Question

Two rings have their moments of inertia in the ratio $2:1$ and their diameters are in ratio $2:1$. The ratio of their masses will be

• A. $2:1$
• B. $1:2$
• C. $1:4$
• D. $1:1$

Answer 1

The correct option is B $1:2$

consider the problem

Moment of inertia of the ring is given by

$I=m{r}^2$

Where $m$ is the mass of the ring and $r$ is the radius of the ring in terms of diameter

$I=\frac{m{d}^2}{4}$

For first ring

$\begin{array}{c}{I}_1=\frac{m_1{d_1}^2}{4}\\ I_2=\frac{m_2{d_2}^2}{4}\\ \frac{I_1}{I_2}=\frac{2}{1}\\ \frac{d_1}{d_2}=\frac{2}{1}\end{array}$

$\begin{array}{c}\frac{I_1}{I_2}=\frac{\frac{m_1{d^2}_1}{4}}{\frac{m_2{d^2}_2}{4}}=\frac{m_1}{m_2}{\left(\frac{d_1}{d_2}\right)}^2\\ 2=\frac{m_1}{m_2}{\left(2\right)}^2\\ \frac{m_1}{m_2}=1:2\end{array}$

Hence, Option $B$ is the correct answer.