Two loops P and Q are made from a uniform wire. The radii of P and Q are r1 and r2 respectively, and their moments of inertia are I1 and I2 respectively. If I2=4I1 , then (r2/r1) equals

Question

Two loops P and Q are made from a uniform wire. The radii of P and Q are r1 and r2 respectively, and their moments of inertia are I1 and I2 respectively. If I2=4I1 , then (r2/r1) equals (A) 4(2/3) (B) 4(1/3) (C) 4(- 2/3) (D) 4(- 1/3)

Tardigrade Admin · Accepted Answer

Moment of inertia of a loop is, I=Mr2=(2 π r A ρ )r2 Therefore I propto r3 or r propto I(1/3) For the two loops (r2/r1)=((I2/I1))(1/3) As I2=4I1 (r2/r1)=((4/1))(1/3)

Q. 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 $I_{2} = 4 I_{1}$ , then $\frac{r _{2}}{r _{1}}$ equals

$4^{\frac{2}{3}}$

$4^{\frac{1}{3}}$

$4^{\frac{- 2}{3}}$

$4^{\frac{- 1}{3}}$

Moment of inertia of a loop is,
$I = M r^{2} = (2 π r A ρ) r^{2}$
Therefore $I \propto r^{3}$
or $r \propto I^{\frac{1}{3}}$
For the two loops
$\frac{r _{2}}{r _{1}} = (\frac{I _{2}}{I _{1}})^{\frac{1}{3}}$
As $I_{2} = 4 I_{1}$
$\frac{r _{2}}{r _{1}} = (\frac{4}{1})^{\frac{1}{3}}$

Q. Two loops P and Q are made from a uniform wire. The radii of P and Q are r1​ and r2​ respectively, and their moments of inertia are I1​ and I2​ respectively. If I2​=4I1​ , then r1​r2​​ equals

432​

431​

43−2​

43−1​