Question

Three identical thin rods each of length $l$ and mass $m$ are joined together to form a letter $H$ . What is the moment of inertia of the system about one of the side of $H?$

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

Answer 1

Answer: (D)

Situation is as shown in the figure. Let us calculate moment of inertia $\left(MI\right)$ about side $AB.$

$MI$ of side $AB$ about $AB$
$I_1=0$
$MI$ of rod $CD$ about $AB$
$I_2=\frac{m{l}^2}{3}$
And, $MI$ of side $EF$ about $AB$
$I_3=m{l}^2$
(as each point of $AB$ is at a distance $l$ from $AB$
$\therefore$ Net moment of inertia about $AB,$
$I_{net}=I_1+I_2+I_3$
$\Rightarrow I_{net}=0+\frac{m{l}^2}{3}+m{l}^2$
$\Rightarrow I_{net}=\frac{4m{l}^2}{3}.$