Question

Two loops $P$ and $Q$ are made from a uniform wire. The radii of \)P\) and $Q$ are $r_1$ and $r_2$, respectively, and their moments of inertia are $I_1$ and $I_2$, respectively. If $\frac{I_2}{I_1}=4$, then $\frac{r_2}{r_1}$ is equal to

• A. $4^{\frac{-1}{3}}$
  
• B. $4^{\frac{-2}{3}}$
  
• C. $4^{\frac{2}{3}}$
  
• D. $4^{\frac{1}{3}}$
  

Answer 1

The correct option is D $4^{\frac{1}{3}}$


Given, $\frac{I_2}{I_1}=4$
$\frac{I_2}{I_1}=\frac{m_2{r}_2^2}{m_1{r}_1^2}$
$\because m\alpha 2\pi r\Rightarrow m\alpha r$
$\frac{I_2}{I_1}=\frac{r_2^3}{r_1^3}$
$\frac{4}{1}={\left(\frac{r_2}{r_1}\right)}^2$
$\frac{r_2}{r_1}={\left(4\right)}^{\frac{1}{3}}$
Final Answer: $\left(b\right)$