Question

The ratio of kinetic energy of two bodies is $2:1$ and their angular momentum are in the ratio of $1:2$. Then the ratio of their moment of inertia will be (assume pure rotational motion)

• A. $1:4$
• B. $4:1$
• C. $8:1$
• D. $1:8$

Answer 1

The correct option is D $1:8$

Let, kinetic energy of the first body is $K{E}_1$ and second body is $K{E}_2$
$L_1\&L_2$ and $I_1\&I_2$ are angular momentum and moment of Inertia of the first body and second body respectively.
Given, $\frac{K{E}_1}{K{E}_2}=\frac{2}{1},\frac{L_1}{L_2}=\frac{1}{2}$
Now,
$\frac{K{E}_1}{K{E}_2}=\frac{\frac{1}{2}{I}_1{\omega }_1^2}{\frac{1}{2}{I}_2{\omega }^2}=\frac{\frac{1}{2}\frac{(I_1{\omega }_1{)}^2}{I_1}}{\frac{1}{2}\frac{(I_2{\omega }_2{)}^2}{I_2}}$
$\Rightarrow \frac{K{E}_1}{K{E}_2}=\frac{\frac{(L_1{)}^2}{I_1}}{\frac{(L_2{)}^2}{I_2}}$ as $(I_1{\omega }_1=L_1\text{and}{I}_2{\omega }_2=L_2)$
$\Rightarrow 2=\frac{1}{4}\frac{I_2}{I_1}$
$\Rightarrow \frac{I_2}{I_1}=\frac{8}{1}\Rightarrow \frac{I_1}{I_2}=\frac{1}{8}$