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

Answer 1

Answer: (D)

Situation is as shown in the figure. Let us calculate moment of inertia $\left(M I\right)$ 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 $AB$
$I_{3}=ml^{2}$
(as each point of $AB$ is at a distance $l$ from $AB$
$\therefore $ Net moment of inertia about $AB,$
$I_{net}=I_{1}+I_{2}+I_{3}$
$\Rightarrow I_{net}=0+\frac{m l^{2}}{3}+ml^{2}$
$\Rightarrow I_{net}=\frac{4 m l^{2}}{3}.$