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Q.

Two circular loops A and B are made of the same wire and their radii are in the ratio 1 : n. Their moments of inertia about the axis passing through the centre and perpendicular to their plane are in the ratio 1 : m. The relation between m and n is

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a

$m = n^{3}$

b

m=n

c

$m = n^{2}$

d

$m = n^{4}$

answer is C.

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Detailed Solution

Here $M_{A} = 2 π μ R$ and $M_{B} = 2 π n μ R$ . Their moments of inertia are
$I_{A} = M_{A} R_{A}^{2} = 2 π μ R \times R^{2} = 2 π μ R^{3} a n d I_{B} = M_{B} R_{B}^{2} = 2 π n μ R \times (n R)^{2} = 2 π n^{3} μ R^{3} ∴ \frac{I_{B}}{I_{A}} = n^{3}; \frac{I_{B}}{I_{A}} = m (g i v e n)$
Thus, $m = n^{3}$

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