Question

Consider a uniform square plate of 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. $\frac{7}{12}m{a}^2$
  
• B. $\frac{1}{12}m{a}^2$
  
• C. $\frac{2}{3}m{a}^2$
  
• D. $\frac{5}{6}m{a}^2$
  

Answer 1

The correct option is C $\frac{2}{3}m{a}^2$


$I_x=I_y=\frac{m{a}^2}{12}$
$I_z=I_x+I_y=\frac{m{a}^2}{6}\left({\perp }^{ar}\text{axes theorem}\right)$
$OA=\frac{a}{\sqrt{2}}$

M.I. about an axis passing through $A$ and ${\perp }^{ar}$ to the plane

$I_A={\left(I_z\right)}_{c.m.}+m{\left(OA\right)}^2$
$=\frac{m{a}^2}{6}+m{\left(\frac{a}{\sqrt{2}}\right)}^2$

$=m{a}^2(\frac{1}{6}+\frac{1}{2})=\frac{2}{3}m{a}^2$
Final Answer: $\left(c\right)$