Question

Consider a uniform square plate of side $a$ and mass $M$. The moment of inertia o f this plate about an axis perpendicular to its plane and passing through one of its comers is

• A. $\frac{5}{6}M{a}^2$
• B. $\frac{1}{12}M{a}^2$
• C. $\frac{7}{12}M{a}^2$
• D. $\frac{2}{3}M{a}^2$

Answer 1

Answer: (D)

For a rectangular sheet moment of inertia passing through $O$, perpendicular to the plate is
$I_O=M\left(\frac{a^2+b^2}{12}\right)$
For square plate it is $\frac{M{a}^2}{6}$ (as $a=b)$

$r=\sqrt{\frac{a^2}{4}+\frac{a^2}{4}}=\frac{a}{\sqrt{2}}$
$\therefore r^2=\frac{a^2}{2}$
$\therefore$ Moment of inertia about B parallel to the axis through O is
$I_B=I_O=M{r}^2=\frac{M{a}^2}{6}+\frac{M{a}^2}{2}=\frac{2M{a}^2}{6}$
$I=\frac{2}{3}M{a}^2$