Question

A thin wire of length 'L' and uniform linear mass density $\rho$ is bent into a circular loop with centre at O as shown in the figure. 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. $\frac{5\rho L^3}{16{\pi }^2}$
• D. $\frac{3\rho L^3}{8{\pi }^2}$

Answer 1

The correct option is D $\frac{3\rho L^3}{8{\pi }^2}$

Mass of the loop $=M=L\rho$
Further if r is the radius of the loop, then $2\pi r=L$
$\Rightarrow r=\frac{L}{2\pi }$
Moment of inertia about XX' is $I=\frac{3}{2}M{r}^2$
$\Rightarrow I=\frac{3}{2}\left(L\rho \right)\frac{L^2}{\left(2{\pi }^2\right)}=\frac{3\rho L^3}{8{\pi }^2}.$