Question

Two disc have the same mass and thickness . their materials are densities $d_1$ and $d_2$. the ratio of their M.I about their central axis is

• A. $d_2:d_1$
• B. $d_1:d_2$
• C. $d_1^2:d_2^2$
• D. $d_2^2:d_1^2$

Answer 1

The correct option is A $d_2:d_1$

$\begin{array}{c}\text{Let , the two discs be}A\text{and}B\text{having densities}{d}_1\text{and}{d}_2\\ \text{respectively,}\\ \text{Given,}{m}_A=m_B\text{and both have same thickness, (t)}\\ \text{Let, their radii be}{r}_A\text{ard}{r}_B\text{.}\end{array}$

$\begin{array}{c}\text{Then,}\left(\pi r_A^2\right)\times t\times d_1=\left(\pi r_B^2\right)\times t\times d_2\\ \therefore \cdot \frac{r_{A^2}}{r_{B^2}}=\frac{d_2}{d_1}\end{array}$

$\begin{array}{c}\text{We know, moment of inertia of disc}=\frac{m{r}^2}{2}\\ \text{Thus, required ratio}=\frac{r_{A^2}}{r_B^2}=\frac{d_2}{d_1}\end{array}$