Question

A thin square plate of side $4\text{cm}$ and mass $m$ has a moment of inertia $I_{\text{Sq}}$ about the line $AB$ as shown in figure. It is then reformed into a rectangle of length $8\text{cm}$ and breadth $2\text{cm}$ with a moment of inertia $I_{\text{Rect}}$ about the line $CD$ as shown in the figure. What is the value of the ratio $\frac{I_{\text{Sq}}}{I_{\text{Rect}}}?$

• A. $1$
• B. $2$
• C. $\frac{1}{2}$
• D. $\frac{1}{4}$

Answer 1

The correct option is A $1$

The axis passes through one of the sides of the square
$\therefore$ Moment of inertia of square about the axis $AB$,
$I_{AB}=\frac{m\times a^2}{3}$
$a=4\text{cm}=0.04\text{m}$
$\therefore I_{\text{Sq}}=\frac{m\times (0.04{)}^2}{3}$
The axis $CD$ passing through the mid point of the length of the rectangle with length $L=8\text{cm}=0.08\text{m}$

$\Rightarrow I_{\text{Rect}}=\frac{m{L}^2}{12}=\frac{m\times (0.08{)}^2}{12}$

$\therefore \frac{I_{\text{Sq}}}{I_{\text{Rect}}}=\frac{\frac{m\times (0.04{)}^2}{3}}{\frac{m\times (0.08{)}^2}{12}}=\frac{4}{4}=1$