Question

The moment of inertia of a uniform semicircular wire of mass M and radius r about a line perpendicular to the plane of the wire through the centre is

Answer 1

Given,

Mass of semicircular wire, $m$

Linear mass density of semicircle, $\sigma =\frac{m}{\pi r}$

For small length,$dl=rd\theta$, mass is $dm=\sigma dl=\frac{m}{\pi r}rd\theta =\frac{md\theta }{\pi }$

Moment of inertia small mass, $dI=dm.r^2=\frac{md\theta }{\pi }{r}^2$

On integrating, $\left(0to\pi \right)$

$I=\frac{m{r}^2}{\pi }{\theta |}_0^{\pi }=m{r}^2$

Hence, inertia of semicircular wire is $m{r}^2$