question_answer

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 an axis XY is

A) \[\frac{\rho {{L}^{3}}}{8{{\pi }^{2}}}\]

B) \[\frac{\rho {{L}^{3}}}{16{{\pi }^{2}}}\]

C) \[\frac{5\rho {{L}^{3}}}{8{{\pi }^{2}}}\]

D) \[\frac{3\rho {{L}^{3}}}{8{{\pi }^{2}}}\]

Correct Answer: D

Solution :

[d]   \[L=2\pi r\Rightarrow r=\frac{L}{2\pi },m=\rho L\] \[{{l}_{xy}}=\frac{3}{2}m{{r}^{2}}\] \[=\frac{3}{2}\rho L{{\left( \frac{L}{2\pi } \right)}^{2}}=\frac{3}{8}\frac{\rho {{L}^{3}}}{{{\pi }^{2}}}\]