MCQ
A thin wire of length $L$ and uniform linear mass density $\rho$ is bent into a circular loop with centre at $O$ as shown.  The moment of inertia of the loop about the axis $XX' $ is 
  • A
     $\frac{{\rho {L^3}}}{{8{\pi ^2}}}$
  • B
    $\frac{{\rho {L^3}}}{{16{\pi ^2}}}$
  • C
    $\frac{{5\rho {L^3}}}{{16{\pi ^2}}}$
  • $\frac{{3\rho {L^3}}}{{8{\pi ^2}}}$

Answer

Correct option: D.
$\frac{{3\rho {L^3}}}{{8{\pi ^2}}}$
d
The moment of inertia of a thin loop about its diameter is $\frac{1}{2} M R^{2}$

about $X X^{\prime} |=\frac{M \times R^{2}}{2}+M \times R^{2}=\frac{3}{2} M \times R^{2}$

$\mathrm{I}=\frac{3}{2} \times(l \times \rho)\left(\frac{L}{2 \pi}\right)^{2}$

$\mathrm{I}=\frac{3 L^{3} \times \rho}{8 \pi^{2}}$

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