Question

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

• A. $\frac{10m{l}^2}{3}$
• B. $\frac{4m{l}^2}{3}$
• C. $\frac{2m{l}^2}{5}$
• 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 of individual rod about the axis passing through $\text{O}\left(I_0\right)$ = M.O.I of individual rod about its C.O.M + $m{d}^2$
Here, $d=$ perpendicular distance of each rod from $\text{O}=\frac{a}{2}$
i.e $I_0=\frac{m{l}^2}{12}+m{\left(\frac{l}{2}\right)}^2=\frac{m{l}^2}{3}$

Now considering all four rods as a system, C.O.M of system will be at $\text{O}$.
$\therefore$ MOI of the system about $\text{O}$
$I_{com}=4\times I_0=\frac{4m{l}^2}{3}$


Using parallel axis theorem again, MOI of system about axis passing through corner $\text{A}$
$I_z=I_{com}+\left(4m\right)d^2$
Here, $d=\frac{\text{AC}}{2}=\frac{l\sqrt{2}}{2}$
$\therefore I_z=I_{com}+\left(4m\right){\left(\frac{l\sqrt{2}}{2}\right)}^2$
$=\frac{4m{l}^2}{3}+\frac{4m{l}^2}{2}$
$=\frac{10m{l}^2}{3}$