) are same..
$I_3=I_4=M(\frac{L}{2}{)}^2$
$AD\left|\right|PQ$ and $BC\left|\right|PQ$ and all points on AD are at same distance from the axis PQ, all points on BC are at same distance from the axis PQ,
$\therefore I=I_1+I_2+I_3+I_4=2\times \frac{M{L}^2}{12}+2\times M(\frac{L}{2}{)}^2=\frac{2}{3}M{L}^2$