Physics - Rotation Question with Solution | TestHub

${\mathrm{I}}_0$ is moment of inertia of block about an axis through $\mathrm{O}$ and perpendicular to the plane of block.

If ${\mathrm{I}}_{\mathrm{c}}$ is moment of inertia about center of mass then ${\mathrm{I}}_{\mathrm{c}}=\frac{{\mathrm{Ma}}^2}{6}$

hence from parallel axis theorem ${\mathrm{I}}_0={\mathrm{I}}_{\mathrm{c}}+{\mathrm{Mr}}^2$ and ${\mathrm{r}}^2={\left(\frac{\mathrm{a}}{2}\right)}^2+{\left(\frac{\mathrm{a}}{2}\right)}^2=\frac{{\mathrm{a}}^2}{2}$

${\mathrm{I}}_0=\frac{{\mathrm{Ma}}^2}{6}+\frac{{\mathrm{Ma}}^2}{2}=\frac{2{\mathrm{Ma}}^2}{3}$

From conservation of angular momentum about $\mathrm{O}$

$\mathrm{Mv}\frac{\mathrm{a}}{2}=\frac{2}{3}{\mathrm{Ma}}^2\mathrm{\omega }$

$\mathrm{\omega }=\frac{3\mathrm{v}}{4\mathrm{a}}$