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

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 $\frac{Ma^{2}}{6}$ (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
$I_{B}=I_{O}=Mr^{2}=\frac{Ma^{2}}{6}+\frac{Ma^{2}}{2}=\frac{2Ma^{2}}{6}$
$I=\frac{2}{3}Ma^{2}$