Question

Consider a uniform square plate fo side a and mass M. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners 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

The correct option is D $\frac{2}{3}M{a}^2$

For a rectangular sheet moment of inertia passing through O, perpendicular to the plate is $I_0=M\left(\frac{a^2+b^2}{12}\right)$

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
$I_B=I_o+M{r}^2=\frac{M{a}^2}{6}+\frac{M{a}^2}{2}=\frac{4M{a}^2}{6}$
$I=\frac{2}{3}M{a}^2$