From a given a simple uniform wire, two circular loops P and Q are made, P of radius r and Q of radius nr. If the M.I of Q about its axis is four times that of P about its axis (assuming the wire to be of diameter much smaller than either radius), the value of n is

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${(4)}^{\frac{1}{2}}$

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

${(4)}^{\frac{1}{4}}$

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

answer is B.

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Detailed Solution

Given that $I_{Q} = 4 I_{P}$

Here $I = mass \times {(radius)}^{2} .$ If $σ$ is the linear mass density (mass/length) of the wire,
then $σ \times 2 π (mr) \times n^{2} r^{2} = 4 (σ \times 2 πr \times r^{2})$

$\Rightarrow n^{3} = 4 \Rightarrow n = {(4)}^{\frac{1}{3}}$

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