A thin wire of length $l$ and uniform linear mass density $\rho$ is bent into a circular loop with c

Question

A thin wire of length $l$ and uniform linear mass density $ho$ is bent into a circular loop with centre O and radius $r$ as shown in figure. The moment of inertia of the loop about the axis $\mathrm{XX}^{\prime}$ is\

Professor PC Thomas Classes · Answer

Since $2 \pi \mathrm{r}=\ell, \mathrm{r}=\ell / 2 \pi$ Mass $\mathrm{M}=\rho \ell$\ Moment of inertia of the loop about the xis XX' $=$ moment of inertia about its diameter $+\mathrm{Mr}^{2}$\ (by theorem of parallel axes)\ $=\frac{\mathrm{Mr}^{2}}{2}+\mathrm{Mr}^{2}=\frac{3}{2} \mathrm{Mr}^{2}=\frac{3}{2}(\rho \ell)\left(\frac{\ell}{2 \pi}\right)^{2}$\[Using (i) and (ii)]\ $=\frac{3 \rho \ell^{3}}{8 \pi^{2}}$\