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Standard XII

Physics

Moment of Inertia of a Rod

Two loops P a...

Question

Two loops P and Q are made from a same uniform wire. The radii of P and Q are $r_{1}$ and $r_{2}$ respectively and their moments of inertia are $l_{1}$ and $l_{2}$ respectively. If $\frac{l_{2}}{l_{1}}$ = 4, then $\frac{r_{2}}{r_{1}}$ is equal to:

4^{2 / 3}

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4^{1 / 3}

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4^{- 2 / 3}

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4^{- 1 / 3}

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Solution

The correct option is B $4^{1 / 3}$
$\frac{I_{2}}{I_{1}} = 4$
$\Rightarrow \frac{M_{2} R_{2}^{2}}{M_{1} R_{1}^{2}} = 4$ .....(i)
$λ ∴$ linear mean density
$M_{1} = (2 π R_{1}) λ, M_{2} = 2 π R_{2} λ_{2}$
by eq (i)
$\frac{2 π R_{2} λ R_{2}^{2}}{2 π R_{1} λ R_{1}^{2}} = 4$
$\frac{R_{1}^{3}}{R_{1}^{3}} = 4 \Rightarrow \frac{R_{2}}{R_{1}} = (4)^{1 / 3}$

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