Question

A rectangular plate of mass $m=3\text{kg}$ having sides $a=4\text{m}$ and $b=2\text{m}$ is placed on XY plane as shown in figure. Find the moment of inertia of rectangular plate about an axis passing through the point ${}^{\prime }{O}^{\prime }$ and perpendicular to its plane.

• A. $5{\text{kg-m}}^2$
• B. $4{\text{kg-m}}^2$
• C. $1{\text{kg-m}}^2$
• D. $3{\text{kg-m}}^2$

Answer 1

The correct option is A $5{\text{kg-m}}^2$


Given:
$a=4\text{m}$
$b=2\text{m}$
Mass of rectangular plate $m=3\text{kg}$
$I_x$ and $I_y$ be the moment of inertia of rectangular plate about an axis passing through the point ${}^{\prime }{O}^{\prime }$ in the plane of plate.
So, we have
$I_x=\frac{m{a}^2}{12}=\frac{3\times 4^2}{12}=4{\text{kg-m}}^2$
$I_y=\frac{m{b}^2}{12}=\frac{3\times 2^2}{12}=1{\text{kg-m}}^2$

Applying perpendicular axis theorem
$I_z=I_x+I_y$
$=(1+4){\text{kg-m}}^2$
$=5{\text{kg-m}}^2$
So, moment of inertia of rectangular plate about an axis passing through the point ${}^{\prime }{O}^{\prime }$ and perpendicular to its plane is $5{\text{kg-m}}^2$