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 $I_{2}=4I_{1}$ , then $\frac{r_{2}}{r_{1}}$ equals

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

Answer 1

Answer: (B)

Moment of inertia of a loop is,
$I=Mr^{2}=\left(2 \pi r A \rho \right)r^{2}$
Therefore $I \propto r^{3}$
or $r \propto I^{\frac{1}{3}} \, $
For the two loops
$\frac{r_{2}}{r_{1}}=\left(\frac{I_{2}}{I_{1}}\right)^{\frac{1}{3}}$
As $I_{2}=4I_{1}$
$\frac{r_{2}}{r_{1}}=\left(\frac{4}{1}\right)^{\frac{1}{3}}$