Question

A thin wire of length l and mass m is bent in the form of a semicircle as shown in the figure. Its moment of inertia about an axis joining its free ends will be

• A. ${ml}^2$
• B. zero
• C. $\frac{{ml}^2}{{\pi }^2}$
• D. $\frac{{ml}^2}{{2\pi }^2}$

Answer 1

The correct option is D $\frac{{ml}^2}{{2\pi }^2}$

Length $=$ Circumference of semicircle

$l=nAr=\frac{l}{\pi }$

Moment of inertia of half ring about

$=m\left(\frac{r^2}{2}\right)=m\left(\frac{l^2}{2{\pi }^2}\right)$ replacing r with $\frac{l}{\pi }$

Moment of Inertia$=\frac{m{l}^2}{2{\pi }^2}$