Two loops ' $A$ ' and ' $B$ ' of radii ' $R_1$ ' and ' $R_2$ ' are made from uniform wire. If moment of…

Two loops ' $A$ ' and ' $B$ ' of radii ' $R_1$ ' and ' $R_2$ ' are made from uniform wire. If moment of inertia of ' $A$ ' is ' $I_A$ ' and that ' $B$ ' is ' $\mathrm{B}_{\mathrm{B}}$ ', then $R_2 / R_1$ is $\left[\frac{I_A}{I_B}=27\right]$
  1. $1: 6$
  2. $1: 4$
  3. $1: 3$
  4. $1: 2$

Solution

$\mathrm{I}_{\mathrm{A}}=\mathrm{M}_1 \mathrm{R}_1^2, \mathrm{I}_{\mathrm{B}}=\mathrm{M}_2 \mathrm{R}_2^2...(i)$
If $m$ is mass per unit length then $\mathrm{M}_1=2 \pi \mathrm{R}_1 \mathrm{~m}$ and $\mathrm{M}_2=2 \pi \mathrm{R}_2 \mathrm{~m}$ $\therefore \quad \frac{\mathrm{M}_1}{\mathrm{M}_2}=\frac{\mathrm{R}_1}{\mathrm{R}_2}...(ii)$ $\therefore \quad \frac{\mathrm{I}_{\mathrm{A}}}{\mathrm{I}_{\mathrm{B}}}=\frac{\mathrm{M}_1}{\mathrm{M}_2}\left(\frac{\mathrm{R}_1}{\mathrm{R}_2}\right)^2 \ldots$ [From(i)] $\frac{I_A}{I_B}==\frac{R_1}{R_2}\left(\frac{R_1}{R_2}\right)^2 \ldots$ [From(ii)]
$\frac{I_A}{I_B}=\left(\frac{R_1}{R_2}\right)^3$ $\begin{array}{ll}\therefore \quad 27=\left(\frac{R_1}{R_2}\right)^3 \quad \ldots\left(\text { given, }, \frac{\mathrm{I}_A}{\mathrm{I}_{\mathrm{B}}}=27\right) \\ \therefore \quad & \frac{\mathrm{R}_2}{\mathrm{R}_1}=\frac{1}{3}\end{array}$

Asked in: MHT CET 2024 (03 May Shift 2)

Practice more Rotational Motion questions on Aicharya