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
(a) $M{r}^2$

(b) $\frac{1}{2}M{r}^2$

(c) $\frac{1}{4}M{r}^2$

(d) $\frac{2}{5}M{r}^2$.

Answer 1

(a) $M{r}^2$

Consider an element of length, dl = rdθ.
$dm=\frac{M}{\mathrm{\pi }r}dl=\frac{M}{\mathrm{\pi }r}rd\theta$
MOI of semicircular wire = ${\int }_0^{\mathrm{\pi }}{r}^{2}dm$
$$\begin{gathered} I={\int }_0^{\mathrm{\pi }}{r}^2\frac{m}{\mathrm{\pi }r}rd\theta \\ \Rightarrow I=m{r}^2 \end{gathered}$$