Physics - Rotation Question with Solution | TestHub

If$M$mass of the square plate before cutting the holes, then mass of portion of each hole,

$m=\frac{M}{16R^2}\times \mathrm{\pi }{R}^2=\frac{\mathrm{\pi }}{16} M$

$\therefore$Moment of inertia of remaining portion

$I=I_{\mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}}-4I_{\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{e}}$

$=\frac{M}{12}\left(16R^2+16R^2\right)-4\left[\frac{m{R}^2}{2}+m{\left(\sqrt{2}R\right)}^2\right]$

$=\frac{M}{12}\times 32R^2-10m{R}^2$

$=\frac{8}{3}M{R}^2-\frac{10\mathrm{\pi }}{16}M{R}^2=\left(\frac{8}{3}-\frac{10\mathrm{\pi }}{16}\right)M{R}^2$