Question

Four identical thin rods of mass $m$ and length $l$ form a square frame as shown in fig. Find the MOI of the frame about an axis passing through one of its corner and perpendicular to its plane.

• A. $\frac{10m{l}^2}{3}$
• B. $\frac{4}{3}\frac{m{l}^2}{3}$
• C. $\frac{2}{5}m{l}^2$
• D. $\frac{m{l}^2}{3}$

Answer 1

The correct option is A $\frac{10m{l}^2}{3}$


Using parallel axis theorem
M.O.I $\left(I\right)$ of individual rod about $I_0=$ M.O.I of individual rod about it's C.O.M+$m{d}^2$
$I=\frac{m{l}^2}{12}+m{\left(\frac{l}{2}\right)}^2=\frac{m{l}^2}{3}$

Now considering all four rod as system C.O.M of system will be at $O$.
$\therefore I_{com}=4\times I=\frac{4m{l}^2}{3}$

Using parallel axis theorem,
$I_z=I_{com}+\left(4m\right)d^2$
Here $d=\frac{\text{AC}}{2}=\frac{l\sqrt{2}}{2}$
$I_z=I_{com}+\left(4m\right){\left(\frac{l\sqrt{2}}{2}\right)}^2=\frac{4m{l}^2}{3}+\frac{4m{l}^2}{2}=\frac{10m{l}^2}{3}$